annalen der a - Wiley Online Library · 2017. 11. 7. · COVER PICTURE Sample Issue 2011 Contents EDITORIAL Page A6 Page A7-A12 Page A13-A14 Highlights from recent Annalen der Physik

1© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

www.ann-phys.org

Multi-orbital band structure of iron-pnictide superconductorsby O. K. Andersen and L. Boeri

annalen der ad

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ISSN 0003-3804 · Ann. Phys. (Berlin), Sample Issue (2011)

physik

New in 20

12

Sample Issue 201 1

www.ann-phys.org © 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim2

d

Editor-in-ChiefGuido W. Fuchs

Section EditorChristian Joas

Max Planck Institute

for the History

of Science, Berlin,

Germany

Editorial Team

Regina Hagen

Lars Herrmann Cornelia Wanka

Dietmar Reichelt

Sonja Hoffmann

Editorial Offi ce

Wiley-VCH Verlag GmbH & Co. KGaA

Rotherstr. 21, 10245 Berlin, Germany

Phone: +49 (0) 30 – 47 03 13 21

Fax: +49 (0) 30 – 47 03 13 99

E-mail: [email protected]

Advisory BoardD. D. Awschalom, Santa Barbara, USA

C. W. J. Beenakker, Leiden,

The Netherlands

K. Blaum, Heidelberg, Germany

I. Bloch, Munich, Germany

C. Bruder, Basel, Switzerland

A. Caldwell, Munich, Germany

F. Capasso, Cambridge, USA

I. Cirac, Garching, Germany

G. Dvali, Munich, Germany R. Fazio, Pisa, Italy

R. Frésard, Caen, Frankreich

N. Gisin, Genève, Switzerland

T. Hänsch, Munich, Germany S. Hell, Göttingen, Germany

A. Imamoglu, Zurich, Switzerland

C. Kiefer, Cologne, Germany

P. Kim, New York, USA

J. Mannhart, Stuttgart, Germany

G. Schön, Karlsruhe, Germany

M. Schreiber, Chemnitz, Germany

Y. Tokura, Tokyo, Japan

V. Vedral, Oxford, UK

H. Zohm, Garching, Germany

P. Zoller, Innsbruck, Austria

Honorary Advisory Board

U. Eckern, F. W. Hehl, B. Kramer,

G. Röpke, A. Wipf, I. Peschel

How to citeAnnalen der Physik is cited as follows:Ann. Phys. (Berlin) vol. no. (issue no.), page(s) (year). Example: Ann. Phys. (Berlin) 524 (3), 320-334 (2012).Please make sure to always use Ann. Phys. (Berlin) as journal abbreviation.

Submission

Online manuscript submission: http://mc.manuscriptcentral.com/andp

www.ann-phys.org

annalen der ad

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Founded in 1790 by F. A. C. Gren

Continued by L. W. Gilbert,

J. C. Poggendorff, G. and E. Wiedemann,

P. Drude, W. Wien, M. Planck,

E. Grüneisen, F. Möglich, H. Kopfermann,

G. Richter, H.-J. Treder, W. Walcher,

B. Mühlschlegel, U. Eckern

Annalen der Physik publishes original

work in modern physics, overview articles

on topics of special interest, as well as

short letter contributions of exceptional

relevance. As a general physics journal

it covers theoretical, experimental and

applied physics and related areas of

physical sciences.

Scope

The journal covers the physics of � Condensed Matter / Solid State /

Materials� Optics / Photonics / Quantum

Information� Cosmology / Gravitation / Relativity� High Energy / Particles / Nuclear� Atoms and Molecules / Plasma � Biophysics / Biological and Medical

Applications� Geo / Climate / Environment

24 (3) 320 334

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e Ann. Phys. n.

Heft_Korrektur 4.indd 2Heft_Korrektur 4.indd 2 19.08.2011 12:03:1219.08.2011 12:03:12


annalen der ad

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New in 2012

For more information visit: www.ann-phys.org/Relaunch2012

Browse this issue and discover the new Annalen der Physik


HIGHLIGHTS

Page A5

COVER PICTURE

Sample Issue 2011

Contents

EDITORIAL

Page A6

Page A7-A12

Page A13-A14

Highlights from

recent Annalen der Physik issues

E. Pavarini

Lattice distortions in KCuF3:

a paradigm shift?

PHYSICS FORUM

A. H. Romero and M. J. Verstraete

A theoretical approach to iron-based

superconductors

Page A15

G. W. Fuchs

Annalen der Physik –

a brief history of a living legend

Full text on our homepage at www.ann-phys.org

ContentsCo

nten

ts

A2

ADVISORY BOARD

Page A4

G. W. Fuchs

Annalen der Physik –

refreshed and renewed

The new Advisory Board

EXPERT OPINION

EXPERT OPINION

THEN & NOW

RETROSPECT

The Review Article by O. K. Andersen

and L. Boeri on p. 1 of this sample issue

investigates the electronic structure and

magnetic stripe order in iron-pnictide

superconductors. The cover picture

shows details of the magnetic energy in

k-space which are clearly related to the

peculiar propeller-shaped Fermi surface of

LaOFeAs.


Ann. Phys. (Berlin) A3 (2011)

Contents

A3

ORIGINAL PAPER

Sample Issue 2011

RAPID RESEARCH LETTER

Page 43 – 46

J. Deisenhofer, M. Schmidt, Zhe Wang,

Ch. Kant, F. Mayr, F. Schrettle,

H.-A. Krug von Nidda, P. Ghigna, V. Tsurkan,

and A. Loidl

Lattice vibrations in KCuF3

This paper reports on polarization depend-

ent refl ectivity measurements in KCuF3 in

the far-infrared frequency regime. The ob-

served IR active phonons at room tempera-

ture are in agreement with the expected

modes for tetragonal symmetry. A splitting

of one mode at 150 K is observed as is the

appearance of a new mode in the vicinity

of the Néel temperature.

Page 37 – 42

N. J. Popławski

Cosmological constant from quarks and

torsion

A simple and natural way to derive the ob-

served small, positive cosmological con-

stant from the gravitational interaction

of condensing fermions is presented. In

Riemann-Cartan spacetime, torsion gives

rise to the axial–axial vector four-fermion

interaction term in the Dirac Lagrangian for

spinor fi elds. This nonlinear term acts like a

cosmological constant if these fi elds have a

nonzero vacuum expectation value.

REVIEW ARTICLE

Page 1 – 36

O. K. Andersen and L. Boeri

On the multi-orbital band structure and

itinerant magnetism of iron-based super-

conductors

This paper explains the multi-orbital band

structures and itinerant magnetism of

the iron-pnictide and chalcogenide su-

perconductors. The presence of iron in a

superconductor implies that there is an

interplay between magnetism and super-

conductivity - a strange combination be-

cause the two effects should be exclusive.

Annalen der Physik is indexed in Chemical Abstracts Service/SciFinder, COMPENDEX, Current Contents®/Physical, Chemical & Earth Sciences, FIZ Karlsruhe Databases, INIS: International Nuclear Information System Database, INSPEC, Journal Citation Reports/Science Edition, Science Citation Index Expanded™, Science Citation Index®, SCOPUS, Statistical Theory & Method Abstracts, VINITI,Web of Science®, Zentralblatt MATH/Mathematics Abstracts

Recognized by the European Physical Society

www.ann-phys.org © 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim6A4

David D. Awschalom

California Nanosystems

Institute, University of

California, Santa Barbara,

CA, USA

Carlo W.J. Beenakker

Instituut-Lorentz,

Universiteit Leiden,

The Netherlands

Klaus Blaum

Max Planck Institute for

Nuclear Physics,

Heidelberg, Germany

Immanuel Bloch

Quantum Many-Body Sys-

tems Division, Max Planck

Institute for Quantum

Optics, Garching, Germany

Christoph Bruder

Department of Physics,

University of Basel,

Switzerland

Allen Caldwell

Max Planck Institute of

Physics, Munich, Germany

Federico Capasso

Harvard University,

Cambridge, MA, USA

Ignacio Cirac

Max Planck Institute

for Quantum Optics,

Maximilians University

Munich, Garching, Germany

The new Advisory Board

Georgi Dvali

Faculty of Physics,

Theoretical Physics, Ludwig


Munich, Germany

Rosario Fazio

Scuola Normale Superiore,

Faculty of Sciences, Pisa,

Italy

Raymond Frésard

Laboratoire Crismat-

ENSICAEN, France

Nicolas Gisin

GAP-Optique, Université

de Genève, Geneva,

Switzerland

Theodor Hänsch

Faculty of Physics, Ludwig


Munich, Germany

Stefan Hell


Biophysical Chemistry,

NanoBiophotonics,

Göttingen, Germany

Atac Imamoglu

Institute for Quantum

Electronics, ETH Zurich,

Switzerland

Claus Kiefer

Institute for Theoretical

Physics, Cologne University,

Cologne, Germany

Philip Kim

Department of Physics,

Columbia University,

New York, NY, USA

Jochen Mannhart


Solid State Research,

Stuttgart, Germany

Gerd Schön


Solid State Physics,

Karlsruhe, Germany

Yoshinori Tokura

Department of Applied

Physics, University of Tokyo,

Japan

Vlatko Vedral

Department of Atomic &

Laser Physics, Clarendon

Laboratory, University of

Oxford, UK

Hartmut Zohm


Plasma Physics, University

Munich, Garching, Germany

Peter Zoller


Physics, University of

Innsbruck, Austria


Dear Reader,

The time has come for a change and Annalen der Physik – AdP will ap-pear completely renewed in 2012. With this sample issue you get a fi rst impression of what the AdP will look like. The new AdP will have a com-pletely different and modern look, with monthly changing front cover pictures and a new journal and arti-cle layout. As in the past, the scope of the journal involves all aspects of physics. The aim is to attract scien-tists to publish about important and timely topics at the forefront of mod-ern physics. This will also include topics from applied physics. AdP will report about fast evolving areas and will organize special issues in re-search areas that leading scientists expect to be relevant in the future. For that, Annalen der Physik enables a new generation of young, talented or particularly creative physicists to participate more closely in the de-velopment of the journal by becom-ing Advisory Board members. With this new team new topical priorities can be set. It is an honor and a pleasure for me to build on Ulrich Eckern’s suc-cessful leadership as Editor-in-Chief of Annalen der Physik. I have served as Managing Editor for AdP since 2009, and have had the opportunity to gain insights into the journal that form the basis of the restructuring that now is taking shape.

EDITORIAL

Annalen der Physik –refreshed and renewed

For AdP authors there will be some changes, too. As can be seen in this sample issue, Rapid Research Letters will be introduced and Review Arti-cles will be solicited [1,2]. The manu-scripts have to be submitted via an online manuscript system [3]. AdP allows the online submission of vid-eo abstracts and supporting materi-al. The Einstein Lectures series, i.e. special invited articles from interna-tionally leading scientists, will be re-vived and continued. A new section Then & Now will be introduced, or-ganized by the Max Planck Institute for the History of Science (Berlin/Germany). Furthermore, there will be an Expert Opinion section where researchers comment on recent or co-published articles in AdP in a brief essay form. The peer-review system will continue to be fast and sound and all articles will be quickly available as Early View online publi-cations shortly after their accept-ance. Shaping the future of a journal is not completely in the hands of the editor. The most crucial part comes from the scientifi c community, i.e. scientists and interested researchers and enthusiasts like you. As author and reader you can signifi cantly con-tribute to the success of Annalen der Physik by submitting your fi rst-class scientifi c work and most distin-guished manuscripts to the journal.

It is only with your contributions that AdP can be attractive and keep its status as a highly respected jour-nal. I invite you to be a part of the new AdP!

For details please visit our webpage www.ann-phys.org/Relaunch2012.

With kind regards

Guido W. FuchsEditor-in-Chief 2012

HighlightsEditorial

Ann. Phys. (Berlin) A5 (2011)

References

[1] See author guidelines for letter articles on the journals homepage, www.ann-phys.org [2] There is a review proposal sheet available for this purpose on www.ann-phys.org (Wiley Online Library) [3] Submit your manuscript here: http://mc.manuscriptcentral.com/ andp

A5


Hig

hlig

hts

RETROSPECT – Highlights of recent Annalen der Physik issues

Volume 523 | Issue 6 | Pages 439–449 (2011)

High precision thermal modeling of complex systems with application to the fl yby and Pioneer anomaly B. Rievers and C. Lämmerzahl

DOI 10.1002/andp.201100081

In the 1970s, the spacecrafts Pioneer 10 and 11 were launched for the exploration of

the outer solar system. Ever since J.C. Anderson (Jet Propulsion Laboratory, Pasade-

na, California) et al. in 1998 reported on a mysterious small acceleration of Pioneer

10 and 11 of about 10-9 m/s2 toward the Sun, attempts were made to explain this

“anomaly’’. Was it “new physics’’ (modifi ed general relativity, vacuum fl uctuations,

dark energy,...) or simply an effect overlooked in the analyses of the experiments?

You will fi nd the answer to this question in this remarkable paper of B. Rievers and

C. Lämmerzahl from ZARM in Bremen, Germany.

Volume 523 | Issue 1-2 | Pages 1-190 (2011)

Topical Issue: Optical and Vibrational Spectroscopy

Special issue in honor of the 75th birthday of Manuel Cardona

Eds.: Aldo H. Romero and Jorge Serrano

http://onlinelibrary.wiley.com/doi/10.1002/andp.v523.1/2/issuetoc

Relevant review papers are reported here on the areas of superconductivity in iron

pnictides, optical properties of nanostructures, electron holography in nitrides, and

the quantum Boltzmann equation. The special issue has been prepared on the oc-

casion of the seventy fi fth birthday of Manuel Cardona.

Volume 522 | Issue 7 | Pages 467-519 (2010)

Probabilistic observables, conditional correlations, and quantum physics C. Wetterich

DOI: 10.1002/andp.201010451

The authors discuss the classical statistics of isolated subsystems. Only a small part

of the information contained in the classical probability distribution for the subsys-

tem and its environment is available for the description of the isolated subsystem.

The “coarse graining of the information” to micro-states implies probabilistic ob-

servables. Furthermore the classical statistical realization of entanglement within a

system corresponding to four-state quantum mechanics is discussed. It is concluded

that quantum mechanics can be derived from a classical statistical setting with infi -

nitely many micro-states.

f

e-

r

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A6

RETROSPECT – Highlights


THEN & NOW

Annalen der Physik – a brief history of a living legendGuido W. Fuchs

Just in time for its 222 anniversary the journal Annalen der Physik will experience a metamorphosis – crowned by its relaunch in January

2012. From next year onwards readers can expect a new content, a new look, and a new editorial team. And all this happens for good

reasons. Today, the publishing world is under constant pressure to change. This also holds for academic journals and consequently for

Annalen der Physik, too.

Ann. Phys. (Berlin) A7-A12 (2011)

Annalen der Physik – AdP1 appears as a landmark of modern physics, an institution, distinguished by works from Einstein, Planck and other ex-traordinary talents. But what makes it unique today? The age of print media is said to be over. Our time is dynamic and digital. Nowadays, science arti-cles are delivered electronically, e.g. as pdf or XML documents, right to the researchers offi ce desk or laboratory. Literature search is done via special-ized databases or general search en-gines like Google and the like. In addi-tion, successful journals advertise their content and use the internet for marketing purposes, e.g. utilizing news portals, newsletter, rss-feeds, fa-cebook, twitter, etc. Over the last 200 years the amount of published phys-ics articles has increased exponen-tially. None of the established and well-recognized journals can deal with the vast fl ood of information. Fil-tering out the essence, i.e. informa-tion that most likely will advance physics, is the main task of editorial manuscript selection and article compiling. High peer-review quality standards will be more and more im-portant. This especially holds for the general physics journal Annalen der Physik, which no longer claims to be a comprehensive manuscript archive, as in the early times, but rather focus-es on key aspects of modern physics.

This change in scope is good reason to refl ect on the past events of this journal.

Birth of a legend (1790–1824)

The story begins with Friedrich Gren, a natural scientist born in 1760 in Bernburg/Saale, who held a position as professor in Halle, Germany. The general progress in natural sciences in Europe at the end of the 18th century was remarkable. In comparison to other European coun-tries Germany was only scientifi c province at this time. Most new fi nd-ings were published in minutes of so-ciety meetings or academy reports,

e.g. in Paris, London or Saint Peters-burg, or were propagated by private communications in the form of let-ters. Thus, ideas could only circulate within small and elitist communities. Gren believed that the lack of a suita-ble communication and publication medium was jointly responsible for the weak performance of German re-search. Inspired by the 1778-founded chemical journal of his teacher Lorenz von Crell (“Crell’s Annalen”) Gren started his own journal “Journal der Physik” in Halle. In the preface of the fi rst issue in June 1790 he wrote: “My purpose of publishing this jour-nal is to make acquainted with the discoveries in mathematics and chemistry of the foreigners and na-

Editors-in-Chief of Annalen der Physik from 1790 until 1947

(top, from left) Friedrich Albert Carl Gren, Ludwig Wilhelm Gilbert,

Johann Christian Poggendorff, Gustav Heinrich Wiedemann,

(below, from left) Eilhard Wiedemann, Paul Drude, Wilhelm Wien, Max Planck

1 Engl.: Annals of Physics

A7

HighlightsPhysics Forum

www.ann-phys.org © 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim1 0

tives in the fi eld of the natural scienc-es […]“. Accordingly, translations of foreign works built an important base for the new journal. With his journal Gren also aimed to give amateurs (“Privatmann”) access to recent re-search, both as reader as well as au-thor. Important topics were thermo-dynamics, electricity and mag-netism. Gren died in 1798, only 38 years old. His journal marked the be-ginning of a legend – the home jour-nal of the most distinguished minds in physics. Ludwig W. Gilbert, born in 1769 in Berlin, was a mathematician and geo-grapher and in 1801 succeeded Gren as university professor in Halle. Al-ready, Gren had worked towards a re-launch of his journal under the new name Annalen der Physik. Now in 1799, Gilbert continued his mentor’s work and the journal appeared for the fi rst time under the new name. It was denoted as series1 number one out of eight series until today. Gilbert trans-lated, edited, and enriched many original foreign articles with didactic fi nesse to the benefi t of the German reader. He edited 76 volumes and served AdP for a full quarter-century. During his time physics did not have the rather well-defi ned topical focus and conformity that it has to-day. Thus, many articles appeared that nowadays belong to meteorolo-gy, climatology, geography, nautical science, or even biology. Still, most publications were in the core area of physics, like the translations of works by David Brewster, Michael Faraday or Joseph Gay-Lussac. Electrochem-

istry was a timely topic as well, with contributions from Johann Wilhelm Ritter, Sir Humphrey Davy and oth-ers. AdP reported on Ampere’s work about magnetic phenomena and their relation to moving electricity. However, the fi nal Ampere’s law from 1824 did not appear in the journal.From the fi rst issue of Annalen der Physik until volume 30 (1808) the journal appeared at the publisher Rengersche Buchhandlung (Renger’s Bookstore) in Halle. From volume 31 (1809) onwards, i.e. from series two of the AdP, Johann Ambrosius Barth (1760–1813) became the publisher of the journal [1]. As it turns out, the journal was published by the pub-lisher J.A. Barth for more than 180 years until 1992. Originally, Gilbert aimed for physics articles only, but this proved to be diffi cult and from 1819 to 1824 the name of the journal was extended to Annalen der Physik und der physikalischen Chemie 2.

The Poggendorff Era (1824–1877)

In 1820 Johann C. Poggendorff (1796–1877) studied natural sciences at the university in Berlin. Already in 1823 the young man was thinking about starting his own chemical-physics journal. After the unexpected death of Gilbert in 1824, Poggendorff real-ized that there was a chance of be-coming the editor of AdP and con-tacted the publisher Verlag J.A. Barth. Poggendorff knew exactly what he wanted: Either become editor of AdP or start his own journal that he in-tended to become the leading journal in physics and chemistry in Germany. He spoke with leading scientists to ensure their willingness to publish with him and used these connections to put pressure on the J.A. Barth pub-lisher. Surprisingly, his plan worked out. Poggendorff, only 28 years old, boldly managed to become editor of AdP and published his fi rst issue in

1824 under the name Annalen der Physik und Chemie3. The new editor arranged for the new name because he thought that both physics and chemistry could not be separated in a meaningful way. In the second half of the 19th cen-tury translations of foreign works lost their importance for the journal to the benefi t of original contributions and appeared only infrequently from that time onwards. This development was due to new physics institutes at Ger-man universities that were founded in the course of the Prussian university reform. These institutes had an ex-plicit order to promote and perform research, as opposed to the previous “Cabinette”. The fi rst institutes were founded in Leipzig and Göttingen and were inspired by the French Ecole Po-lytechnique. The amount of new fi nd-ings made Poggendorff publish in to-tal seven supplementary volumes in addition to the regular AdP volumes. In 1874, at the 50th anniversary of “his” Annalen, Poggendorff out-lined what he considered to be the most important topics: “Electrodynamics, induction, diamagnetism, photo-magnetism, thermochrosy, telegra-phy, photography, diffusion, fl uores-cence, spectral analysis, and me-chanical theory of thermodynamics”. From today’s point of view not all scientifi c-editorial decisions from Poggendorff were free of errors. For example, the works by Julius Mayer (1841) and Hermann v. Helmholtz (1847) about the principle of energy conservation were not accepted for publication in AdP. Also, Philip Reis’s invention of the telephone was not appreciated, neither was Sadi Car-not’s work from 1824 about the heat engine. Only nine years later, Benoît Clapeyron’s article appeared in AdP, where he explained Carnot’s thoughts in a more practical way. On the other hand, Rudolf Clausius’s concept from 1850 containing the fi rst and second law of thermody-

1 The term series is a translation of the

German ‘Folge’. However, in some cases a

‘Folge’ was subdivided into ‘Serien’, i.e.

subseries. In this essay series always

refers to ‘Folge’.2 Engl.: Annals of Physics and Physical

Chemistry3 Engl.: Annals of Physics and Chemistry

G.W. Fuchs: Annalen der Physik – a brief history of a living legendPh

ysic

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rum

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1 1© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

namics, which basically equals the concepts of Joule and Carnot, was well received and is one of the high-lights of AdP. In the area of electricity AdP re-ported about Ohm’s current–voltage law, as well as Faraday’s numerous experimental investigations that, for instance, contained his discovery of magnetic induction and its visualiza-tion through magnetic lines of forces, the behavior of dielectrics, the intro-duction of electric lines of forces and the discovery of diamagnetism. Poggendorff took care of the AdP for 53 years. He was the editor of 160 volumes. His Annalen der Physik und Chemie were simply called the Poggendorff Annalen. The heyday of the journal was between 1850 and 1920. During that time AdP devel-oped into one of the leading physics journals in Europe, if not the leading journal. With only a few exceptions, reading AdP suffi ced to keep oneself up-to-date in physics [2]. In addition, AdP was very popular and was not only subscribed by university librar-ies, but also by many secondary3 and technical schools.

The Heydays (1877–1914)

After Poggendorff’s death in 1877, the publisher Hans Barth appointed the 51-years-old Gustav H. Wiede-mann (1826–1899) as the new editor of AdP. Right at the beginning, AdP cooperated closely with the Physical Society of Berlin (PGzB)4, which was realized by appointing Hermann

Helmholtz as coeditor of the journal. The “extraordinary increase of the material [… ] and the fact, that in ad-dition to the Annalen also other comprehensive journals have been founded” (Helmholtz 1893), led to fewer and fewer articles being pub-lished in AdP that belonged to chem-istry, mineralogy, metrology and physical chemistry. As a conse-quence of this ongoing differentia-tion of the natural sciences into sub-fi elds, AdP began to focus on pub-lishing articles solely from physics. In addition, it became more diffi cult for authors to publish in AdP due to in-creased editorial selection criteria, as Helmholtz (1893) states: “The number of German manuscripts has gradually increased over time, so that by now, only a selection of those can be considered.” In 1893, Gustav’s son, Eilhard Wiedemann (1852–1928), became Co-Editor-in-Chief of AdP. After Helmholtz’s death in 1895 Max Planck (1858–1947) took over his role and supported the AdP as co-editor on behalf of the PGzB. Only four years later, in 1899, Gustav Wie-demann passed away in Leipzig. His son Eilhard, although himself a pro-fessor of physics at the University Er-langen, declined to take over full re-sponsibilities for the journal and fi -nally resigned as editor from AdP. Paul Drude (1863–1906), who nowa-days is considered as one of the pio-

neers of solid-state physics, became the new Editor-in-Chief of AdP. Thir-ty-six years old, talented, dynamic and already in permanent position as professor, Drude seemed to be an ide-al choice. In addition, he resided in Gießen, a medium-sized German town in Hesse, and thus did not be-long to the Berlin physicists commu-nity that in the rest of the German Reich was often perceived as too dominant. In 1900, with the change of editorship, Annalen der Physik und Chemie were renamed to Annalen der Physik and the journal has kept that name ever since. It was now the fourth series of AdP. One novelty was the in-troduction of an Advisory Board5 of fi ve professors, with Max Planck be-ing one of them. Despite Drude’s sci-entifi c brilliance he seemed to be rather less determined and less criti-cal in editorial matters. Very much to the displeasure of Planck, many man-uscripts of low quality passed Drude’s judgment. But it was also the time when legendary papers of modern physics were published in AdP. For example, Planck’s work about the en-ergy density distribution of the black-body radiation6 was published in AdP, where he also introduced the quan-tum of action ħ, the constant now named after him. This work was gen-erally regarded as the beginning of quantum theory. In 1905, Albert Ein-stein published seminal papers in

3 Here Secondary School is used as a

translation for the German Gymnasium.4 Orig.: Physikalische Gesellschaft zu

Berlin, PGzB5 Orig.: Kuratorium6 M. Planck, “Ueber das Gesetz der

Energieverteilung im Normalspektrum”,

Ann. Phys. (Berlin), 309(3), 553–563 (1901)

Hermann Helmholtz Albert Einstein (Credit: ÖND/Wien,

Bildnummer LSCII 0081-C)

Ann. Phys. (Berlin) 1-80 (2011)

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AdP that built the base for his later fame [3]. At that time, Einstein was mostly unknown to the relevant aca-demic circles and was not working in academia. It is thanks to Drude that Einstein’s works were published in AdP. However, Planck was one of the fi rst who realized Einstein’s genius and the importance of his work. Also, Planck helped this 26-year old patent offi ce clerk to start an academic ca-reer and later made great efforts in at-tracting Einstein to Berlin. On July 5, 1906 Drude committed suicide. The reasons remain un-known. Planck and Wilhelm Wien (1864–1928) became the new Edi-tors-in-Chief of equal rights. Howev-er, Wien was more involved in the day-to-day business of the journal, whereas Planck reserved the right to be consulted in all critical cases like manuscript rejections or revisions. At that time, Planck was already a well-known scientist and an accepted long-standing editor of the AdP. Planck appeared as an author of AdP already in 1881 and published pre-dominantly in this journal during his whole life. His last article in AdP ap-peared in 1941. In 1947, after the war,

he was involved in the reactivation of the journal [4]. Thus, Planck accom-panied AdP over 66 years – truly mo-mentous years of the journal.

