Con - UNIGRAZphysik.uni-graz.at/~uxh/diploma/ladstaedter.pdf · 2008-03-06 · Chapter 1 In tro duction Ov er man y decades the understanding of metals as protot ypical y-particle

Ab Initio Cal ulation of Hot Ele tronLifetimes in Metals:Ele tron-Ele tron S attering and its Impli ationsfor Ballisti Ele tron Emission Spe tros opy

d-orbitals of PdDiplomarbeitzur Erlangung des akademis hen Grades einesMagistersan der Naturwissens haftli hen Fakult�at derKarl-Franzens-Universit�at Grazvorgelegt vonFlorian Ladst�adterbetreut vonProf. Claudia Ambros h-DraxlundProf. Ulri h HohenesterInstitut f�ur Theoretis he PhysikKarl-Franzens-Universit�at GrazUniversit�atsplatz 52001

ii

Contents1 Introdu tion 12 BEEM 32.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Theoreti al Treatment . . . . . . . . . . . . . . . . . . . . . . . . 52.2.1 A Four Step Model . . . . . . . . . . . . . . . . . . . . . . 52.2.2 Introdu ing the Band Stru ture . . . . . . . . . . . . . . . 72.3 The Pd=Si System . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Lifetimes of Hot Ele trons 133.1 The Self-Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.1.1 Semi lassi al Self-Energy of an Ele tron . . . . . . . . . . 143.1.2 Quantal Self-Energy . . . . . . . . . . . . . . . . . . . . . 153.2 Ab Initio Cal ulation of Lifetimes . . . . . . . . . . . . . . . . . . 173.2.1 Extension to Periodi Crystals . . . . . . . . . . . . . . . . 183.2.2 The Diele tri Fun tion . . . . . . . . . . . . . . . . . . . 193.2.3 The Basi Formulae . . . . . . . . . . . . . . . . . . . . . . 223.3 Density Fun tional Theory (DFT) . . . . . . . . . . . . . . . . . . 233.4 The LAPW Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 Results 294.1 BEEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.1.1 Details of the Cal ulation . . . . . . . . . . . . . . . . . . 304.2 Ab Initio Lifetimes . . . . . . . . . . . . . . . . . . . . . . . . . . 334.2.1 Convergen e . . . . . . . . . . . . . . . . . . . . . . . . . . 344.2.2 Lifetimes of Metals - Results . . . . . . . . . . . . . . . . . 404.3 Combining BEEM & Ab Initio Data . . . . . . . . . . . . . . . . 465 Summary 47A Mathemati al Details 49List of Figures 51iii

iv CONTENTSBibliography 53A knowledgements 57

Chapter 1Introdu tionOver many de ades the understanding of metals as prototypi al many-parti lesystems has led to numerous theoreti al e�orts. Only sophisti ated many-bodyte hniques, whi h were originally developed in the �eld of quantum ele trodynam-i s, �nally allowed to understand the basi hara teristi s of ele tron dynami s inmetals within the normal state: while ele trons with ex ess energies only slightlyabove the Fermi level an be approximately treated as independent parti les withalmost in�nite lifetimes, states at higher energies be ome short-lived be ause ofCoulomb renormalizations, whi h an be e�e tively des ribed by ele tron-ele trons atterings.It is somehow surprising that, despite of this long standing history, the detailsof ele tron lifetimes are still not fully understood. On the one hand, there seemto be large variations in measured lifetimes depending on the hosen experimen-tal te hniques, thus indi ating the diÆ ulty to uniquely extra t su h transportquantities. On the other hand, the theoreti al treatment has simpli�ed detailsof the metal band stru ture over many years. Only re ently, with the advent ofmodern omputer te hnology and the progress of many-parti le methods basedon density fun tional theory (DFT), a �rst-prin iples al ulation of lifetimes inreal metals has be ome feasible. These al ulations have revealed quite large de-viations from the free-ele tron-gas predi tions for metals with more ompli atedband stru tures.A detailed understanding of these transport properties in metals is of ru ialimportan e for the design of ele troni devi es and of future nanos ale te hnology,whose performan e will be ultimately limited by su h s attering pro esses. Of fur-ther interest in this respe t is the ele tron transport a ross metal/semi ondu torinterfa es, whi h is governed by the potential barrier (S hottky barrier) and theabrupt hange of material parameters. In parti ular, for non-epitaxial interfa esthe transport properties show strong spatial variations whi h are diÆ ult to re-solve experimentally. With the invention of ballisti ele tron emission mi ros opy(BEEM) by Bell and Kaiser in 1988, a te hnique based on s anning tunnelingmi ros opy, it be ame possible to investigate su h buried interfa es with nanome-1

2 CHAPTER 1. INTRODUCTIONter resolution. Quite generally, the theoreti al des ription of BEEM spe tra hasturned out to be fairly involved and has onsequently led to a number of ontra-di ting on lusions. A proper theoreti al model, taking into a ount the full bandstru ture e�e ts, has been developed in the last ouple of years by Flores and oworkers. Not only that this model ould resolve several puzzling observations,it also revealed that BEEM spe tra provide an extremely sensitive measure ofinelasti ele tron lifetimes.In this work, we will develop a framework for the ab initio al ulation ofele tron-ele tron s atterings in thin metal �lms and will show how to in ludethese results into realisti BEEM al ulations. In Chapter 2 we introdu e thenon-equilibrium Green's fun tion te hnique for the al ulation of BEEM spe tra.Chapter 3 presents the framework for the ab initio al ulation of the inelasti ele tron-ele tron s attering rates and dis usses details of the LAPW implemen-tation used within this work. Ele tron lifetimes are shown in Chapter 4 alongwith preliminary results for BEEM spe tra in luding realisti transport parame-ters. Finally, in Chapter 5 we draw some on lusions and give a short outlook tofuture developments.

Chapter 2BEEMBallisti Ele tron Emission Mi ros opy (BEEM) [BK88, KB88℄ is a te hnique,based on s anning tunneling mi ros opy, whi h allows the study of potential stepsat interfa es with high lateral resolution. In this respe t, espe ially metal/semi- ondu tor jun tions are of great te hni al interest, and a good understanding oftheir ele troni and stru tural properties, in parti ular of the potential barrierat the interfa e (the S hottky barrier height), is needed. A large variety of te h-niques, su h as urrent-voltage measurements or photoele tron spe tros opy, hasbeen used to investigate these properties, but most of them either show a la k ofspatial resolution or are too surfa e sensitive [Pri95℄. BEEM, however, providesan alternative approa h for the investigation of buried interfa es with nanometerresolution of the jun tion up to a metal substrate thi kness of around 100{300�A.This hapter gives a short introdu tion to the te hnique, aiming to our �nal goalof des ribing how to theoreti ally analyse experimental BEEM data.2.1 Experimental SetupS anning Tunneling Mi ros opy (STM) has proven to be a perfe t tool for in-vestigating surfa es with atomi resolution. BEEM is an extension to the STMte hnique, whi h makes possible the investigation of buried interfa es with highspatial resolution. Although this te hnique allows to investigate many types ofinterfa es, hitherto it has been mostly applied to metal/semi ondu tor jun tions.Figure 2.1 shows the s hemati al setup of a BEEM experiment. It in ludesa thin metal �lm deposited on top of a semi ondu tor substrate, and an STMtip whi h a ts as an inje tor of hot ele trons into the metal. Here, the STM hara teristi s is exploited in two di�erent ways: First, ele trons are introdu edin the metal through a very small, atomi -resolution limited area, and se ondtheir energeti position is tuned between the Fermi level and the va uum level,with a very narrow energy distribution (i.e. BEES, Ballisti Ele tron EmissionSpe tros opy). Typi al ex ess energies eVt for the inje tion of hot ele trons are3

4 CHAPTER 2. BEEM

semiconductor

metal film

tip

I t

I BEEM

V t

Figure 2.1: S hemati setup of a BEEM experiment. The metal �lm has a typi althi kness of 10{300�A.V t

(a) (b)

Tip Metal SemiconductorTip Metal Semiconductor GapGap

b V

Figure 2.2: S hemati energy diagram for the BEEM experiment. (a) Zero tunnelbias is applied, (b) Tunnel bias greater than barrier height, Vt > Vb, where Vt and Vbde�ne the applied tunnel voltage and the S hottky barrier height, respe tively.in the range of 1{2 eV. The two layers are onta ted separately so that, for anapplied tip voltage Vt, two types of urrents an be measured: The tip-metal urrent It, whi h is just the STM tunnel urrent, and the tip-semi ondu tor urrent IBEEM. Figure 2.2 shows the energy diagram for the BEEM pro ess:Without any external tip-sample bias applied, the Fermi level of tip and sampleare aligned through the onta t between the tip and the metal, and there is notunneling urrent [Fig. 2.2(a)℄. In reasing the tip voltage auses the onset of atunneling urrent; for the ase Vt > Vb [Fig. 2.2(b)℄ ele trons an over ome theS hottky barrier and are measured as the BEEM urrent.By keeping the tip at a �xed position and varying the tip voltage Vt one ob-tains the IBEEM(Vt) spe trum, whi h provides information about the hot ele trontransport through the metal �lm and a ross the interfa e. On the other hand,keeping the tip voltage onstant and s anning over the surfa e, one obtains ami ros opi image of the interfa e IBEEM(x; y). Thus, IBEEM-Vt spe tra provide

2.2. THEORETICAL TREATMENT 5a dire t probe of the interfa e stru ture, in luding a spatially resolved pi ture ofthe S hottky barrier height, re e tions at the interfa e, and transport propertiesof the metal �lm.2.2 Theoreti al Treatment2.2.1 A Four Step ModelIn the standard model used for des ribing BEEM, it is assumed that the BEEM urrent results as a produ t of four independent steps [Pri95℄: (1) Tunnelingfrom the STM-tip to the metalli surfa e, (2) transport through the metal, (3)transmission a ross the metal-semi ondu tor interfa e, and (4) transport insidethe semi ondu tor. Within this approximation s heme, one an derive the fol-lowing general expression for the BEEM urrent as a fun tion of the tip voltageVt [AGRF01℄:IBEEM(Vt) = Z VtVb dE ZZIBZ dkkD(Vt; E;kk)P (E;kk)T (E;kk)S(E): (2.1)The integration is performed over the irredu ible 2D Brillouin zone parallel to thesurfa e (interfa e) layer, spe i�ed by the omponent of the wave ve tor parallelto the surfa e, kk.The four steps are in luded in this formula via the following fa tors: (1)D(Vt; E;kk) is the distribution of the ele trons after the tunneling from the tipto the metal; (2) P (E;kk) takes into a ount the e�e ts due to the transportthrough the metal; and the transmission oeÆ ient T (E;kk) takes are of thetransmission to the semi ondu tor (3), assuming kk onservation at the interfa e;�nally (4), S(E) a ounts for transport e�e ts inside the semi ondu tor, su has phonon ba ks attering of ele trons. Pro esses like multiple re e tions in themetal �lm are not in luded here and will be dis ussed later. The above mentionedkk onservation at the interfa e is still matter of intense dis ussion, and there havebeen indi ations in favour [SL91, Bel96, SSHS01℄ and against [LB93, SLN98℄.However, it has been re ently demonstrated, that details of kk onservation orviolation at the interfa e are not of ru ial importan e for the al ulation ofBEEM spe tra, provided that the metalli band stru ture is properly taken intoa ount [GAF96℄. This will be treated in further detail below. For the moment,we shall assume perfe t kk onservation in the spirit of Refs. [Reu98, RAG+98℄.In the following, we will derive expressions for the four above-mentioned steps.For simpli ity, as a preliminary simpli� ation we shall employ a quasi-1D modelfor the tip-metal and the metal/semi ondu tor transport:1. The inje ted tunneling urrent is derived from a simple one-dimensionalmodel, justi�ed by the large tip-sample distan es of the order of 5{10�A

