Fermi Surfaces of Metals - Iowa State Universitycmp.physics.iastate.edu/canfield/course/EM2_25.pdfFermi Surface group to start a new group in Ames. Andrew Gold was first and he worked

Novel Materials and Ground States

Fermi Surfaces of Metals –magnetoresistance

and quantum oscillations

590B S09

Sergey L. Bud’ko(Сергей

Леокадьевич

Будько)

[part of the slides (blue) are taken from the presentation of Ilya

Sheikin

(Grenoble) –

http://mfs-cargese.grenoble.cnrs.fr/Sheikin.pdf]

http://mfs-cargese.grenoble.cnrs.fr/Sheikin.pdf


Experimental studies of FS: -

why?

(from Bruce Harmon’s lecture, Feb. 23)

Optimist would say:

And we want to keep theorists honest

Quantum oscillations are fascinatingly simple and (to a large extent) do not require a lot of assumptions to understand


Magnetoresistance

and Fermi surface topology

Requirements:-

large magnetic fields (ωτ

>> 1)

-“normal”

metal (no magnetic scattering or field-

induced transitions)

ω

–

cyclotron frequency, ω

= eH/m*c

τ

– relaxation time

“usually”

magnetic scattering effects are stronger than FS effects

does not tell a w

hole

lot about th

e orbit


Magnetoresistance


Weak magnetic field (ωτ

<< 1)

Δρ/ρ

~ H2

δ

= 2rφ

– 2rsinφ ≈ 1/3 rφ3

If we assume rφ

~ λ

(mean free path),

δ/ λ

~ (λ/r)2 => Δρ/ρ

~ (λ/r)2

r ~ v/ω; λ

~ vτ

=> Δρ/ρ

~ (ωτ)2

~ H2

2φr

No help here…


Magnetoresistance


Strong magnetic field (ωτ

>> 1)

closed trajectories

open trajectories


Magnetoresistance


Strong magnetic field (ωτ

>> 1)


Magnetoresistance

and Fermi surface topology Strong magnetic field (ωτ

>> 1)

More formally for the tensor:

CLOSED ORBITS

OPEN ORBITS

(γ

~ 1/H)

ne

≠

nh

ne

= nh

Au


Applications: Cd

under pressure, mid 1960s

P=0 P~20 kbar

H||c: closed orbits, Δρ/ρ

~ H2 H||c: open orbits, Δρ/ρ

~ const

Gaidukov, Voronovskii, Itskevich, … ~1965




measure of the FS curvature


per Lifshitz

& Pogorelov

(1954) if the FS is convex and has a center of symmetry there is an analytical procedure to calculate FS shape from oscillations’

frequencies and the velocities from the effective masses. (NEED ROTATION)



be careful with “large”

field intervals and FFTs


for a mundane material shouldn’t be very different for different FS sheets


need to know the effective mass



Quantum oscillations

HOW TO RECOGNIZE THEM:

Periodic in 1/H

Amplitude increases in higher fields

Amplitude decreases at higher temperatures


Quantum oscillationsNEED: low temperatures, high magnetic fields, high quality single crystals, decent sensitivity of the measurement techniques

GET:precise areas of extremal

orbits,effective masses,scattering time

PARAMETERS (note: much more freedom than ARPES/2DACAR):temperature (e.g. oscillations through TN

, TK

),magnetic field (e.g. metamagnetic

transitions, field dependent masses),angle,pressure/stress


Quantum oscillations in “everything”

Magnetization/susceptibility (dHvA) (|dM/dH| = 2πF/H M/H)

Torque (Tq

= -1/F dF/dθ

MHV)

Resistivity (Δρ/ρ

~ (m*F/H)2

dM/dH)

Magnetostriction

(Δli

/l

~ dlnF/dpi

MH)

Magnetothermal

oscillations

(|ΔT| ~ T2H2/F2

|dM/dH|

Specific heat

Elastic constants and sound velocity

…(oscillating amplitudes are written)

NOTE: different relative amplitudes of the oscillations in different techniques (can see in one but not in another…)

Can learn more using more than one technique.


Magnetic breakdown

MgTunneling probability:

H ~ U2/EF

U ~ 10-2

eV, EF

~ 1 eV

=> H ~ 104

Oe

Need to keep in mind if there are differences with band structure.


