JOU RNAL OF RESEARCH 01 the Nationa l Bureau of Standards - A. Physics and Chemistry Vol. 74A, No.3, May-June 1970

Beyond the One-Electron Approximation: Density of States for Interacting Electrons*

L. Hedin, B. I. Lundqvist, and S. Lundqvist

Chalmers University of Technology, Goteborg, Sweden

(October 10, 1969)

The concept " dens it y of states" can be given many different meanings wh e n we go beyond the o ne­ e lectron approximation. In this s urvey we concentrate on the de fin iti o n ti e d to excitati on processes, where one electron is added or re moved from the solid. We discuss the one-parti cle s pectral (uncti on for conduction a nd core e lectrons in me tals, how it c an be approx imately ca lc ulate d, and ho w it can be re­ late d to diffe re nt types of expe ri me nts like x-ray photoemi ssion, x-ray e mi ssion and absorpti on , photoemission and optica l absorpti on in th e ultraviolet, and the CO)nplOn effect. We a lso di scuss the form of the exc hange-corre lation pote ntia l for use in band structure calcu lations_

Key words: De nsity of states: interacting e lectrons; one- partic le Green function; oscillator stre ngths; quas i particle d ensity of states; x-ray e mission and absorption_

1. Introduction functions. The quantity which is most easy to define is the true density of states of the fully inte racting In the one-electron approximation the concept "elec- systems tronic density of states" is unique and simple and is defined by the formula N(w) = L o(w - EII)' (3) n

I (1) where Ell are the energy levels of the syste m. Although l being a key quantity in thermodynamics it is of no use for the description for e.g. optic al, x-ray or e nergy loss where Ei denotes the single-particle energies_ Similarly spectra. We rather need the proper extensions of eqs the optical density of states is gi ven by the joint density (1) and (2) to describe one- and two-electron properties of state for electrons and holes, thus of the system. A particularly simple case is that of parti­ cles having a Lorentzian energy distribution (2)