6 CHAPTER 2. BEEMwhi h leads to weak oupling. Using furthermore a free-ele tron model forthe des ription of the tip and the metal, one arrives for the dependen eof the tunneling urrent on the tip-sample separation d and the ele tronenergy E at [Pri95, Reu98℄:D(Vt; E;kk) / e�2�d; (2.2)with � =q2W � (2E � k2k) and W the work fun tion with a typi al valueof W � 4{5 eV.2. In the original Bell-Kaiser model, the transport in the metal base has beentreated within a free ele tron model (below we will demonstrate that su h ades ription is too simple for a realisti hara terization of BEEM urrents).The thi kness of the metal �lm is usually smaller than typi al attenua-tion lengths for those materials. For this reason, ele trons are treated asballisti arriers, giving the te hnique its name. In general, the inje tedele trons may su�er elasti or inelasti s attering with di�erent ex itationsof the solid. S atterings with phonons usually result in minor energy losses(� 10 meV) and are therefore energeti ally of no great interest. Of greater on ern is, however, that su h quasi-elasti s attering events result in a hange of the kk-distribution of the ele trons, whi h, in a free-ele tronmodel, is otherwise determined at the tunneling step; note that the kk-distribution is the de isive parameter for the entran e into the semi ondu -tor, as will be dis ussed below.The main sour e for inelasti s atterings is regarded to be the intera tionbetween the hot ele tron and the ele trons of the metal �lm. For the lowenergies used in BEEM experiments, the inelasti mean free path is usuallyquite large (� 100�A) and larger than the typi al thi kness of the metal�lm, as pointed out before. Consequently, there is a good probability forthe inje ted ele trons to propagate through the metal without su�erings atterings and to have therefore enough energy to over ome the S hottkybarrier (see Figure 2.2). An ele tron-ele tron s attering event auses theele tron to loose about half its ex ess energy. For the energies onsideredhere the hot ele tron will have no han e to over ome the S hottky-barrieranymore. Thus, the e�e ts of inelasti s attering is the attenuation of thetotal signal and an be des ribed by a simple exponential damping term:P (E;kk) / e�l=�: (2.3)3. For modelling the transmission a ross the MS-interfa e, one an onsiderthe quantum-me hani al expression for the transmission/re e tion at a 1Dpotential step, whi h leads to [CDL77℄T (E;kk) = 4kMkS(kM + kS)2 ; (2.4)

2.2. THEORETICAL TREATMENT 7V

b

∆Ζ

Figure 2.3: S hemati view of the shift of the absolute barrier maximum inside thesemi ondu tor due to the image for e and of opti al phonon ba ks attering.where kM and kS represent the wave ve tors perpendi ular to the interfa ein the metal and the semi ondu tor, respe tively. In this expression oneshould keep in mind that the hot ele trons (in the low-energy regime) getintrodu ed into the semi ondu tor only in the vi inity of the ondu tionband minima (CBM) of the semi ondu tor. The energies and wave ve torsin the semi ondu tor are therefore referred to the CBM, not ne essarilylo ated at the � point (kk = 0). In the metal the wave ve tors are alwaysmeasured with respe t to �.4. For the low energy region of interest the only important transport e�e to urring inside the semi ondu tor is the ba ks attering of ele trons to-wards the metal by ele tron-phonon intera tion, Figure 2.3. In addition,the maximum of the barrier at the interfa e gets shifted well inside thesemi ondu tor due to the image harge e�e t of the ele tron. This shift �z an be as large as 50�A, leading to a high probability for s attering eventsto o ur before the ele tron has over ome the maximum of the barrier. Inthe theory this an be introdu ed by a simple fa tor S(E), adjusting thetransmission oeÆ ient T (E;kk).2.2.2 Introdu ing the Band Stru tureA series of experimental data has revealed that the free-ele tron model introdu edabove is not suitable for des ribing BEEM experiments, e.g., the high lateral res-olution mentioned above. In a free-ele tron model one would expe t for a metal�lm of 100�A a resolution of approximately 100�A in ontrast to the observed res-olution of about 10{15�A [Pri95, MMK+92℄. Other puzzling observations are thevery similar results in the Au=Si system obtained for the interfa es with Si(111)and Si(100): The (111) orientation possesses a proje ted CBM at the � point,while the (100) does not (Figure 2.4). The ele trons enter the semi ondu tor only

8 CHAPTER 2. BEEM

Figure 2.4: Constant energy surfa e of the Si ondu tion band and its proje tion ontothe (001) and (111) orientation. Shown are the CBM lo ations (+) and the proje tionfor 0:2 eV (dashed line) and 0:6 eV (dotted line) above the minimum (illustration takenfrom [RAG+98℄).in the vi inity of the CBM, as pointed out before; sin e in a free-ele tron modelthe urrent in the metal is on entrated around the � point, one would thereforeexpe t a pronoun ed di�eren e in the total BEEM urrent. Experimentally thisdi�eren e is not observed. The solution to these two astonishing observations(the high lateral resolution and the weak dependen e on the Si orientation) liesin the intera tion of the hot ele trons with the periodi latti e stru ture. Thestandard model sket hed in 2.2.1 ompletely negle ts the details of the bandstru ture of the metal �lm. To in lude the e�e ts of the band stru ture, Floresand oworkers have developed a Green's fun tion formalism [AGRF01, GAF96℄,whi h will be outlined brie y in the following:The model is based on a Hamiltonian written in an LCAO (linear ombinationof atomi orbitals) basis: H = HT + HS + HI ; (2.5)where HT = P ��n� +P T�� y� � de�nes the tip, HS = P �ini +P Tij yi j themetal layer, and HI = P T�m y� m des ribes the oupling between the tip andthe surfa e in terms of a hopping matrix T�m. Greek indi es (�; �) label sites onthe tip, Latin indi es (i; j) refer to sites in the metal. The n�, �, and y� are thefermioni number, annihilation, and reation operators.The band stru ture is introdu ed into this s heme by the tight-binding param-eters Tij, whi h an be obtained by a �t to ab initio band stru ture al ulations[SK54℄; tabulated values an be found, e.g., in the book by Papa onstantopou-los [Pap86℄. The hopping matrix T�m is also expressed in terms of the atomi orbitals of the atom(s) m in the metal and the atom(s) � in the tip. Here, onlythe oupling between s orbitals needs to be onsidered be ause of the quite large

2.2. THEORETICAL TREATMENT 9tip-sample distan es (5{10�A) in a typi al BEEM experiment.Using this basis, an expression for the urrent between two planes i and j inthe metal an be derived:Jij(kk; V ) = 4e�h Im Tr Xm��n[Tij(kk)gRjm(kk; V )Tm��(kk; V )T�ngAni(kk; V )℄; (2.6)where the indi es �; � and m;n run over layers in the tip and the sample respe -tively. The gR;A denote the retarded and advan ed equilibrium Green's fun tions,as shown in Appendix A. For further details of the Keldysh Green's fun tionte hnique used in this derivation see the appendix.On e the urrent distribution Eq.(2.6) has been al ulated, it is only a matterof applying the transmission oeÆ ient T (E;kk)S(E) and integrating over allenergies from the S hottky barrier Vb to the applied tip voltage Vt to obtain theinje ted urrent in the semi ondu tor:1IBEEM(Vt) = Z VtVb dE ZZIBZ dkk Jn�1;n(E;kk)T (E;kk)S(E): (2.7)Temperature e�e ts in form of the orresponding Fermi distributions of the tipand the metal, respe tively, are negle ted (T ! 0) throughout this work.Up to now this approa h is free of adjustable parameters. Inelasti e�e ts anbe in luded by introdu ing a self-energy � into the Green's fun tion:gR(E) = 1E � H + i� : (2.8)An arbitrarily small � is used to ensure mathemati al onvergen e, but hoosinga �nite � introdu es a damping to the wave �eld, whi h mimi s the e�e ts ofinelasti ele tron-ele tron s atterings.The primary goal of this work will be to study the e�e t of �| omputedwithin a �rst-prin iples approa h|on BEEM spe tra, whi h then an be dire tly ompared to the experiment.Now the main task is to al ulate the equilibrium GF gR;Aij for an isolatedmetalli layer. This is done by a de imation te hnique [RAG+98, GTFL83℄,whi h is an iterative pro edure for al ulating the surfa e GF and the propagatorgRj1 of a semi-in�nite slab. The semi-in�nite rystal is in this te hnique omposedof a sta king sequen e of \superlayers" (a slab onsisting of reasonable groupedlayers of the rystal), intera ting only with their nearest neighbours, and assumingperfe t periodi ity in both dire tions parallel to the surfa e. The de imationte hnique starts with two isolated superlayers and the Green's fun tion of thesuperlayers. The GF of the joint two-slab system is then onstru ted by means ofDyson's equation. This new system is again oupled in an iterative way to obtain1Note that the tunneling distribution e�2�d is in luded in the urrent distribution Jn�1;n.

10 CHAPTER 2. BEEMa four-layer system and so forth. After a number of iterations the two surfa esde ouple e�e tively (when introdu ing some dissipation in the al ulation), thusallowing the des ription of semi-in�nite systems.The results are the GF of the surfa e, gR11, and a relation between this surfa eGF and the propagator of a semi-in�nite slab:gRj1(E;kk) = T (j�1)gR11(E;kk); (2.9)where T is the so- alled transfer matrix of the system [AG�SF96℄. The propagatorgRj1(E;kk) des ribes the propagation of an ele tron with energy E and momentumkk from the surfa e to the layer j inside the sample. With this we have allbasi ingredients to perform a full quantum me hani al al ulation of the BEEM urrent.2.3 The Pd=Si SystemThe Green's fun tion method outlined above has been used in the past few yearsfor analyzing the Au=Si and the CoSi2=Si systems [Reu98℄. For these systems,the puzzling observations mentioned in Se tion 2.2.2 ould be su essfully ex-plained by introdu ing the band stru ture of the materials. E.g., the observedla k of di�eren e in the Au=Si system between the Si(111) and Si(100) surfa eswas tra ed ba k to the metalli band gap, leading to a forbidden propagation ofhot ele trons along the f111g dire tion; the ele trons annot propagate throughthe Au(111) layers along this dire tion but get de e ted sidewards, resulting ina ringlike k-spa e distribution where kk = 0 is ex luded. This learly di�ersfrom the free-ele tron model where the kk = 0 ontribution would be the mostdominant one. The propagation gap of hot ele trons around the �� point makesthe existen e or non-existen e of a proje ted Si CBM at �� (see Fig. 2.4), wherethe ele trons ould enter the semi ondu tor, a ompletely irrelevant fa t. It thusturned out that, at least for this system, additional s attering hannels for provid-ing the ele trons with kk-momentum, like the above-mentioned violation of thekk onservation at the interfa e, are not ne essary to explain the experimentalobservations [GAF96℄.In this work we will fo us on a system similar to the Au=Si one, namely onPd=Si(111). Pd and Au are both transition metals sharing a very similar bandstru ture (Fig. 2.5), with one ru ial di�eren e: The d-bands of Pd ross theFermi level, the ones of Au do not. As a onsequen e, one expe ts the e�e tsof the d-bands for Au only to appear in an enhan ed s reening, while for Pdthere are also ele troni transitions a ross the Fermi level asso iated with the d-bands. This should enhan e the e-e s attering rate in Pd and noti eably redu ethe lifetime of the hot ele trons. A �tting of transport parameters, su h as thelifetime of hot ele trons, an be obtained by omparing BEEM data and theGF results. BEEM seems to be parti ularly quali�ed for hara terizing these

2.3. THE PD=SI SYSTEM 11

Figure 2.5: The band stru ture of Pd (left) and Au (right) as omputed with theLAPW method (WIEN97 ).parameters. A BEEM experiment an be trimmed su h that the predominants attering pro ess will be, on average, one ele tron-ele tron s attering. This anbe a hieved by ooling down the system so that the e-p intera tion be omes weakand by using only very thin �lms up to about 100�A [LB93℄, still below the typi almean free path lengths of metals. After one e-e s attering event the ele tron willhave lost about half its energy and will therefore not be able to ross the S hottkybarrier anymore.In the next hapter we will set out for providing an ab initio des ription of thee-e s attering pro esses. Using the results obtained by the ab initio al ulations,we an ultimately he k on the expe tations raised in this se tion and on theBEEM data respe tively the �tting of the transport parameter to the GF results.The a tual status of this \work in progress" will be presented in Chapter 4.