Times of crisis (1914–1945) The political and economic crisis did not go unnoticed by the Annalen der Physik. As opposed to 3800 published pages in the year 1914, the number quickly reduced to 2100 pages in 1918 induced by the war – a decrease by 45%. In the following years, this number further decreased reaching a trough of 1800 pages in 1921. Besides the AdP and the 1899-founded jour-nal Physikalische Zeitschrift, a new journal Zeitschrift der Physik came into existence. In 1920, with the ac-ceptance of the German Physical So-ciety (Deutsche Physikalische Ge-sellschaft, abbrev. DPG), the publish-er Vieweg launched the new journal that soon turned out to be a strong competitor for AdP. The new journal managed to publish many important works of young researchers in the booming fi eld of quantum physics, like those from Max Born and Werner

Heisenberg. In view of these young scientists Annalen der Physik ap-peared no longer timely, because it was known that Planck and Wien were skeptical towards the new trends in quantum mechanics. Here the young Erwin Schrödinger was an exception. His four ground-breaking articles about the “Quantization as Eigenval-ue Problem”, published in 1926, ap-peared in AdP. In these works, Schrödinger outlined his wave-me-chanical approach that nowadays is a cornerstone of quantum mechanics. But there were also other examples of important contributions in quantum theory that appeared in AdP, like the works from Maria Göppert-Mayer from 1931 “Elementary processes with two quantum transitions”, Max Born and Robert Oppenheimer’s 1927 article “About the quantum theory of molecules” or Wolfgang Pauli’s 1922 contribution “On the model of the hy-drogen molecule ion”. In addition to the domestic competitors, interna-tional journals also appeared on the scene, like the Physical Review that al-ready used the modern peer-review procedure to quality check its manu-scripts. In 1928 Wien died. His successor Eduard Grüneisen (1877–1949) start-ed to serve the AdP in 1929 and this marks the start of a new AdP series, the fi fth. It was the time of Hitler’s rise and with the advent of the Nazis a turning point in German history had been reached, not only for the Anna-len der Physik but also for German science in general. A fl ood of German emigrants, persecuted because of their political opinion or their race, left the country, among them many authors of AdP. The myriads of per-sonal tragedies that took place was not directly refl ected in the AdP, how-ever, the absence of many important articles was signifi cant – mostly those from Jewish authors. Accordingly, the decrease in page numbers until 1939, i.e. already in prewar time, was signifi -

Editors-in-Chief of Annalen der Physik from 1947 until today

(top, from left) Eduard Grüneisen, Friedrich Möglich, Hans Kopfermann, Gustav Richter

(Photo: Th. Richter),

(below, from left) Hans-Jürgen Treder (Photo: bpk/Gerhard Kiesling), Wilhelm Walcher,

Bernhard Mühlschlegel, Ulrich Eckern


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cant. Planck and Grüneisen managed to keep the politics out of the AdP dai-ly business as much as possible. Thus, even in the late 1930s works from Lise Meitner, Rudolf Ladenburg, Paul Ewald, and Richard Gans appeared in Annalen der Physik. But over time, fewer and fewer emigrants published in German and preferred foreign, mostly English language journals. In spring 1938 Debye, together with oth-er guest editors, organized a special issue in honor of Arnold Sommer-feld’s 70th birthday. In this context, the publisher approached the organ-izers with the request to only publish articles from “Arian” authors. This un-paralleled case of open racism trig-gered protests and outrages by many physicists and most prominently by Wolfgang Pauli. But in the end, the special issue appeared as wanted by the publisher. With the outbreak of the second world war in 1939 the number of con-tributions again reduced signifi cant-ly. Shortly after the assassination at-tempt on Hitler on July 20, 1944 all companies and factories that had no direct war-relevance were closed and large parts of the population were obliged to work in the armaments in-dustry. In the course of these ‘total warfare’ measures the publishing house J.A. Barth in Leipzig was also closed down and with that Annalen der Physik de facto ceased to exist.

Divided but together (1946–1990)

Germany was broken, the war lost. The severity of destruction, ongoing resentments and loss of manpower made a restart diffi cult. Grüneisen, who lived in Marburg, which was sit-uated in the Western zone of oc-cupation, had no hope to get permis-sion from the Western allied powers to relaunch the Annalen der Physik. After Max von Laue’s discharge from

Farm Hall (U.K.), where he had been detained as a prisoner of war, and his return to Germany in 1946, he imme-diately started with the rebuilding and organization of the German sci-ence and physics program. Laue was involved in many projects East and West of the ideological border and in-dependent of the political landscape within Germany. He recommended his former PhD student Friedrich Möglich (1902–1957) as co-Editor-in-Chief of AdP. The appointment of a Western and an Eastern Editor-in-Chief of AdP remained common practice for the journal until 1992, i.e. until shortly after the German reuni-fi cation. Annalen der Physik adhered to the conviction that both parts of Germany belong together. Only dur-ing a short period, between 1950 and 1951, was this new tradition disrupt-ed. On August 1, 1946 the Soviet mili-tary administration granted permis-sion for the restart of AdP: “For the benefi t of German science and for the benefi t of humanity and internation-al understanding”. But the license came with some restrictions. AdP was not allowed to publish certain branches of physics like nuclear physics, semiconductor physics, high-frequency technology and elec-tronics. It was now series six of the Annalen der Physik. Grüneisen and Möglich, as well as Planck who died in 1947 shortly before the fi rst new is-sue appeared, were the fi rst editors after the war. The journal clearly emerged debilitated from the past crisis and published only 500 pages per year, only occasionally were 1000 pages realized. Though AdP still pub-lished articles of high quality and long-term importance until the 1940s and 1950s – at least from time to time – the heydays were clearly over. AdP lost ground compared to other inter-national journals like Physical Review that now took the lead. Relevant na-tions in the fi eld of physics were the

USA and the Soviet Union. As op-posed to the prewar situation there were almost no contributions from non-German authors in AdP which, continued to publish in German lan-guage. The successor to Grüneisen, who died in 1949, was Hans Kopfermann (1895–1963) who started to serve AdP in 1952 as the West German editor. After Friedrich Möglich’s death in 1957 Gustav Richter (1911–1999) be-came the new East German editor. This marked the beginning of a new series, series seven of Annalen der Physik. In East Germany the Socialist Unity Party was worried about its se-curity due to an allegedly ongoing mass migration of its citizens into the Western zones. It was decided to for-tify the national borders and the building of the Berlin Wall in August 1961 marked the beginning of a thir-ty-year physical isolation and separa-tion of West and East Germany. The AdP publishing house remained in Leipzig, i.e. in East Germany. Regard-less of the new political situation AdP followed its policy to appoint two Ed-itors-in-Chief from East and West Germany, respectively. The successor to Kopfermann was Wilhelm Walcher (1910–2005) for the Western Federal Republic of Germany and Hans-Jür-gen Treder (1928–2006) for the East-ern German Democratic Republic (GDR).

Times of reunifi cation (1992–2011)

With the end of the cold war and of communist regimes in the former Eastern Bloc a new era began for AdP. The journal was restructured in 1992. The Hüthig GmbH, Heidelberg/Ger-many took over AdP from the former J.A. Barth publishing house. At that time the J.A. Barth publisher was al-ready state property of the GDR since 1988. However, Hüthig published

Ann. Phys. (Berlin) 1-80 (2011)

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AdP still under the former publisher’s trademark ‘J.A. Barth Verlag’ until 1998. Once again, a new series start-ed: Series No. 8. Bernhard Mühl- schlegel (1925–2007) from Cologne/Germany became the new Editor-in-Chief of the AdP in 1992. The Adviso-ry Board was dissolved and replaced by three international co-editors, among them Alexei Abrikosov who later won the 2003 Nobel Prize in Physics. From that time onwards AdP published solely in English. The jour-nal intended to be globally more visi-ble, and attractive for young scien-tists. However, the manuscript sub-missions fell behind expectations and gave rise to concerns about the future of the journal. In 1998, Mühlschlegel was 73 years old and handed over editorship to the theoretical solid state physi-cists Ulrich Eckern from Augsburg. In the same year, in the course of a re-structuring measure Hüthig sold ten of its academic journals to the pub-lisher Wiley-VCH (Weinheim/Berlin) Among these journals was AdP. At the same time, the 120-year old coopera-tion with the DPG (and PGzB) ended. Now, for the fi rst time, and in addi-tion to the print issues, the journal appeared in electronic form. Further-more, old print issues from 1799 on-wards were scanned and refurbished, and have been available in electronic form since 2006. In the year 2005, in honor of Einstein’s annus mirabilis AdP introduced the Einstein Lectures. With this distinction contributions from important prize winners, like the Nobel Prize winners Theodor Hänsch (MPQ Garching), Roy J. Glau-ber (Harvard), or Peter Grünberg (FZ Jülich) were highlighted.

The new AdP (2012)

Ulrich Eckern will end his term as AdP Editor-in-Chief at the end of 2011, after 14 years of successful lead-

ership. From January 2012 onwards the journal will be run directly by the publisher with an in-house editorial team with Guido W. Fuchs as the new Editor-in-Chief. The former Editorial Board members Friedrich Hehl, Bernhard Kramer, Gerd Röpke and Andreas Wipf will continue to serve for AdP as Honorary Advisory Board members. They will be joined by Ul-rich Eckern and Ingo Peschel, who is currently an Advisory Board member. Ingo Peschel, for example, had been crucial and supportive in diffi cult times, right after the German reunifi -cation, during the start of the 8th se-ries in 1992, but also later when Bern-hard Mühlschlegel handed over edi-torship to Ulrich Eckern in 1998. The scope of the journal is still to publish general physics – in all its as-pects. This also includes topics in ap-plied physics. Letter articles will be newly introduced. The Einstein Lec-tures will be revived and continued. The section ‘100 years ago’ will be ceased. Translations of originally German articles into English will not be published as separate articles in current issues but appear as addi-tional material to the original articles. Historical essays Then & Now will be introduced with contributions from the Max Planck Institute for the His-tory of Science (Berlin/Germany). In addition, there will be an Expert Opinion section where authors can comment on recent or copublished articles in AdP in a brief essay form. In 2012, AdP will have changed completely. It is set up to serve an in-ternational readership, and although AdP has experienced several renam-ings, it is not considered to change the title into an English one because meanwhile the name Annalen der Physik belongs to our international cultural heritage. To emphasize the continuation of scientifi c publishing from its fi rst ap-pearance as Annalen der Physik in 1799 until today, already in 2009, the

volume counts were offi cially changed [5]. Previously, volume numbers restarted with the begin-ning of a new series (Folge), e.g. the latest series No. 8 started with vol-ume 1 in 1992. Now, all volumes are counted from the fi rst one in 1799, so that volume 523 refers to the volume published in 2011. The relaunch of AdP in 2012 will not trigger a new se-ries. Instead this concept of series is abandoned. The progress of AdP will simply be denoted by its volume number. Shaping the future of a journal is not completely in the hands of the editor or publisher. The most crucial input has to come from the scientifi c community. It remains to hope that AdP will be well received by the phys-icists and interested scientists. An-nalen der Physik has good intentions. It wants to serve the physics commu-nity. It has always been a journal act-ing as a mirror of current research and this is also its guiding theme for the future: Knowing Annalen der Physik means knowing physics.

Acknowledgements. The author thanks Dieter Hoffmann and Ulrich Eckern for their kind support and sugges-tions.

References

[1] K. Wiecke, 200 Jahre Johann Ambrosius Barth (Verlag Johann Ambrosius Barth, Leipzig, 1980), pp. 17–30. [2] F. Hund, Die Annalen im Wandel ihrer Aufgabe, Ann. Phys. (Berlin), 502(4), 289–295 (1990).[3] J. Renn (ed.), Einstein’s Annalen Papers (Wiley-VCH, Weinheim- Berlin, 2005).[4] D. Hoffmann (ed.), Max Planck: Annalen Papers (Wiley-VCH, Weinheim-Berlin, 2008).[5] G. W. Fuchs and U. Eckern, Ann. Phys.

(Berlin) 522(6), 371 (2010).


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Ann. Phys. (Berlin) 523, No. 7, 580–581 (2011) / DOI 10.1002/andp.201110469

EXPERT OPINION

A theoretical approach to iron-based superconductors*

Aldo H. Romero and Matthieu J. Verstraete

Very recently, in 2008, Hosono and collaborators [1] reported a new su-perconducting material with a tran-sition temperature of 26 K. Besides setting the right tone in renewing the race to increase the supercon-ducting critical temperature, Ho-sono gained attention because the material was based on a layered iron arsenide compound. By substitut-ing some oxygen in the starting iron arsenide structure, giving LaAsOFe, and after fl uorination doping, the new superconducting material is obtained. This composition is strange - the presence of iron in a superconductor implies there is an interplay between magnetism and superconductiv-ity. As we all learn in undergradu-ate physics, the two effects should be exclusive: magnetism tries to break Cooper pairs and inhibits the appearance of superconductivity, whereas superconductivity tends to expel magnetic fi elds. The fact that now a single material is able to show both of these phenomena is quite a revolution: the case for magnetic ef-fects (spin fl uctuations) in cuprate superconductors was much less clear-cut. In only 3 years the paper has received almost 2000 citations, an average of 700 per year - clearly a very hot topic! At low fl uorine doping the iron atoms couple antiferromagnetically,

but as the doping concentration is increased the magnetic coupling decreases; it is the weakening of the antiferromagnetic order which allows the rise of superconductiv-ity. Very quickly, a full family of iron compounds was reported in the lit-erature, with the same basic structure and reaching critical temperatures of 38K [2]. Since the report by Hosono, the related compounds were con-structed by trying to keep the layer arrangement, but with distortions, or using other chemical species. The original work was based on arsenic, but by now we know that similar be-havior can be obtained by exchang-ing As with P or Se or Te. In most of those structures, the iron plane is not perfectly fl at and binds to the period-VI element in the formulae, creating tetrahedra. This is in stark contrast to the octahedra in cuprate supercon-ductors. In particular, it was demon-strated that the higher the symmetry in the tetrahedral structure the larger the increase of the superconducting temperature [3,4]. Most calculations indicate that electrons fl ow along the planes formed by the irons, which probably points (surprise!) to elec-trons coupling to spin fl uctuations. It was initially speculated that the physics of this new superconductor is related to that of the cuprates, but

soon demonstrated that the situa-tion was much more complex. Both systems are antiferromagnetic, but LaOFeAs is also a spin-density wave metal, with electrons which are much more delocalized than in the cuprates. Another difference appears in experimental measurements in-dicating that the bands around the electronic gap are nicely symmetric in the pnictides, while they are not in the cuprates. The asymmetry in the cuprate gap was explained using the d-wave argument [5], which does not apply to this new system. The sym-metry around the energy gap is still under discussion, but it is strongly system dependent. What is clear is the interrelation between the elec-tronic structure, specifi cally the to-pology of the bands around the Fer-mi energy, and the superconducting behavior. To make the puzzle more complicated, the electron-phonon interaction is fairly small and clearly insuffi cient to explain the global be-havior.Since the discovery of superconduc-

* Published in Ann. Phys. (Berlin) 523, No. 7,

580–581 (2011); slightly modifi ed for the

sample issue, with permission from the

authors.

Figure 1 The set of fi ve Fe d-like Wannier

orbitals (downfolded and orthonormalized

NMTOs) which span the fi ve LaOFeAs bands

extending from -1.g eV below to 2.2 eV above

the Fermi level. For more information see [6].

A13



A.H. Romero / M. J. Verstraete: A theoretical approach to iron-based superconductorsPh

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tivity in pnictides, a large number of theoretical papers have appeared, trying to explain the properties of these new superconductors, and in particular trying to reconcile the dif-ferent theoretical approaches with the experimental observations. It is very hard to follow the full literature because of the plethora of different systems and approximations. A com-plete understanding of this problem has yet to come out. The paper by Andersen and Boeri in the present volume [6] presents a comprehen-sive approach of the band structure by using the same methodology and explaining, quite pedagogically, the contributions of the different orbitals to the energy bands. They also gener-ate the corresponding tight binding parameters, which allows them to get deeper insight into the spin wave behavior and the interplay with the

structural symmetry. They fi nish the paper with a discussion on the con-tribution of the spin polarization to some of the observed properties, in particular considering spin spirals. This remarkable contribution gives the most comprehensive theo-retical overview to date of the elec-tronic structure of the iron pnictides compounds and how their properties could give rise to superconductivity.

Aldo H. Romero

CINVESTAV, Unidad Querétaro, Libramiento

Norponiente 2000, Real de Juriquilla,

Querétaro CP 76230, Mexico


Matthieu J. Verstraete

Department of Physics, Universite de Liege,

Av du 6 Aout, 17, B-4000 Liege, Belgium

E-mail: [email protected].

References

[1] Y. Kamihara, T. Watanabe, M. Hirano, and H. Hosono, J. Am. Chem. Soc. 130, 3296 (2008).[2] M. Rotter, M. Tegel and D. Johrendt, Phys. Rev. Lett. 101, 107006 (2008).[3] C.H. Lee, A. Iyo, H. Eisaki, H. Kito, M.T. Fernandez-Diaz, T. Ito, K. Kihou, H. Matsuhata, M. Braden and K. Yamada, J. Phys. Soc. Jpn. 77, 083704 (2008). [4] D.C. Johnston, Adv. Phys. (Berlin) 59, 803 (2010).[5] A.D. Christianson, E. A. Goremychkin, R. Osborn, S. Rosenkranz, M.D. Lumsden, C.D. Malliakas, I.S. Todorov, H. Claus, D.Y. Chung, M.G. Kanatzidis, R. I. Bewley and T. Guidi, Nature 456, 930 (2008).[6] O.K. Andersen and L. Boeri, Ann. Phys. (Berlin) 523, 8 (2011).

A14


EXPERT OPINION

Lattice distortions in KCuF3: a paradigm shift?Eva Pavarini

KCuF3 is the paradigmatic compound for the co-operative Jahn-Teller effect.

But do we really know its structure?

Co-operative Jahn-Teller distortions are ubiquitous. But do we really know where they come from? Usually they go along with other distortions and ordering phenomena, which make it diffi cult to identify the actual driving mechanism. KCuF3 is believed to be a beautiful exception, the cleanest re-alization of a co-operative Jahn-Teller system. Its CuF6 octahedra are slightly compressed along z, and have a long (l) and a short (s) CuF bond in the xy plane. The electronic confi guration of Cu is 3d

9, a single hole in the eg states;

the distortions split the otherwise de-generate eg orbitals, and the hole goes into the |s2−z

2 ⟩ state. The spatial alter-

nation of l and s bonds in all direction produces the orbital pattern shown in Fig. 1. Despite its simple structure, the origin of the co-operative Jahn-Teller distortion in KCuF3 remained a puz-zle. Is it driven by electron-phonon coupling or by many-body superex-

change [1]? Early-on static mean-fi eld LDA+U calculations showed that the stability of the Jahn-Teller distor-tion (total energy gain) is strongly enhanced by Coulomb repulsion, a result recently confi rmed by dynami-cal mean-fi eld (DMFT) calculations [2]. Does this mean that manybody super-exchange is driving the co-operative distortion? The answer remained elusive for half a century. Only recently, by using DMFT and a new approach to separate super-exchange and electron-phonon cou-pling effects, the puzzle was fi nally solved. It was shown [3,4] that many-body super-exchange alone gives a critical temperature of about 350 K, very large, but far too low to explain experimental facts: the co-operative Jahn-Teller distortion persists up to TOO ∼ 800 K, and probably at even higher temperatures. Hence the static Jahn-Teller distortions must be driv-en by electron-phonon coupling. But, resourceful as ever, KCuF3 has new surprises in store for us. Far below TOO, a new, dynamic, phase appears to manifest itself. The work of Dei-senhofer et al. [5] reports a splitting of the infrared-active phonon mode Eu(3) at ∼ 150 K, indication of symme-try lowering; an analogous splitting of the Eg(2) mode is seen in Raman scat-tering slightly above the Neel tem-perature, TN = 40 K. This makes KCuF3 remarkable in yet another way – as a compound whose low temperature properties are dominated by dynam-ics. Indeed, the evolution of the Eg(2) and Eu(3) modes with temperature

could be understood if strong lattice fl uctuations around a lower symme-try orthorhombic lattice were present. Remarkably, above TN KCuF3 behaves as a one-dimensional anti-ferromag-netic Heisenberg chain, characterized by strong spin fl uctuations. Strong lattice fl uctuations can couple to these spin fl uctuations, and give rise to novel spin-lattice effects, even in the paramagnetic phase. On the time-scale of lattice fl uctuations, spin orbit could give rise to a dynamical Dzya-loshinsky-Moriya interaction, which on lowering the temperature eventu-ally becomes static and probably cru-cial in establishing three-dimensional magnetic order. New experiments and theoretical work will tell us if this dynamical scenario is correct.

Eva Pavarini

Institute for Advanced Simulation,

Forschungszentrum Jülich, 52425 Jülich,

Germany


References

[1] K.I. Kugel and D.I. Khomskii, Zh. Eksp. Teor. Fiz. 64, 1429 (1973) [Sov. Phys. JEPT 37, 725 (1973)].[2] I. Leonov et al., Phys. Rev. Lett. 101, 096405 (2008).[3] E. Pavarini, E. Koch, A.I. Lichten- stein, Phys. Rev. Lett. 101, 266405 (2008)[4] E. Pavarini and E. Koch, Phys. Rev. Lett. 104, 086402 (2010).[5] J. Deisenhofer et al., Ann. Phys. (Berlin) 523 (8-9), 645 (2011)Figure 1 Orbital order in KCuF3 [3].


A15

Ann. Phys. (Berlin) 523, No. 7, 580–581 (2011) / DOI 10.1002/andp.201110469

www.ann-phys.org © 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim1 8A16


On the multi-orbital band structure and itinerant magnetism of iron-based superconductors* Ole Krogh Andersen** and Lilia Boeri

Received 7 November 2010, accepted 26 November 2010,

Published online 4 January 2011

* Published in Ann. Phys. (Berlin) 523, No. 1–2, 8–50 (2011);

modifi ed for the sample issue, with permission from the

authors.** Corresponding author:


Max-Planck-Institute for Solid State Research,

Heisenbergstrasse 1, 70569 Stuttgart, Germany

Ann. Phys. (Berlin) 523, No. 1-2, 8-50 (2011) / DOI 10.1002/andp.201000149

This paper explains the multi-orbital band structures and itin-

erant magnetism of the iron-pnictide and chalcogenide super-

conductors. We fi rst describe the generic band structure of a

single, isolated FeAs layer. Use of its Abelian glide-mirror group

allows us to reduce the primitive cell to one FeAs unit. For the

lines and points of high symmetry in the corresponding large,

square Brillouin zone, we specify how the one-electron Hamil-

tonian factorizes. From density-functional theory, and for the

observed structure of LaOFeAs, we generate the set of eight

Fe d and As p localized Wannier functions and their tight-bind-

ing (TB) Hamiltonian, h(k). For comparison, we generate the set

of fi ve Fe d Wannier orbitals. The topology of the bands, i. e.

allowed and avoided crossings, specifi cally the origin of the d6

pseudogap, is discussed, and the role of the As p orbitals and

the elongation of the FeAs4 tetrahedron emphasized. We then

couple the layers, mainly via interlayer hopping between As pz

orbitals, and give the formalism for simple tetragonal and

body-centered tetragonal (bct) stackings. This allows us to ex-

plain the material-specifi c 3D band structures, in particular the

complicated ones of bct BaFe2As2 and CaFe2As2 whose interlayer

hoppings are large. Due to the high symmetry, several level

1 Introduction

The first report of superconductivity in an iron pnictide, specifically in F-doped LaOFeP below 5K in 2006 [1, 2], was hardly noticed and only two years later, when F-doped LaOFeAs was reported to su perconduct below 28 K, the potential of iron pnictides as high-temperature superconducing materials was realized [3]. Following this discovery, more than 50 new iron superconductors with the same basic structure were discovered [4] with Tc reaching up to 56 K [5]. This structure is shown in Fig. 1 for the case of LaOFeAs. The common motive is a planar

inversions take place as functions of kz or pressure, and linear

band dispersions (Dirac cones) are found at many places. The

underlying symmetry elements are, however, easily broken by

phonons or impurities, for instance, so that the Dirac points are

not protected. Nor are they pinned to the Fermi level because

the Fermi surface has several sheets. From the paramagnetic

TB Hamiltonian, we form the band structures for spin spirals

with wavevector q by coupling h(k) and h(k+q). The band struc-

ture for stripe order is studied in detail as a function of the ex-

change potential, Δ, or moment, m, using Stoner theory. Gap-

ping of the Fermi surface (FS) for small Δ requires matching of

FS dimensions (nesting) and d-orbital characters. The interplay

between pd hybridization and magnetism is discussed using

simple 4×4 Hamiltonians. The origin of the propeller-shaped

Fermi surface is explained in detail. Finally, we express the

magnetic energy as the sum over band-structure energies and

this enables us to understand to what extent the magnetic en-

ergies might be described by a Heisenberg Hamiltonian, and to

address the much discussed interplay between the magnetic

moment and the elongation of the FeAs4 tetrahedron.