Exam

ples


Exam

ples



Osc

illat

ion

s th

rou

gh A

FM t

ran

siti

on


WHERE IS IT? WHAT IS IT?


The sculpture made of intersecting rings of rusted Fe, was machined in the Physics Shop by Dick Brown based on a calculation by Terry Loucks

for the iron actinides and it was called "Iron-Actin"

Gordon Danielson, our Semiconductor guy in the early 1960s spent

a sabbatical at Cambridge UK and hired three students from the Shoenberg

Fermi Surface group to start a new group in Ames.

Andrew Gold was first and he worked mostly on the Fermi Surface

of Pb

in the early going doing de Haas van Alphen with pulsed fields

and a giant capacitor bank in Tringides

current lab.

Allan Mackintosh came a year later and he was an idea guy more than a nuts and bolts guy.

Allan collaborated with experimentalists all over the world doing phenomenology to sort out their data.

Eventually Allan started a positron annihilation program for Fermi surface work

doing a lot with the rare earths.

Bob Young was the third Cambridge guy, I have forgotten his technique.

Terry Loucks

from Penn State was a theorist calculating everything in sight

and he wrote a book. John Stanford started doing radio frequency size effect, the John went into atmospheric physics looking for Tornados.

Sunny Sinha

was the last of the Cambridge group.

Sunny did mostly neutron scattering, but he did Fermi surface work as well.

My guess is that Gold came in 1958 or 1959, Mackintosh in 1960, and Young in 1961.

All three were here when I came in 1962.

Loucks

came in about 1964 and Stanford from Maryland in 1965.

Between 1975 and 1980, Gold went to British Columbia, Mackintosh to Riso, Denmark, Young to Birmingham, Loucks

to North American in California, and Stanford to tornados.

Fermiology

in the Physics Department

Doug Finnemore remembers


Fermiology

in the Physics Department

David Lynch remembers

I arrived in Ames in the Fall of 1960.

The Fermi-surface people arrived starting the next year: Allan Mackintosh 1960, Andrew Gold

1961?, Bob Young

1963?, all from Cambridge.

Then Terry Loucks, a theorist, and some postdocs

and visitors:

Bob Chambers (Cambridge or Bristol), John Collins (Australia, ex-

Cambridge), George Dheer

(Cambridge), and Sunny Sinha

(Cambridge).

The latter came as Allan?s

postdoc, but stayed to run much of the CMP research on the new Ames-Lab reactor.

Allan set up a spectrometer for correlation of gammas from positron annihilation

and he also measured transport property, especially of Cr doped with V and with Mn.

Andrew built a pulsed dHvA

system.

One of the things done on it is still cited a lot: the Fs of Pb, which did not fit a free-electron model at all until spin-

orbit splittings

were added.

Another was the FS of Fe and Co.

One result from that was that the force on electrons in these metals was v x B, not v x H. Bob Young measured magnetoresistance, looking for open orbits.

Ames and Cambridge were the centers of Fermi surface research in

the 1960s.

I think the sculpture seen from the department office was made by Dick Brown, the foreman of the department machine shop, but I don’

t know who designed it or got him to do it.

It looks like a Fermi surface in the extended zone scheme, but I think it was not intended to be that of any metal whose Fs was known at that time.

Maybe something has been determined since then for a real metal.


Fermiology

in the Physics Department – these days (in addition to the world class ARPES): 1999 -

now

dHvA-SdH: resistivity, magnetization, torque, TDR, magnetostriction, magneto-TEP

(more curiosity driven)


Reading materials

Electron Theory of Metals Authors: I.M. Lifshits, M.Ya. Azbel’, M.I. Kaganov; Consultants Bureau, 1973

Fundamentals of the theory of metals Author: A.A. Abrikosov; North-Holland, 1988

Electrons at the Fermi surfaceEditor: M. Springford; Cambridge University Press, 1980

Magnetic oscillations in metals Author: D. Shoenberg; Cambridge University Press, 1984

Band theory and electronic properties of solidsAuthor: John Singleton; Oxford University Press, 2001

Documents

Fermi Surfaces of Metals - Iowa State Universitycmp.physics.iastate.edu/canfield/course/EM2_25.pdfFermi Surface group to start a new group in Ames. Andrew Gold was first and he worked