12 CHAPTER 2. BEEM

Chapter 3Lifetimes of Hot Ele tronsInelasti s atterings of ele trons in metals are known to play an important role in avariety of physi al and hemi al phenomena. The hot ele trons in the low-energydomain are sensitive to the ele troni stru ture of the solid, and al ulationstaking into a ount the full band stru ture are ne essary to quantitatively obtainlifetimes for a real metal. Ab initio al ulations of su h lifetimes have been arriedout only re ently [CSP+99, SKBE99℄.The theory for al ulating BEEM urrents as outlined in Chapter 2 an intro-du e su h s attering events via an imaginary self-energy, provided that they arethe main loss me hanism for low ex ess energies. The Green's fun tion method(Se tion 2.2.2) fully a ounts for the metalli band stru ture, and it is there-fore reasonable to laim the same a ura y for omputing the lifetimes of hotele trons.In this hapter we set up an ab initio method for al ulating relaxation life-times of ex ited ele trons in real solids [CSP+99℄. In parti ular, we are aim-ing to quantitatively demonstrate the expe ted di�eren e between Pd and Au(see Chapter 2.3), thus investigating the in uen e of the d-bands on su h trans-port properties. The results will be presented in Chapter 4. Atomi units(e2 = �h = m = 1) will be used throughout the hapter.3.1 The Self-EnergyIn this se tion we will present two approa hes for deriving the self-energy ofan ele tron moving in an uniform ele tron gas [EFR90℄, whi h will serve as amotivation for our �nal expression Eq.(3.29), that fully a ounts for band stru -ture e�e ts. The semi lassi al one is useful to understand the on ept of theself-energy of a parti le intera ting with a polarizable medium. The quantumapproa h gives the lassi al result for high momenta.Note that the usual derivation uses a Green's fun tion approa h [EPCR00℄,�rst arried out by Quinn [QF58℄, whi h we will avoid for brevity.13

14 CHAPTER 3. LIFETIMES OF HOT ELECTRONS3.1.1 Semi lassi al Self-Energy of an Ele tronThe starting point is Poisson's equation, linking the total s alar potential � withthe total harge density �(r; t):r2� = �4��(r; t); (3.1)where �(r; t) = �0(r; t) + �ind(r; t). �0 indi ates the harge density of the \ex-ternal" parti le, i.e. the ele tron moving in the ele tron gas and the positively harged homogeneous ba kground, and �ind is the harge density indu ed in theele tron gas by the external parti le. The external potential satis�es a similarequation: r2�0 = �4��0(r; t): (3.2)Now we express all quantities as Fourier integrals of the formf(r; t) = Z d3q(2�)3 Z 1�1 d!2� ei(q�r�!t)fq;!: (3.3)For an uniform ele tron gas we assume that �q;! and �0q;! are linearly related via�0q;! = �(q; !)�q;!; (3.4)where �(q; !) indi ates the diele tri fun tion of the medium. The Fourier trans-forms of Eq.(3.1) and Eq.(3.2) together with Eq.(3.4) give:�q;! = 4��0q;!q2�(q; !) : (3.5)An ele tron moving with onstant velo ity v may be onsidered to give rise to adensity �0(r; t) = Æ(r�vt), where the Fourier transform gives �0q;! = 2�Æ(!�q�v).Considering that �indq;! = �q;! � �0q;! one obtains for the indu ed s alar potential:�indq;! = 8�2q2 Æ(! � q � v)� 1�(q; !) � 1�: (3.6)The rate of energy loss per unit time, i.e. the power loss _W , is now obtained fromthe indu ed ele tri �eld Eind = �r�ind at the ele tron position:_W = �v �Eind(r = vt; t): (3.7)Inserting Eq.(3.6) into Eq.(3.7) and taking into a ount that the real and imagi-nary parts of the response fun tion �(q; !) are an even respe tively odd fun tionwith respe t to ! yields:_W = Z d3q(2�)3 Z 10 d!2� 2!Im[��indq;!℄: (3.8)

3.1. THE SELF-ENERGY 15For brevity, we will from now on use the abbreviationZ dx � Z d3q(2�)3 Z 10 d!2� : (3.9)The probability P (q; !) to loose energy ! and momentum q per unit time anbe de�ned using Eq.(3.8) as: _W = Z dx !P (q; !); (3.10)where P (q; !) = 2Im[��indq;!℄ = 16�2q2 Imh �1�(q; !)iÆ(! � q � v); (3.11)des ribes the probability for reating a real ex itation in the ele tron gas. This an be used to de�ne a mean inelasti ollision time � as1� = Z dx P (q; !): (3.12)The e�e t of inelasti s attering an be represented by introdu ing an imaginaryopti al potential �I in the S hr�odinger equation. This leads to a temporal de ayof the ele tron probability density as e2�I t, where�I = � 12� : (3.13)For ompleteness, we state that the real part of the self-energy is found to be:�R = 12�ind(r = vt; t) = Z dx Re[�indq;!℄: (3.14)The omplex self-energy � is then given by:� = Z dx �indq;! = Z dx8�2q2 Æ(! � q � v)� 1�(q; !) � 1�: (3.15)3.1.2 Quantal Self-EnergyThe Hamiltonian des ribing an external parti le intera ting with an ele tron gasis given by: H = H0 + T0 +HI; (3.16)where H0 denotes the Hamiltonian of the ele tron gas, T0 is the kineti energyoperator for the external parti le with harge Z and mass M .1 HI represents theintera tion between the external parti le and the ele tron gas:HI =Xj Zjrj �RIj = Z d3r Z�(r)jr�RI j : (3.17)1The spe ial ase of an ele tron being the external harge will be dis ussed later.

16 CHAPTER 3. LIFETIMES OF HOT ELECTRONSHere RI denotes the oordinate of the in ident harge, rj is the oordinate of thejth ele tron in the ele tron gas, and �(r) is the parti le density operator of theele tron gas �(r) =Pj Æ(r� rj).The probability for an ele troni transition per unit time is given by �rst-ordertime-dependent perturbation theory. We des ribe the initial state of the externalparti le by a plane wave jii = eiki�RI , where ki is the initial momentum. Thestates of the ele tron gas are denoted by fjnig, su h that H0jni = "njni. Thetransition probability from an initial state jiij0i to a �nal state jfijni = eikf �RI jni an then be written as:Pi!f = 2�Xn jhnjhf jHijiij0ij2Æ�"n � "0 + k2f � k2i2M �= 2�Xn Z d3r d3r0 h0j�y(r)jnihnj�(r0)j0iDeiki�RI �� Zjr�RI j��eikf �RIE�� Deikf �RI �� Zjr0 �RIj ��eiki�RIE Æ�"n � "0 + k2f � k2i2M � : (3.18)Denoting q = ki�kf and !n0 = "n�"0 and realizing that the matrix elements ofZjr�RI j in this formula are given by the Fourier transforms of the potential vqe�iq�r,and vq = �4�Z=q2, one an write for Pi!f :Pi!f = 2�Xn h0j�y(q)jnihnj�(q)j0i�4�Zq2 �2Æ�!n0 � k2f � k2i2M � ; (3.19)where �(q) = Pj e�iq�rj is the Fourier transform of �(r). Thus Pi!f is propor-tional to the Fourier transform of the stati density-density orrelation fun tionjhnj�(q)j0ij2 [Hov54℄. For positive ! the imaginary part of the inverse of theresponse fun tion of the ele tron gas an be written as [NP58℄:Imh �1�(q; !)i = 4�2q2 Xn jhnj�(q)j0ij2Æ(! � !n0): (3.20)Using this relation, Pi!f be omesPi!f = Z 10 d!2�P 0(q; !); (3.21)where P 0(q; !) is given byP 0(q; !) = 16�2q2 Imh �1�(q; !)iÆ�! + k2f � k2i2M � : (3.22)This formula is not valid if the external parti le is an ele tron be ause of thefermioni nature of ele trons, where the �nal state kf = k � q, with k = ki,

3.2. AB INITIO CALCULATION OF LIFETIMES 17is subje t to the restri tion that kf has to be greater than kF , the Fermi waveve tor of the ele tron gas; i.e., the external ele tron an s atter only to a stateoutside the Fermi sphere of the ele tron gas. Thus we have to orre t Eq.(3.22)to: P 0(q; !) = 16�2q2 Imh �1�(q; !)iÆ(! + 12([k� q℄2 � k2))�(jk� qj � kf): (3.23)Note that for high in ident momenta k� q Eq.(3.23) oin ides with the lassi alexpression of Eq.(3.11).Eq.(3.23) an then be used to de�ne an imaginary part of a self-energy, in thesame way as we have done in Eq.(3.12) and Eq.(3.13):�I = � 12� = �12 Z dx P 0(q; !): (3.24)It an be shown [EFR90℄ that the total omplex self-energy an be written as� = Z dx8�q2 � 1�(q; !) � 1�Imh 1! � i� +�i; (3.25)where � = 12([k� q℄2 � k2) and � is a positive in�nitesimal onstant.In obtaining Eq.(3.23) we have negle ted ex hange pro esses, interfering withthe dire t pro ess. In a dire t pro ess, the external ele tron s atters from a statek to k� q and an ele tron from inside the Fermi sphere makes a transition fromk0 to k0 + q. The ex hange pro ess orresponds to another way of rea hing thesame �nal state, namely the transition of the external ele tron from a state kto k0 + q, while the ele tron in k0 goes to the state k � q. These two pro essesinterfere. In our approximation we negle t ex hange pro esses ompletely. Thisis equivalent to the GW approximation used in the Green's fun tion formalism[EPCR00℄, whi h is based on the random-phase approximation (RPA) [PN66℄where pro esses of higher order are negle ted, assuming independent propagationof the external harge and the indu ed polarization.For a homogeneous ele tron gas Eq.(3.24) is already the result wanted. In themore realisti s enario of a real rystal one works with Blo h wave fun tions anda more sophisti ated band dispersion. The resulting modi� ations are shown inthe next se tion.3.2 Ab Initio Cal ulation of LifetimesIn the last se tion we have motivated the existen e of a omplex self-energy for anele tron of momentum greater than the Fermi momentum, where the imaginarypart leads to an attenuation of the one-ele tron state, with a mean lifetime of� = � 12�I . After this time the ele tron will have transferred most of its initial

18 CHAPTER 3. LIFETIMES OF HOT ELECTRONSex itation energy to the ele tron gas. This is a onsequen e of the fa t that asingle-ele tron ex itation is not a stationary state of the ombined system ele tronplus medium, and results from inelasti s attering of the external ele tron withthe ele trons of the gas. This Coulomb s attering is s reened by the ele trons inthe gas, whi h is represented by the diele tri fun tion �.Below we will present an extension to the formulae presented so far whi hallows to treat the s attering pro ess in the metal in an ab initio manner, fullyin luding band stru ture e�e ts.3.2.1 Extension to Periodi CrystalsIn a periodi rystal, the ele troni wave fun tions are des ribed by Blo h states�k;n(r) = 1peik�ruk;n(r); (3.26)where n is the band index.Using Fourier expansions appropriate for periodi rystals, Campillo et al.have derived the following equation for the inverse lifetime [CSP+99℄:��1 = 1�2 Xf ZBZ dqXG XG0 B�0f (q+G)B0f (q+G0)jq+Gj2 Imh� 1�GG0(q; !)i; (3.27)where B0f(q) = Z d3r �0(r)e�iq�r f (r); (3.28)��1GG0(q; !) represents the Fourier omponents of the inverse diele tri fun tion,and the G's are re ipro al latti e ve tors. The integration over q is extendedover the �rst Brillouin zone (BZ). ! is the energy transfer ! = "0� "f , where theindi es 0 and f denote the initial and the �nal state, respe tively. The sum overf is extended over a omplete set of �nal states with energies "Fermi � "f � "0.2A simpli� ation an be introdu ed by negle ting the so- alled lo al-�eld ef-fe ts. The lo al-�eld approximation states that a perturbation with q and there ipro al latti e ve tor G orresponds only to a hange in the density at thesame re ipro al ve tor G; i.e., the ele tron-density variations in real rystals arenegle ted. Thus, if these ouplings between the wave ve tor q +G to the waveve tor q+G0 are negle ted, one arrives at:��1 = 1�2 Xf ZBZ dqXG jB0f(q+G)j2jq+Gj2 Im[�GG(q; !)℄j�GG(q; !)j2 ; (3.29)where we have also rewritten the imaginary part of the inverse of the diele tri fun tion of Eq.(3.27) in an obvious manner. If we take plane waves instead of2Here the onsidered transitions are the ones of the hot-ele tron, "0 being its initial energy