FeAs layer in which the Fe atoms form a square lat tice, tetrahedrally coordinated with As atoms placed alternat-ingly above and below the hollow centers of the squares.

1

HighlightsReview

Article


Instead of As, the ligand could be another pnictogen (P) or a chal-cogen (X=Se or Te), but for sim-plicity, in this paper we shall refer to it as As. These superconduc-tors are divided in four main fam-ilies depending on their 3D crys-tal structure [6]: The iron chal-cogenides are simple tetragonal (st) with the FeX layers stacked on top of each other (11 family). The iron pnictides have the FeAs layers separated by alkali met-als (111 family), or by rare-earth oxygen/fluoride blocking layers (1111 family as in Fig. 1), in st stacking, or by alkali-earth met-als (122 family) in body-centered tetragonal (bct) stacking. Iron-based superconductors share some general physical properties, although the de-tails are often specific to families, or even to compounds. With the exception of LiFeAs, the undoped compounds are spin-density wave (SDW) metals at low tempera-ture with the Fe spins ordered anti-ferromagnetically between nearest neighbors in the one direction and fer-romagnetically in the other, thus forming stripe or dou-ble-stripe (FeTe) patterns. The values of the measured magnetic moments range from 0.4 μB/Fe in LaOFeAs [7], to ∼ 1μB in BaFe2As2 compounds, to over 2 μB in doped tellurides [9–11]. At a temper ature above or at the Neel temperature, which is of order 100K, there is a tetragonal-to-orthorhombic phase transition in which the in-plane lattice constant contracts by 0.5-1.0% in the direction of ferro magnetic order. Superconductivity sets in when the magnetic order is suppressed by pressure, electron or hole doping, or even isovalent doping on the As site, and at a much lower temperature. Both super-conductivity and magnetism are found to depend cru-cially on the details of the crystal structure; for example is it often observed that the highest Tc s occur in those compounds where the FeAs4 tetrahedra are regular [8]. Critical temperatures range from a few K in iron-phos-phides to 56 K in SmOFeAs. The variations in the phonon spectra are, however, small and seem uncorrelated with Tc . This, together with the proximity of magnetism and superconductivity in the phase diagram, was a first in-dication that the superconductivity is unconventional. A stronger indication seems to come from the symmetry of the superconducting gap, which is currently a strongly debated issue [12]. Depending on the sample, and on the experimental technique, multiple gaps with s symmetry

and various degrees of anisotropy – but also of nodes – have been reported [9–11]. It now seems as if the gap symmetry is not universal, but material specific in these compounds. Current understanding of the basic electronic struc-ture has been reached mainly by angle-resolved pho-toemission (ARPES) [13–22], quantum oscillation, and de-Haas-van-Alphen (dHvA) experiments [23– 29] in combination with density-functional (DFT) calcula-tions [30–41]. All parent compounds have the electronic configuration Fe d6 and are metallic. In all known cases, the Fermi surface (FS) in the para magnetic tetragonal phase has two concentric hole pockets with dominant dxz /dyz character and two equivalent electron pockets with respectively dxz /dxy and dyz /dxy character. A third hole pocket may also be present, but its character, dxy or d

3z2

−1, as well as the sizes and shapes of all sheets, vary

among different families of compounds, and, within the same family, with chemical composition and pres sure. In all stoichiometric compounds, the volumes of the hole sheets compensate those of the electron sheets. The magnetically stripe-ordered phase remains metallic, but

Figure 1 The layered structure of simple tetragonal LaOFeAs. The 3D

primitive cell contains one Fe2As2 and one La2O2 layer, each contain-

ing three sheets: a square planar Fe (red) or O (blue) sheet sand-

wiched between two planar As (green) or La (yellow) sheets. c =

874pm. The coordination of Fe with As, or O with La, is tetrahedral. x

and y are the vectors between the Fe-Fe or O-O nearest neighbors

(separated by α = 285 pm) and X and Y are those between As-As or

La-La nearest neighbors in the same sheet. The directions of those

vectors we shall denote x, y, X , and Y .

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the FS becomes much smaller and takes the shape of a propeller [16] plus, possibly, tiny pockets [23]. Given the strong tendency to magnetism, and the low value of the calculated electron-phonon in teraction [42–44], spin fluctuations are the strongest candidate for mediating the superconductivity. Al ternative scenarios have been proposed, in which superconductivity is due to magnetic interactions in the strong-coupling limit, polarons, or orbital fluctuations [45]. Models for spin fluctuations are based on the weak-coupling, itinerant limit, with superconductivity related to the presence of strong nesting between hole and electron sheets of the paramagnetic Fermi surface, which is also held respon-sible for the instability towards magnetism [40, 41, 46]. This possibility has been investigated using more ore less sound models of the band structure, combined with dif-ferent many-body methods (RPA, FLEX, frG, model ME calculations) which do seem to agree on a picture with competing instabilities towards mag netism and super-conductivity [40, 41, 47–58]. The superconducting phase should be characterized by multiple gaps, with s and d symmetries almost degenerate. Modifying the shape and orbital characters of the different sheets of the Fermi sur-face by doping, pressure, or chemistry can influence the leading instability and affect the structure of the gap. As a result, a reasonable, qualitative picture of the materi als trend, such as the dependence of Tc and gap symmetry on the tetrahedral angle, has evolved [50,55]. Most ex-perimental evidence seems to support this picture, but several points remain controversial. A badly understood issue is how to include 3D effects, which is particularly serious for the bct 122 com pounds. Another problem concerns the magnetism: While it is true that spin-polarized DFT (SDFT) calcula tions repro-duce the correct atomic coordinates and stripe-order of the moment, the magnitude of the moment is, except in doped FeTe, at least two times larger than what is mea-sured by neutron scattering, or inferred from the gaps measured by ARPES [28, 59], dHvA, and optics [60], albeit much smaller than the saturation moment of 4 μB /Fe. Suppressing the too large moments in the calculations will, however, ruin the good agreement for the structure and the phonon spectra [43, 61–64]. This over-estimation of the moment is opposite to what was found 25 years ago for the superconducting cuprates where the SDFT gave no moment, but is typical for itinerant magnets close to a magnetic quantum critical point (QCP) [61]. The magnetic fluctuations in time and space have been described [65] using a localized Heisenberg model with competing ferro-and antiferromagnetic interactions be-tween respectively first and second-nearest neighbors, but to reconcile this model with the partly metallic band

structure is a problem [46, 66–69]. Another possible solu-tion of the moment problem in SDFT is that moments of the predicted size are present, but fluctuate on a time scale faster than what is probed by the experi ments [61]. In fact, two recent studies of realistic, DFT-derived multi-band Hubbard models solved in the dynamical mean-field approximation (DMFT) show that the magnetism has two different energy scales [70, 71]. It is therefore possible that the electronic correlations after all do play a role in these multi-band, multi-orbital materials [72, 73]. Experiments and calculations have revealed a marked interplay between the details of the band structure and the superconducting properties. Most of these observa-tions are empirical and we feel that there is a need to ex-plain the origin of such details. In this paper, we therefore attempt to give a self contained, pedagogical description of the paramagnetic and spin-polarized band structures. Specifi cally, we discuss the Fe d As p band-structure to-pology, causing the pseudogap at d6 as well as numer ous Dirac cones, the interlayer hopping in the simple-tetrag-onal and body-centered-tetragonal struc tures, the spin-spiral band structures, and the band-resolved magnetic energies. In all of this, the co valency between Fe d and As p is found to play a crucial role. Applications to super-conductivity are beyond the scope of the present paper. In Sect. 2 we explain the structure of a single, isolated FeAs layer and use the glide mirror to reduce the primi-tive cell to one FeAs unit and have k running in the large, square Brillouin zone (BZ) known from the cuprates. Halving the number of bands will prove important when it comes to understanding the multi-orbital band struc-ture. In Sect. 3 we show that this band structure may be generated and un derstood from downfolding [74], of the DFT Hilbert space for LaOFeAs to a basis set consisting of the five Fe d, localized Wannier orbitals, or – as we prefer – including explicitely also the three As p orbitals. Even the latter 8 × 8 tight-binding (TB) Hamiltonian, h (k), has long-ranged pp and pd hoppings due to the diffuseness of the As p orbitals, and its accurate, analytical matrix elements are so spacious that they will be published at a different place [75]. The crucial role of the As p orbit-als for the low-energy band structure, the electron bands in particular, and the presence of a d 6 pseudogap is em-phasized. The different sheets of the FS are discussed. In Fig. 2 we show the factorization of the Bloch waves along the lines and points of high symmetry in the large BZ. The high symmetry of the single, tetragonal layer allows many bands to cross and leads to linear dispersions, and even to Dirac cones. Our understanding of this generic band structure of a single layer then allows us to discuss standard DFT calculations for specific materials. This is done in Sect. 4, where we first see that increasing the As


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Figure 2 Upper-right quarter of the large Brillouin zone (BZ) for the

glide mirror space group of the single FeAs layer (black), the factori-

zation of the band Hamiltonian (blue), and the LaOFeAs Fermi sur-

face (red). The BZ for merely the translational part of the space group

has half the area and is folded-in as indicated by the dashed black

lines. In order to distinguish the corners, M, and edge mid points, X,

of these two square BZs, we use an overbar for the large BZ. Hence

Γ= Γ , M= X , and X is the com mon midpoint of the XY and ΓM

-lines. The folding causes all three hole pockets to be centered at Γ

and the two electron pockets be centered at M with their axes

crossed. The blue boxes along the lines of high symmetry contain

the orbitals whose Bloch-sums may hybridize (belong to the same

irreducible representa tion). At the high-symmetry points, this fac-

torization of the Hamiltonian into diagonal blocks is as follows: Γ[xy][XY] [Xz, X] [Yz,Y][zz,z], X [xz][xy,y] [yz, z], [XY, zz,x], and M [XY ][zz][xy, z][Xz, X ][Yz, Y ]. With of ten used notations [41],

the inner and outer sheet of the M-centered xz/yz-like hole pock-

ets are respectively α1 and α2 while the X and Y -centered xy/xz and xy/yz-like electrons sheets are respectively β1 and β2, and the Γ -centered xy-like hole pocket is γ.

height moves an anti bonding pz /dxy level down towards the degenerate top of the dxz /dyz hole bands, with which it cannot cross, and thereby causes the inner, longitudinal band to develop a linear dispersion. In-terlayer hopping is shown to pro-ceed mainly via the As pz orbital and to have a strength and (kx, ky)

-dependence which depends on the material family. This hopping is strongest for the bct structure where the As atoms in neighboring layers face each oth-er. In st SmOFeAs and for kz at the edge of the 3D BZ, the antibonding pz /dxy level reaches the top of the hole bands and forms a Dirac cone together with the longitudinal hole band. In LiFeAs and FeTe the Dirac point is inside the BZ. The interlayer hopping not only causes the As pz -like 2D bands to disperse with kz , but also folds the bands into the conventional, small BZ, i. e. it cou-ples h (k)and h (k + πx + πy). The formalism for interlayer hopping is given in Sect. 4.2, and its increasing

influence on the band structures of BaFe2As2, CaFe2As2, and collapsed CaFe2As2 is shown and explained, for the first time, we believe. In CaFe2As2, we find that the nearly linear dispersion of the dxy /pz -like electron band has de-veloped into a full Dirac cone. The effects of spin polarisation on the generic 2D band structure are discussed in Sect. 5. We con sider spin spirals which have a translationally invariant magnitude but a spiralling orientation which is given by q. Their band Hamiltonian possesses translational symmetry both in configurational and in spin-space, but indepen-dently of each other as long as spin-orbit coupling is ne-glected. The spin spiral therefore simply couples h (k) to h (k + q) , regardless of whether q is commensurable or not. For h (k) we use the DFT pd Hamiltonian derived in Sect. 3. In order to keep the analysis transparent and amenable to generalization, we shall treat the exchange coupling using the Stoner model rather than full SDFT. This has the avantage that it decouples the band struc-ture and self-consistency problems, so that we can study the band structure as a function of the exchange poten-tial, Δ. In Sect. 5.2 we discuss the bands and FSs for the observed stripe order. As long as the moment is a linear function of Δ, gapping requires matching of d-orbital

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O. K. Andersen / L. Boeri: Band structure and magnetism of iron-based superconductors


characters as well as FS dimensions (nesting). For larger moments, and ferromagnetic order in the x direction, the FS is different and shaped like a two-blade propeller in the ky direction. It is formed by crossing dxy /pz -dyz↓ and dzz↓/dXY ↑ bands, which cannot hybridize along the line through the blades and the hub. The resulting Dirac cone has been predicted before [76] and also observed [29, 77]. The interplay between pd hybridization and magne-tism is discussed using sim ple, analytical 4 × 4 models. In Sect. 5.3 we first show the static spin-suceptibility, m (Δ)/Δ, calculated for stripe and checkerboard orders as functions of the electron doping in the rigid-band approxima tion. The low-moment solution – maybe for-tuitously – resembles the behaviour of the observed mo-ment as a function of doping and q. We then discuss the electronic origin of the magnetic energies and first show how the magnetic energy may be interpreted as the dif-ference between double-counting corrected magnetic and non-magnetic band-structure energies. This directly relates the magnetism to the band structure and we specifically look at the origin of the magnetic energy. We find that the mag netic energy gain is caused by the cou-pling of the paramagnetic dxy hole and dxy /pz electron bands, as well as by that of the dxz parts of the two other electron and hole bands. The Fermi-surface contribu-tions to the magnetic energy are comparatively small. We can then explain why increasing the distance between the As and Fe sheets increases the stripe-ordered mo-ment, and vice versa. At the end, we compare our results with those of fully self-consistent SDFT spin-spiral calcula tions of moments and energies as functions of q and doping in the virtual-crystal approximation, for LaO1−xFxFeAs and Ba1−2yK2yFe2As2.

2 Structure

The basic structural unit for the iron-based supercon-ductors is a planar FeAs layer consisting of three sheets: (Fig. 1). In the high-temperature paramagnetic tetrago-nal phase, the iron atoms form a square sublattice (a ≡ 1) with each Fe tetrahedrally coordinated by four As li-gands. The latter thus form two 2 × 2 square lattices above and below the Fe plane at a vertical distance of ap-proximately half the a-constant of the Fe sublattice. The Fe and As positions are thus described by respectively :

where x and y are the orthogonal vectors between the Fe nearest neighbors and nx and ny take all integer val-ues. z is perpendicular to x and y, and has the same length. For perfect tetrahedra, η = 1, and for LaOFeAs, η = 0.93. Instead of η ≡ 2cot θ/2 ≡ 2 2zAs, it is customary to specify the As-Fe-As tetrahedral angle, θ, or the inter-nal parameter, zAs. While t are the translations of the Fe sublattice, T ≡ nX X + nY Y are those of the As sublattice whose primitive translations are X ≡ y + x and Y ≡ y − x. The latter are turned by 45° with respect to x and y, and

2 longer. The translation group of the FeAs layer is T and has two FeAs units per cell. These are, however, re-lated by a glide mirror. Rather than using the irreducible representations of the 2D translation group, it is there-fore simpler to use those of the group generated by the primitive Fe-translations, x and y, combined with mir-roring in the Fe-plane. These glide-mirror operations (“take a step and stand on your head”) generate an Abe-lian group with only one FeAs unit per cell and irreduc-ible representations, exp (ik · r) , which are periodic for k in the reciprocal lattice, hx 2πx+hy 2πy, with hx and hy integer. The corresponding Brillouin zone (BZ) shown in Fig. 2 is a square, centered at the Γ -point k = 0, with corners at the M -points, k = πy ±πx and−πy

±

πx, i. e. at ±πX and ±πY, and edge-centers at the X and Y-points, k =±πx and ±πy. In this paper we shall use this more heavy notation instead of e.g. (π, π) for M and (π,0) for X as done for cuprates, because for the iron superconductors, no consensus exists about whether to use the (x, y) or the (X , Y ) coordinate system. The overbar is used to designate the high-symmetry points in the 2D reciprocal space for the glide-mirror group. In conclusion, use of the glide-mirror group reduces the number of bands by a factor of two, and this is impor-tant when attempting to understand the intricacies of the band structure. In Fig. 3 we sketch the antibonding Bloch sums of the Fe dxy (top) and dxz (bottom) orbitals, and realize that with the glide-mirror notation the former has k = 0 and the latter k · x = π. Accordingly, the top of the pure Fe dxy

band is at Γ , while the degenerate top of the pure dxz and dyz bands is at M . We shall often return to this. (Authors who unfold without reference to the glide-mirror group, may have Γ and M interchanged, with the result that the xy hole pocket and the two xz/yz hole pockets are re-spectively at M and Γ . In order to avoid this confusion, it is useful to remember that the two xz/yz hole pockets are those towards which the electron superellipses at Xand Y are pointing). The real 3D crystals consist of FeAs layers stacked in the z-direction with other layers intercalated, although the iron chalcogenides, FeX, have no intercalation. Fig 1


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specifically shows LaOFeAs, for which all our Wannier-orbital (3D) calculations were done, unless otherwise stated. The interlayer coupling is weak but not negligible, and it depends on the material. Although the 2D glide-mirror may take the 3D crystal into itself, as is the case for LnOFeAs, FeX, and LiFeAs, we do want to use kz to enumerate the states in the third direction. For the 3D crystals we shall therefore use the standard 3D transla-tion group according to which only the X and Y transla-tions, combined with an out-of plane translation, leave the crystal invariant. The corresponding 2D reciprocal lattice is hX πX+hY πY = hX + hY πy+hX − hY πx. Hence, the 3D Brillouin zone is as shown by the dashed lines in Fig. 2 (for kz =π/2c), with M falling onto Γ and with corners at X and Y , now named M. Interlayer hopping may thus couple the glide-mirror states at k with those at k+πx+πy. This material-dependent coupling will be considered in Sect. 4 after we have explained the generic electronic structure of a single FeAs layer.

Spin-orbit interaction also invalidates the glide-mirror symmetry, but the splitting of states degen erate at k and k +πx+πy is at most 3

2 ζFe3d 0.1 eV, and this only oc-curs if all three xy, yz, and xz states happen to be degen-erate and purely Fe d-like.

3 Paramagnetic 2D band structure

In this section we shall describe the generic 2D band structure of an isolated FeAs layer. We start by observing that the bands are grouped into full and empty, separat-ed by a pseudogap. We then discuss the grouping of the bands into Fe 3d and As 4p, and derive two sets of Wan-nier orbitals from DFT, one set describing merely the five Fe d-like bands and another set describing the eight Fe d-and As p-like bands. Armed with those sets, we can return to a detailed description of the low-energy band-structure, i. e. the one which forms the pseudogap at d6 and the Fermi surface. This is done in subsection 3.3 where we shall see that the hybridization between – or covalency of – the As p and the Fe d orbitals is crucial for the band topology. Bringing this out clearly, was in fact our original reason for deriving the eight-orbital pd set, although the five-orbital d set suffices to describe the low-energy band structure. For FeTe and LaOFeAs the formal ionic states are respectively Fe2+Te2− and La3+O2−Fe2+As3−. In fact, for all parents of the iron-based superconductors, the nomi-nal electronic configuration is ligand p6 Fe d6. The ge-neric 2D band structure is shown in Fig. 4 for energies ranging from 4.5 eV below to 2.5 eV above the Fermi lev-el and along the high-symmetry lines of the BZ (Fig. 2). In the energy range considered, there are eight bands which are seen to separate into three low-energy and five high-energy bands. They may be called respectively the ligand p-and iron d-bands, and the correspond-ing electron count is as written on the figure. At p6d6

the two uppermost bands are seen to be detached from the rest, except at one (Dirac) point along the XM -line where two bands cross, because their Bloch functions are respectively even and odd with respect to reflection in a vertical mirror parallel to XM and containing near-est-neighbor As atoms. If the energy of this crossing could be moved up, above the relative band maxima atΓ and M, it would drag the Fermi level along and the material would transform into a zero-gap semiconduc-tor. For the iron-based superconductors, however, the Fermi level is merely in a pseudogap and the Fermi sur-face (FS) consists of a Γ -centered hole pocket, two M-centered hole pockets, and two compensating electron pockets centered at respectively X and Y (Fig. 2).

Figure 3 Sketch of the anti bonding Bloch sum of Fe dxy orbitals in

the xy-plane (top) and of the antibonding Bloch sum of Fe dxz orbit-

als in the xz-plane (bottom). A Bloch sum is formed by adding the

glide-mirrored π orbital multiplied by exp i k · t , where the glide, t, is

a primitive translation, x or y, and the mirror is the Fe plane. The

antibond ing Bloch sum of dxy orbitals has k=0 and that of dxz or-

bitals has k·x=π. That the lobes of the real Wannier orbitals avoid the

As sites (Fig. 6) is indicated by enhancing the countours of the lobes

pointing towards the reader.

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3.1 Fe d five-orbital Wannier basis

Characterizing the five upper bands as Fe d is sound, be-cause they can be spanned exactly by five Wan nier func-tions [41] which behave like Fe d-orbitals. This can be seen in Fig. 5. Our Wannier functions were constructed [74] to have d character on the central Fe site and no d character on any other Fe site. This makes them localized Wannier

orbitals. The five bands of course have characters other than Fe d, and those characters are mixed into the Fe d Wannier orbitals. This by-mixing follows the point sym-metry in the crystal. Specifically, the Fe dxy Wannier orbital has on-site Fe pz character breaking the horizontal-mirror symmetry of the pure dxy orbital, as well as strong off-site pz character on all four As neighbors. The sign of the As pz character is antibonding to Fe dxy because the As p hy-bridization pushes the Fe d band up in energy. The cor-responding nodes between the Fe d and As p tails make neighboring lobes difficult to see in the figure. Hence, only the As pz lobes pointing towards the La lay ers are big. Sim-ilarly, the Fe dXz Wannier orbital antibonds with pX on the two As neighbors in the X direction, and Fe dY z antibonds with pY on the two As neighbors in the Y direction. If the Fe-site sym metry had been exactly tetragonal, the three above-mentioned Wannier orbitals would have been de-generate and transformed according to the t2 irreducible representation. However, the non-tetrahedral environ-ment, e.g. flattening of the tetrahedron (η < 1) , increases the energy of the dxy orbital above that of the d orbitals be-longing to t, i. e. dXz and dY z or, equivalently, dxz and dyz. In LaOFeAs, the energy of dxy is ∼0.1 eV above that of dt . The two remaining Wannier orbitals, d

3z2−1≡dzz and dy2−x2 ≡dXY ,

antibond less with As p because their lobes point between the arsenics. Fe dzz is seen to antibond with pz on the four As neighbors and Fe dXY antibonds with pY on the two As neighbors in the X direc tion, and with pX on the two As neighbors in the Y direction. In tetrahedal symmetry these two orbitals would transform according to the e represen-tation, and that holds quite well also in the real materials where the orbitals are degenerate within a few meV. Their energy is ∼0.2 eV below that of the dXz and dYz orbitals. This e-t2 splitting of a central d shell in a tetrahedron having p orbitals at its corners is an order of magnitude smaller than the t2g-eg splitting in an octahedron which allows for better alignment of the p and d orbitals. The ∼0.2 eV e-t2 splitting in LaOFeAs is 20 times smaller than the width of the Fe d-band structure in Fig. 4 and does not cause sepa-ration into two lower e and three higher t2 bands with a pseudogap at d4. Nevertheless, the t2 and e orbitals do play quite different roles in forming the band structure near the Fermi level, as we shall see later. Whereas in cubic perovskites, including the cu-prate superconductors, the effective dd hopping in the separated t2g and eg bands proceeds almost exclusively through the p tails, which are placed be tween the near-est-neighbor d orbitals, the effective dd hopping in the iron-based superconductors proceeds directly between nearest-neighbor d orbitals on the square lattice as well as via the p tails lying above and below the plane of the d orbitals.

Figure 4 Band structure of paramagnetic, tetragonal pure LaOFeAs

with the experimental structure near the Fermi level (≡ 0) and for k

along the high-symmetry lines in the large 2D Brillouin zone (Fig. 2).

Band energies are in eV. These DFT-GGA bands were calculated with

the NMTO method and a basis of Fe d and As p downfolded NMTOs.

Transformation to real space yields the eight Wannier functions

shown in Fig 6. The 2D bands were obtained by neglecting the inter-

layer hoppings and forming glide-mirror Bloch sums of the Wannier

orbitals on a single FeAs layer, i. e. by appropriately flipping the signs

of the intra-layer hopping integrals. The large gaps in the figure are

labelled by an electronic configuration which corresponds to a p-set

and a d-set of Wannier orbitals which span respectively the three

lowest and the five highest bands. This d-set is illustrated in Fig 5.

Upon electron doping in the rigid band approximation, the Γ-cen-

tered hole pocket fi lls once the doping exceeds 0.1 e/Fe, and when it

exceeds 0.3 e/Fe, also the M -centered hole pockets fill.


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Tabulations of the hopping integrals for the d-orbital Hamiltonian may be found in [41] and [57].