3.2. AB INITIO CALCULATION OF LIFETIMES 19the Blo h states in Eq.(3.29), this equation redu es again to the equation for anuniform ele tron gas within RPA, Eq.(3.24).The imaginary part of the diele tri fun tion Im[�GG(q; !)℄ is a measure ofdensity-of-states available for real transitions involving a given momentum q+Gand an energy transfer !. The fa tor j�GG(q; !)j�2 a ounts for the s reening ofthe ele trons in intera tion with the external ele tron. The overlap of initial and�nal states of the hot ele tron enters through the matrix elements B0f (q+G).3.2.2 The Diele tri Fun tionNow we need to al ulate the mi ros opi diele tri fun tion �GG(q; !). To moti-vate the formula we shortly outline an approa h by Adler [Adl62℄, who extendedthe des ription of the diele tri response to real rystals within the RPA. Thederivation �rst sets out to ompute the harge density n indu ed by the external harge, while this harge density is related to the total potential (i.e., externaland indu ed) via the polarization of the medium whi h, in turn, determines thediele tri fun tion.The starting point is the single-parti le Liouville von Neumann equationi��t = [H; �℄; (3.30)with H being the Hamiltonian of the perturbed system, onsisting of the Hamil-tonian of the unperturbed system H0 plus the perturbation �, whi h onsists ofthe externally applied potential �ext and the s reening potential �s indu ed bythe external ele tron. We write the density operator of the perturbed system �as � = �0 + �1; (3.31)where �0 is the density operator of the unperturbed system and �1 denotes thedensity hange due to the perturbation. A ting with �0 on the eigenstates of theunperturbed system jnki (n being the band index and k the wave ve tor) yields:�0jnki = f0("nk)jnki; (3.32)with "nk the single-parti le energies and f0 the Fermi-Dira distribution fun tion.We an now write a linearized form of the Liouville equation for the density hange�1 by negle ting a term [�; �1℄ of higher order in the perturbation:i��1�t = [H0; �1℄ + [�; �0℄: (3.33)Assuming a time dependen e for the external potential �ext = e�i!t, where ! isassumed to ontain an in�nitesimal imaginary part iÆ to ensure ausality, and

20 CHAPTER 3. LIFETIMES OF HOT ELECTRONStaking the matrix elements of Eq.(3.32) between the states jlki and jmk + qi,one arrives at:hlkj�1jmk+ qi = f0("mk+q)� f0("lk)"mk+q � "lk � ! hlkj�jmk+ qi: (3.34)This gives a relation between the indu ed harge density and the total perturbingpotential �.Next, we expand the perturbing potential in a Fourier series:�(r) = 1Xq XG �G(q)ei(q+G)r; (3.35)where is the rystal volume. The in lusion of re ipro al latti e ve tors G inthe Fourier sum takes are of the lo al �eld e�e ts, i.e., the variation of theexternal potential on a mi ros opi , atomi s ale. Introdu ing the Fourier seriesinto Eq.(3.34) leads to:hlkj�1jmk+ qi = f0("mk+q)� f0("lk)"mk+q � "lk � ! XG �G(q)MGlm(k;q); (3.36)with the matrix elements M de�ned as:MGlm(k;q) = hlkje�i(q+G)rjmk+ qi: (3.37)To get the desired indu ed harge density, one uses the basi relationn(r) = Tr[Æ(r� re)�1℄ =Xq Xlmk �mk+q(r) lk(r)hlkj�1jmk + qi; (3.38)with the wave fun tions lk(r) = hrjlki, whi h transforms in the re ipro al spa erepresentation to nG(q) =Xlmk �MGlm(k;q)��hlkj�1jmk+ qi: (3.39)Comparing Eq.(3.39) and Eq.(3.36) �nally gives the wanted relation between theindu ed harge density n and the total perturbing potential �:nG(q; !) =XG0 PGG0(q; !)�G0(q; !); (3.40)where PGG0(q; !) is the polarization (to be de�ned below). A simpli� ation ofthis formula an again be made by negle ting lo al-�eld e�e ts in the diele tri response as explained in Se tion 3.2.1 [see Eq.(3.29)℄. The indu ed harge density

3.2. AB INITIO CALCULATION OF LIFETIMES 21depends in this approximation only on the perturbing potential at the samere ipro al latti e ve tor G:nG(q; !) = PGG(q; !)�G(q; !); (3.41)where PGG(q; !) is given by:PGG(q; !) = 2Xlmk f0("mk+q)� f0("lk)"mk+q � "lk � ! ��MGlm(k;q)��2 (3.42)and the fa tor 2 takes into a ount spin degenera y. Finally, using Poisson's equa-tion and Eq.(3.4) as the de�nition for the diele tri fun tion, one an easily showthe following relation between the polarisation P and the diele tri fun tion �:�GG(q; !) = 1� v(q+G)PGG(q; !); (3.43)where v(q) = 4�q2 is the Coulomb potential. This equation is similar to theLindhard equation for the diele tri fun tion of an ele tron gas but in ludesthe hara teristi s of real metals via the band stru ture "mk and the matrixelements M .For the al ulation of the inverse lifetime, Eq.(3.29), we need the imaginaryand real part of the diele tri fun tion, Eq.(3.43). Remembering that ! ontainsan imaginary part iÆ, as explained before, we split � in its real (�1) and imaginary(�2) part by making use of the following relation:lim 1z � i0+ = }�1z�� i�Æ(z): (3.44)} �1z� denotes the prin ipal value of 1z . Using this relation for the polarisation Pone �nds:Re[�GG(q; !)℄ = 1� 2v(q+G) Xlmk ��MGlm(k;q)��2 1"mk+q � "lk � ! (3.45)Im[�GG(q; !)℄ = �2�v(q+G) Xlmk ��MGlm(k;q)��2 Æ("mk+q � "lk � !)(3.46)where ! denotes a real number from now on and we have taken the limit T ! 0.For metals one should introdu e an additional broadening in Eq.(3.45) to a ountfor the �nite lifetime of ex ited ele tron states and to avoid numeri al problemsdue to the divergent denominator for intraband ontributions. In this work wethus use instead of Eq.(3.44):1x + i = xx2 + 2 � i x2 + 2 : (3.47)

22 CHAPTER 3. LIFETIMES OF HOT ELECTRONSTypi al lifetimes of 100 fs orrespond to an imaginary energy i of about 0:04 eV(a value whi h has proven to give good results).We on lude this se tion by stating the free-ele tron gas result for the realpart of the diele tri fun tion and for q = 0:�1(!) = 1� !2p!2 : (3.48)Here, !p is the plasma frequen y for olle tive os illations of the ele tron gas. Atypi al plasma frequen y for metals is of the order of 10 eV. Thus for the energies onsidered (0:5{2 eV) we are far below resonan e, allowing us to take the stati approximation for the real part of the diele tri fun tion Eq.(3.45), ! ! 0. Thismeans that for the \slow" hot ele trons onsidered, the system has suÆ ient timeto s reen e�e tively.3.2.3 The Basi FormulaeAt this point let us olle t all the basi formulae used for the ab initio al ulationsof hot ele tron lifetimes.The entral equation is Eq.(3.29) but using stati s reening as dis ussed above.For the stati ase ! ! 0 the imaginary part of the diele tri fun tion vanishes;thus we approximate the s reening by using only the stati real part of the di-ele tri fun tion, jRe[�GG(q; ! = 0)℄j2.If we furthermore repla e the integral over q by a sum, R dq � (2�)3 Pq, weobtain: ��1 = 8� Xf XqG jB0f (q+G)j2jq+Gj2 Im[�GG(q; !)℄jRe[�GG(q; ! = 0)℄j2 (3.49)For the real and imaginary part of the diele tri fun tion, Eqs.(3.45) and (3.46)respe tively, we employ the Lorentzian line shape of Eq.(3.47) and arrive at:Im[�GG(q; !)℄ = � 8�jq +Gj2 BZXlmk ��MGlm(k;q)��2 ("mk+q � "lk � !)2 + 2Re[�GG(q; ! = 0)℄ = 1� 8�jq+Gj2 BZXlmk ��MGlm(k;q)��2 "mk+q � "lk("mk+q � "lk)2 + 2(3.50)Finally, the matrix elements B0f (q+G) and MGlm(k;q) read:B0f(q) = Z d3r �0(r)e�iq�r f (r)MGlm(k;q) = Z d3r �lk(r)e�i(q+G)�r mk+q(r) (3.51)Note the similarity between the two expressions, allowing for a al ulation ofboth matrix elements in the same manner.

3.3. DENSITY FUNCTIONAL THEORY (DFT) 233.3 Density Fun tional Theory (DFT)The basis fun tions used for the al ulation of the matrix elements Eqs.(3.51)were al ulated employing the LAPW formalism (see Chapter 3.4). The theoret-i al foundation of this method is the density fun tional theory (DFT), and this hapter aims to give a short introdu tion into this formalism. For a more exten-sive des ription of the theory and some of its appli ations, see e.g. the review byJones et al. [JG89℄.Density fun tional theory is based on two theorems by Hohenberg andKohn [HK64℄. We onsider a system of N ele trons in an external potentialVext(r). The �rst theorem states that the total ground state energy E of thesystem is a fun tional of the total ground state ele tron density nE[n(r)℄ = F [n(r)℄ + Z d3r n(r)Vext(r): (3.52)F [n(r)℄ is an unknown fun tional of the ele tron density n, independent of Vext(r).In other words, the potential is uniquely, up to a onstant, determined by theground state density n(r). Note that the ele troni density also determines thenumber of ele trons in the system N , and the kineti and potential energy ofthe isolated ele troni system. Thus, knowing n(r) �xes the total HamiltonianH and therefore the full many-parti le ground state of the system.The se ond theorem states that the a tual ele tron density minimizes thistotal energy fun tional. Kohn and Sham [KS65℄ showed that the fun tionalF [n(r)℄ an be separated into several parts, whi h are partially known expli itly:E[n(r)℄ = T0[n(r)℄ + EH [n(r)℄ + Uext[n(r)℄ + EXC[n(r)℄ + Eion: (3.53)T0 is the kineti energy of the non-intera ting system, EH is the Hartree energyresulting from the dire t Coulomb intera tions of the ele trons:EH [n(r)℄ = 12 ZZ d3r d3r0 n(r)n(r0)jr� r0j ; (3.54)and Uext is the energy from the external potential Vext of the positively hargednu lei: Uext[n(r)℄ = � Z d3r n(r)Vext(r): (3.55)EXC denotes the ex hange- orrelation energy and Eion des ribes the Coulombintera tion between the nu lei.The diÆ ult part is the ex hange- orrelation energy EXC[n(r)℄ whi h is notknown expli itly, and we need to �nd an approximative expression. This ex- hange energy takes into a ount the fermioni hara ter of the ele troni sys-tem, i.e., that ele trons arrying the same spin are not moving independently dueto Pauli's prin iple. The orrelation energy a ounts both for the fa t that the

24 CHAPTER 3. LIFETIMES OF HOT ELECTRONSa tual solution of the many-parti le S hr�odinger-like equation is not a olle tionof single-parti le states, and for the deviations of the kineti energy from the T0term.So far, no approximations have been employed. However, the exa t fun tionalEXC[n(r)℄ is not known. The most su essful approximation made to obtain theex hange and orrelation energy is the lo al density approximation (LDA) [Sla51℄.Within this s heme, the non-lo al fun tional is approximated by a lo al fun tion:ELDAXC [n(r)℄ = Z d3r n(r)"XC(n(r)); (3.56)where "XC(n(r)) is the ex hange- orrelation energy per ele tron of an uniformele tron gas of density n.3 For the a tual form of "XC(n(r)) there exists a numberof parameterizations [PW92℄.Kohn and Sham [KS65℄ introdu ed single parti le wave fun tions i (Kohn-Sham orbitals), from whi h the ele tron density an be onstru ted by summingup all o upied orbitals: n =Xo . j i(r)j2: (3.57)For the kineti energy we an write:T0[n(r)℄ = �12Xo . Z d3r �i (r)r2 i(r): (3.58)Finally substituting the pre eding equations into Eq.(3.53) and minimizing thetotal energy subje t to the onstraint that the Kohn-Sham orbitals i are or-thonormal yields: ��12r2 + vKS(r)� i(r) = "i i(r): (3.59)This is a single parti le S hr�odinger-like equation [KS65℄, where vKS(r) denotesthe Kohn-Sham potential:vKS(r) = Z d3r0 n(r0)jr� r0j + Vext(r) + vXC(r) (3.60)For the fun tional derivation of the ex hange- orrelation potential vXC(r) we em-ploy the LDA approximation [Eq.(3.56)℄, yieldingvXC(r) = Æf(n(r)"XC(n(r))gÆn(r) : (3.61)3The LDA is obviously exa t for the homogeneous ele tron gas. Therefore best results anbe expe ted for nearly free ele tron systems.