3.2 Fe d As p eight-orbital Wannier basis and its Hamiltonian

In order to explain the band structure, and in particular the interlayer coupling in Sect. 4, we find it useful to ex-hibit the As p characters explicitly. We therefore choose not to downfold the As p channels, but span the entire eight-band structure in Fig. 4 by the eight As p and Fe d Wannier orbitals; all other channels remain downfolded [74]. These eight orbitals are shown in Fig. 6. Due to the lack of As p tails, the Fe d orbitals of this pd set are more localized than those of the d set and the integrals for hop-ping between them have a shorter range. This basis set is also more suited for including the on-site Coulomb cor-relations. The As p orbitals are, on the other hand, quite diffuse and give rise to strong and long-ranged pp and pd hoppings inside the layer. For the pz orbital, this is partly

due to its La d and O p tails. This situation is very different from the one found in the cuprates, where long-ranged pp hopping is blocked by the presence of Cu in the same plane. Although the orbitals of the AsFe pd set resemble atomic orbitals more than those of the d set, they do tend to avoid the space covered by the other orbitals in the set: The Fe d orbitals avoid the As sites and the As p orbitals avoid the Fe sites. This distorts in particular the Fe t2 or-bitals. The on-site energies, εα, and the nearest-neighbor hopping integrals, t

nx ,ny

α,β are given in the table below. Here, all energies are in eV and the hopping integral is the ma-trix element of the Hamiltonian between Wannier orbit-als α and β with nx x + ny y being the vector from α to β, projected onto the Fe plane. All hopping integrals needed to obtain converged energy bands together with their ana-lytical expressions will be published in [75]. For LaOFeAs, a = 285pm (and, within a few per cent, the same for the other iron-based superconductors). The energies of the p and d orbitals are respectively −1.8 and −0.7 eV. This 1.1 eV pd separation is merely a fraction of the 7 eV pd-band width and it therefore seems fair to claim that the band structure is more covalent than ionic. Nevertheless, it does split into three lower As p-like and five upper Fe d-like bands as noted above. The band structure fattened by the weight of each of the eight Wannier orbitals of the pd set is shown in Fig. 7. Here and in the following we write xy for Fe dxy, t for Fe dt , z for As pz , a.s.o.. The strange wiggles of some of the bands may be seen to have strong z char-acter and this tells us that the reason for those wiggles is intra-layer hopping via the LaO layers, whose orbitals are downfolded mainly into the As z orbital.

3.3 2D Bands and Fermi surface

We now follow the bands around the d6 pseudogap and begin with the Fermi surface near Γ . The Γ -centered hole pocket is seen to have xy charac-ter and, as sketched at the top of Fig. 3, its Bloch function

Figure 5 The set of five Fe d-like Wannier orbitals (downfolded and

orthonor malized NMTOs) which span the five LaOFeAs bands extend-

ing from -1.8 eV below to 2.2 eV above the Fermi level. Shown are the

positive and negative contours, χm (r) = ±|c| , with the former in red

and the latter in blue. Orienta tion and coloring (Fe red, As green, La

yellow, and O blue) as in Fig. 1. The three orbitals to the left and the

two to the right woul belong to respectivel the t2 and e representa-

tions, had the point symmetry been tetragonal. Now, t2 split into a (dxy) and (t dxz, dyz ). Note that the t orbitals dxz ≡ (dXz − dYz / 2 and (dyz ≡ dX z + dYz) / 2, whose Bloch sums form the proper linear

combinations for k along ΓX and XM (Fig. 2), are not simply

45°-turned versions of dX z and dY z shown here, in particular because the p tails of the latter are on

different pairs of arsenics. The p tails are thus always directed to-

wards the nearest As neighbors in the same plane, i. e. they are X or Y, and they antibond with the t head.

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dd-antibonds with all four nearest Fe neighbors. If in this figure we imagine inserting the xy orbital of the d set (Fig. 5), we realize that the sum of the As pz tails cancel. In fact, none of the other 8 orbitals in the pd set can mix with the Bloch sum of xy orbitals at Γ . This we have stated in the caption to Fig. 2 together with the selection rules for all other high-symmetry points. The selection rules for the high-symmetry lines are given in blue on the figure. For k moving from Γ towards X , the xy band is seen to dis-perse downwards because in the x-direction, the char-acter of the wavefunction goes from dd antibonding to bonding and, at the same time, the band gets repelled by the above-lying xz/y band whose xz orbitals point in the direction of the k-vector, i. e. the longitudinal t band. The

corresponding inter-band matrix element increases lin-early with the distance from Γ , whereby the downwards curvature of the xy band is enhanced by about 10%. The resulting hole band mass is about twice that of a free-electron. At X, the xy Bloch sum bonds between nearest Fe neighbors in the x direction and antibonds be tween those in the y direction. In addition, weak hybridization with a 2 eV lower-lying As y band pro vides y character to antibond between the xy orbitals in the y direction. This pushes the xy band up at X by 0.2 eV to −0.4 eV. As k now moves on from X towards M , the Bloch sum of xy orbitals becomes bonding between Fe nearest neighbors in the y direction as well, whereby the As z tails (Fig. 5)

Figure 6 Same as Fig. 5, but

for the set of eight As p-and Fe

d-like Wannier functions

which span the entire band

structure shown in Fig. 4.


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no longer cancel and their antibonding contribution in-creases linearly with ky . This causes the band to disperse strongly upwards, to a maximum 1eV above the Fermi level. As can be seen from Fig. 7, this strong change of band character may also be explained as the result of strong pd hybridization and avoided crossing of a pure xy band dispersing downwards from X to M and a pure z band dispersing strongly upwards due to long-range hopping, partly via La and O (see Fig. 6). Corresponding to the 1.3 eV up wards dispersion of the xy/z antibonding band along XM, we see a downwards dispersion the z/xy bonding band. At M, xy and z hybridize, but only with each other: the pure xy level is at −2.0eV, the pure z level at −0.4eV, and the xy-z hybridization is 2 eV, thus pushing

the antibonding, predominantly z-like level up to +0.9 eV and the bonding, predominantly xy-like level down to −3.4 eV. The t-bands which form the M -centered hole pock-ets exhibit a very similar behavior as the xy band when, instead of going along the path Γ -X -M (or Γ-Y -M), we go along the path M- X -Γ for the xz band and along M -Y -Γ for the yz band. The avoided pd crossing is now bet-ween a pure xz band dispersing downwards from X to Γand a pure y band dispersing strongly upwards; these dispersions are strong because k changes in the direction of strong hoppings, ddπ and ppπ, respectively. At X , the xz band is pure and merely 0.1 eV below the Fermi level. At Γ , the pure xz level is at −1 eV, the pure y level is at +0.5 eV, and the hybridization between them is over 2 eV thus pushing the antibonding, predominantly y-like level to +2.2 eV above the Fermi level and the bonding, predomi-nanly xz-like level down to −2.7 eV. Whereas along ΓX, the xy and xz/y bands hybridize and therefore cannot cross, along XM, they do cross because the Bloch sums with kx =π of Fe xy and Fe xz orbitals are respectively even and odd upon reflection in the As-containing vertical mirror perpendicular to the x direction. In other words, they belong to different irreducible representations (Fig.

Figure 7 Band structure from Fig. 4 fattened by the character of each

of the eight Wannier orbitals in the pd set (Fig. 6). A fatness was ob-

tained by perturbing the on-site energy of the orbital in question.

That two bands share the same fatness means that they hybridize. In

order to concentrate the fatness onto as few bands as possible we

chose the appropriate linear-combination of t orbitals (see Fig. 2) for

each high symmetry line . The ΓM -line was chosen as the one in the

X direction.

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2). Since outside the XM-line, hybridization between them is no longer forbidden, the accidental degeneracy is at a Dirac point in 2D (a line in 3D). To separate the two bands would require moving the xy/y above the xz level at X . Near X , the xy-and xz-like bands are close in ener-gy and both have minima. The minimum of the xy-like band curves steeply upwards towards M due to strong hybridization with z, and that of the xz like band curves steeply upwards towards Γ , due to strong hybridization with y, but is flat towardswhich is the direction transver-sal to the dominating ddπ hopping. These two bands hy-bridize weakly with each other, except on the XM-line. As a result, they form a lower and an upper band of which the latter cuts the Fermi level at an X -centered electron pocket. As long as the Fermi level is well above the band crossing along M, the shape is that of a 4th-order super-ellipse. This ellipse points towards M where its character is mainly xy/z and is flat towards Γ , where its character is predominantly xz/y . The two bands forming the X -centered electron pocket can be modeled by the Hamiltonian,

in the basis of two effective (downfolded) Bloch orbitals, xy and xz, and where the origin of (kx , ky) is taken at X . Note that the dispersion of the effective xy band towards M is nearly linear for energies not too close to εxy due to the avoided crossing of the pure xy and z-bands and the linear increase of their hybridization (see Fig. 7; we shall return to this). In expression (2), εxy = −0.44 eV and εxz = −0.13eV are the levels at X with respect to the Fermi level, v y

xy,xy = 0.53 and v xxy,xz =−0.29 eV·a are band slopes, i.

e. group velocities, and mxxy,xy = 2.9, mddπ = 2.3, and mddδ =

−7 are the band masses relative to that of a free electron. Negative masses are those of holes. Moreover, τ ≡ ( a0/a)

2

Ry=0.47 eV with a0 the Bohr radius.The numerical values are for pure LaOFeAs with the experimental structure and were obtained by fitting the result of an LAPW calcu-lation near the Fermi level. Like everywhere else in this paper, k is in units of the inverse Fe-Fe nearest-neigh-bor distance, 1/a = 1/285 pm. By symmetry, the xy-and yz like bands give rise to a Y -centered electron pocket which also points towards M , where the character is mainly xy/z, and is flat towards Γ with predominant yz / x character. This is illustrated in Fig. 8 by projection of the Fermi surface onto the various orbitals. Although none of the sheets of the paramagnetic Fermi-surface have major e character, the two e orbitals

play a decisive role in the formation of the d6 pseudogap. The pure zz band is centered at −0.6 eV and disperses so little that it lies entirely below the Fermi level. At M, the zz band is pure, and at Γ hybridization with the z band pushes it up by 0.6 eV, an amount larger than the width of the pure zz band, to −0.4 eV. The pure XY band, on the other hand, is broad because the lobes of the nearest -neighbor XY orbitals point directly towards each other. This band has its minimum at Γ , saddlepoint at X , and maximum at M. Both extrema, at respectively −1.8 eV and +0.6 eV, are pure. At intermediate energies, the XY band is however gapped in large regions centered at Xand Y by avoided crossings with the zz band. Near the ΓM -lines, where the XY and zz bands can-not hybridize, the XY band has an avoided crossing near M with the upper, transversal hole band, Yz/Y (Xz/X ) in the X (Y ) direction. The hybridization between the trans-versal hole band and the downwards-dispersing pure XY band vanishes at M, but increases linearly with the distance from M, and with a slope proportional to the pdπ-like hop ping integral between the XY and Y orbit-als. This means that, if at M, the XY level at 0.6 eV could be lowered by 0.4 eV such as to become degenerate with the degenerate top of the hole bands, then the transverse hole band and the XY band would form a Dirac cone. The trace of this cone can still be seen in Fig. 7, in particular at the low-energy edge of the fat XY band. If the singly degenerate XY level at M had been below the degenerate Yz/Y level, then this lowest level would be the singly-de-generate top of an XY -like hole band and the higher-ly-ing, doubly-degenerate level would be the bottom of the transversal Yz/Y electron band. The degenerate partner does not hybridize with XY /Y and is therefore indepen-dent of the position of the XY -band. Hence, the actual band structure of the LaOFeAs has the XY and Yz/Y lev-els, at respectively 0.6 and 0.2 eV, inverted.

The model Hamiltonian for a Dirac cone is:

where for the above-mentioned example k is the distance from M. The zero of energy is midway between the two M levels, XY and t/p, which are separated by g. In view of the approximately circular shape and isotropic XY char-acter of the transversal (outer) M-centered hole sheet seen in Fig. 8, the isotropic 2 × 2 Hamiltonian (3) is a rea-sonable representation and may be obtained by limiting k to one of the four ΓM directions where only three trans-versal orbitals can mix (Fig. 2), and then downfolding the transversal p orbital, i. e. Y if k is along X. Since the un-


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hybridized XY band disperses downwards towards

Γ , its mass, m2, is negative, and since the unhybridized trans-versal band disperses upwards, its mass, m1, is positive. Finally, the coupling vk is proportional to the product of the Yz-Y hybridization at M, which is in fact responsible for moving the pure Yz band up by 0.4 eV, and the XY -Y hybridiza tion, which is proportional to k. These hybrid-izations are clearly seen in the pictures of the d Wannier orbitals in Fig5. For this incipient XY −Yz/Y Dirac cone in LaOFeAs, g= 0.4eV, m2=−1.4, m1= 1.4, and v 0.5eV a = 1.4 eV A˚= 1.4c/1973, i. e. about thousand times less than the velocity of light, c. Now, for k ≪ 1

2 g /v = 0.4 ∼ |ΓM|/10, the

Hamiltonian (3) yields the cone: ε (k) = ±vk, and for k ≪ 12

g /v, it yields parabolic Yz/Y and XY -like bands gapped by g and with inverse masses given by respectively

Later in this paper, we shall meet not only incipient-but real Dirac cones. The final gap needed to complete the d6 gap in the central part of the ΓXMY square is the one produced by the avoided crossings along ΓM of the downwards-dispersing upper z/xy band with the upwards-dispersing upper, longitudinal X z/X band. Here again, none of

these bands are allowed to hybridize at M, and the ma-trix elements between them increase linearly with the distance from M in the X direction. Specifically, the pd matrix elements Xz-z, X -z, and X -xy are all linear in k. Also the zz band at −0.6 eV mixes in, with the weak 2nd-nearest neighbor dd hopping integral between Xz and zz orbitals providing the slope of the linear matrix element. At M, the z/xy and Xz/X levels are thus inverted, but by being at respectively 0.9 and 0.2 eV, they are too far apart to make the Dirac cone vis ible in Fig 7. This will however change when, in the following section, we consider other materials and include the kz -dispersion. In conclusion, the d6 pseudogap is caused by the XY and uppermost z/xy levels being above the degenerate t/p levels at M. Had the opposite been the case, a situation with the two t/p bands entirely above the three z/xy and e bands, i. e. that of a d6 insulator, could be imagined. Having sorted out the intricacies of the band struc-ture and thereby understood the subtle origins of the X-centered electron pockets and the d6 pseudogap, we shall finally return to the M- centered hole pockets using Figs. 2, 7, and 8. These hole pockets have fairly compli-cated shapes and orbital characters. Although the t char-acter dominates, p hybridization pushes the top of the band up by 0.4 eV, to 0.2 eV above the Fermi level, as has

Figure 8 Fermi surface for LaOFeAs

(see also Fig. 2). The fatness was

obtained as in Fig. 7 and thus gives

the orbital weight times the inverse

of the Fermi velocity projected onto

the plane. The fatness of the domi-

nating xy, xz, and yz orbitals have

been reduced by a factor 5. Here,

h and e refer to, respectively, hole

and electron sheets.

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been mentioned before. Departing from M, the two bands split into a steep X-centered electron pockets and the ¯one with relative mass numerically smaller than one and a shallow one with mass numerically larger than one. They give rise to respectively the inner and the outer hole pockets. As long as the character of the band is pre-dominantly t-like, the steeper, inner band will be the one for which the ddπ hopping is along the vector distance from M, that is, the longitudinal band. Accordingly, we see in Fig. 7 from M to X , the yz-like band stay intact and disperse strongly downwards. From M towards Γ , we see the Xz-like band disperse downwards and stay intact until it suffers an avoided crossing with the zz band. The inner band is in fact steeper towards Γ than towards X and Y, and this is partly because the p hybridization of the longitudinal band has a node along ΓM , as can be seen for the Xz-like band in Fig. 7. The further reason for the small mass of the inner hole pocket is the gapped Di-rac cone formed with the 0.7 eV higher-lying z/xy band. The outer, transversal hole band has a large mass not given by the weak ddδ hopping integral, but as discussed above in connection with Eq.(3), by its hybridization pro-portional to k, with the g = 0.4eV higher-lying XY band. Along ΓM this transversal hole band is mainly xz/x hy-bridizing proportional to k, not only with XY but also with zz. The latter gives the anisotropy seen in Fig. 8. In the next section we shall see how these details are modified by the material-dependent height of As above the Fe plane and interlayer coupling.

4 Influence of As height and interlayer hopping

Until now we have discussed the generic 2D band struc-ture for an isolated FeAs layer. This band struc ture was obtained by (i) downfolding the proper 3D bands of LaOFeAs with k in the small BZ to a 16×16 pd TB Hamil-tonian, (ii) neglecting the interlayer hoppings and (iii) re-ducing the resulting Hamiltonian to an 8 × 8 by transfor-mation to the glide-mirror Bloch representation with k in the large BZ. The un derstanding of this relatively simple, generic, 2D band structure obtained in Sect. 3.3 enables us now to explain the material-dependent, complicated, 3D bands obtained by standard DFT calculations in the small BZ. Specifically, we shall present and discuss the 3D band structures of simple-tetragonal (st) LaOFeAs and SmOFeAs in Fig. 9 and in Sect. 4.1, mentioning those of FeTe and LiFeAs en passant, and then in Sect. 4.2 move on to the band structures of body-centered tetragonal (bct) BaFe2As2 and CaFe2As2, in the normal as well as the collapsed phase. The band structure of bct BaRu2As2 will finally be mentioned. The interlayer hopping is mainly

between As z orbitals. In the st LnOFeAs materials this hopping is fairly weak and the material dependence of the band structures is caused more by the vary ing height of As above the Fe plane than by interlayer hopping. This we shall see in Sect. 4.1. For st FeTe and LiFeAs, and in particular for the bct materials, the interlayer hopping is dominating, and since its effects are non-trivial, we have derived the formalism and shall present it in Sect. 4.2. It turns out that the folding of the bands into the small BZ and subsequent interlayer hybridization at general k-points cause many bands to have nearly linear disper-sions and in some cases to form full Dirac cones.

4.1 Simple tetragonal LnOFeAs, FeX, and LiFeAs

Since in these st crystals, the FeAs and LnRO layers (see Fig. 1) are simply translated in the z-direction by a multi-ple of c and then stacked on top of each other, the primi-tive translations in real and reciprocal space are respec-tively

The 3D BZ is therefore simply a rectangular box whose cross-section is the 2D zone, folded-in as shown by the dashed lines in Fig. 2. The midpoints of the vertical faces, 12 g1 and 1

2 g2, are labelled X = ( 12 M, kz = 0), those of the

vertical edges, 12 (g1 ± g2) = πy and πx, are labelled M = (

Y, kz = 0) and (X ,0), those of the horizontal faces, ± 12 g3,

are labelled Z = ( Γ , ± πc ), those of the horizontal edges, ±

12 (g1 + g3) and ± 1

2 (g2 + g3), are labelled R =( 12 M, ± π

c ), and those of the corners, 1

2 (± g1 ± g2 ± g3), are labelled A = (Y, ± π

c ) and (X , ± πc ).

In order to compare with our familiar 2D bands in Fig. 4, we first consider them along XM , where they are the same as along YM, and then then translate YM by −g2 to ΓX , which is ΓM in Fig. 9. Now we can easily rec-ognize the M-centered, doubly-degenerate top of the t-like hole bands, the above-lying XY and z/xy bands, and the zz band at −0.6 eV. Next, we consider the ΓM bands in Fig. 4 and translate this line by –g2 to MY , which is ΓM in Fig. 9. This time, we recognize the Γ-centered xy hole band, the zz/z band at −0.4 eV, and the Γ-centered bot-tom of the XY band at −1.8 eV. Near M = X and Y, we also


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recognize the bands responsible for the X -centered electron super-ellipse, along the direction towards Γ and as well as towards M. Finally, we consider the ΓM bands in Fig. 4 and superpose those translated by g1, to

ΓM , onto them. Those bands are symmetric around M/2, and their fi rst half, from Γ to M/2, is placed from Z to R in Fig. 9. Here again, we can easily recognize the bands. After having obtained an understanding of the 3D band structure of LaOFeAs from merely placing ε α(k + g2) on top of ε α( k ), we now search – and subse-quently explain – the effects of interlayer hopping. kz is 0 along MΓ and π/c along ZR. Along the vertical path ΓZ, we see the kz-dispersion at M and Γ . From this, it may be realized that only bands with As z charac-ter disperse signifi cantly with z, i. e. that the interlayer hopping proceeds mostly from As z to As z. The bands seen to disperse in Fig. 9 are the upper z/xy band near M and the upper zz/z band near Γ (Fig. 7). This inter-layer hopping simply modulates the energy of the z or-

bital, εz(kz) = t ⊥cos ckz. Now we see something very in-teresting: For z near π/c, the upper z/xy band has come so close to the top of the M-centered hole pockets that the inner, longitudinal band takes the shape of a Dirac cone over an energy region of 0.4 eV around the Fermi level. The inner hole cylinder, as well as its radius, thus become warped due to this incipient Dirac cone. This is even more pronounced for SmOFeAs because here, the z/xy band lies nearly 0.3 eV lower than in LaOFeAs, in fact so low that the 2D band in the kz =π/c plane is nearly a complete Dirac cone at 0.2 eV and with slope v ∼ 0.3eV·a (see expression (3)). To bring the Fermi level up to the cusp would, however, require electron doping beyond 30%. The reason why the z/xy band lies lower in SmOFeAs than in LaOFeAs is that As lies higher above Fe η=0.98 in the former than in the latter compound η=0.93. The z-xy hybridization is therefore smaller, and that moves the upper z/xy band down at M. This is clearly seen along all directions in Fig. 9, but whereas this flattening of the upper z/xy-like band increases the mass at the X-centered electron pocket towards M and makes it more d-like, it decreases the mass of the inner M-centered t/p like hole pocket due to the incipient Dirac cone. In-creasing η, generally decreases the pd hybridization, whereby pd antibonding levels move down in energy with respect to those of pure d character and become more d-like. Important effects of this are the lowering of the top of the t/p hole band at M with respect to that of the xy hole band at Γ and the lowering of the bottom of the xy-like electron band with respect to the pure xz

Fig. 9 3D band structures of simple tetragonal LaOFeAs (left) and

SmOFeAs (middle), as well as body-centered tetragonal BaFe2As2

(right). For the two former, the BZ is a rectangular box whose cross-

section is the 2D folded-in zone shown by the dashed lines in Fig. 2.

The MΓ-line is in the kz=0 plane, the ΓZ-line is along the kz -direction,

and the ZR-line is in the kz =π/c plane. The 2D notation for the projec-

tion onto the (kx, ky) -plane is given on the top. The BaFe2As2 band

structure is plotted along those same lines, now labelled XΓ, ΓZ,and

ZΓ/2, as may be seen from Fig. 10. The computational scheme was

GGA-LAPW ( [78]).

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level at X . These changes are clearly seen in Fig. 9: With the Fermi level readjusted, the size of the Γ -centered xy hole sheet is increased for SmOFeAs and is now simi-lar to that of the outer M-centered hole sheet. Finally, we may note that the decreased t/p hybridization at M decreases the coupling linear in k to the XY band, so that this band becomes less steep in the Sm than in the La compound. Of all known iron-based superconductors, SmOFeAs has the highest Tc max (55 K) and the most reg ular FeAs4 tetrahedron, i. e. its η is closest to 1. For nearly all Fe-based superconductors, Tc max versus η seems to follow a parabolic curve [13], a correlation which has been ex-tensively studied, but is not understood. For LaOFeAs, Tc max =27K. Also LiFeAs and the iron chalcogenides, FeX, have the st structure and calculations [21, 32, 33] yield: η=1.12 for LiFeAs, while for X = S, Se, Te: η=0.87, 0.97, 1.16, respectively [33]. In LiFeAs and FeTe, the upper z/xy band thus sits considerably lower in energy. More-over, since in LiFeAs the perpendicular As z hopping is enhanced by hopping via Li s, and since the per-pendicular Te 5pz hopping is stronger than the As 4pz hopping in LnOFeAs, the z/xy band disperses 3 and 4 times more along ΓZ in respectively LiFeAs and FeTe, than in LnOFeAs. As a consequence, the z/xy-like band crosses the degenerate t/p band already when kz ∼π/2c, and here, it forms a Dirac cone with the inner, longitu-dinal t/p band, at 0.1 eV above the Fermi level in LiFeAs and at 0.2 eV in FeTe. For π/2c kz 3π/2c, the band which at M has longitudinal t/p character disperses up-wards and the other band, which at M has z/xy char-acter, downwards. Accordingly, the inner hole sheet of the Fermi surface is not a cylinder, but extends merely a bit further than from −π/2c to π/2c where the z/xy band along ΓZ dips below the Fermi level. The mass of this sheet vanishes when kz is at the Dirac value, ∼π/2c. Here, the slope of the cone in the (kx, ky) plane is v ∼0.5 eV·a. The Γ -centered xy pocket is a straight cylinder, whose cross-section in FeTe has about the same size as that of the outer M-centered t/p hole sheet at kz =0, i. e. like in SmOFeAs, and in LiFeAs is even a bit larger. LiFeAs is a non-magnetic superconductor with Tc =18 K.

4.2 Body-centered tetragonal BaFe2As2, CaFe2As2, and BaRu2As2

In the body-centered tetragonal (bct) structure, the FeAs layers are translated by x before they are stacked on top of each other. This means that the As atoms of adjacent layers are directly on top of each other. More-

over, the interlayer As-As distance, d = 379 pm is about the same as the intralayer As -As distances, 2a =396pm and 2aη. It is therefore conceivable that the interlayer hopping vertically from As z to As z (ppσ) is very strong. This is in fact the reason for the 2 eV dispersion seen along ΓZ in the band structure of BaFe2As2 on the right-hand side of Fig. 9. Since η=0.97 for BaFe2As2, the position of this z-like band is not as low as in FeTe, but more like in SmOFeAs. At kz 3π/4c, the band crosses the degenerate t/p band and forms a Dirac cone with its longitudinal branch in the (kx, ky)-plane, as we shall see explicitly later. Note that the longitudinal branch disperses downwards from Γ towards X in the kz =0 plane, but upwards from Z in the kz =π/c plane.Ba is intercalated in the holes between the neighboring As sheets and thus has 8 nearest As neigh bors. Also Ba orbitals can be vehicles for interlayer coupling and, in fact, a Ba 5dxz/yz band lying above the frame of Fig. 9 repels the top of the doubly degenerate t/p band near Γ with the result that there, the latter is only slightly above the top of the dispersionless Γ xy band, whereas at Z, it is 0.2 eV above. Clearly visible in the figure is also a Ba 5dxy band starting at 1.0 eV at X and then dispers-ing downwards towards Γ , which is reached at 0.3 eV after an avoided crossing with the y/xz band decreas-ing from its maximum at Γ . From Γ towards Z, the 5dxy band then disperses upwards to 1.0 eV and, from there, continues in the kz =π/c plane towards Γ , but soon suffers an avoided crossing with the hybridized z/xy-longitudinal-t/p band. But before we continue our discussion of the BaFe2As2 bands we need to write down a formalism for the interlayer coupling which is strong – and poorly un-derstood – in the bct structure. We start from the 2D Bloch waves, | r; α, k ⟩, of a sin-gle FeAs layer with α labelling the state (e.g. the band) and k the irreducible representation of the glide-mir-ror group. These 2D Bloch waves are ex pressed as linear combinations of localized Wannier orbitals. For the 3D crystal, we now use its out-of-plane translations, n3T3, to stack the 2D Bloch waves in the 3rd direction and form the corresponding Bloch sums:

which we shall then use as basis functions. Here and in the remainder of this chapter, an overbar is placed on the 2D Bloch vector in order to distinguish it from the 3D one, k ≡ k +kz z. Since in the bct structure, the As atoms in a top sheet are vertically below those in the bottom sheet of the lay-er above, the corresponding vertical interlayer hopping


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via As z is particularly simple and strong, so we first spe-cialize to this case. The bct primitive translations in real and reciprocal space are respectively:

where c is the distance between the FeAs layers in units of the nearest-neighbor Fe-Fe distance, i. e. half the bct c-lattice constant. The bct Brillouin zone and the stack-ing between neighboring Wigner-Seitz cells (BZs) of the reciprocal g-lattice is shown in Fig. 10. Using that T3 · k = kx + ckz , it is now a simple matter to form the 3D Bloch sums (5). For two states α and β with the same k , the in-terlayer coupling caused by the vertical z-z hopping is easily found as:

where t⊥ is the ppσ hopping integral ( 0) between an As z orbital in the top sheet to the As z orbital vertically above, in the bottom sheet of the next layer. cz,β ( k ) is the eigenvector coefficient to the As z orbital in the 2D β-state. Note that we have not missed a factor 2 in (7), because only one lobe of the z orbital is used for inter-layer coupling. Now, from the mere knowledge of a k-function in a single – bottom or top – As sheet, the kand k + g 1

(or k + g2) translational states are indistin-

guishable. Their difference is that they have opposite parity upon the glide-mirror interchanging the top and bottom sheets. Interlayer hopping can therefore mix states with k and k + g

1. For the corresponding inter-

layer coupling, off -diagonal in the 2D Bloch vector, we then find :

where the different parities of the k and k + g1 states causes the sin ckz -dispersion. Finally, before cou pling the k + g1 state (see Fig. 10) to that with k, the former must be brought back to the central zone, and that re-quires shifting kz in (7) back by π/c. As a consequence,

In order to get a first feeling for this formalism, let us as-sume that we have nothing, but interlayer hopping. That is, we have pure z states which only couple between – but not inside – the layers. The Hamiltonian for this problem with two z-orbitals per cell is:

Diagonalization yields two dispersionless bands with energy ±t⊥, and this is because this system with out in-tra-layer coupling is merely an assembly of As2 dimers (dangling bonds). The same kind of thing happens at the non-horizontal boundaries of the bct BZ, where the states with energies εα ( k ) and εα ( k + g 1) are degener-ate, because if there are no further degeneracies and if t⊥

is so small that we only need to consider those two states, their energies simply split by ±t⊥ |cz,α ( k )|2. We thus see, that neglecting interlayer coupling does not simply cor-respond to taking kz =π/2c; this merely makes the diago-nal couplings vanish. In simple tetragonal FeX, the interlayer coupling pro-ceeds mainly from an As z orbital to its 4 nearest As z or-bitals in the next layer, with a hopping integral t ∠ . From this follows that the diagonal and off diagonal interlayer couplings are given by respectively

Since the simple tetragonal reciprocal lattice vectors g1 and g2 in Eq. (4) have no kz component, there is no kz -translation leading to an Eq. (9). Instead, the prefactor cos kx +cos ky provides the sign-change for k going to the next BZ. In addition, the prefactor makes the inter-layer coupling vanish on the vertical faces of the 3D st BZ, that is on ( XY, kz). This is different from the bct case. Finally, for the system with only z orbitals and no intra-layer coupling the st fomalism yields two dispersionless bands with energies ±4t

∠corresponding to isolated As-

As4 molecules with no coupling between the 4 atoms in the same plane. Having deepened our understanding of the interlayer coupling via the As z orbitals, we can now return to our description of the bct band structures for which this in-terlayer hopping is particularly strong and – most notice-ably – gives rise to the 2eV dispersion of the z-like band

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seen along ΓZ for BaFe2As2 in Fig. 9. This dispersion is seen to be five times larger than in the LnOFeAs com-pounds, and it turns out that this is not even the entire interlayer dispersion, 2t⊥ |cz,z/xy ( M)|2: From our previous discussion of the 2D bands, we may recall that, at M, the z orbital can only mix with xy and the level of interest is the antibonding z/xy level which in LaOFeAs is at 0.9 eV and has about 70% z char-acter, specifically, cz,z/xy ( M) =− 0.83. At Γ , z can only mix with zz and there again, the level of interest is the anti-bonding zz/z level, which in LaOFeAs is at −0.4 eV and 85% d-like, cz,zz/z ( Γ ) = − 0.39. If we now just couple those two LaOFeAs antibonding bands with the bct interlayer coupling given by Eqs. (7), (8), and (9), and adjust the one parameter t⊥ to the BaFe2As2 ΓZ band, thus yield-ing t⊥∼ −2eV, we get the two bands shown in Fig. 11. The good agreement of the upper band with the ΓZ band in BaFe2As2, dispersing from 2.1 to 0.1 eV in Fig.9, hints that this band does result from an avoided crossing of the downwards-dispersing M z/xy band and the upwards-dispersing Γ z/zz band, such that the 0.1 eV state at Z is not M z/xy, but Γzz/z. The M z/xy state at Z must then be the top of the lower band which is seen to have energy −0.9 eV in BaFe2As2 (this includes a push-down by a high-lying Ba 5dzz band). The bottom of the lower band is then the Γ z/zz state at Γ, which in BaFe2As2 is seen acciden-tally also to have energy −0.9 eV. After this estimate, let us briefly recall the proper way of including the interlayer z-z coupling along ΓZ. First of all, from the caption to Fig. 2 we learn that z can only mix with xy at M and with zz at Γ . Secondly, from the 2D bands in Fig. 7, we see that at M, as well as at Γ , the

z-like levels are separated by as much as 4eV. This is due to strong pd hybridization; the z, xy matrix element at M is 2 eV and the z, zz element at M is 1.2 eV. For an inter-layer hopping, t⊥, as large as 2eV, we now ought to solve a 4 × 4 eigenvalue problem (that the d states at k and k + g are different is irrelevant). However, for producing Fig. 11, we got away with neglecting the pd-bonding levels. On the other hand, t⊥∼−2eV was obtained by fitting to bands which include the Ba 5dzz hybridization and this leads to an overestimation of t⊥, as we shall see later. We now discuss further aspects of the bct band structure computed for BaFe2As2. The first two top pan-els of Fig. 12 show respectively the As z and Fe zz pro-jected bands for kz =0 and along the path Γ ΓXM , famil-iar from Fig.7 but labelled ΓXZΓ in the bct reciprocal lattice (Fig. 10). This path takes us from the center, Γ, of the central BZ to the center, X, of a vertical BZ face, from there to the center, Z, of the bottom face of the neigh-boring BZ, and finally back to the origin, Γ. The piece outside the central BZ may of course be translated back to the bottom face of the central zone by −g1, or to the top face by g3 − g1. As a result, not only the eight ΓXZΓ= (Γ ΓXM ,0) -bands, but also the eight ΓXZΓ= ( ΓMY M, π/c )-bands are obtained in such a standard calculation. For kz =0 and π/c, the k -states are pure because sin ckz =0, and the Fe zz projection confirms that the 0.1 eV state at Z has zz character.

Figure 11 Interlayer coupling of the ( M , kz − π/c) z/xy (grey dashed)

and the ( Γ , kz) z/zz (grey dotted) bands along ΓZ for bct BaFe2As2

(schematic). We used Eq.s (8) and (9) together with the 2D LaOFeAs

parameters and t⊥= 2 eV.

Figure 10 Central bct Brillouin zone and the one translated by the recip-

rocal-lattice vector g1 = πx + πy + πz/c, see Eq. (6). Γ is the center of

the zone, Z = ± g3/2 = ± πz/c are the centers of the 2 horizontal faces,

N are those of the 8 large slanting faces, e.g. g1/2, X are the centers of

the 4 vertical faces, e.g. (g1 − g2 + g3)/2 = πx and (g1 + g2)/2 = πy, and

P are the 8 corners between the vertical X neighbors, e.g. πx + πz/2c.

Note that Z in the zone translated by g1 is g1 − g2/2 = ± πx + k z


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The other band with zz character, i. e. the M zz band, is seen to have energy −0.6 eV at Γ and −1.0 eV at Z. For rea-sons of symmetry, this band can have no As z character, and nevertheless disperses with kz (as seen directly along ΓZ in Fig. 9). This is due to repulsion from the Ba 5dxy band. When, in Fig. 9, we observed that the longitudinal branch of the doubly-degenerate t/p band curves down-wards at Γ in the kz =0 plane, but upwards at Z in the kz =π/c plane, both in the central BZ (Fig. 10), this was seen as a consequence of the Dirac cone in the kz =3π/4c plane at the crossing of the t/p band with the upper z-like band between Γ and Z. In the meantime, we have learned

that the upper z-like band cannot hybridize with the t/p band in the kz =π/c plane where sin ckz = 0, because the former is a Γ-state and the latter an M-state at Z. The upwards curvature at Z is therefore due to repulsion from the lower z-like band, the M z/xy band near −0.9 eV. This repulsion is substantial and causes another, but merely incipient Dirac point at −0.4 eV at Z. However, as we now depart from the kz =nπ/c plane, also the Γ z/zz band hybridizes with the longitudinal t/p band – propor-tional to sin ckz and to the horizontal distance from M – and this causes a Dirac cone to be formed at the cross-ing of the upper z-like and the degenerate t/p bands at kz =3π/4c and 0.25 eV. At the bottom four panels of Fig. 12 we therefore show the band structure in the kz = π/4c + mπ/c and kz = 3π/4c + nπ/c planes extending over the two BZs, projected onto the As z and relevant Fe d partial waves. The Dirac cone at ( M,π/4c) inthe BZ at g1 is seen to have z, zz, longitudinal t/p, and xy characters in agree-ment with what was said above, and to have a low-energy slope and a steeper high-energy slope. The low-energy slope v = 0.4 eV·a is of course due to the hybridization of the t/p band with the crossing, upper z-like band and the high-energy slope v = 0.8eV·a is due to hybridization with the lower z-like band, which is around −0.9 eV. (The hole

Figure 12 3D band structure of bct BaFe2As2, fattened by various par-

tial-wave characters and calculated with the GGA-LAPW method [78].

The k ≡ ( k ,kz) paths chosen are: ( Γ ΓXM ,0) = ( ΓMY M, π/c) in the

two top left panels, (Γ ΓXM , π/2c) = ( ΓMY M, π/2c) in the two top

right panels, and( Γ ΓXM , π/4c) = ( ΓMY M , 3π/4c) in the four bot-

tom panels. Note that these paths extend over the two BZs shown in

Fig. 10. The fatness of As z has been enhanced by a factor 5 compared

with those of Fe d in order to account approximately for the fact that

the As z Wannier orbital has most of its charge density outside the

atomic sphere used in the LAPW method.

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band seen in Fig. 12 to have the strongest xy character is irrelevant for the Dirac cone because it is the Γ-centered hole band which, due to lack of z character, cannot mix with k + g1 states). As kz is now increased above π/4c, e.g. to kz =π/2c as shown in the last two top panels of Fig. 12, the upper z-like band moves above the t/p band at M to 1.0 eV, whereby the longitudinal branch of the lat-ter curves downwards, like the transversal branch. For kz =3π/4c, the behaviour can be seen near M in the four bottom panels: The upper z-like band is now at 2.1 eV and has no z/zz but only z/xy character. The repulsion from this band steepens the longitudinal, inner t/p hole band. The outer, transversal t/p hole band attains its hole character mainly from repulsion by the XY band, as was explained in connection with Eq.(3) and seen in Fig. 7 (but excluded in Fig.12). Finally, for kz =π/c, the behav-iour can be seen near M in first panel, which in fact shows what we have already observed without orbital projections on the right-hand side of Fig. 9. The hole part of the Fermi surface thus has a strongly warped, cylindrical sheet, centered around the vertical ΓZ line (see Fig. 10). In most of the zone, this sheet has longitudinal M t/p and some M z/xy character, i. e. it is the small-mass, inner hole sheet. Going from Γ towards Z, this sheet narrows down to a neck for the Dirac value, kz =kDz , where the Γ z/zz character starts to dominate. Finally, close to Z, the character becomes purely Γz/zz and the cylinder bulges out. Unlike in FeTe and in LiFeAs, this sheet remains a cylinder because the upper z-like band remains above the Fermi level for all kz . Concentric with this longitudinal t/p hole cylinder is the transversal one, whose mass is dominated by its XY character. That cylinder has little warping and lies outside the longitu-dinal cylinder, except near Z where the latter bulges out. The third hole sheet has nearly pure Γxy character and is a straight cylinder. Also this cylinder is centered along ΓZ, but it does not hybridize with the two M cylinders and it is as narrow as the Dirac neck of the longitudinal t/p cylinder. Next, we turn to the electron sheets, which in 2D are the X and Y-centered super-ellipses pointing towards M (Fig. 2). They are formed by the xy-like band together with the transverse xz-like band for the X sheet and with the transverse yz-like band for the Y sheet (see Eq. (2)). Figs. 7 and 8 show that the only part which has As z char-acter and may therefore disperse with z, is the one point-ing towards M. The corresponding band is the z/xy band which we studied above and which at M was found to disperse by 3 eV, from 2.1 eV for Γ to −0.9 eV for Z. Being centered at respectively X and Y , the electron cylinders are however far away from M, and since the xy band can

have no z character at X and Y, it hardly disperses with kz there. But the z character increases linearly with the distance ky from X – and with the distance kx from Y– towards M, and so does the upwards kz-dispersion. As a consequence, the latter is strongest towards Γ = ( M, π/c and weakest towards Z = ( M,0), where the z/xy band eventually bends over and becomes part of the longitu-dinal t/p band dispersing upwards from Z (see bottom panels of Fig. 12). This diagonal interlayer coupling (7) thus modulates the long axis, F, of the super-ellisoidal cross section such that it becomes minimal towards Γ and maximal towards Z (see Fig. 10). Specifi cally, for the long axis of the Y cylinder: 2kFx (kz) 2kF + δ kF cos ckz

, and the same for the long axis of the X cylinder, Fyz. Taking then the coupling of the X and Y cylinders into account, we fi rst translate the X cylinder to the Y site and upwards by π/c, and then couple the two cylinders by the matrix elements (8). The coupling has no effect in the kz = n π / c planes containing the Γ, X, and Z points, but in the kz= π/2c planes containing the P and N points, the two cross sections mix to become identical around P, as can be seen in the last two top panels of Fig. 12. As a result, the double cylinder twists and follows the shape of the string of X-centered rhombic BZ faces (Fig. 2), i. e. it stretches out towards the zz/z-bulge around Z of the lon-gitudinal hole sheet. Finally we should mention that the other part of the electron double cylinder is made up of the X xz/y and Y yz/x bands which have no z character and no kz-dispersion. In the two last two top panels of Fig. 12, we observe a Dirac point at P and −1.4eV. Its upper cone in this kz =π/2c plane stays intact over an energy range of nearly 1.5 eV and over a distance of almost π (v = 0.5eV·a) where-after it develops into the 4 neighboring maxima of the mixed Mt/p-longitudinal and Γ zz/z hole band. The lower cone cone extends merely over 0.2 eV. In addition, the Dirac point has a second, upper cone which slopes by as much as v = 1.3eV·a and extends several eV above the Fermi level, but is truncated slightly below. This is the xy/z electron band. From the cross sections of the band structure with the planes shifted by multiples of π/4c in Fig. 12 one can see that the two upper Dirac cones at the P points are fairly 3D. This seems to differ from the previ-ously discussed Dirac point at 0.25eV and ( M,kDz) whose cones merely extend in a particular kz =kDz 2D plane. In that case, the mechanism is that kz tunes the relative position of two bands, which have different symmetries at a 2D high-symmetry point, k D, and a hybridization increasing linearly with the distance from that point, to be degenerate at k = ( k D, kDz) . Referring to expression (3): kz tunes the gap, which vanishes at the Dirac point, g (kz =kDz = 0. For the low-energy z/zz-t/p cone at 0.25


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eV and ( M, π/4c) , g (kz) is the kz dispersion of the upper M z/xy - Γzz/z band seen in Fig. 11 which is larger than that of the cone, v = 0.4eV·a. For the Dirac cones at P, the mechanism is actually the same, but the kz dispersions of the two relevant z-like levels are merely from −1.8 eV and −1.4eV at P to −1.6eV and −1.2eV at X (Fig. 12), and this amounts to less than the Dirac slopes, v = 0.5 and 1.3 eV·a. The reason for the small kz -dispersion along XPX, compared with that along ΓZ, is simply that XPX is at the zone boundary where interlayer coupling is between the degenerate X and Y states. Hence, to first order in t⊥, the two degenerate α levels split by ±t⊥ | cz,α (X )|2, which is in-dependent of kz. But if t⊥ were as large as 2eV, first order might not suffice. Therefore, once again, we first consult the caption to Fig. 2 to learn that at X , z may only mix with one other state, yz, which is the bottom of the M longitudinal t/p band. The interlayer coupling of X with Y therefore requires merely the solution of a 4 × 4 matrix, which after exact Löwdin downfolding of the d block re-duces to:

These two second-order equations for ε (X , kz) can be solved exactly and give no kz-dependence. The above-mentioned 0.2 eV kz -dependence seen in Fig.12 must therefore be due to interlayer hopping via orbitals other than As z, but this is negligible. Secondly, we confirm from Fig. 7 that at X , there is essentially only one level with z character (because tz,yz (X ) = 0.1eV). This level is at −2.1 eV and is non bonding between nearest neighbors separated by x±ηz or y ± ηz. The yz level is at −1.3 eV and is the one seen in Fig.12 for BaFe2As2 to be at −1.4eV at P and −1.2eV at X. So if BaFe2As2 were merely 2D LaOFeAs with added bct interlayer coupling, the shift from −2.1 to −1.6 eV should be ∼ |t⊥ |, but that is too inaccurate. In or-der to find t⊥, we therefore seek the lower z-level along XPX in BaFe2As2 and find that it is dispersionless and lies 2.0 eV below the upper z level (and below the frame of the figure). Hence t⊥=−1.2eV, assuming a 20% bymixing of yz character to the upper z-like level due to the level separa-tion of merely 0.4eV in BaFe2As2. We remark that although the bands do not disperse along XPX, the wave-function characters of course do, e.g. in the kz =nπ/c planes, the upper and lower z-like levels have respectively purely X z/yz and Yz/xz char-acters, while in the kz =π/2c plane, they are completely mixed. Secondly, we remind that the z states along XPX are intra-layer non-bonding and interlayer ppσ bonding and antibonding.

Going away from the XPX line, the upper z-like band hybridizes linearly with the nearby zz/z and z/xy bands and thereby form the P-centered Dirac cones discussed above. That the cones are centered at the kz =π/2c plane, i. e. the one containing the high-symmetry points N and P (see Fig. 10), is due to the fact that, in this plane, the bct bands are periodic for k in the small BZ, whereby the bands along ΓX equal those along MY, which by tetrag-onality equal those along MX . This higher symmetry is clearly seen in the last two top panels of Fig 4. In CaFe2As2, η=1.04 so that the intra-layer z/xy and z/zz hybridizations are smaller than in Ba, whereby the centers of the antibonding z/xy and z/zz bands lie lower. However, of greater importance is that the smaller size of the Ca ion makes the As-As interlayer distance 70 pm – i. e. nearly 20% – shorter than in in BaFe2As2. This substan-tially increases the interlayer hopping t⊥. As a result, the splitting of the z-band at X, ∼ 2t⊥, which was 2 eV for Ba, is 3.2 eV for Ca. As seen in the first two panels of Fig. 13, this splitting is from −3.9 to −0.7 eV. Hence, the upper z-level is above the X yz and zz/z levels at X and merely 0.1 eV below the xy-like level at −0.6 eV. The consequence is that in Ca, the xy-like electron band forms a complete, upper Dirac cone in the kz =π/2c plane. This cone slopes by about 3.5 eV over the distance π (v = 1.1eV · a) and is thereby twice as steep as the Yz/XY cone considered after Eq.(3). Contrary to the case in BaFe2As2, the xy-like band now forms the inner electron cylinder. The second consequence of the upper z-level at X ly-ing as high as −0.7eV, is that the Γz/zz hole band lies higher than in Ba. At Z, this amounts to 0.4 eV and places the Γz/zz level 0.2 eV above the degenerate M t/p level. Going out in top face of the central BZ from Z, this Γ z/zz hole band now crosses the longitudinal t/p electron band which, like in BaFe2As2, curves upwards due to re-pulsion from the M z/xy band (hybridized with Ca4dzz ). This high-lying Γz/zz hole band crosses the Fermi level far outside the transverse Mt/p hole band. For kz de-creasing below π/c, the crossing between the Γz/zz hole band and the M z/xy-hybridized longitudinal M t/p electron band gap proportional to sin ckz . The resulting lowest band is thus shaped like a volcano with a wide foot of Γz/zz character and a caldera of Mt/pz/xy char-acter around Z and 0.5 eV above the Fermi level. As kz decreases towards 3π/4c, this gap increases so much that the rim is washed out and the caldera develops into a flat hilltop. Eventually, the characters of the z-like bands (Fig. 11) gapped around the degenerate M t/p band along ZΓ change back to normal order with the upper band being M z/xy like and the lower band Γz/zz like. As a conse-quence, the flat hill continuously transforms into the in-ner, longitudinal M t/p sheet. This can be seen for kz

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=π/2c in the second panel of Fig. 13. Since the two z-like bands are gapped around the M t/p band along ZΓ, there is no Dirac point along ΓZ. The corresponding sheet of Fermi surface is thus a warped, ΓZ-centered cylinder with a very broad, Γz/zz-like base near Z and a long nar-row piece around Γ. Outside of this, except near Z, lies the transversal M t/p hole cylinder. The top of the Γxy hole band is slightly above that of the M t/p band and its straight cylindrical FS sheet lies outside the transversal Mt/p hole cylinder. Pure CaFe2As2 becomes superconducting with Tc

max∼12K without doping but by the application of hydro-static pressure in the range 2–9 kbar [81]. At 5.5 kbar there is a first-order phase transition into a collapsed bct non-magnetic and possibly superconducting phase [82] with η marginally smaller and with the interlayer As-As dis-tance decreased by an additional 27pm. In this collapsed phase, whose bands are shown in the two last panels of Fig. 13, the interlayer hopping is increased so much that the splitting of the z-band at X, ∼ 2t⊥, is now 4.5 eV. As a result, the upper level is 0.2 eV above the Fermi energy. This, first of all means that the electron cylinder has lost one sheet, essentially the xy/z sheet, so that there is no Dirac cone at P. On the other hand, at Z, the Γz/zz level is now 1 eV above the Fermi level and the Mz/xy level is 0.1 eV below the doubly-degenerate Mt/p level. The lat-ter creates a prounced, slightly gapped Dirac cone with

v = 0.8 eV·a. Moreover, the doubly degenerate t/p level is essentially at the Fermi level. As kz decreases below π/c, the Γz/zz hole band and the upper Dirac cone, which has mixed longitudinal t/p and z/xy character, gap at their crossing, which is at 0.7 eV. This volcano thus has a cone-shaped caldera. As kz decreases towards π/2c, the rim and the Dirac caldera are flattened away and this flat hilltop sinks below the Fermi level. The holes are thus in a large Γz/zz M xy/z like sheet shaped as a disc cen-tered at Z. At this center, there may be a non-occupied pin-hole with Dirac character. There are no Γ xy holes because that band is slightly below εF . As mentioned above, the electrons are in an XP-centered cylinder of xz/yz character. This band structure is thus very different from the standard one, but quite interesting. The band structure of BaRu2As2 [83] is similar to that of (non-collapsed) CaFe2As2 shown in the first two panels of Fig.13, including a Dirac point at P. But it differs in two respects: The Γxy hole band is en tirely below the Fermi level and the doubly degenerate top of the hole bands disperses like in BaFe2As2 due to hybridization with Ba 5dxz/yz near Γ. This causes the top of the t/p bands to sink below the Fermi level near Γ and the corresponding in-ner and outer Fermi-surface sheets to truncate.

5 2D Spin-spiral band structure

At low temperature and normal pressure, the parent compounds of the Fe-based superconductors (except LiFeAs) become orthorhombic, antiferromagnetic met-als. Superconductivity seems to appear, once these spin

Figure 13 3D band structure of bct CaFe2As2, fattened by the As z

character and along the paths ( Γ ΓXM ,0) = ( ΓMY M, π/c) and

(Γ ΓXM , π/2c) = ( ΓMY M, π/2c). First two panels: normal pressure.

Last two panels: Collapsed phase at 0.48 GPa. [80] Otherwise as Fig. 12.