3.4. THE LAPW BASIS 25Thus, solving the single parti le equation (3.59), one an onstru t the densityfun tion n(r) using Eq.(3.57). As mentioned above, the true ground state densityis the one that minimizes the total energy.4Equations (3.59), (3.57), (3.60) and (3.61) are alled Kohn-Sham equationsand have to be solved self- onsistently: starting with an initial guess for the totalele tron density fun tion n, e.g. onstru ted by a superposition of atomi densi-ties, the Kohn-Sham equation (3.59) is solved, yielding the Kohn-Sham energies"i and wave fun tions i(r). These are used to onstru t a new harge densityn, whi h serves as a new input density for the next iteration. The Kohn-Shampotential is omputed from this density using Poisson's equation and adding theex hange- orrelation potential. These steps are iterated until onvergen e.The so obtained Kohn-Sham energies "i and orbitals i(r) have to be in-terpreted with are. This is be ause the energies "i enter in DFT as Lagrangeparameters in the variation pro edure of Eq.(3.59), thus la king dire t physi almeaning. Furthermore, DFT is designed to yield ground state properties, su h asele tron densities or the total energy. For quantities involving ex ited states, likethe al ulation of the diele tri fun tion, DFT misses, stri tly speaking, a rigor-ous justi� ation. The band stru ture is de�ned as the ele tron ex itation energiesin the many-parti le system and involves in this sense ex ited states. However,it has turned out in numerous al ulations that the interpretation of "i as theband stru ture is a valid and extremely su essful on ept. Note that even theinterpretation of the uno upied states as ex ited states (like opti al properties)turned out to be su essful. Thus the appli ation of DFT to the al ulation ofthe diele tri response fun tion is well grounded.3.4 The LAPW BasisTo use the DFT formalism outlined in the last se tion, one has to think about afeasible, yet a urate method for solving the Kohn-Sham equations for periodi rystals. One of the most a urate ones is the linearized augmented plane wave(LAPW) method, whi h will be introdu ed in this se tion. The ode that has beenused for all al ulations within this work (WIEN97 ) is a full-potential LAPW ode developed by Blaha and oworkers [BSL99℄.To start with, the unknown Kohn-Sham orbitals are expanded into a set ofgiven basis fun tions �j(r) i(r) =Xj aij�j(r): (3.62)Substituting this expansion into the Kohn-Sham equation (3.59) onverts theproblem from a partial di�erential equation to a generalized matrix eigenvalue4If approximations like LDA are made, this is not pre isely true anymore. Anyhow, a goodapproximation for the energy fun tional should give good values for energy and density.

26 CHAPTER 3. LIFETIMES OF HOT ELECTRONS

��

��

Figure 3.1: Dividing the unit ell volume into the interstitial (hat hed area) and thespheres region.problem (Ritz's variational prin iple)Xj (Hkj � "iSkj)aij = 0: (3.63)The Hamiltonian matrix Hkj is given byHkj = Z d3 ��k(r) ��12r2 + vKS(r)��j(r) (3.64)and Skj is the overlap matrixSkj = Z d3 ��k(r)�j(r): (3.65)A simple hoi e for the basis fun tion, ompatible with the latti e periodi ity,would be plane waves of the form ei(k+G)r. This basis set works well for the regionin between the atomi nu lei; however, near the nu lei, where the potential variesstrongly, an una eptable number of plane waves would be needed to a uratelydes ribe the wave fun tions. Slater [Sla37℄ introdu ed a method to augmentthe plane waves by atomi -like fun tions near the nu lei, the APW method. Thismethod and its su essors have been among the the most popular methods forsolving the Kohn-Sham equations.The idea of the APW and similar methods is to split the unit ell volumeinto two distin t regions [Figure (3.1)℄: The interstitial, where plane waves give agood des ription of the physi al onditions, and non-overlapping spheres aroundthe positions of the nu lei. These two regions are onsequently des ribed bytwo sets of basis fun tions: In the interstitial plane waves are used, while insidethe spheres (also alled muÆn-tin spheres) atomi radial fun tions and spheri alharmoni s are employed to give a better hara terization of the region lose tothe nu leus.

3.4. THE LAPW BASIS 27The LAPW basis set, whi h has been used throughout this work, onsists onsequently of the following split of basis fun tions:�k+G(r) = 8>><>>: 1pei(k+G)r r 2 interstitiallmaxPl=0 lPm=�l �a�lmul + b�lm _ul�Ylm(r�) jr� r�j � R� (3.66)The radial wave fun tions ul are solutions of the radial part of the Kohn-Shamequation5 (3.59), entered at atom a at the position ra and with radius Ra ( alledmuÆn-tin radius). is the unit ell volume. In prin iple the sum over the an-gular momentum quantum numbers is unlimited; however, to perform numeri al al ulations one has to introdu e some ut-o� parameter, denoted in Eq.(3.66)as lmax (with a typi al value of 10).Unlike the situation for a free atom, the radial solutions ul are no eigenfun -tions of the system. This omes from the fa t that the volume of the muÆn-tinspheres is �nite, and one an no longer require that the fun tions ul(r) van-ish for r ! 1. This implies that the solutions will be ome energy-dependent,ul = ul(r; "l), thus ompli ating the solution pro edure for the se ular equation(3.63). A solution to this problem is given by the LAPW method, where a �xedvalue for the energies "l is hosen beforehand.6 The drawba k of this pro edure learly is that the radial fun tions are not anymore able to des ribe the wavefun tion over a broader energy range. This problem an be relaxed by addingthe energy derivative of ul, denoted by _ul in Eq.(3.66). The result is a linearapproximation for the energy dependen e, giving the name linearized augmentedplane wave method [And75℄.The oeÆ ients alm and blm are determined by the ondition that the basisfun tions and their �rst derivatives are ontinuous at the sphere boundaries. Thetypi al number of basis fun tions needed is about 100 per atom.7 The de isiveparameter ontrolling the onvergen e of this basis is the uto� parameterRKmax,whi h is the produ t of the smallest muÆn-tin radius RMT and the magnitudeof the largest re ipro al latti e ve tor in Eq.(3.62), here alled Kmax. A typi alrange for this parameter is 6{9.In addition, lo alized basis fun tions (lo al orbitals) an be added [Sin94℄ toa ount for semi- ore states and to improve on the linearization. It should bementioned that the potential and the ele tron density are also expanded into planewaves in the interstitial and spheri al harmoni s within the muÆn-tin spheres,respe tively (see, e.g., Ref. [Sin94℄ for details).The matrix elements MGlm(k;q) and B0f (q) were al ulated using the LAPWbasis outlined above [Pus02℄. Only the basi prin iples of this al ulation shall5In the a tual implementation the fully relativisti form of Eq.(3.59) is used.6Usually these energy values are hosen to lie at the enter of the orresponding band withl-like hara ter.7Of ourse this number depends on the given problem.

28 CHAPTER 3. LIFETIMES OF HOT ELECTRONSbe mentioned here. The matrix elements MGlm(k;q) take the form8 [Eqs.(3.51)℄:MGlm(k;q) = Z d3r �lk(r)e�i(q+G)�r mk+q(r); (3.67)where the wave fun tions mk+q(r) are expanded in the LAPW basis set: nk(r) =Xi anki�ki(r): (3.68)Insertion of this equation into Eq.(3.67) yieldsMGlm(k;q) =Xij a�nkiamkj+qmGij (k;q) (3.69)with mGij (k;q) = Z d3r ��ki(r)e�i(q+G)�r�kj+q(r): (3.70)This is the matrix to be al ulated taking into a ount the di�erent basis fun tionsin the spheres and the interstitial. For the interstitial part, the omputation ofEq.(3.70) is quite simple be ause one just has to deal with plane waves; however,for the spheres the al ulation is quite tri ky, as one has to expand the planewave e�i(q+G)�r in spheri al harmoni s and spheri al Bessel fun tions using theso- alled Rayleigh expansion. A detailed des ription of the omputation of thematrix elements will be published in [Pus02℄.

8The matrix elements B0f (q) of the hot ele trons are basi ally of the same form, but on-sidering that both the initial and the �nal state need to be above the Fermi level.

Chapter 4ResultsIn this hapter we present results of our al ulations performed within the frame-work outlined in Chapters 2 and 3. We have broken this hapter into three majorparts: First, we present results of BEEM urrent distributions in Pd=Si, usinga preliminary guess for the attenuation length; this will allow us to dis uss someof the basi features expe ted. Se ond, lifetimes obtained within our ab ini-tio approa h are omputed for various transition and simple metals; we present onvergen e riteria for the LAPW implementation and dis uss the lifetime har-a teristi s with spe ial fo us on d-band e�e ts. Finally, preliminary results forBEEM spe tra al ulated by use of these ab initio lifetimes are shown.4.1 BEEMIn Se tion 2.3 the similarities and di�eren es between the Au=Si and the Pd=Sisystems have been outlined. Here we will ontinue to investigate the propertiesof the Pd=Si system, resuming some details of the al ulations performed andpresenting some results.Fig. 4.1 shows the onstant energy surfa e of Pd for an energy of 1 eV abovethe Fermi energy. One an learly identify the forbidden dire tions f111g, wherene ks open up, representing the la k of propagating states. Apart from thesedire tions, where the in uen e of the metalli band stru ture be omes noti eable,the onstant energy surfa e is quite similar to a free-ele tron like sphere. Bothof these features, the opening of ne ks as well as the otherwise free-ele tron likebehaviour, are very similar to the ones observed for Au [Reu98℄. Therefore oneexpe ts similar results for the k-spa e distribution of the hot ele trons in the metallayer. As a matter of fa t, the al ulations we performed |with reasonable valuesfor the attenuation � indi ated in the �gure aptions| for this system show thesame kind of distribution, Fig. 4.2. The main features observed are the de e tionof the ele trons away from the �� point to the boundary of the proje ted Brillouinzone (dark areas in Fig. 4.2), and the hange of symmetry when the damping is29

30 CHAPTER 4. RESULTS

Figure 4.1: The onstant energy surfa e for Pd at E = Ef +1:0 eV, omputed withina tight-binding parameterization.in reased.The latter e�e t is an interesting possibility to dire tly observe the transitionbetween a semi lassi al regime for a larger self-energy � [Eq.(2.8)℄ and a fullyquantum-me hani al system for small �'s [RAG+98℄. In a semi lassi al distribu-tion, only the k-ve tors representing transport towards the interfa e have to be onsidered. The overall symmetry of the urrent distribution re e ts the symme-try of the latti e [Fig. 4.2(a)℄, threefold for an f material. A ording to quantumme hani s, for a very small damping the k-ve tors representing transport in theopposite dire tion have to be onsidered additionally, sin e they are not anymore lassi ally separated from the \dire t" ones. Thus one more symmetry operationis provided and the overall symmetry hanges to sixfold, Fig. 4.2(b).4.1.1 Details of the Cal ulationIn this se tion we set out to dis uss the basi ingredients for the al ulation ofBEEM spe tra, whi h are based on Eq.(2.7). For the transmission oeÆ ientT (E;kk), we adopt the previously al ulated values for Au=Si [RAG+98℄. Thisseems reasonable be ause of the similarity of the Pd=Si and Au=Si systems.The fa tor had been al ulated within a 3D approximation s heme, employinga free ele tron band for the metal and a so- alled Jones-zone approa h for thesemi ondu tor.The ba k inje tion of ele trons in the metal, represented by the fa tor S(E)as outlined in Se tion 2.2.1, was al ulated by Reuter et al. [RHA+00℄, whoevaluated the s attering with a ousti and opti al phonons in the semi ondu tor,using a deformation potential approximation. The resulting fun tion T (E)S(E)

4.1. BEEM 31(a) (b)Figure 4.2: k-spa e urrent distribution after propagating through ten Pd(111) layers(E = Ef+2:0eV ). The ontrast has been strongly enhan ed to emphasize the features.Dark areas indi ate high urrent. Also indi ated is the border of the 2D Brillouin zone.(a) � = 0:2eV , semi lassi al regime outside the oheren e region; (b) � = 0:03eV , insidethe oheren e region.