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and charge orders are suppressed, e.g. by doping (elec-tron or hole) or pressure. This superconductivity is pre-sumably mediated by spin-fluctuations. In this section we shall study the interplay between the band structure and the spin order. Ab initio calcula-tions employing spin-density-functional theory (SDFT) tend to yield the proper spin or der, which is striped with spins on the iron rows along x aligned and along y alter-nating. The mo ments, albeit still considerably smaller than the saturation magnetization of 4 μB /Fe, are much larger (∼2 μB /Fe) than those obtained by neutron scat-tering or muon spin rotation; the latter are ∼0.4 μB /Fe for LaOFeAs [7] and twice as large for BaFe2As2 [10]. Even worse, only with the calculated large mo ments do the spin-density-functional calculations yield the correct structure (η=0.93 and 0.5% con traction in the ferromag-netic direction for LaOFeAs); the structures calculated without allowing for spin-polarization differ much more from the observed ones than is normal for density-func-tional cal culations (η=0.81 and no orthorhombicity for LaOFeAs) [36, 61]. It thus seems that the large moments exist, but fluctuate on a time scale shorter than what can be resolved with neutrons or muons [70]. Below, we shall first study how the spin-polarization modifies the band structures discussed in the pre vious section which were calculated for the experimental structures. Thereafter we shall consider the energetics of the spin spirals.

5.1 Formalism

SDFT involving d-electron spins reduces approximately to a Stoner model [84, 85]. This reduction has the con-ceptual advantage of cutting the SDFT self-consistency loop into a band-structure part which for a given site and orbital-dependent exchange splitting, Δ, yields the site and orbital-dependent spin moment, m (Δ), plus a self-consistency condition which simply states that Δ = mI, where I is the Stoner (∼ Hund’s rule) interaction param-eter. The band-structure part gives insight into the spin response of the non-interacting system, and not only in the linear regime. The spin arrangements which we shall consider are simple spin spirals. For these, the moment lies in the (x, y)-plane and has a constant magnitude, but rotates from site to site by an angle, ϕ (t) = q · t, proportional to the projection of the lattice translation, t in Eq. (1), onto the spin-spiral wave vector, q. Hence, the spin spiral with q at Γ produces FM order and the one with q at Y produc-es stripe order because the moment rotates by π upon y-translation, and by 0 upon x-translation. Finally, the spin spiral with q at M produces checkerboard order because

the moment rotates by π upon y as well as upon x-trans-lation. These spin spirals with q at high-symmetry points are collinear and commensurate but with the formalism which we now explain any q can be treated. In order to solve the band-structure problem in the presence of such a spin spiral, we use a basis set of lo-calized Wannier orbitals times pure spin-functions, |↑⟩ and |↓⟩, with quantization direction chosen along the lo-cal direction of the moment. In this representation, the one-electron Hamiltonian is trans lationally invariant, albeit with q-dependent hopping integrals, so that there is no coupling between Bloch sums with different wave vectors. As a consequence, the band-structure prob-lem can be solved for any q without increasing the size of the primitive cell [86]. When merely seeking insights in this section, we shall neglect the interlayer coupling and use the 2D bands in the large BZ. So in this case, configuration space is invariant to the t-mirror group, and spin space is invariant to the t-spinrotation group, which both have the same irreducible representations. As long as spin and orbital spaces remain uncoupled (spin-orbit coupling neglected), the one-electron Hamil-tonian therefore factorizes down to the orbital and spin degrees of freedom for a primitive cell of the t-group. This, together with SDFT, enables simple calculation of spin-spiral band structures, moments, and magnetic en-ergies. The Hamiltonian turns out to be simply:

in the local ↑, ↓ representation and with the origin of k shifted to q/2. If the two paramagnetic Hamiltonians, h (k) and h (k + q), are identical (not merely their eigen-values), this form is block diagonal. This is also the form appropriate for Δ larger than the bandwidths. For small Δ, it is more practical to transform to the (↑

±

↓)/ 2 rep-resentation in which

In these expressions, h (k) is the paramagnetic 8 × 8 pd Hamiltonian whose eigenvalues, εα (k), are the 2D bands discussed in the previous section, and Δ is an8 × 8 di-agonal matrix whose diagonal elements have the same value, Δ, for all five Fe d orbitals, and 0 for all three As p orbitals. The approximation that only like orbitals couple goes back to the assumption that the spin density on Fe

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is spherically symmetric, and it is justified by the fact that, using this simple form for Δ, we find good agree-ment with the results of full calculations using SDFT for the spin-polarized bands. Of course, a form with the proper point symmetry on Fe could be used, but that would require more parameters. Although it takes the spin-spiral representation to see that the Hamiltonian above is general, we do note that, for an ↑-electron in a commensurate antiferromagnet, a Hamiltonian of the form (12) is obtained by elementary means using the (⇑, ⇓)-sublattice representation, global spin directions, and purely spatial Bloch functions. The Hamiltonian obtained for a ↓-electron is the same, but with ⇑ and ⇓ interchanged. Diagonalization of the 16×16 Hamiltonian (13) yields energy bands, εβ (k) , and corresponding eigenvectors, {c(↑−↓)l,β (k), c(↑+↓)l,β (k) } ≡ {cl,β (k) , cl,β (k + q) }, with l enu-merating the 8 orbitals. (Here, cl,β (k) is a simplified nota-tion for one of the 16 eigenvector components; for small Δ, this equals one of the 8 eigenvector components of the paramagnetic Hamiltonian, times 1/ 2 ). We can now find the orbital-projected spin polarization of state βk as:

and summing this over the Fe d orbitals, pβ (k) = ∑l=1,5 pl,β (k), and over the occupied – or empty – states, we ob-tain the Fe moment:

This is the magnetic output of the spin-spiral band-structure calculation.

5.2 Stripe 2D band structure

In order to demonstrate how the spin-spiral formal-ism works for the 2D band structure of LaOFeAs, we start from the paramagnetic bands, εα (k) , decorated with the weight of each of the eight Wannier orbitals in Fig. 7. We consider a Y stripe, and thus prepare for the Δ-coupling as shown in the upper half of Fig. 14: On top of the bands at k (in green) we place those at k + πy (in grey). Specifically, on top of the green ΓX bands we place the grey YM bands (which are the same as the XM bands with xz and yz exchanged), on top of the green XM bands we place the grey MX bands, and on top of the green ΓYbands (which equal the ΓX bands with xz and yz ex-changed) we place the grey ΓY band. The ΓM bands couple with the XY bands, but since the latter were not shown in Fig. 7, this line is not shown in Fig. 14 either.

Now, the effect of Δ is to split degeneracies, εα (k) = εβ (k + q) , by Δ times the geometrical average of the d charac-ters, ∑l=1,5 c*l,α(k) cl,β (k + q). This of course only holds as long as Δ is so small that no fur ther bands get involved. States without common d-character therefore do not split. We note that states throughout the band structure split, independent of the position of the Fermi level, i. e. of the doping, but for small Δ only those states which gap around the Fermi level contribute to the magnetization and the magnetic energy, so this is how doping enters. The paramagnetic bands are seen to be linear inside an energy window of ±0.1 eV, at the most, around the Fer-mi level, and this means that effects of the exchange po-tential ± 1

2 Δ can be treated with linear-response theory only when Δ 0.2 eV.

5.2.1 Bands and Fermi surface in the linear-response region, Δ 0.2 eV

A close inspection of the top xy panel of Fig. 14 reveals that the crossing of the purely xy-like band with itself halfway between Γ and Y is 0.3 eV below the undoped Fermi level and thus requires Δ > 0.6eV to gap around εF . On the other hand, the crossing of the xy hole band along ΓY with the yz/x electron band along ΓY occurs only slightly below εF , meaning that the Γ -centered hole pocket and the Y-centered electron superellipse almost nest along ΓY. This can be seen in Fig. 15 where we show the Y-folded Fermi surfaces in brown lines. However, the yz/x band has only very weak xy character caused by its weak hybridization with the below-lying xy band, as was explained in connection with Eq. (2). Hence, due to lack of common orbital characters, these two states gap by much less that Δ. Finally, the xy hole band along ΓX crosses the xy/z electron band along YM at ∼0.1 eV below εF , but due to the reduced xy charac-ter of the xy/z band, Δ must exceed ∼0.3 eV to gap that part of the Fermi surface. As a result, for the value Δ = 0.18eV which via Eq.(15) produces the same moment as the one observed experimentally, m (0.18eV) = 0.3 μB /Fe, the xy hole pocket does not gap. The Fermi surface calculated for Δ = 0.18 eV is shown by black lines in Fig. 15. For the Γand Y-centered sheets, it only differs from the one calculated for Δ = 0 and shown in brown lines, because the Fermi level is slightly shifted due to gapping of the other sheets, i. e. those centered at M and X . That gapping, which causes the small moment of 0.3 μB /Fe, takes place as follows: The side of the X -centered electron superellipse which is normal to the x direction, i. e. which points to-wards Γ , matches the inner, longitudinal M-centered


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hole pocket both in Fermi-surface dimension (nesting) and in orbital character, xz. Those two bands therefore gap around the Fermi level, while the outer, transversal yz hole sheet stays intact. Near the X and Y directions, the X electron sheet however matches the outer, trans-versal hole sheet in size and orbital character, Yz and Xz, respectively. So near those directions, the outer, trans-versal hole sheet is gapped while the inner, longitudinal sheet is intact. Finally, due to lack of common charac-ters near the y direction, where the xy/z electron band does not hybridize with the lower-lying xz band, no gapping occurs. As a consequence, small paramagnetic electron pockets with xy/z and transversal, xz/XY /zz/x characters occur. Such electron pockets will remain at the Fermi level, also for large Δ, as we shall see below. Note that the FS parts not gapped away are essentially not spin-polarized; this will not be the case for larger Δ. Since the Y stripe has antiferromagnetic order in the y direction and ferromagnetic order in the x direction one might expect higher conductivity in the x than in the y direction. However, most of the FS has a predominantly y-directed group velocity and predominantly yz-electron

Figure 14 Top: 2D paramagnetic bands decorated like in Fig. 7 and pre-

pared (folded) for stripe order with q = πy : On top of the ΓY bands in

green we place the ΓY bands in grey, on top of the ΓY bands in green

we place the YM bands in grey, and on top of the XM bands in

green we place the MX bands in grey. Bottom: 2D stripe band struc-

ture decorated with the orbital-projected spin-polarizations as given

by Eq.(14) and with positive and negative polarizations in respectively

dark and light blue. For the exchange potential, the value Δ = 1.8 eV

was used, which by Eq. (15) yields the moment m (1.8 eV) = 2.2 µB /Fe

and corrresponds to the value I =0.82 eV of the SDFT Stoner parameter.

The dashed line is the Fermi level, which has moved up by 0.5 eV. Note

that the paramagnetic and spin-spiral band structures are lined up

with respect to the common paramagnetic potential, i. e. h k in the TB

Stoner calculation. The 2D stripe Fermi surface is shown in Fig. 16. For

stripe order, As p projections cannot be spin-polarized and have there-

fore been omitted.

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or yz longitudinal-hole character. Where the velocity is in the x direction, the character is xy/z electron or yz trans-verse hole. The xy hole pocket is isotropic in the plane. In conclusion, the exchange potential needed (Δ=0.18 eV) to give the observed moment with the Stoner model, is smaller than the fine structure of the bands. The gap-ping of the Fermi surface and the susceptibility, m (Δ)/Δ, therefore depend crucially on the details of the k-and-orbital nesting.

5.2.2 Bands and Fermi surface beyond the linear- response region, 0.2eV : < 3 eV

The exchange potential obtained selfconsistently from the SDFT and yielding the proper crystal struc ture, is ten times larger: Δ=1.8eV, and thereby has the same scale as the structure of the subbands, i. e. this Δ is intermedi-ate and linear-response theory invalid. The SDFT value of the Stoner parameter is I=0.82eV and the moment is m (1.8eV) = 2.2 μB /Fe. Compared with the maximum mo-ment of 4 μB /Fe for the d6 configuration, the value 2.2 μB /Fe is intermediate. The lower part of Fig. 14 now shows the stripe bands for this situation. These bands are com-plicated, because the gapping and spin-polarization de-pend on the energies and the p and d orbital characters of those bands at k and k + q which are sep arated by less than ∼ Δ. In the present section, we shall describe those bands and their Fermi surface and calculate specific, im-portant levels analytically.

In Fig. 14, the paramagnetic and the spin-spiral band structures are lined up with respect to the common paramagnetic potential, i. e. h k in the TB Stoner calcula-tion. We see, as was pointed out before, that bands with d character split irrespective of their position relatively to the Fermi level, but those in the lower half the d-band structure generally shift downwards (dark blue) while those in the upper generally half shift upwards (light blue). We recall that the shift upon an increase of the exchange potential is the negative of the spin-polariza-tion: ∂εβ (k /∂ (Δ/2) =−pβ (k) , by 1st-order perturbation theory. In fact, there happens to be a fairly well-defined dividing line between positively and negatively spin-polarized bands around 0 eV. Moreover, on this dividing line, the non-hybridizing yz and zz/XY bands along ΓY are nearly degenerate and dispersionless, so increas-ing Δ beyond 1.8 eV will open up a gap which extends throughout the BZ and makes any correponding d5 ma-terial (e.g. LaOMnAs or LaOFeN [87]) an antiferromag-netic insulator. We now discuss the intermediate-moment stripe bands for 2D LaOFeAs in detail. Starting again on the left-hand side of Fig. 14 with the crossing of the para-magnetic, nearly pure xy band with itself, halfway be-tween Γand Y at −0.3 eV, we see the bands split to the energies −0.3 ± 0.9eV =−0.3eV ± Δ/2 around the Fermi level, which has now moved up to 0.5eV, with the lower and upper bands fully spin polarized. This gap extends in a large region around the ΓY -line. So, whereas for small Δ, the Γxy hole and Yxy-yz electron sheets did not

Figure 15 Right: Nesting of orbital-projected Fermi-surface

sheets for undoped 2D LaOFeAs (from Fig. 8) for stripe order with

q = πy (green arrow). Left: Partly gapped Fermi surface resulting

from the small exchange potential Δ=0.18 eV which yields the

small moment m=0.31 µB /Fe. This corresponds to the Stoner

parameter I=0.59 eV.


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gap at all, for intermediate Δ, these two FS sheets no lon-ger exist. It is however only the xy-parts which are gapped away: Due to strong yz/x hybridization, the yz-like bands near Γ (green) has no partner at Y (grey) within the ±Δ/2-range with which it can couple. This band therefore only splits midways between Γand Y, i.e. near 1

2 Y, but hardly closer to Γ and towards X (grey). Hence, the reason for the disappearance of the yz-part of the superellipse at Y is not gapping, but the 0.5 eV upwards shift of the Fermi level. This shift is due to the fact that with configuration d6, the Fermi level lies in the upper half of the d-like band where most bands are shifted upwards by the exchange potential and drag the Fermi level along with them. The shift ∂εF (Δ)∂ (Δ/2) is upwards, if at εF (Δ) the density of ↓ states exceeds that of ↑ states. It may be noted that the crossing of the grey xy/z and yz bands along YM, which is at d 6 for the para-magnetic bands, still occurs for the stripe bands, but far below the Fermi energy. The bands expected to gap mostly for Y-stripe order are d bands dispersing less than Δ/2 in the y direction, i. e. the xz and zz bands. The xz band is dispersionless near

the XM line where it forms the M-centered transverse hole band and the bottom of the X -centered electron band. Further towards the ΓY line, however, the ky -dis-persion of the xz band becomes large due to by-mixing of y character towards Γ , and also the concomitant di-lution of d character reduces the exchange coupling. As a consequence, near the XM line, the xz electron and hole bands split to ∼±Δ/2, whereby the ↓ band is above the shifted Fermi level and the ↑ band is far below. This emptying of antibonding xz states yields the observed 0.5 % orthorhomic contraction in the x direction, along which the spins are aligned ferromagnetically [35]. In the remainder of the zone, there are no bands at the Fermi level. The paramagnetic zz band has a width of about 1eV ∼ Δ/2 and is centered at ∼−0.6eV, so we expect the exchange splitting to shift the zz ↓ band up to – or above – the shifted Fermi level. Fig. 14 shows that this is rough-ly the case, but the details are more complicated, due to strong hybridization with the XY band, as we shall see later in connection with Eqs. (17) and (18). The paramag-netic and stripe bands, albeit merely for positive energies and the exchange splitting, Δ = 1.1 eV, may be seen more clearly in the left-hand side of Fig. 18. We shall return to this figure. Only two bands remain at the Fermi level: (1) the longitudinal yz ↓ band dispersing downwards from M, crossed by and hybridizing with a weakly spin-polarized xy/z ↑ band dispersing upwards from X , and (2) the zz

Figure 16 Fermi surface for stripe order (q = πy) resulting from the

large exchange potential Δ = 1.8 eV which yields the large moment

m = 2.2 µB /Fe corresponding to to the SDFT Stoner value I = 0.82 eV.

See also Figs. 15 and 8.

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with the XY /zz/x-block at X as the notable exception (see caption to Fig. 2). We thus start explaining the spin and orbital characters of band 1 by using (16) to couple the levels at k = X to those at k + q = M: the paramag-netic xy-like antibonding and bonding levels at X (see Figs.7 and 14) are the strongly xy-like bottom of the elec-tron band (green), for which εa(X ) = −0.46 eV and c(X ) =0.95, and the strongly y-like level at εb (X ) = −2.79eV. These levels couple to the xy-like levels at M (grey) of which the antibonding one at εa( M)= 0.95 eV is mostly z-like, and the bonding one at εb( M)= −3.36 eV is mostly xy-like, s( M)= 0.83. The four xy-like stripe levels at X are thus the eigenvalues of the Hamiltonian:

These eigenvalues (seen in Fig. 14) are: 1.12, −0.45, −2.73, and −3.59eV when Δ/2 = 0.9 eV, and hence little perturbed by the stripe order. The reasons are that the paramagnet-ic levels are separated by more than Δ/2, and that the p hybridization reduces the geometrical averages of the d characters far beyond unity, except for two levels which are, however, separated by as much as 3 eV. In particular the state of interest, the one at −0.45eV, has been pushed down by the level at 0.95 and up by the one at −3.36, both at M, and as a result, has moved by merely 0.01eV. For the same reason, its spin polarization is only about 50%. This state thus remains essentially the (green) bottom of the electron band at X . The paramagnetic yz-like antibonding and bonding levels at X are the strongly yz-like bottom of the longitudi-nal hole band, for which εa (X )= −1.26 eV and c (X ) =0.998, and the strongly z-like level at εb (X ) = −2.12 eV. These levels couple to the yz-like levels at M (grey), which are top of the doubly degenerate hole band, for which εa ( M)= 0.21eV and c ( M) = 0.90, and the y-like level at εb ( M) = −1.97 eV. With these values, the yz-like eigenvalues of the 4 × 4 stripe Hamiltonian (16) becomes 0.58, −1.39, −2.11, and −2.21eV. Here the uppermost level, being near the Fermi level, is our band 1. It is described to a good ap-proximation by using merely the antibonding paramag-netic states, i. e. by the 2×2 Hamiltonian


↓ band dispersing downwards from M, hybridized with more weakly spin-polarized XY ↑ band. Being respective-ly even and odd by reflection in a vertical mirror contain-ing nearest-neighbor As atoms and the q vector (see Fig. 2), bands 1 and 2 cannot hybridize and therefore cross at a Dirac point along the XM line. This pins the d6-Fermi level and gives rise to a Fermi surface shaped like a pro-peller [16], with two electron blades and a hole hub (Fig. 16). The hub is yz ↓ like and the inner parts of the blades are mixed zz ↓ -XY ↑ like, while the outer edges are mixed yz ↓ -xy/z ↑ like. Due to the pinning, the propeller shape is robust, e.g. not sensitive to Δ. Hole doping by a few per cent will bring the Fermi level to the Dirac point and make each electron blade shrink to a point. Upon further hole doping, the blades will reappear as hole sheets. The velocity of the xy/z ↑ and the zz ↓ parts of the anisotropic cone are respectively ∼ 1eV · a = 2.9 eV Å and ∼ 0.4eV · a = 1.1eV Å. These Dirac cones in the stripe-ordered SDW state have been predicted [76] and later observed using respectively quantum oscillations [29] and ARPES [77]. Compared with ours, the experimental velocities seem to be renormalized by a factor ∼ 1/4. This Dirac cone will be gapped by any lattice imperfection breaking the above-mentioned mirror symmetry and is therefore not “protected”. In order to explain how the complicated spin and or-bital characters of conduction bands 1 and 2 arise from the paramagnetic band structure, let us consider the simple case that the paramagnetic TB Hamiltonian h (k) in Eq.(13) is a 2 × 2 matrix. Its eigenvalues are the bond-ing, εb (k) , and antibonding, εa(k) , paramagnetic bands with the respective eigenvectors, {– sin φ (k),cos φ(k)} and {cos φ(k), sin φ (k)} ≡{c (k), s (k)} (we have chosen the phases of the orbitals such that the Hamiltonian is real). Taking the first orbital is a d and the second as a p orbital, transformation of the spin-spiral Hamiltonian (13) to the bonding-antibonding representation yields:

because the pp and pd elements of the exchange block, 12 Δ, vanish. Here, c2 and s2 = 1 − c2 are the d characters of

the antibonding and bonding levels, respectively. Note that only the d – but not the p− characters need to be the same at k and k + q. This 4 × 4 form (16) is exact when k, and thereby k + πy, is at a high-symmetry point, because here, h (k) factorizes into blocks of dimension 2 × 2,

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because these states are the only ones with substantial yz character, as may also be seen from Fig. 14. The upper-most state thus has about 80% M yz and 20% X yz char-acter, and its spin polarization is −75%. As we now move from X towards M, via 1

2 XM as in Figs. 14 and 18, the X xy/z ↑ band disperses up wards from −0.45eV like the paramagnetic (green) electron band, and the yz ↓ band disperses down wards from 0.58eV like the paramagnetic (grey) longitudinal M yz hole band. At about 1/4 the distance to M, these two bands suffer an avoided crossing. From there on, the xy/z ↑ band contin-ues towards M like the paramagnetic band, apart from the facts (a) that it gets folded and split at 1

2 XM with a covalency reduced Δ, and (b) that it continues to hybrid-ize with the longitudinal yz ↓ band whose downwards dis-persion (grey) is halted near 1

2 XM due to repulsion from the upcoming (green) yz band. The xy/yz hybridization matrix element along XM , −2

�2(t 10

x y,X z + t 11x y,X z + t 21

x y,X z

)sin ky , would have vanished, if the xy and Xz Wannier orbitals of the pd set shown in Fig. 6 had been respec-tively symmetric and an tisymmetric with respect to the Fe plane. Band 1 obtained from the pd Stoner model is somewhat more shallow than the one obtained from a standard LAPW calculation, which yields a bandwidth of 0.6 eV. This can be traced back to the gap between the downwards-dispersing longitudinal yz ↓ band and the upwards dispersing xy/z ↑ band, being 0.5eV in the mod-el but merely 0.2 eV in the LAPW calculation, thus caus-ing the model band to be 0.3 eV more narrow. Reducing t 10

x y,X z + t 11x y,X z + t 21

x y,X z does not entirely remove this dis-crepancy, which might also be due to our assumption of a spherically symmetric exchange potential. We now come to band 2. It was pointed out in Sect. 3.3, and can clearly be seen in Figs. 7 and 14, that the paramagnetic XY and zz bands hybridize strongly, ex-cept along the ΓM lines, and have avoided crossings around X and Y causing them to gap around the Fermi level. They also gap around the Fermi level along ΓM, but due to avoided crossings with other bands. In order to understand the effect of a stripe potential, let us use a 4 × 4 model like (16), but now for two d orbitals. In this case, the exchange block is constant and the spin-spiral Hamiltonian in the bonding-antibonding representation becomes:

Clearly, if the bonding and antibonding linear combina-tions of the two d orbitals were the same at k and k + q, i.

e. if φ (k) =φ (k + q) , then the bonding-antibonding repre-sentation (17) would be identi cal with the dd representa-tion. In that representation, the off-diagonal block is 1

2

Δ times the unit matrix because we have assumed the exchange potential to be spherical. If now also εa (k) = εa (k + q) and εb (k) =εb (k + q) , as would be the case at the zone boundary, then we could transform to the local spin representation (12) and would then immediately realize that the 4 stripe eigenvalues (in eV) are:

This fits well with Fig. 14, from where we have taken the values 0.8 and −0.8 eV, for the paramagnetic antibond-ing and bonding levels at 1

2 XM. The stripe level belong-ing to band 2 is the bonding, minority spin level at 0.1 eV. To be honest, the energies ±0.8 eV of the paramagnetic levels do include weak hybridizations with the xz and x orbitals, which have been neglected in the 4 × 4(XY , zz) model (17). We should also warn that although h (k) and h (k + q) have the same eigenvalues at the zone bound ary, these matrices are generally not identical; off-diagonal elements may have different signs, e.g.

hX Y ,zz

(−π

2 y)= hX Y ,zz

(π2 y

)but hxz,zz

(−π

2 y)=−hxz,zz

(π2 y

)

.

As seen from the figures, the paramagnetic bonding and antibonding XY /zz bands disperse less than their separation, δ ∼ 1.6 eV, so in order to be able to diagonalize the 4 × 4 stripe Hamitonian (17) let us stay with the as-sumption that εa (k) = εa (k + q) ≡ δ/2 and εb (k) = εb (k + q) ≡ −δ/2, but drop the assumption that φ (k) = φ (k + q). The 4 stripe bands are then given by:

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For ϕ = 0, the zone-boundary case (18), the bonding and antibonding d orbitals are identical and each of the four levels, ± δ/2 ± (Δ/2) , have pure spin and pure bonding or antibonding character. For ϕ = π/2, the bonding and antibonding bands have orthogonal d characters and therefore only have off diagonal exchange coupling. This does not split the spin-degenerate bonding and anti-bonding levels, but separates them to ±1

2

�δ2 +Δ2. The

realistic case at X (green) is that the antibonding and bond ing levels have roughly the same XY and zz charac-ters. We therefore take φ (X)= π/4, and the XY orbital as the first orbital. At M (grey) the “antibonding” level has pure XY and the “bonding” level pure zz character, so φ ( M) = 0. As a result, the bands have dispersed from the levels given by (18) to ±1

2

√δ2 ±�

2δΔ+Δ2 = ±1.57 and ±0.66 eV at X = M. This agrees well with what is seen in Fig. 14 and explains why band 2 disperses upwards from 0.1 at M to 0.66eV eV at X . The latter level is essentially bonding zz ↓ /XY ↑. Armed with the detailed understanding of the inter-layer hopping provided in Sect. 4 and that of the generic 2D stripe band structure provided above, the interested reader should be able to digest the complicated 3D stripe

bands for specific materials found in the literature and recently reviewed in [9, 10].

5.3 Magnetization and magnetic energy

Apart from the spin-spiral band structure described above in the case of stripe order, the output of a band-structure calculation with an imposed exchange poten-tial, Δ, is the Fe-magnetization, m (Δ). In Fig. 17 we now give the results obtained for stripe (q = πy) and checker-board (q = πx + πy) orders for various electron dopings, x, in the rigid-band approximation. With the Stoner ap-proximation, we have been able to afford sampling the spin-polarization over a very fine k-mesh so that nesting features are resolved. Since the magnetization increases linearly with Δ, when it is small, we plot the static spin suceptibility, χ (m) ≡ m (Δ)/Δ, and as a function of m rath-er than of Δ, since the m is an observable. Note that with 4 − x empty bands, 4 − x is the value of the saturation magnetization and χ (m) therefore vanishes for m larger than this. Given a value of the Stoner exchange-coupling con-stant, I, the self-consistent value of the mag netization is the solution of the equation χ (m) = 1/I, and we see that for the SDFT value, I=0.82 eV, m∼2.2 μB /Fe for both stripe and checkerboard order, and that m decreases with elec-tron doping. The reason for the latter can be understood by considering the stripe bands for Δ = 1.8 eV at the bot-tom of Fig. 14: The magnetization along the upwards-sloping line χ = m/(1.8eV) in Fig. 17 is the sum over the empty bands of their spin polarizations, taken with the opposite sign according to Eq(15), i.e. of the light-blue fatness. Since light-blue is seen to dominate over dark-blue, at every energy above the Fermi level, moving the

Figure 17 Non-interacting, static spin susceptibilities χ (m) ≡ m/Δ

for 2D LaOFeAs calculated from spin-spiral band-structure calcula-

tions, i. e. from diagonalizing Hq(k) in Eq. (13) and finding m from the

eigenvectors according to Eq.(15). The paramagnetic TB pd Hamilto-

nian, h (k), was calculated for the observed structure. The electron

dopings, x (in e/Fe), were varied in the rigid-band approximation. For

a given value of the Stoner interaction parameter, I, the self-consistent

moment is given by χ (m) = 1/I. The SDFT value of I is 0.82 eV and the

value fitting the experimental moment and its doping dependence is

0.59 eV.


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latter up, as electron doping will do in the rigid-band sp-proximation, must decrease the moment. Coming now to the small-moment part of Fig. 17, the linear response, χ (0), is seen to be particularly large for stripe order and no doping. This is due to the good nest-ing shown in Fig.15 between the xz part of the X -centered electron superellipse and those of the M-centered hole pockets. This nesting is, however, sensitive to the relative sizes of electron and hole sheets and is therefore rapidly destroyed with electron (or hole) doping, thus causing χ (0) to decrease. Also, increasing Δ beyond 0.2/1.8 ∼ 0.1

eV is seen to make χ (m) decrease rapidly. We should re-member (Fig. 4) that the top of the Γ-centered xy-like hole pocket is merely 0.06 eV above the Fermi level for the pure material and that this pocket disappears once the doping exceeds 0.1 e/Fe. We recall also, that the top of the M-centered hole pockets is merely 0.2 eV above the pure Fermi level and that these pockets disappear, as well, once the electron doping exceeds 0.3e/Fe. The Γ-centered hole pocket and the xy-part of the Y-centered electron superellipse start to gap when Δ exceeds 0.2 eV, and for larger Δ, the xy moment becomes as large as the xz moment. The experimentally observed moment in LaOFeAs is ∼0.4 μB /Fe, stripe ordered, and vanishes for x 3%.This would be consistent with our 2D bands and the Stoner model if I = 0.59 eV. However, only by virtue of its large moment, ∼2μB , does the SDFT yield the ob-served large value of the As height, η = 0.93, and the ob-served 0.5% orthorhombic contraction in the direction of ferromagnetic order. For checkerboard order, q is at the M point and this places the Γand M-centered hole sheets – as well as the X and Y-centered electron sheets on top of each other. Nesting of sheets with the same (electron or hole) char-acter is not optimal for gapping, and χ (0) is therefore neither very high nor very doping dependent. For moments so low that for all possible spin ori-entations m(Δ) is linear and the magnetic energy qua-dratic, the effective coupling between the spins can be expressed in terms of the (Stoner enhanced) linear, static

Figure 18 Band-resolved magnetic energy for stripe order (q at Y ).

Red: Unoccupied part of the folded paramagnetic band structure, i. e.

εα (k) and εα (k+q) for Δ=0. This is the upper part of Fig. 14, but

only for positive energies and without fatness. Black left: Band struc-

ture, εβ Δ, k , obtained by applying an exchange potential of inter-

mediate strength, Δ=1.1 eV, yielding a magnetization of 1.5 µB /Fe and

thus corresponding to I = 0.73 eV. The Fermi level (dashed line), εF

(Δ), has moved up by 0.3 eV. The domi nant d-orbital characters have

been written onto the bands. Black right: Unoccupied part of the

band structure, eβ (Δ, k) ≡ εβ (Δ, k) + 14 pβ (Δ, k) Δ, corrected for

double counting such that the magnetic energy gain per β k-hole is

eβ (Δ, k) − εβ (0, k), i. e. black minus red. Note that only the unoc-

cupied part of the bands are shown and, in par ticular, that the dou-

ble-counting corrected e-bands are truncated by the ε-band Fermi

level, εF (Δ), a truncation which does not occur at the same energy

for all e-bands.

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spin susceptibility. For moments so large that the system is insulating, on the other hand, the electronic degrees of freedom can be integrated out, whereby the coupling between the spins is given by a Heisenberg model. That model, with 1st and 2nd-nearest neighbor antiferro-magnetic cou plings and J1 2J2 has, in fact, often been used to describe the magnetism of the iron pnictides. This is, however, hardly justified because the iron pnic-tides are metals, presumably with intermediate mo-ments. Full SDFT calculations, like the spin-spiral calcu-lation described at the end of this section, do however account well for many experimental observations and as a first step towards deriving better exchange models we shall therefore try to explain the origin of the magnetic energies using the Stoner model. The relation between the magnetic energy for a par-ticular spin spiral, a Y stripe, and the underlying band structure is illustrated in Fig. 18. Its left-hand side shows the paramagnetic bands for positive energies, i. e. the un-occupied bands, as well as the stripe bands for the some-what reduced value Δ = 1.1 eV of the exchange splitting. This corresponds to I = 0.73 eV and to an SDFT calculation for BaFe2As2 adjusted to the experimental dHvA FS [59]. The dashed line shows the Fermi level, εF (Δ) = 0.3 eV. Note that, like in Fig.14, the paramagnetic and spin-spiral band structures are lined up with respect to the common para-magnetic potential. We clearly see that the Δ = 1.1eV stripe perturbs all bands, but the p-like ones the least. In SDFT, the total energy is a stationary functional of the electron and spin densities. For densities which can be generated by occupying the solutions of a single-par-ticle Schrödinger equation for a lo cal potential according to Fermi-Dirac statistics, the value of this functional is simply the sum of the occupied single-particle energies, minus corrections for double-counting of the Hartree and exchange correlation energies. This holds when the potential is the self-consistent one, i. e. the one which minimizes the energy functional. For our Stoner model with h (k) describing the self-consistent param agnetic bands and for Δ taking the self-consistent value, m (Δ) I, the double-counting correction of the magnetic energy is simply 14 m (Δ)Δ = ∑occ

βk14 pβ

(Δ,k

)Δ. For d6 materials, we

shall sum over empty states, because of those there are only 4/Fe, and nearly all have negative spin polarization. The double counting correction of the stripe bands has now been performed on the right-hand side of Fig. 18 from where we realize that these (black) bands, eβ (Δ, k )≡ εβ (Δ, k) + 1

4 pβ (Δ, k) Δ, are far less perturbed than the real stripe bands, εβ (Δ, k) . This means, that the state-resolved magnetic energy gain (black minus red), eβ (Δ, k) − εβ (0, k) , is concentrated near the exchange gaps and near the paramagnetic and spin-spiral Fermi surfaces. Note that

each of these empty bands, eβ (Δ, k) and εβ (0, k) , should be defined as 0 ≡ εF (0), if that band is occupied. This means that the empty paramagnetic bands, εβ (0, k) , are continuous, but truncated with a kink at the lower figure frame. The empty, corrected magnetic bands, eβ (Δ, k), should be truncated discontinuously. Due to the 0.3 eV upwards shift of εF (Δ), all empty parts of the corrected bands are above the frame of the figure so that all empty magnetic bands are visible. In order to see which states contribute to the energy of the Y stripe, let us now once again start from Y and move to the right in the right-hand figure. The zone-boundary gapping around 1.2 eV of the corrected XY /zz/xz/y band is small and fairly localized near Y, and there, its positive and negative contributions nearly cancel. The gapping around 0.7 eV of the yz/xy/x band is much larger due to the dominating yz character of this band, but here again, the negative and positive contributions essentially cancel, until k gets closer to Γ . There, the character of the lower band is xy from the Γ -centered hole band and xy/z from the Y-centered superellipse electron band. This band contributes positively to the energy of the stripe in the large region of the BZ around Γ= Y where the FS is completely gapped, a contribution which in-tegrates up to about half the stripe energy. Between Γand X , the black xy/z band is seen to suffer an avoided crossing with the band formed from the X -centered xz/y electron band and the longitudinal M-centered xz hole band. Near the avoided crossing the magnetic energy density becomes negative, but is essentially cancelled by the positive contribution from the upper band. Clos-er to X, the contribution from the lower band becomes positive again, and its positive magnetic energy density is seen to extend over the large region around X = M where the xz part of the FS is completely gapped. This part of the band is formed by coupling of the paramag-netic, flat X -centered xz electron band, which is occu-pied and therefore lies below the frame af the figure, and the paramagnetic transversal M-centered xz hole band, which becomes occupied outside the transversal hole pocket. So whereas the empty, black xy-xz band extends smoothly throughout the BZ, its paramagnetic partner, against which we measure the band-resolved magnetic energy, goes to zero at a few places in the BZ, such as out-side the transversal M-centered hole band along M. The magnetic one-electron energy of the xz-like band thus reaches +0.5 eV between X and 1

2 XM. Also the two uppermost black bands, which are de-generate at Γand then split into longitudinal and trans-versal p/t bands, stay empty, i. e. they extend smoothly throughout the BZ. The lower of these bands, the one which is longitudinal p/t-like near Γ , becomes M z/xy


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like near X and is seen to contribute negligibly to the magnetic energy in LaOFeAs (but possibly not in the bct materials). The same holds between 1

2 XM and ∼ 13

XM for its lower partner, which is essentially the para-magnetic X -centered xy/z electron band extending in the direction towards M. At 1

3 XM , where the magnetic band becomes the tip of the propeller blade, the cor-rected xy/z band jumbs to 0, whereby the one-electron magnetic energy jumps from 0 to −0.25 eV. Also the M-centered yz hole band coincides with the corrected magnetic yz band, so that only the Fermi-surface trun-cations contribute to the magnetic energy, which in this case amounts to a small negative energy from the region between the propeller hub and the yz-part of the M-cen-tered hole pockets. The higher of those two bands which are degenerate at Γ , i. e. the one which is transversal p/t near Γ , de-velops into the uppermost of the four bands formed by the coupling of the paramagnetic X XY /zz and M XY bands described in connection with Eq.(19), albeit for a larger Δ. This band is seen to contribute positively to the magnetic energy, a contribution which is, however,

overwhelmed by a large, negative contribution from the second of the four bands, which is part of the XY /zz band decreasing from 1.3 eV at Y to 0.6 eV at X , and finally to 0.2 eV around 1

4 X . At this point the zz/XY band is on the inner part of the propeller blade, and therefore jumps discontinuously to 0, where is stays until reach ing the zone boundary at 1

2 XM . The magnetic one-electron energy of the zz/XY -like band thus jumps from −0.6 to −0.8eV at 1

4 XM . It is simpler to eyeball the balance of magnetic one-electron energies along the line between X and 1

2 XM, where we found large cancellations, if we connect the magnetic XY /zz and xz bands according to energy. In fact, the real zz/XY and xz bands may cross along ΓX and YM, but not between X and 1

2 XM . When doing so, we see that the magnetic energy loss from the XY /zz-xz band lying near 0.5 eV is nearly balanced by the gain pro-vided by the upper XY /zz band lying near 1 eV. The lower xz-zz/XY band lies ∼0.2 eV above the xz part of the para-magnetic M-centered hole band and thus gains consid-erable magnetic energy. The Fermi-surface contribution seems to be small because kFy for the outer, transversal M-centered hole surface is about the same dimension as the distance from the center of the hub to the inner, zz/XY part of the blade. Finally, by joining this xz-zz/XY band across the blade to the xy/z band, we see that there is a loss of one-electron magnetic energy of about 0.2 eV inside the blade. This is seen quite clearly in the left-hand part of Fig. 19 where we show the k-resolved magnetic en ergy, which is the state-resolved magnetic energy considered above, summed over empty bands. What stabilizes stripe order when its moment is 1 μB /Fe, is then, first of all, cou-pling of the paramagnetic Γ-centered xy hole and the Y-centered xy/z electron bands over a large part of k-space

Figure 19 k-resolved magnetic Y-stripe energy,

, as in Fig. 18 but for Δ= 2.3eV and summed over the emp-

ty bands and shown throughout (kx, ky )-space. Magnetic energy

gains are red and losses blue. The discontinuities caused by the pro-

peller sheet are clearly seen. While the left-hand figure is for 2D

LaOFeAs with the observed structure (η=0.93), the right-hand figure

shows the result obtained by reducing from 0.52 to 0.30 eV.

With the latter value, the 2D paramagnetic z/xy band becomes de-

generate with the top of the hole bands at M, and thus forms a Dirac

cone with the longitudinal t/p band. This corresponds to the elonga-

tion η ∼1.2.

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centered at Γ . The second, almost as large contribution comes from coupling of the xz part of the paramagnetic M-centered hole band to that of the X -centered elec-tron band over a smaller part of the zone centered at X . These Γ and X -centered red regions do not overlap. The Fermi-surface contributions to the magnetic energy are relatively small, and the positive (red) contribution from the hub tends to cancel the nega tive (blue) contribution from the blades. This being the case, it should be possi-ble to derive a Heisenberg model which fairly accurately describes the change of the magnetic energy for pertur-bations of q around commensurable stripe order. We can now address the interplay between the dis-tance between the As and Fe sheets and the stripe mag-netism: The main effect of increasing this distance, η, is to decrease the z-xy hybridization, as was discussed in Sect. 4.1, and thereby to decrease the splitting between the paramagnetic z/xy antibond ing and xy/z bonding levels near M, and thus to move the empty M z/xy level down. The z/xy electron band from Y= Γ to M = X , as well as the ones folded from X to 1

2 X M (see Fig.14), will be less steep and more xy-like upon increasing η. This, in turn, will increase the polarization and the gapping of the xy-like stripe bands and thereby increase the fixed-Δ moment, m (Δ), and the differential suceptibility, χ ≡ dm (Δ)/dΔ. The increase of the self-consistent moment will finally be enhanced over the increase of m (Δ) by the Stoner factor 1 − Iχ

−1 . At the same time as the moment

increases, so does the gapping, the stripe energy, and, hence, the magnetic energy. Also the mere flattening of the z/xy bands increases the magnetic energy by extend-ing the regions around Γand X of positive magnetic one-electron energy. As seen in the right-hand part of Fig. 19, the Γregion increases in the kx direction and the X region in the ky direction. This explains why spin po-larization tends to increase the vertial Fe-As distance. In conclusion, what stabilizes stripe order when its moment is 1 μB /Fe, is – first of all – the coupling of the paramagnetic Γ-centered xy hole and Y-centered xy/z electron bands over a large part of k -space centered at Γ . The second, almost as large contribution comes from cou-pling of the xz part of the paramagnetic M-centered hole band to that of the X -centered electron band over a smaller part of the zone centered at X . These Γ and X -centered re-gions do not overlap. The Fermi-surface contributions to the magnetic energy are comparatively small and mostly negative. This being the case, it should be possible to derive fairly accurate Heisenberg models describing the change of the magnetic energy for perturbations of q around com-mensurable stripe order. For a start, one might compute the spin-spiral energy dispersions as a function of q, using the simple TB Stoner model (13) and then analyse which

one-electron states are reponsible for the energy changes, like we did for q = πy. The results shown so far were obtained using the spherical Stoner model, which allowed us to sim plify the calculation of the spin-spiral band structures and mag-netic energies so much, that we might understand the results by solving simple analytical problems. Although approximate, this model is in many respects more general than SDFT calculations. Now, coming to the end of our tu-torial paper, we show in Fig. 20 results of SDFT spin-spiral calculations for LaO1−xFxFeAs and Ba1−2yK2yFe2As2 of self-consistent moments and energies as funtions of q and doping in the virtual-crystal approxima tion (VCA) [46]. Note that in this figure, q takes the usual Γ ΓXM path, so that the spin-spiral patterns near X correspond to an X stripe. For LaOFeAs, the moment is seen to be ∼1.3 μB /Fe for both stripe and checkerboard order, which is some-what smaller than what we obtained in Fig. 17, presum-ably be cause the moment is calculated by integration of the spin-polarization in an Fe sphere with radius ∼a/2 in the SDFT calculation, rather than being summed over Fe Wannier orbitals. Other causes could be our use of the Stoner aproximation with too high an I and a spherical Δ. More significant is, however, that whereas we found, and understood, that in the rigid-band approximation the large moment de creases with electron doping, and increases with hole doping, the behaviour seems to be the opposite in the VCA approximation, both for electron-doped LaO1−xFxFeAs and hole-doped Ba1−2yK2yFe2As2. The VCA approximates O1−xFx (or Ba1−2yK2y) by a virtual atom having a non-integer number of pro tons. Such anomalies have recently been discussed for Co and Ni-substitution at the Fe site using su percell calculations [88], but not for substitution in the blocking layers. We may speculate that at least for Ba1−2yK2yFe2As2, the strong Ba 5d hybrididiza-tion near the Fermi level found in Sect. 4.2 could make substitution of Ba by K a non-trivial doping. Nevertheless, the spin-spiral energies shown in the lower half of Fig. 20 agree with experiments to the extent that the stable spin order is the stripe for low doping, but shifts to in-commen-surable order for higher doping, x > 5% in LaOFeAs. This is consistent with the onset of superconductivity in this ma-terial, but is a completely different, and presumably more accu rate scenario than the one suggested by Fig. 17. The black dashed and dotted lines are fits by a simple Heisen-berg nearest and next-nearest neighbor J1, J2 model to the calculated spin-spiral dispersions for respectively the undoped and 20% doped compounds. These fits are not bad, although exchange interactions of longer range are needed to fit the LaOFeAs incommensurability. However, the SDFT en ergies of other spin arrangements, such as starting from stripe order and then rotating the spins on


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one Fe sublattice rigidly with respect to those on the other, could not be reproduced by the (Jm1, Jm2) model.

Summary

We hope to have given a pedagogical, self-contained description of the seemingly complicated multi orbital band structures of the new iron pnictide and chalcogen-ide superconductors. First, we derived a generic Fe d As p TB Hamiltonian by NMTO downfolding of the DFT band structure of LaOFeAs for the observed crystal structure. By use of the glide mirror symmetry of a single FeAs layer, its primitive cell was reduced to one FeAs unit, i. e. the TB pd Hamiltonian, h (k) , is an 8×8 matrix whose converged analytical expressions are, however, too long for the pres-ent paper. We specified how h (k) factorizes at points – and along lines of high symmetry in the 2D BZ and pointed to the many band crossings and linear dispersions (“in-cipient Dirac cones”) caused by the factorizations. Their role, together with that of Fe d -As p hybridization for

the presence of the d6 pseudogap at the Fermi level and the details of the shapes and masses of the electron and hole pockets were subsequently explained. Thereafter we included interlayer coupling, which mainly proceeds via the As z orbitals, and showed how the st and bct 3D band structures can be obtained by coupling h (k) and h (k+πx+πy), i. e. by folding the 2D bands into the small BZ. This formalism allowed us to explain, for the first time, we believe, the complicated DFT band structures of in particular SmOFeAs, bct BaFe2As2, bct CaFe2As2, and collapsed bct CaFe2As2. What causes the complications are the material-dependent level inversions taking place as functions of kz . We found several Dirac points near the Fermi level. Whether these points have any physical im-plications remains to be seen. They do not pin the Fermi level, because there are also other FS sheets, and they are not protected, because they are merely caused by crystal symmetries, such as the vertical mirrorplane containing the nearest-neighbor As atoms, which are easily broken by phonons, impurities a.s.o. We then studied the generic band structures in the presence of spin spirals, whose Fe moment has a con-stant value, m, but whose orientation spiral along with wavevector q. The formalism simply couples h (k) to h (k + q) and does not require q to be commensurate. We used the Stoner approximation to SDFT, because it is simple and allows one to calculate the spin-spiral band struc-tures as functions of the strength, Δ, of the exchange po-tential and impose the selfconsisteny condition, Δ = m (Δ) I at a later stage. We limited ourselves to using this for-

Figure 20 Magnetic moments (upper panels) and energies (lower pan-

els) per Fe of spin spirals as functions of q for different electron (x) and

hole (y) dopings in the virtual-crystal approxi mation. These results

were obtained by self-consistent SDFT-LMTO calculations [46]. The

energies of the J1, J2 Heisenberg model for x(y) = 0 and 0.2 are given

by respectively dashed and dash-dotted lines. Representative real-

space spin structures are shown at the bottom right for the q-vectors

denoted by dots (adapted from [46]).

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malism to explain the 2D band structures for stripe order as a function of Δ, but in quite some depth, often using simple analytical theory. What complicates the magnetic band structure is the simultaneous presence of As-Fe co-valency and Stoner-exchange cou pling. That the latter is only between like Fe d orbitals, gave the structure of the small-moment SDW-gapping of the paramagnetic FS as well as the intermediate-moment propeller-shaped FS. With the goal of eventually understanding – and possi-bly simplifying the calculation of – the spin spiral energy dispersions, we expressed the magnetic energy as the difference between magnetic and nonmagnetic band-structure energies, whereby the magnetic band structure should be the one corrected for double counting of the exchange interaction. This formalism was then applied to the large-moment stripe or der and we found its stabi-lization energy to have two main sources (1) the coupling of the paramagnetic xy hole and xy/z electron bands and (2) the coupling of the other electron band and the xz part of the doubly degenerate hole band. The Fermi-sur-face contributions to the magnetic energy were found to be comparatively small, and that gave some hope for de-veloping a suitable Heisenberg Hamiltonian. We also ex-plained the much discussed coupling in SDFT between the stripe moment and the As height above the Fe plane. In the end, we showed and discussed the self-consistent spin-spiral moment and energy dispersion obtained from a SDFT calculation, co-authored by one of us [46].

Acknowledgements. We would like to thank Alexander Yaresko for pointing out to us the beauty of spin spirals. Dmytro Inosov convinced us about the non-triviality of interlayer coupling in BaFe2As2. Maciej Zwierzycki pro-vided us with computer programs using the Overlapping-Muffin-Tin-Approximation (OMTA) [89]. We are grateful to Claudia Hagemann for proof-reading the manuscript, and to Ove Jepsen for being extremely helpful, as usual, and for pointing out an error in the manuscript. O. K. A. thanks to Guo-Qiang Liu for discussions of his spin-orbit coupled calculations. L. B. would like to thank Alessandro Toschi for many useful discussions. Finally, to all those colleagues who have published results related to ours, but of which we are not aware, we apologize for having not given reference. This research was supported in part by the National Science Foundation under Grant No. PHY05 51164 (KITP UCSB) and by the DFG SFP1458.

Key words Fe-based superconductors, electronic structure, tight-

binding, downfolding, magnetism, in terlayer coupling.

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* Published in Ann. Phys. (Berlin) 523, No. 4, 291-295 (2011); modi-

fi ed for the sample issue, with permission from the authors. ** Corresponding author E-mail: [email protected] Department of Physics, Indiana University, Swain Hall West,

727 East Third Street, Bloomington, Indiana 47405, USA

Cosmological constant from quarks and torsion*

Nikodem J. Popławski **

Received 6 December 2010, revised 27 December 2010,

accepted 31 December 2010

Published online 1 February 2011

We present a simple and natural way to derive the

observed small, positive cosmological constant from the

gravitational interaction of condensing fermions. In the

Riemann-Cartan spacetime, torsion gives rise to the axial–

axial vector four-fermion interaction term in the Dirac

Lagrangian for spinor fi elds. We show that this nonlinear

term acts like a cosmological constant if these fi elds have a

nonzero vacuum expectation value. For quark fi elds in QCD,

such a torsion-induced cosmological constant is positive

and its energy scale is only about 8 times larger than the

observed value. Adding leptons to this picture could lower

this scale to the observed value.