Figure 4.3: The produ t of the transmission oeÆ ient T (E) and S(E), as taken from[Reu98℄.is only weakly energy dependent, as an be seen in Fig. 4.3.As mentioned in Chapter 2, the hot ele trons an enter the semi ondu toronly in the vi inity of the proje ted CBM. In Fig. 4.4 the resulting ellipses aredrawn. Sin e the �rst Brillouin zone of Si is larger than the one for Pd, alsoproje tions in the se ond zone (whi h are still part of the �rst zone of Pd) haveto be onsidered through remapping via a re ipro al latti e ve tor of Si (seeouter ellipses in Fig. 4.4). Altogether, there are basi ally two restraints for thehot ele trons rossing the S hottky barrier: First, there is an energeti restri tion,whi h follows from the ondition that the ele tron must have enough energy toover ome the S hottky barrier with a given kk. This ondition reads, for theele troni states inside the semi ondu tor:k2?2m�? � 0) 2(E � Vb) � j kk � kCBM j22m�k ; (4.1)where kCBM indi ates the position of the CBM in the semi ondu tor Brillouinzone (for Si, there are six symmetry-related CBM at 85% of �-X), Vb is the


Figure 4.4: k-spa e urrent distribution after propagating through 20 layers ofPd(111) (E = Ef + 1:3eV , � = 0:1eV ). The borders of the 2D Brillouin zone forSi and Pd are plotted. The ellipses represent the proje tion of the CBM of Si ontothe (111) orientation. The remapping of these ellipses by re ipro al ve tors from theSi Brillouin zone is also shown (outer ellipses)S hottky barrier height, E indi ates the ele tron energy in the metal, m� is thee�e tive mass in the semi ondu tor, and k? (kk) are ve tors inside the semi on-du tor perpendi ular (parallel) to the interfa e, respe tively. This de�nes theellipses drawn in Fig. 4.4, keeping in mind that an ele tron with energy E = eVbenters exa tly at the position of the CBM, so that the energy is referred to thispoint in this respe t.Se ond, the quantum me hani al probability for re exion/transmission at theinterfa e has to be onsidered, expressed in the above-mentioned fa tors T (E)and S(E) [Reu98℄.For the a tual al ulation of BEEM urrents one has to take into a ountanother e�e t negle ted so far: multiple re exions at the interfa e and the surfa e[Bel96, Reu98℄. This be omes important espe ially for very thin �lms (whi h weare interested in, as mentioned before), where a onsiderable number of ele trons an travel several times ba k and forth in the metal layer, without su�eringinelasti s atterings. The generalized form of the BEEM urrent in luding thise�e t an be written as:IBEEM(Vt) = 1Xi=0 Z VtVb dE ZZIBZ dkk Jm(E;kk; i)T (E;kk)S(E): (4.2)Jm(E;kk; i) indi ates the ith ontribution to the urrent at the interfa e, i.e., the urrent distribution after i re exions at the interfa e and the surfa e. Jm(E;kk; 0)is the dire t ontribution. Assuming kk onservation at the interfa e, the urrentdistribution after i passes is simply given by Jm(2i+1)(E;kk; 0) at the m(2i+1)thlayer, where m is the number of layers orresponding to the metal layer thi knessof interest. Additionally, one has to a ount for the redu tion in the number ofele trons whi h have already su essfully passed the barrier after the ith step.

4.2. AB INITIO LIFETIMES 33The fa tor (1 � T (E;kk)S(E)) represents the probability for the ele tron to bere e ted at the interfa e. Thus, the urrent distribution at the barrier after ipasses yields Jm(E;kk; i) = Jm(2i+1)(E;kk; 0)�1� T (E;kk)S(E)�i; (4.3)to be put into Eq.(4.2).Using this methods we have al ulated I-Vt spe tra for the Pd=Si system.The aim is to ompare our results with experimental data [LB93℄ and to dedu elifetimes of the hot ele tron arriers from omparison with experiment.To he k if the theory for al ulating BEEM urrents, in luding inelasti e�e ts, is appropriate, we set up an ab initio method for evaluating lifetimesof hot ele trons as des ribed in Chapter 3. For the experimental onditions onsidered (thin �lms � 100�A, low temperature, and low voltages), s atteringme hanisms others than ele tron-ele tron will be negligible. Otherwise it wouldintrodu e a large un ertainty how to ombine the di�erent mean free paths forele tron-ele tron, ele tron-phonon and ele tron-defe t s atterings. In the nextse tion we will present results of the ab initio al ulations for various metals.4.2 Ab Initio LifetimesThe ab initio omputation of lifetimes of hot ele trons has been performed us-ing the formulae summarized in Se tion 3.2.3. First, a self- onsisten y y le is arried out to yield a self- onsistent Kohn-Sham ele tron density (see Se tion3.3), using the WIEN97 ode. Using this onverged density, the wave fun tionsand eigenvalues, and subsequently the matrix elements (3.51) between two statesseparated by a wave ve tor q are al ulated on the grid hosen. Sin e we aresumming over all possible q ve tors, two grids have to be provided. To obtainbetter onvergen e, these two meshes are shifted against ea h other [Sin94℄. Thusthe q = 0 ve tor, whi h is unimportant for our purposes, is not in luded; in do-ing so, we are able to onsider a smaller minimum q ve tor as ompared to theunshifted ase.1The omputation of the eigenve tors and eigenvalues in the LAPW methodis quite time onsuming, and so is the al ulation of the matrix elements inthis basis set. To redu e the amount of eigenfun tions and matrix elements tobe al ulated, the symmetry properties of the Brillouin zone and of the matrixelements MGlm(k;q) are fully exploited [Pus02℄.2 Still, the omputation of allmatrix elements is a matter of weeks rather than days, even when using severalpowerful omputers in parallel.1Here we used a shifting of half the grid size, providing a minimum q ve tor of half thedistan e between two grid points.2E.g., for a grid of 20� 20� 20 points and an f metal, the number of q ve tors for whi hone has to al ulate the matrix elements redu es from 8000 to below 200.

34 CHAPTER 4. RESULTS4.2.1 Convergen eMost onvergen e tests have been made for Pd, being the metal with the most omplex band stru ture of the ones onsidered. In the following, we will showresults of onvergen e tests for this material to support the hoi e of parametersused in the al ulations.On a ount of the time-expensive omputation we are obliged to hoose pa-rameters arefully, for neither wasting omputational time nor risking un on-verged results. The onvergen e of the imaginary part of � with the used k-meshfor Pd and for two arbitrarily hosen ombinations of q and G ve tors is shownin Fig. 4.5. The real part of � is shown in Figs. 4.6 and 4.7, where we use abroadening for imaginary and real part, respe tively, of 0:14 eV and 0:04 eV.3One observes that Re[�℄ shows only a weak dependen e on the grid size, even forthe smallest q ve tor onsidered.Another de isive parameter is the number of plane waves in the basis set,whi h is basi ally ontrolled by RKmax, as explained in Se tion 3.4. Figs. 4.8and 4.9 show the onvergen e with RKmax of Im[�℄ and Re[�℄, respe tively, forspe i� q's and G's. For both Im[�℄ and Re[�℄ we �nd strong variations for thesmallest RKmax values onsidered, while good onvergen e is observed for valuesabove or equal to 8. For the ut-o� parameter lmax of the angular momentumquantum number l, needed for the basis fun tions inside the muÆn-tin spheres[Eq. (3.66)℄, we show the orresponding �gure in Fig. 4.10. It turns out that theresults do not depend de isively on lmax.Furthermore, to al ulate the inverse lifetime ��1 we need to sum over re ip-ro al latti e ve tors G [Eq. (3.49)℄. The limiting parameter for this summationis alled Gmax and denotes the maximum absolute value of q+G (in units of theBohr radius) to be in luded in the sum.Altogether, the tests made support the following hoi e of parameters fora urate but yet feasible al ulations:� A grid size of 20� 20� 20 k-points;� RKmax = 8 for the parameter to ontrol the number of plane waves used;� lmax = 10 for the omputation of the energy eigenvalues and the waveve tors, lmax = 3 (see Fig. 4.10) for the al ulation of the matrix elements;� a maximum value for jq+Gj of Gmax = 2.3In prin iple, it would be logi al to employ the same value for the real and the imaginarypart. Anyhow, the onvergen e tests made gave better results using di�erent parameters,without hanging mu h the overall results.

4.2. AB INITIO LIFETIMES 35

0 0.5 1 1.5 2 2.50

2

4

6

8

10

Im(ε

)

63

163

203

243

263

303

0 0.5 1 1.5 2 2.5

ω(eV)

0

0.1

0.2

0.3

0.4

0.5

0.6

Im(ε

)

63

163

203

243

283

323

Figure 4.5: The onvergen e of the imaginary part of � for Pd with the grid size.For the lower �gure, q = (0:2; 0; 0) and G = (�1; 1; 1), and for the upper �gure,q = (0:5; 0:25; 0) and G = (0; 0; 0) (in units of 2�a , a being the latti e onstant).


63

163

203

243

283

303

0

5

10

15

20

Re(

ε)

q=(0.5,0.25,0), G=(0,0,0)

63

163

203

243

283

323

0

0.5

1

1.5

2

2.5

3

Re(

ε)

q=(0.2,0,0), G=(-1,1,1)

Figure 4.6: The onvergen e of the real part of � for Pd with the grid size for di�erent ombinations of q and G.


63

163

203

243

283

323

k-points

0

200

400

600

800

1000

1200

1400

1600R

e(ε)

q=(1/20,1/20,1/20), G=(0,0,0)

Figure 4.7: The onvergen e of the real part of � for Pd with the grid size for a smallq ve tor.

0 2 4 6 8 10

ω(eV)

0

2

4

6

8

10

12

14

16

Im(ε

)

RKmax

: 5.0

6.07.08.09.010.0

Figure 4.8: The onvergen e of the imaginary part of � for Pd with the parameterRKmax. The grid size is 163 points, q = (0:5; 0:25; 0) and G = (0; 0; 0).


4 5 6 7 8 9 10

RKmax

0

5

10

15

20

25

Re(

ε)

q=(0.5,0.25,0), G=(0,0,0)

Figure 4.9: The onvergen e of the real part of � for Pd with the parameter RKmax.The grid size is 163 points.

2 3 4 5 6

lmax

311.3

311.4

311.5

311.6

311.7

311.8

311.9

312

Re(

ε)

q=(0.2,0,0), G=(0,0,0)

Figure 4.10: The onvergen e of the real part of � for Pd with the parameter lmax.The grid size is 203 points.


0 0.5 1 1.5 2

Gmax

(Bohr)

0

50

100

150

200

τ(fs

)

1.24eV0.50eV2.34eV

Figure 4.11: The onvergen e of the lifetime � for Pd with Gmax along the line �-K.The grid size is 203 points.


0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

ω (eV)

0

20

40

60

80

100

120

140

τ (f

s)Γ-XΓ-LFEG, r

s=2.07

Campillo et al.