A positive cosmological constant in the Einstein equa-tions for the gravitational field is the simplest form of dark energy, a yet unexplained energy that causes the observed current acceleration of the Universe [1]. Quan-tum field theory predicts that the corresponding vacuum energy density is on the order of mPl

4 ,where mPl is the re-duced Planck mass, which is about 120 orders of mag-nitude larger than the measured value ρΛ =(2.3 meV)4. This cosmological-constant problem is thus the worst problem of fine-tuning in physics. Zel’dovich argued, us-ing dimensional analysis, that the cosmological vacuum energy density should be on the order of ρΛ ∼m 6/mPl

2 , where m is the mass scale of elementary particles [2, 3]. However, some theoretical arguments have been used to show that the cosmological constant must vanish [4]. It is possible that the huge value of a cosmological constant from the zero-point energy of vacuum may be cancelled out by an effective cosmological term arising from spin-ning fluids in the Riemann-Cartan spacetime [5] or re-duced through some dynamical processes [6]. It is also

possible that the observed osmological constant is sim-ply another fundamental constant of Nature [7]. A model of a cosmological constant caused by the vacuum expectation value in quantum chromody namics (QCD) through QCD trace anomaly from gluonic and quark condensates gives ρΛ ∼ Hλ3

QCD [8], where H is the

Hubble parameter and λQCD 200 MeV is the QCD scale parameter of the SU(3) gauge coupling constant [9]. If a cosmological constant is caused by the vacuum energy density from the gluon condensate of QCD then ρΛ ∼ λ6

QCD /m2Pl

[10], which resembles the formula of Zel’dovich [2]. Another QCD-derived model of a cosmological con-stant gives ρΛ ∼ Hmq⟨qq ⟩/mη′ , where ⟨qq ⟩ is the chiral quark condensate [11]. A cosmological constant may be also caused by the vacuum energy density from the elec-troweak phase transition, giving ρΛ ∼ E 8

EW/m4 Pl, where

EEW is the energy scale of this transition [12]. The cosmic acceleration could also arise from a Bar-deen-Cooper-Schrieffer condensate of fermions in the presence of torsion, which forms in the early Universe [13], or from dark spinors [14]. In this paper, we present a simple and natural way to derive the small, positive cosmological constant from fermionic condensates and the Einstein-Cartan-Sciama-Kibble theory of gravity with torsion. Such a constant arises from a vacuum expectation value of the Dirac-Heisenberg-Ivanenko-Hehl-Datta four fermion interac-tion term in the Lagrangian for quark (and lepton) fields. Thus the cosmological constant may simply originate from particle physics and relativistic gravity with spin.

Ann. Phys. (Berlin) 523, No. 4, 291-295 (2011) / DOI 10.1002/andp.201000162

Highlights

Rapid Research Letter

37


The Einstein-Cartan-Sciama-Kibble (ECSK) theory of gravity [15] naturally extends Einstein’s general relativity to include matter with intrinsic half-integer spin, which produces torsion, providing a more com plete accountof local gauge invariancewith respect to the Poincaré group [16,17].The Riemann spacetime of general relativity is generalized to the Riemann-Cartan spacetime with tor-sion. The ECSK gravity is a viable theory, which differs significantly from general relativity only at densities of matter much larger than the density of nuclear matter. Torsion may also prevent the formation of singularities from matter with spin [18,19], averaged as a spin fluid [20], and appears to introduce an effective ultraviolet cutoff in quan tum field theory for fermions [21]. Moreover, torsion fields may cause the current cosmic acceleration [22]. In the Riemann-Cartan spacetime, the Dirac La-grangian density is given by

, where the semicolon denotes a full covariant derivative with respect to the affine con-nection. Varying Lwith respect to spinor fields gives the Dirac equation with a full covariant derivative. Varying the total Lagrangian density −R

√−g2κ + L with respect to

the torsion tensor gives the relation between the torsion and the Dirac spin density which is quadratic in spinor fields [16,17]. Substituting this relation to the Dirac equation gives the nonlinear (cubic) Dirac-Heisenberg-Ivanenko-Hehl-Datta equation for (in the units in which � = c = 1, κ = m−2

Pl )[16,17]:

where the colon denotes a covariant derivative with re-spect to the Christoffel symbols. This equation and its adjoint conjugate can also be obtained directly by vary-ing, respectively over (ψγ and , the following effective La-grangian density [16]:

without varying it with respect to the torsion. The cor-responding effective energy-momentum tensor Tik =

2√−gδLe

δgik

( )

is, using the identity δγj

δgik = 12 δj

(iγk) (which results from the definition of the Dirac matrices, γ(iγk) = gikI), given by:

Substituting (1) into (3) gives

The first term on the right of (4) is the energy-momen-tum tensor for a Dirac field without torsion while the second term corresponds to an effective cosmological constant [5,19,23],

or a vacuum energy density,

Such a torsion-induced cosmological constant de-pends on spinor fields, so it is not constant in time (it is constant in space at cosmological scales in a homogene-ous and isotropic universe). However, if these fields can form a condensate then the vacuum expectation value of Λ will behave like a real cosmological constant. Quark fields in QCD form a condensate with the nonzero vacu-um expectation value for ψψ ,

In the Shifman-Vainshtein-Zakharov vacuum-state-dominance approximation, the matrix element 〈0|ψΓ1ψψΓ2ψ|0〉,where Γ1 and Γ2 are any matrices from the set {I, γi, γ[iγk], γ5, γ5γi}, can be reduced to the square of 〈0|ψψ|0〉 [24]:

For quark fields, we have Γ1 = γiγ5t a and Γ2 = γiγ5ta, where

t a are the Gell-Mann matrices acting in the color space and normalized by the condition tr (t a t b ) = 2δab. Thus we obtain

corresponding to a positive cosmological constant. This formula resembles celebrated Zel’dovich’s relation [2], with the mass scale of elementary particles m cor-responding to (−〈0|ψψ|0〉)1/3. Combining this relation with the expression for ρΛ in [8] gives Hm2

Pl ∼ λ3QCD. In-

terestingly, using a Lorentz-violating axial condensate instead of the QCD quark vacuum condensates leads to

(1)iγkψ:k = mψ − 3κ

8(ψγkγ5ψ)γkγ5ψ,

Le =i√−g

2(ψγiψ:i − ψ:iγ

iψ) − m√−gψψ

+3κ

√−g

16(ψγkγ5ψ)(ψγkγ5ψ), (2)

Tik =i

2(ψδj

(iγk)ψ:j − ψ:jδj(iγk)ψ)

− i

2(ψγjψ:j − ψ:jγ

jψ)gik + mψψgik

− 3κ

16(ψγjγ

5ψ)(ψγjγ5ψ)gik. (3)

Tik =i

2(ψδj

(iγk)ψ:j − ψ:jδj(iγk)ψ)

+3κ

16(ψγjγ

5ψ)(ψγjγ5ψ)gik. (4)

Λ =3κ2

16(ψγjγ

5ψ)(ψγjγ5ψ), (5)

ρΛ =3κ

16(ψγjγ

5ψ)(ψγjγ5ψ). (6)

〈0|ψψ|0〉 ≈ −(230 MeV)3 ∼ −λ3QCD. (7)

〈0|ψΓ1ψψΓ2ψ|0〉=

1122

((trΓ1 · trΓ2) − tr(Γ1Γ2)

)× (〈0|ψψ|0〉)2. (8)

〈0|(ψγjγ5taψ)(ψγjγ5taψ)|0〉 =

169

(〈0|ψψ|0〉)2, (9)

N. J. Popławski: Cosmological constant from quarks and torsionRa

pid

Rese

arch

Let

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38


a very similar result [25]. Substituting (7) into (10) gives

The value of the observed cosmological constant would agree with the torsion-induced cosmological con stant presented here if 〈0|ψψ|0〉were –(28 MeV)3, suggest-ing a contribution to from spinor fields with a lower (in magnitude) vacuum expectation value. Such fields could correspond to neutrinos [26]. The presented model combines the ECSK gravity, which is the simplest theory with torsion, and QCD. It predicts a positive cosmological constant due to: the axial-axial form of the four-fermion interaction term in the Dirac Lagrangian (2), the vacuum-state-dominance formula for SU(3) (8), and the nonzero vacuum expecta-tion value for quantum fields (7). The vector-vector form of a four-fermion interaction would give a negative cos-mological constant, but this form does not result from the ECSK theory with minimally coupled fermions. It is pos-sible, however, to modify the form of the quartic term by adding to the Lagrangian density −R

√−g2κ + L two terms:

one proportional to Rijklεijkl , related to the Barbero-

Immirzi parameter [27], and another proportional to √−g2 (ψγiψ;i + ψ;iγ

iψ), measuring the nonminimal cou-pling of fermions to gravity in the presence of torsion [28]. Although the four-fermion interaction in (2) term seems to be nonrenormalizable, we emphasize that this term appears in the effective Lagrangian density Le in which only the metric tensor and spinor fields are dy-namical variables. The original Lagrangian density L, in which the torsion tensor is also a dynamical vari-able, is renormalizable. We also note that the torsion may modify the concept of renormalization by pro-viding an effective ultraviolet cutoff for fermions [21]. Another problem could be: what cancels much larger contributions to the vacuum energy density arising from quantum field theory? It has been argued in [7], however, that vacuum energy does not gravitate; only a shift in vacuum energy (vacuum expectation value of physical fields) produces a gravitational field. Therefore extremely large contributions to the vac uum energy density from quantum field theory should not appear in the Einstein equations. These issues need to be in-vestigated further. The torsion in the ECSK theory is minimally coupled to spinor fields. Thus the only parameter in this simple model of the cosmological constant is the energy of the two-quark condensate (7). This model gives a cosmo-logical constant whose energy scale is only about 23028

8 times larger than that corresponding to the observed cosmological constant. Therefore it provides the sim-plest explanation for the sign (and, to some extent,

magnitude) of the observed cosmological constant. We expect that adding lepton condensates to this picture could lower the average |〈0|ψψ|0〉| such that the result-ing torsion-induced cosmological constant would agree with its observed value. The absolute value of 〈0|ψψ|0〉could also be lowered by introducing the two param-eters considered in [28]. We also emphasize that our model naturally derives Zel’dovich’s formula [2] for the cosmological constant from a fundamental theory (the ECSK gravity coupled to Dirac fields), indicating that the results of this work are not a numerical coincidence.

Acknowledgements. The author would like to thank James Bjorken for very interesting and fruitful discussions on torsion and modified theories of gravity.

References

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〈0|ρΛ|0〉 ≈ (54 meV)4. (11)


39

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[15] T. W. B. Kibble, J. Math. Phys. 2, 212 (1961); D. W. Sciama, in: Recent Developments in General Relativity, (Pergamon, Oxford, 1962) p. 415 ; D. W. Sciama, Rev. Mod. Phys. 36, 463 (1964). [16] F. W. Hehl and B. K. Datta, J. Math. Phys. 12, 1334 (1971). [17] F. W. Hehl, P. von der Heyde, G. D. Kerlick, and J. M. Nester, Rev. Mod. Phys. 48, 393 (1976); V. de Sabbata and C. Sivaram, Spin and Torsion in Gravitation (World Scientific, Singapore, 1994); I. L. Shapiro, Phys. Rep. 357, 113 (2002); R. T. Hammond, Rep. Prog. Phys. 65, 599 (2002); N. J. Popławski, arXiv: 0911.0334. [18] W. Kopczynski, Phys. Lett. A 39, 219 (1972); W. Kopczynski, Phys. Lett. A 43, 63 (1973); A. Trautman, Nature (Phys. Sci.) 242, 7 (1973); J. Stewart and P. Hajıcek, Nature (Phys. Sci.) 244, 96 (1973); J. Tafel, Phys. Lett. A 45, 341 (1973); F. W. Hehl, P. von der Heyde, and G. D. Kerlick, Phys. Rev. D 10, 1066 (1974); M. Tsamparlis, Phys. Lett. A 75, 27 (1979). [19] C. J. Isham, A. Salam, and J. Strathdee, Nature (Phys. Sci.) 244, 82 (1973).

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[20] J. Weyssenhoff and A. Raabe, Acta Phys. Pol. 9, 7 (1947). [21] N. J. Popławski, Phys. Lett. B 690, 73 (2010). [22] I. S. Nurgaliev and V. N. Ponomarev, Fizika 25, 32 (1982); K.-F. Shie, J. M. Nester, and H.-J. Yo, Phys. Rev. D 78, 023522 (2008); C. G. Böhmer and J. Burnett, Phys. Rev. D 78, 104001 (2008); X.-Z. Li, C.-B. Sun, and P. Xi, Phys. Rev. D 79, 027301 (2009); C. G. Böhmer, J. Burnett,D. F. Mota, and D. J. Shaw, arXiv: 1003.3858. [23] G. D. Kerlick, Phys. Rev. D 12, 3004 (1975). [24] M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, Nucl. Phys. B 147, 385 (1979). [25] J. D. Bjorken, arXiv: 1008.0033. [26] D. G. Caldi and A. Chodos, arXiv:hep-th/9903416; J. R. Bhatt et al., Phys. Lett. B 687, 75 (2010). [27] A. Perez and C. Rovelli, Phys. Rev. D 73, 044013 (2006). [28] L. Freidel, D. Minic, and T. Takeuchi, Phys. Rev. D 72, 104002 (2005).

Rapi

d Re

sear

ch L

ette

r


* Published in Ann. Phys. (Berlin) 523, No. 8-9, 645–651 (2011),

modifi ed for the sample issue, with permission from the

authors.** Corresponding author:

E-mail: joachim.deisenhoferqphysik.uni-augsburg.de,

Phone: +49 821 598 3605, Fax: +49 821 598 36491 Experimental Physics V, Center for Electronic Correlations

and Magnetism, Institute of Physics, University of Augsburg,

86135 Augsburg, Germany 2 Institute of Solid State Physics, Vienna University of Technolo-

gy, 1040 Vienna, Austria 3 Dipartimento di Chimica Fisica “M. Rolla”, Università di Pavia,

V. le Taramelli 16, 27100 Pavia, Italy 4 Institute of Applied Physics, Academy of Sciences of Moldova,

2028 Chisinau, Republic of Moldova

Lattice vibrations in KCuF3*

Joachim Deisenhofer1,**, Michael Schmidt1, Zhe Wang1, Christian Kant1,2, Franz Mayr1, Florian Schrettle1 , Hans-Albrecht Krug von Nidda1, Paolo Ghigna3, Vladimir Tsurkan1,4, and Alois Loidl1

Received 11 February 2011, revised 17 March 2011,

accepted 28 March 2011

We report on polarization dependent reflectivity measure-

ments in KCuF3 in the far-infrared frequency regime. The

observed IR active phonons at room temperature are in

agreement with the expected modes for tetragonal sym-

metry. We observe a splitting of one mode already at 150

K and the appearance of a new mode in the vicinity of the

Néel temperature.

1 Introduction

Since the first reports about half a century ago the com-pound KCuF3 has become a paradigm for study ing the effects of a cooperative Jahn-Teller (JT) distortion [1], orbital ordering (OO) [2], and low-dimensional magne-tism [3]. In the paramagnetic state it can be described in terms of a one-dimensional (1D) an tiferromagnetic (AFM) Heisenberg chain [4], which results from the OO in KCuF3, where a single hole alternately occupies 3dx2−z2 and 3dy2−z2 orbital states of the Cu2+ ions with a 3d9 elec-tronic configura tion (see inset of Fig. 1(a)). In agreement with the Goodenough-Kanamori-Anderson rules, the OO leads to a strong AFM superexchange interaction of about 190 K along the crystallographic c-axis and a two orders of magnitude weaker ferromagnetic coupling in the ab-plane [5]. The cooperative JT distortion is characterized by CuF6 octahedra elongated along the a and b axis and arranged in an antiferrodis tortive pattern in the ab-plane. Long-range (A-type) AFM ordering occurs below the Néel temperature TN = 39 K [5], but the reduced ordered mo-ment in the AFM ground state indicates that strong quan-tum fluctuations are still present in KCuF3. Over decades KCuF3 has received special attention from neutron studies

and witnessed most of the experimental achievements in this field [5–13]. Consequently, many of the fingerprints of the magnetic excitation spectrum of quasi-one dimen-sional systems like spinons and longitudinal modes have been discovered in this system [6, 10, 12]. As attractive as this system appeared to experimen-talists, theorists did not want to stay behind and tested their tools to understand the driving forces behind the OO, the JT distortion, and the coupling of orbital and spin degrees of freedom [14–24]. In particular, it could be shown by a combination of ab initio band structure calculations and dynamical mean-field theory that the JT distortion in the para magnetic phase of KCuF3 origi-nates from electronic correlation effects [23, 24]. Despite these intense research efforts, many prop-erties of KCuF3 are still subject to debate or have even not been measured. For example, the room tempera-ture (RT) crystal structure of KCuF3 is still not solved


41

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unambiguously. The established tetragonal symmetry [1, 25] was reported to be actually or thorhombic [26]. Although this claim allowed for a better understand-ing of Raman [27] and electron spin resonance (ESR) properties [28], the observed AFM resonance could not be explained within the proposed symmetries [29]. Only recently, a consistent explanation of the ESR and AFM resonances was proposed by taking into account a dynamical Dzyaloshinsky-Moriya (DM) interaction [30]. This dy namical DM interaction is a result of strong thermal lattice fluctuations which manifest themselves in an anomalous softening of Raman-active phonons, which undergo a splitting prior to AFM ordering in KCuF3 and, thus, indicate a symmetry lowering of the system [31]. In this work we investigate the infrared (IR) active phonons in KCuF3. The temperature dependence of the IR spectra indicates a possible splitting of one mode already at about 150 K far above the Néel temperature and the appearance of a new mode just above TN .

2 Experimental details and characterization

The single crystals (see [32] for details on crystal growth) were oriented by Laue diffraction and cut along the (110)-plane. The heat capacity was measured in a Quantum Design physical properties mea surement

system for a temperature range 1.8 K < T < 300 K. Transmission and polarization-dependent reflectivity measurements were carried out for 10 K < T < 300 K using the Bruker Fourier-transform IR spectrometers IFS 113v and IFS 66v/S with a He-flow cryostat (Cryovac). The dielectric loss ε2 was ob tained from the reflectivity data by the Kramers-Kronig relation. Susceptibility measurements on single crystals obtained by Bridgman method (see inset of Fig. 1(b)) were performed using a SQUID magne tometer (Quantum Design). In Fig. 1(a) we show the temperature dependence of the magnetic susceptibility together with a fit ob-tained by using a rational function which approximates the Bonner-Fisher approach for an antiferro magnetic chain [33,34]. The obtained values for the effective g-factor of 2.26 and an intrachain coupling of 393 K are in good agreement with literature [4]. The antiferromag-netic ordering at about 40 K is clearly seen as a kink-like minimum. The increase below 40 K has been attributed to intergrowth of K2CuF4 traces [35]. The specific heat divided by temperature C/T (Fig. 1(b)) also clearly ex-hibits an anomaly at about 40 K marking the onset of antiferromagnetic ordering [36]. In order to model the lattice contri bution to the specific heat we use a sum of one isotropic Debye (D) and four isotropic Einstein terms (E1,2,3,4). The ratio between these terms was fixed to D : E1: E2: E3: E4 = 1:1:1:1:1 to account for the 15 de-grees of freedom per formula unit [27]. The resulting contribution to the specific heat shown as a solid line in Fig. 1(b) has been obtained with the Debye and Ein-stein temperatures θD = 205.0 K, θE1 = 146.1 K, θE2 = 292.3 K, θE3 = 325.8 K, and θE4 = 590.0 K in agreement with the frequency ranges where optical phonons were identified (see Table 1 and [27]).

Figure 1 (a) Temperature dependence of the dc-susceptibility mea-

sured in a magnetic field of 1 T. The solid line is a fit as described in

the text. Inset: Orbital ordering pattern of KCuF3 [24]. (b) Tempera-

ture dependence of C/T . The solid line indicates the lattice contribu-

tion to the specific heat as described in the text. Lower inset: Re-

sidual specific heat after subtracting the lattice contribution. Upper

inset: KCuF3 single crystal grown by Bridgman method.

J. Deisenhofer et al.: Lattice vibrations in KCuF3

Ori

gina

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42


3 Optical excitations and infrared active phonons

In Fig. 2(a) we show a transmission spectrum of KCuF3 at 294 K to give an overview of the optical exci tations and the corresponding energy scales in this compound. At low energies transmission is strongly suppressed indicating the frequency range of the infrared-active

phonon excitations with an upper limit of about 80 meV. This frequency range and the IR phonon spec-trum will be discussed below in more detail. The next strong absorption region centered at around 1.2 eV has been identified as due to phonon-assisted d-d crystal-field excitations of the Cu2+ ions in a distorted octahe-dral environ ment (see inset of Fig. 2(b)). Just above TN sharp sidebands appear at the onset of A2 and A3 which are attributed to exciton-magnon sidebands [37]. The comparatively weak absorption band centered at 2.4 eV (about twice the energy of the CF excitations) has not been explored in detail up to now but is attributed to exciton-exciton transitions involving two exchange-coupled neighboring ions which are excited simultane-ously to higher-lying orbital states. The strongest ab-

Figure 2 (a) Transmission spectrum of KCuF3 at 294 K. (b) Absorption

spectra showing the crystal-field transition region for polarizations E

|| c and E ⊥ c at 8 K [37]. Inset: Splitting of the Cu d- levels for a dx2−y2

ground state with the longest Cu-F bond defining the local z direc-

tion.

Figure 3 Reflectivity of KCuF3

measured at 12 K (black) and 295

K (red) for directions

E || c (panel a) and E ⊥ c (panel

b). Dielectric loss ε2 of KCuF3

measured at 12 K and 295 K for

directions E || c (panel c) and

E ⊥ c (panel d). The reflectivity

and dielectric loss spectra at

295 K were shifted by an offset

of 0.2 and 4, respectively.


43

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sorption sets in at about 4.5 eV which is assigned to the onset of charge-transfer excitations. In this study we want to discuss the optical excitations in the far-infrared regime. In Fig. 3(a) and (b) we show reflectivity spectra at 295 K and 12 K for polarizations E || c and E ⊥ c, respectively. At 295 K the spectra exhibit three excitations for E || c and five excitations for E ⊥ c which are labeled A2u(i) modes with i = 1,2,3 and Eu ( j) modes with j = 1,2,3,4,5, respectively. This assignment corresponds to the expected normal modes obtained from the analysis of the irreducible representations for the Raman and IR active phonons within the tetragonal structure of KCuF3 at room temperature with space group and two formula units in the primitive cell [1, 25, 38]:

Before we discuss the changes of the phonon spectra with temperature we want to point out that the feature at about 180 cm−1 marked by an asterisk is present for all polarizations and temperatures and corresponds to an experimental artifact. The weak band visible for E || c at 295 K at the low-energy side of A2u(2) is attributed to a polarization leakage and corresponds to mode Eu(3). As a result any change of Eu (3) with temperature will also appear for E || c and has to be distinguished from changes of the A2u modes. The corresponding transverse eigenfrequencies of the modes have been determined directly by the maxima in the corresponding dielectric-loss spectra obtained by Kramers-Kronig transformation (see Fig. 3(c) and (d)) and are listed in Table 1.

The dashed vertical lines drawn across Fig. 3(a) and (b) indicate the additional features in the range of the Eu(3) leakage for E || c. Both features can be explained as a re-sult of the observed two-peak struc ture III of mode Eu(3) at 12 K for E ⊥ c (indicated by two arrows in Fig. 3(b)). This two-peak structure III can be regarded as one of the most evident changes in the IR spectra with decreasing temperature. A further clearly visible change is the dip-like feature II of the A2u(2) mode. Since the A2u(2) mode cor responds already to a one dimensional representa-tion and can not split any further, this could either be an additional mode or the effect of multiphonon processes. However, a shoulder at this frequency is already visible at 295 K and the gradual temperature evolution of this mode shown in Fig. 4(b) favors the latter scenario of mul-tiphonon processes which become more pronounced when the damping of the phonon modes is reduced at lower temperatures. The temperature evolution of the two-peak struc ture III of mode Eu(3) also develops only gradually and becomes visible at about 150 K as shown in Fig. 4(c). No symmetry reduction or other anomalies have been reported in this temperature range, but this feature could indicate a splitting of the doubly degener-ate mode Eu(3) already far above the Néel temperature. Finally, we want to discuss the appearance of the weak mode I at about 107 cm−1 visible for both polariza-tions at 12 K (see Fig. 4(a)). This mode emerges only in the vicinity of the Néel temperature. Whether it corre-sponds to a phonon mode of a possible orthorhombic structure or is of magnetic nature can not be decided at the moment. Altogether we find an experimental situation with an early claim that already at room temperature the sym-

metry is orthorhombic (instead of te-tragonal) and does not change down to 10 K [26]. Our IR data suggests a gradual splitting of a Eu (3) phonon mode which is clearly recognizable at around 150 K and Raman-scatter-ing studies found a splitting of dou-bly degenerate Eg modes to appear slightly above TN = 40 K [31]. The fact that the indications of a symmetry lower than tetragonal are observed at differ ent temperatures might be understood as a result of strong ther-mal lattice fluctuations which mask the effects of the orthorhombic dis-tortion. Depending on the time-scale of the fluctuations at a given tem perature different experimental probes will identify the orthorhom-

Table 1 Transverse eigenfrequencies of the observed IR modes in cm in KCuF

at 12 K and 295 K for polarizations E || c and E⊥c.

J. Deisenhofer et al.: Lattice vibrations in KCuF3

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44


Figure 4 Reflectivity of KCuF3 around the three features I (a), II (b)

and III (c) (see text for details). The spectra are shifted by a constant

offset of −0.01 (a) and −0.03 (b) and (c) for each temperature.

bic distortions in different regimes. Consequently, the paramagnetic phase of KCuF3 is not only characterized by spin fluctuations inher ent to a quasi-one dimensional system, but these spin fluctuations are strongly coupled to the dynamic distortions of the lattice. On the time scale of these lattice fluctuations, spin-orbit coupling leads to a dy namical DM interaction [30]. When these dynamic distortions and the DM contribution become static at low temperatures [29], this additional anisotro-pic contribution to the magnetic exchange might be the decisive factor in suppressing the spin fluctuations and establish three dimensional antiferromag netic order. In summary, we observed all IR active phonon modes expected for the suggested tetragonal symme try of KCuF3 at room temperature. The splitting of one of these modes is found to occur below about 150 K and a new mode was found to appear in the vicinity of TN . These observations are consistent with a scenario of strong lat-tice fluctuations of a possible orthorhombically distort-ed structure.

Acknowledgements. We thank M.V. Eremin, M. A. Fayzullin, and I. Leonov for stimulating discussions. We ac knowledge support by DFG via TRR 80 (Augsburg-Munich).

Key words: Infraredspectroscopy, orbitalordering, Jahn-Teller dis-

tortion.

References

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