Figure 4.12: The lifetime of hot ele trons for Al. The ir les and squares indi ateour ab initio al ulations along dire tions �-X and �-L. The dashed line follows fromEq.(4.4) for a FEG with rs = 2:07. The diamonds represent a re ent al ulation byCampillo et al. [CPC+00℄.4.2.2 Lifetimes of Metals - ResultsWith the parameters dis ussed in the last se tion, al ulations of the lifetimesof hot ele trons in Al, Cu, Au and Pd �lms have been performed and will bepresented in the following. To ompare the data with the free ele tron gas (FEG),the following formula has been used [CSP+99, QF58℄:�QF = 263r� 52s (E � EF )�2eV 2fs; (4.4)valid for the high-density limit (rs � 1).AluminumFig. 4.12 shows the results for Al. These are not in perfe t agreement with re entab initio al ulations by Campillo et al. However, it has to be stressed that thedata by Campillo is an average over all initial states orresponding to waveve tors of the Brillouin zone with the same energy [CPC+00℄, while our dataonly represents two dire tions (X-� and L-�) in k-spa e. It has been shown byS h�one et al. [SKBE99℄ that even for a free-ele tron-like metal like Aluminumthe lifetimes depend strongly on the dire tion in k-spa e. Future work will addressthis point in more detail.


Figure 4.13: The band stru ture of Cu, al ulated within the LAPW method.CopperCopper is a transition metal with the d-bands lying about 2{4 eV below the Fermisurfa e, Fig. 4.13. This should be re e ted in an enhan ed s reening in ompari-son with the FEG, i.e., the lifetime should be noti eably higher. Indeed, this anbe inferred from Fig. 4.14, in good agreement with the ab initio al ulations ofCampillo et al. [CSP+99℄ and S h�one et al. [SKBE99℄.GoldFor gold the band stru ture is shown in Fig. 2.5. Gold and opper share the samevalen e ele tron on�guration, leading to a very similar band stru ture. Thus, weexpe t strong d-band s reening and onsequently larger life times in omparison tothe FEG. Again, our results ompare very well with both the experiment by Caoet al. [CGEA+98℄ and theoreti al al ulations [CPRE00, KSE00℄, see Fig. 4.15.In a ordan e with the results of Campillo et al. we �nd an enhan ement ofour al ulated lifetimes by a fa tor of � 4 as ompared to the FEG model. Ina re ent paper by Reuter et al. [RHA+00℄ a fa tor of two was obtained byexploiting experimental BEES data with the Green's fun tion method outlinedin Chapter 2.


0.5 1 1.5 2 2.5 3

ω (eV)

0

50

100

150

τ (f

s)

Γ-XW-LFEG, r

s=2.67

Campillo et al.Schöne et al.

Figure 4.14: The lifetime of hot ele trons for Cu. The ir les and squares indi ateour ab initio al ulations along two distin t dire tions �-X and W -L. The dashedline follows from Eq.(4.4) for a FEG with rs = 2:67. The diamonds and the uprighttriangles represent re ent al ulations by Campillo et al. [CPC+00℄ and S h�oneet al. [SKBE99℄, respe tively.PalladiumFinally, we present results for Pd. Pd is a transition metal similar to Cu andAu, but with the ru ial di�eren e that here the d-bands ross the Fermi surfa e(as dis ussed in more detail in Se tion 2.3 and 4.1). In ontrast to Cu andAu, where the d-bands just give rise to an enhan ed s reening, in Pd also reald-band ex itations be ome possible; here, e.g., the hot ele tron looses energy bys attering another ele tron from a state below the Fermi surfa e to a d-bandstate above EF . This opens a new s attering hannel whi h redu es the lifetime.This is on�rmed by our al ulations, Fig. 4.16. The resulting lifetimes di�ersubstantially from the free-ele tron like behaviour. A simple phase-spa e estimatepredi ts a 1!2 -like dependen e on the energy, where ! is the ex ess energy of thehot ele tron.4 this is learly not valid in this ase due to the band stru turee�e ts. Furthermore a strong dependen e on the k-spa e dire tion must be noted.In Figs. 4.17 and 4.18 the hot-ele tron lifetimes of Pd and Au are ompared alongtwo di�erent dire tions in k-spa e. As expe ted, the lifetimes for Pd are smallerthan for Au. The strange behaviour of the urves for Pd in W -L dire tion(Fig. 4.16) and for Au in �-K (Fig. 4.18) is not yet understood very well. We4Be ause the available phase spa e for su h transitions in reases with !2


0.5 1 1.5 2 2.5 3

ω (eV)

0

50

100

150

200

τ (f

s)Γ-XW-LCao et al.Campillo et al.Keyling et al.FEG, r

s=3.01

Figure 4.15: The lifetime of hot ele trons for Au. The ir les and squares indi ateour ab initio al ulations along two distin t dire tions �-X and W -L. The dashedline follows from Eq.(4.4) for a FEG with rs = 3:01. The upright triangles representthe experimental measurements by Cao [CGEA+98℄. The diamonds and the + repre-sent re ent al ulations by Campillo et al. [CPRE00℄ and Keyling et al. [KSE00℄,respe tively.should note that in the ase of Pd we had to hoose initial states slightly displa edfrom the W -L dire tion, due to the la k of states around L at energies higherthan � 0:9 eV (see Fig. 4.1); this might be a possible explanation for the sudden hange of the hara ter at about this energy.


0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

ω(eV)

0

20

40

60

80

100

120

140

160

180

200

τ(fs

)Γ-KΓ-XW-LFEG, r

s=2.66

Figure 4.16: The lifetime of hot ele trons for Pd along di�erent symmetry lines. Thedashed line follows from Eq.(4.4) for a FEG with rs = 2:66.

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

ω (eV)

0

50

100

150

200

250

300

τ (f

s)

Pd, Γ-XAu, Γ-XFEG, r

s=2.66 (Pd)

FEG, rs=3.01 (Au)

Figure 4.17: The lifetime of hot ele trons for Pd ompared with Au along the �-Xline. The dashed lines follows from Eq.(4.4) for a FEG with rs adjusted to Pd and Au,respe tively.


0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

ω(eV)

0

20

40

60

80

τ(fs

)

Au, Γ-KPd, Γ-KFEG, r

s=3.01 (Au)

FEG, rs=2.66 (Pd)

Figure 4.18: The lifetime of hot ele trons for Pd and Au. The lifetimes are omparedfor the two metals along the same dire tion in k-spa e, �-K. The dashed lines followfrom Eq.(4.4) for a FEG.


0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5V

t

0

5

10

15

20

I c(pA

)

BEEM (theor.)Ludeke (exp.)

Figure 4.19: Example of a BEEM spe trum for Pd, ompared with experimental databy Ludeke [LB93℄.4.3 Combining BEEM & Ab Initio DataThe lifetimes al ulated with the ab initio method an now be put into the Green'sfun tion formalism to obtain BEES spe tra. Sin e the lifetimes show a strong di-re tional dependen e, they have to be hosen arefully. How to sele t appropriateinitial states an be inferred from the BEEM k-spa e urrent distributions, e.g.,Fig. 4.2. For a three-fold symmetry and Pd(111) a reasonable hoi e seems to bethe �-K dire tion. The resulting lifetimes for Pd and Au are shown in Fig. 4.18.For Pd, it is nearly onstant over the energy range observed, an astonishing re-sult when ompared with the FEG predi tion. The al ulations yield a value of� � 15-20 fs, resulting in a value of about � � 0:022 eV for the imaginary partof the self energy,5 to be put into Eq. (2.8). The resulting BEEM spe trum fora Pd(111) �lm of 20 layers is shown in Fig. 4.19. Here we used a tunneling dis-tribution like eE� ,6 with a relatively large � value of 1:0 eV. We should note thatwith the same set of parameters we were not able to reprodu e the experimentalBEEM spe tra for thi ker metal �lms. Therefore Fig. 4.19 should be seen as anpreliminary example and be onsidered with some are. Future work will explorelifetime e�e ts and the in uen e of � on Pd-Si BEEM spe tra in more detail[LPA+02℄.5Remember that � = 12� , in atomi units.6This expression omes from expanding the square root in Eq.(2.2).

Chapter 5SummaryIn this work, we have developed a framework for the ab initio al ulation ofele tron-ele tron s atterings in thin metal �lms. The analysis, based on therandom phase and GW approximations, has provided us with quite simple ex-pressions, whi h allow to introdu e band stru ture e�e ts of real metals in the al ulation of hot ele tron lifetimes. A pres ription of how to implement theseexpressions within an LAPW s heme has been given, and has proven to be bothfeasible and a urate. We have identi�ed the pertinent parameters for the al- ulation of the inelasti ele tron lifetime � . Our results have revealed a de isivein uen e of d-bands of transition metals on ele tron-ele tron s atterings: Whilefor Cu and Au virtual d-band ex itations lead to a strongly enhan ed s reening,resulting in larger lifetimes in omparison to the free-ele tron-gas (FEG) model,for Pd real d-band ex itations have proven to play a major role. More spe i� ally,a strong redu tion of � has been tra ed ba k to su h ex itations. In addition,we have found a strong dire tional k-spa e dependen e of � , and along ertaink-spa e dire tions � shows very weak energy dependen e|in strong ontrast tothe FEG predi tions.We have dis ussed that Ballisti Ele tron Emission Mi ros opy (BEEM) pro-vides a very sensitive measure of su h inelasti s atterings in thin metal �lms.A re ently developed Keldysh Green's fun tion formalism has been adopted to al ulate su h BEEM spe tra in a �rst-prin iples manner. This formalism allowsto in lude the details of both the band stru ture and the attenuation due to in-elasti s atterings. The general features expe ted from the pe uliarities of thePd band stru ture have been dis ussed. Finally, �rst results for urrent-voltagespe tra in orporating the ab initio lifetimes have been presented.Future work will further explore su h inelasti lifetime e�e ts on BEEM spe -tra, in parti ular for the Pd-Si system. We expe t this analysis to highlight theimportant role of the d-bands on the hot ele tron lifetimes in transition met-als, and to emphasize the strength of BEEM to uniquely extra t su h importanttransport properties. 47

48 CHAPTER 5. SUMMARY

Appendix AMathemati al DetailsTo derive the basi urrent equation Eq.(2.6) one starts with the HamiltonianEq.(2.5) given in an LCAO basis. The urrent between two nodes i; j in thesystem is given by the following general equation:Ji;j = Aj;i yj(t + 0+) i(t)� Ai;j yi(t+ 0+) j(t): (A.1)This equation des ribes the annihilation of an ele tron on site i at time t andthe reation of an ele tron at an in�nitesimally later time t + 0+ on j, plus theopposite pro ess. To �x the onstants A we an use the ontinuity equation��i�t =Pj Ji;j and Heisenberg's equation of motion for the density operator:��i�t = �i[�i; H℄ = �i[ yi i;Xj;k Tjk yj k℄ = �iXj fTij yi j � Tji yj ig: (A.2)The omparison of these two equations immediately gives Aij = iTij.A measurable magnitude is obtained by taking the expe tation value of theoperator Jij whi h is related to the Keldysh Green's fun tion (GF) G+�ij (0+):Jij = h0jJijj0i = i(Tijh0j yi jj0i � Tjih0j yj ij0i) = TijG+�ij (0+)� TjiG+�ji (0+):(A.3)The Keldysh te hnique [Kel65℄ used in this approa h takes into a ount thenon-equilibrium hara ter of the problem. In the equilibrium ase the retardedGreen's fun tion gR fully des ribes the propagation of a parti le. The KeldyshGF G+� is needed as an additional dynami al variable to a ount for the non-equilibrium situation, indu ed by the tunneling pro ess:G+�ij = ih yj(t0) i(t)i:G+� des ribes the o upation of the quasi-parti le states, where hi denotes thestatisti al average over the non-equilibrium distribution.49

50 APPENDIX A. MATHEMATICAL DETAILSIt is onvenient to swit h to the energy domain by Fourier transform, and we�nally arrive at the basi equation for omputing urrents between the two sitesi and j: Jij = 2eh Z TrfTij[G+�ij (E)� G+�ji (E)℄gdE; (A.4)where the tra e (Tr) denotes a summation over the orbitals forming the hosenLCAO basis, e is the ele tron harge, and h denotes Plan k's onstant. Thenon-equilibrium Keldysh GF G+� an be al ulated in terms of the retarded andadvan ed GF GR=GA of the whole intera ting system and the g+� of the isolatedmetal �lm [AGRF01℄:G+� = (1+ GR(E)T )g+�(1+ T GA(E)): (A.5)The g+� again are losely related to the retarded and advan ed fun tions:g+�(E) = f(E)(gA(E)� gR(E));f(E) being the Fermi-Dira distribution. The retarded and advan ed GF an beobtained from a Dyson-like equation using the GF of the un oupled parts of thesystem: GR;A(E) = gR;A + gR;AT GR;A: (A.6)This determines the obje t of interest, G+�(E), in terms of two nonintera tingsystems and their intera tion given by the hopping matrix T .Using the relation gA�� gR�� = 2�i�� for the density of states matrix, onearrives in lowest order perturbation theory and after some algebra at the formulafor al ulating the urrent between two sites i and j in the metal:Jij = 4e�h ImZ 1�1Tr Xm��n[Tij gRjmTm��T�ngAni℄dE: (A.7)The Keldysh GF te hnique has therefore lead to an equation for the urrentthat only in ludes the tight-binding parameters Tij inside the metal, the hoppingmatri es Tm�=T�n between tip and the sample, and the retarded and advan edequilibrium GF gRjm=gAni of the isolated tip and sample. For the urrent distribu-tion per energy unit in re ipro al spa e one arrives at an equation of the samestru ture, but this time i and j refer to two planes in the metal:Jij(kk; V ) = 4e�h Im Tr Xm��n[Tij(kk)gRjm(kk; V )Tm��(kk; V )T�ngAni(kk; V )℄; (A.8)where the indi es �; � and m;n run over layers in the tip and the sample respe -tively and all obje ts are the Fourier transforms of the orresponding obje ts inEq.(A.7) [RAG+98℄. With this results we have arrived at Eq.(2.6), whi h a u-rately des ribes the ele tron transport inside the metal, taking into a ount theband stru ture of the material in a �rst-prin iples manner.

List of Figures2.1 Setup of a BEEM experiment . . . . . . . . . . . . . . . . . . . . 42.2 Energy diagram of a BEEM experiment . . . . . . . . . . . . . . . 42.3 Opti al phonon ba ks attering . . . . . . . . . . . . . . . . . . . . 72.4 Proje tion of the Si ondu tion band . . . . . . . . . . . . . . . . 82.5 Band stru tures of Pd and Au . . . . . . . . . . . . . . . . . . . . 113.1 Dividing of the unit ell in the LAPW method . . . . . . . . . . . 264.1 Constant energy surfa e of Pd . . . . . . . . . . . . . . . . . . . . 304.2 k-spa e urrent distribution in Pd . . . . . . . . . . . . . . . . . . 314.3 Transmission and ba ks attering oeÆ ient . . . . . . . . . . . . . 314.4 Current distribution and interfa e state mat hing . . . . . . . . . 324.5 Convergen e of Im(�) with the grid size . . . . . . . . . . . . . . . 354.6 Convergen e of Re(�) with the grid size . . . . . . . . . . . . . . . 364.7 Convergen e of Re(�) with the grid size for a small q ve tor . . . 374.8 Convergen e of Im(�) for Pd with RKmax . . . . . . . . . . . . . . 374.9 Convergen e of Re(�) for Pd with RKmax . . . . . . . . . . . . . . 384.10 Convergen e of Re(�) for Pd with lmax . . . . . . . . . . . . . . . 384.11 Convergen e of the lifetime � for Pd with Gmax . . . . . . . . . . 394.12 Lifetime of hot ele trons for Al . . . . . . . . . . . . . . . . . . . 404.13 Band stru ture of Cu . . . . . . . . . . . . . . . . . . . . . . . . . 414.14 Lifetime of hot ele trons for Cu . . . . . . . . . . . . . . . . . . . 424.15 Lifetime of hot ele trons for Au . . . . . . . . . . . . . . . . . . . 434.16 Lifetime of hot ele trons for Pd . . . . . . . . . . . . . . . . . . . 444.17 Lifetime of hot ele trons for Pd and Au . . . . . . . . . . . . . . . 444.18 Lifetime of hot ele trons for Pd ompared with Au . . . . . . . . 454.19 BEEM spe trum for Pd . . . . . . . . . . . . . . . . . . . . . . . 4651

52 LIST OF FIGURES

Bibliography[Adl62℄ S. L. Adler, Phys. Rev. 126 (1962), 413.[AGRF01℄ P. L. de Andr�es, F. J. Gar ��a{Vidal, K. Reuter, and F. Flores,Progress in Surfa e S ien e 66 (2001), 3{51.[AG�SF96℄ P. L. de Andr�es, F. J. Gar ��a{Vidal, D. �Sestovi� , and F. Flores,Physi a S ripta Topi al Issues 66 (1996), 277.[And75℄ O. K. Andersen, Phys. Rev. B 12 (1975), 3060.[Bel96℄ L. D. Bell, Phys. Rev. Lett. 77 (1996), 3893.[BK88℄ L. D. Bell and W. J. Kaiser, Phys. Rev. Lett. 61 (1988), 2368.[BSL99℄ P. Blaha, K. S hwarz, and J. Luitz, WIEN97, A Full PotentialLinearized Augmented Plane Wave Pa kage for Cal ulating Crys-tal Properties, Vienna University of Te hnology, 1999.[CDL77℄ C. Cohen{Tannoudji, Bernhard Diu, and Fran k Lalo�e, QuantumMe hani s, Vol. One, Wiley, New York, 1977.[CGEA+98℄ J. Cao, Y. Gao, H. E. Elsayed-Ali, R. J. D. Miller, and D. A. Mantell,Phys. Rev. B 58 (1998), 10948.[CPC+00℄ I. Campillo, J. M. Pitarke, E. V. Chulkov, A. Rubio, and P. M.E henique, Phys. Rev. B 61 (2000), 13484.[CPRE00℄ I. Campillo, J. M. Pitarke, A. Rubio, and P. M. E henique, Phys.Rev. B 62 (2000), 1500.[CSP+99℄ I. Campillo, V. M. Silkin, J. M. Pitarke, A. Rubio, E. Zarate, andP. M. E henique, Phys. Rev. Lett. 83 (1999), 2230.[EFR90℄ P. M. E henique, F. Flores, and R. H. Rit hie, Dynami S reeningof Ions in Condensed Matter, Solid State Physi s, Vol. 43, 1990.[EPCR00℄ P. M. E henique, J. M. Pitarke, E. V. Chulkov, and A. Rubio, Chem-i al Physi s 251 (2000), 1{35.53

54 BIBLIOGRAPHY[GAF96℄ F. J. Gar ��a{Vidal, P. L. de Andr�es, and F. Flores, Phys. Rev. Lett.76 (1996), 807.[GTFL83℄ F. Guinea, C. Tejedor, F. Flores, and E. Louis, Phys. Rev. B 28(1983), 4397.[HK64℄ P. Hohenberg and W. Kohn, Phys. Rev. 136 (1964), B 864.[Hov54℄ L. Van Hove, Phys. Rev. 95 (1954), 249.[JG89℄ R. O. Jones and O. Gunnarsson, Review of Modern Physi s 61(1989), 689.[KB88℄ W. J. Kaiser and L. D. Bell, Phys. Rev. Lett. 60 (1988), 1406.[Kel65℄ L. V. Keldysh, Sov. Phys. JETP 20 (1965), 1018.[KS65℄ W. Kohn and L. J. Sham, Phys. Rev. 140 (1965), A 1133.[KSE00℄ R. Keyling, W. D. S h�one, and W. Ekardt, Phys. Rev. B 61 (2000),1670.[LB93℄ R. Ludeke and A. Bauer, Phys. Rev. Lett. 71 (1993), 1760.[LPA+02℄ F. Ladst�adter, P. Pus hnig, C. Ambros h-Draxl, U. Hohenester,P. L. de Andr�es, and F. Flores, to be published (2002).[MMK+92℄ A. M. Milliken, S. J. Manion, W. J. Kaiser, L. D. Bell, and M. H.He ht, Phys. Rev. B 46 (1992), 12826.[NP58℄ P. Nozi�eres and D. Pines, Il Nuovo Cimento 9 (1958), 470.[Pap86℄ D. A. Papa onstantopoulos, Handbook of the Band Stru ture of El-emental Solids, Plenum, New York, 1986.[PN66℄ D. Pines and P. Nozi�eres, The Theory of Quantum Liquids, W. A.Benjamin, New York, 1966.[Pri95℄ M. Priets h, Physi s Reports 253 (1995), 164.[Pus02℄ Peter Pus hnig, A Theoreti al Des ription of the Ele troni and Op-ti al Properties of Conjugated Mole ular Crystals, Ph.D. thesis (tobe published), Karl-Franzens University of Graz, 2002.[PW92℄ J. P. Perdew and Y. Wang, Phys. Rev. B 45 (1992), 13244.[QF58℄ J. J. Quinn and R. A. Ferrell, Phys. Rev. 112 (1958), 812.

BIBLIOGRAPHY 55[RAG+98℄ K. Reuter, P. L. de Andr�es, F. J. Gar ��a{Vidal, D. �Sestovi� , F. Flo-res, and K. Heinz, Phys. Rev. B 58 (1998), 14036.[Reu98℄ Karsten Ulri h Reuter, Band Stru ture E�e ts in Ballisti Ele tronEmission Mi ros opy, Ph.D. thesis, Friedri h-Alexander-Universit�atErlangen-N�urnberg, 1998.[RHA+00℄ K. Reuter, U. Hohenester, P. L. de Andr�es, F. J. Gar ��a{Vidal,F. Flores, K. Heinz, and P. Ko evar, Phys. Rev. B 61 (2000), 4522.[Sin94℄ D. J. Singh, Planewaves, Pseudopotentials and the LAPW method,Kluwer A ademi Publishers, Boston/Dordre ht/London, 1994.[SK54℄ J. C. Slater and G. F. Koster, Phys. Rev. 94 (1954), 1498.[SKBE99℄ W. D. S h�one, R. Keyling, M. Bandi� , and W. Ekardt, Phys. Rev.B 60 (1999), 8616.[SL91℄ L. J. S howalter and E. Y. Lee, Phys. Rev. B 43 (1991), 9308.[Sla37℄ J. C. Slater, Phys. Rev. 51 (1937), 846.[Sla51℄ , Phys. Rev. 81 (1951), 385.[SLN98℄ D. L. Smith, E. Y. Lee, and V. Narayanamurti, Phys. Rev. Lett. 80(1998), 2433.[SSHS01℄ C. Strahberger, J. Smoliner, R. Heer, and G. Strasser, Phys. Rev. B63 (2001), 205306.

56 BIBLIOGRAPHY

A knowledgements \My spelling is Wobbly: It'sgood spelling, but it Wobbles,and the letters get in the wrongpla es." Winnie the PoohI would like to thank all the people who have been helping me along with thiswork and during my studies.. . . to Claudia Ambros h-Draxl, for providing the ex ellent at-mosphere within her group and for her en ouragement. . . toUlri h Hohenester, for o-advising and orre ting this work,supporting me in everything and tea hing me some \real" physi s. . . to Peter Pus hnig, for patiently and ompetently answeringmillions of questions on a daily basis and for being the mastermindof this diploma thesis. . . to Claudia Sifel, for supporting me in many ways, and for keep-ing me alive during the preparation of this work with regular food-rations. . . to Timo Thonhauser, J�urgen Spitaler, Heinz Auer, Bar-bara J�ager, and Kerstin Weinmeier for helping, supportingand, most of all, having fun. . . a Pedro de Andr�es, Fernando Flores y Peter Ko evar,gra ias por haberme dado la posibilidad de estudiar en Madrid yde ono er esa iudad realmente ex itante. . . a mis ompa~neros de Madrid, nun a olvidar�e mis d��as all�� gra iasa vosotros. . . and to my parents, for giving me the possibility to study withoutfeeling too mu h pressure 57

Documents

Con - UNIGRAZphysik.uni-graz.at/~uxh/diploma/ladstaedter.pdf · 2008-03-06 · Chapter 1 In tro duction Ov er man y decades the understanding of metals as protot ypical y-particle