(4) These functions only describe the cruder aspects of experiments such as soft x-ray emission or optical ab­ The density of states is then given by sorption_ They must be augmented with the proper oscillator strengths, whic h provide important selection N J(w) = LAk(W). (5) '-" rules and as well give rise to deviations from the de ns ity k of state curves. I In cases where the line width r k is small compared with I When one goes beyond the one-electron approxima­ the width of the band, the formulas (5) and (1) obviously ~ tion and considers interactions not included in a self- will give similar results. consiste nt field theory the situation becomes non-trivial The simple case just mentioned does not really carry and one enco un ters a wide varie ty of density of states us beyond the one-electron theory. We have in th e general case to abandon a concept based on single-par­ * An invit ed paper presented at the 3d Material s Researc h Symposium , El.ectronic Density ojSt.ates, Nuve mber 3·6, 1969, Gai t h e r s Lur~. Md . ticle levels and instead use distribution fun ctions or 417 spectral densities of which the function A", (w) given dynamic interaction between the electrons and above is a simple and almost trivial case. Such distribu­ between the electrons and phonons. (' tion functions include the proper oscillator stre ngths, In section 4 we take up the core electron spectrum ;1 and they refer to formally exact many-electron states, and discuss calculations to first order in the dynamic properly weighted according to the physical process interaction with the valence electrons. We also survey considered. They are closely related to correlation the recent work by Langreth who obtained the exact functions and Green functions. An example is the solution to an important model problem. In section 5 we density-density correlation function (p(rt)p(r't')), give a qualitative disc ussion of some experimental \' which correlates density fluctuations at different times results for photon absorption and emission, photoemis- and different positions. This function is of basic im­ sion and Compton scattering. In section 6 finally we portance for describing optical experiments and energy give some concluding remarks. loss spectra. Unfortunately it is very difficult to analyze its structure a nd we will here instead concentrate on a 2. The One-Electron Density of States in an. / simpler entity, the one-particle Green function and its Interacting System and its Connection with X- -~ assoc iated s pectral function [1] . Ray Photoemission and with X-Ray Emission l The one-particle spectral function is a generalization and Absorption of the usual density of states N1(w). It gives an asymp­ In this section we introduce some theoretical techni· totically exac t description of x-ray photoemission, is calities, which however seem unavoidable in order to connected with x-ray emission and absorption and also establish a firm connection between theory and experi­ has some relevance for uv photoemission and absorp­ ment. tion. The one-electron spectral weight function is defined It is, however, not capable to describe edge effe cts in as general, and does not at all account for final state or A (x, x' ; w) = 2: i s (x)ls* (x')a (w - Es). (6) particle-hole interactions. Such effects, which in their simplest form are described by N2(w) or generalizations In the independent-particle approximation the quan­ therefrom provide intricate problems upon which we tities Is and Es are the one-electron wave fun ctions and shall only briefly touch here. These questions will be energies. In an interac ting system the oscillator ~ take n up in de tail in the lecture by Mahan. strength functionls is defined as It has recently been noticed that the one-electron spectral func tion should exhibit some marked and (7) strong structure. This structure is primarily associated with the interaction between the electron and plasmon where t{J(x) is the electron field operator. It gives the r excitations. It should be quite important in x-ray probability amplitude for reaching an excited state "'­ photoemission, influencing the line shape of the core IN - 1, s) , when an electron is suddenly removed from electron peak and giving low energy satellites both to the ground state IN). The quantity Es is the excitation the core electron and conduction band structures. It is energy important in x-ray emission and absorption spectra Es=E(N) - E(N -l,s). (8) showing up at the threshold, and giving rise to satellite structures. Its effect may also be noticeable in uv These definitions apply for w in eq (6) smaller than photoemission and optical absorption spectra. the chemical potential fl. For w larger than fl, states We should also mention the structure in the one-elec­ with N + 1 particles are involved. tron spectrum caused by coupling to phonons. This In applications of the theory it is often practical to structure is limited to a region of the order of the De bye represent the spectral func tion as a matrix in a space energy around the Fermi level, but in that region it is spanned by some suitable complete orthonormal set of quite pronounced, causing e.g. the well-known single·particle states flk(x), thus enhancement of the quasi particle density of states. In the next section we will disc uss the connections A kk , (w) = Jui (x)A (x, x'; w)uk,(x')dxdx' between the one-electron spectrum and x-ray photoemission, and with x-ray emission and absorption. =2: (Nl atIN - 1, s) (N s In section 3 we discuss the spectrum of the conduction electrons from calculations to lowest order in the 418 · The spectral weig ht function is the general di stribution taking eK=E. Using an average momentum matrix ele­ fun c ti on describing one-electron properties, and is a ment and neglecting nondiago nal terms in A we obtain :( generalization of the one-electron density matrix, which the simple result is obtained after an integration over the frequencies. Di stributions with respect to a single variable are in the (15) usual way obtained by summation over all other varia­ The neglect of nondiagonal terms s hould be a good bles. Thus, the one-electron density of states is defined approximation if the one-electron functions are care­ >, as fully chosen Bloch functions and core-electron func­ N(w) =Tr A (w) tions. We note that if the electron is ejected from a core level and if we neglect excitations of the core electron = jA(x, x; w)dx=LAH(w). (10) (13) k system, the operator air in eq must destroy a core electron (air = ad, giving As indicated, the definition is independent of the y' choice of matrix basis. W ~ L I(N:, sINv)p"" 128(w-E,,+Es). (16) The s pectral weight function is a key quantity in the K , S Green function formulation of the many-electron Here INvl denotes the ground state of the valence elec­ problem. A more detailed account is give n In many tron system, and IN,:, Sl are excited states in the texts, e.g. in secti ons 9 and 10 of ref. [1]. presence of the core hole. We recognize eq (16) as th e We next di sc uss the particular cases of x-ray result in the sudden approximation. " photoemi ssion and soft x-ray e mi ssion , where approxi­ We next turn to soft x-ray emi ssion. In this case we mate reductions to the one-electron spectrum can be may write the initial state as [2] made. In the x-ray photoe mi ssion experiment (XPS), an (17) energetic photon of energy w is absorbed, exciting a photoelectron which leaves the system. If the electron wh ere the star on N indicates that the valence electron has a large enough energy we can write the final state system is relaxed towards the core hole. Applying the 7 as Golden rule again we have for the x-ray intensity I'l'J ) = a~ IN -1, s ) , (11 ) I(w) - W L I ( 'l'JI L PIc'kat,akac IN* ) 12 . f klr' where K refers to a Bloch wave function of energy EK = 8(w-Ei+ Ef) h2k 2 /(2m), describing the photoelectron. The probabili­ ty of this event is , by the Golden rule, =w L I (Nv -l, sl L PCkalrlN: )128(w + Ec-Es). k (18) w ~ L I ('l'JI P I'l'i)1 28(w-EJ+ Ed J If we could neglect the relaxation effects in the initial = L I(N -1, sl a" L Pk" rat, akiN) 128(w-E" + Es), (12) state and replace INn by INv ), we would again have K ,S k ,II' the spectral function A involved. The relaxation effects where P is the total momentum and Pick" the momentum can be accounted for in an approximate way by only matrix element. For a fast electron, which is outside the considering particle-hole excitations, thus region of ground state flu ctuations, aKIN) = 0, and the IN: ) = (a+ L aKa;a,,) INv ). (19) expression for W reduces to II ,p

W ~ L I (N - 1, s I L PICk a,.. IN) 128(u.I - E" + Es) Insertion of this expression in eq (18) leads after some K ,S k further approximations to a very simple expression

= A Irk' (E" - w) P:;' PKk ' (13) L L I(w) ~ w L Ip~~12AkIr(w+ Ee), (20) K k ,k' k where peJf involves the coefficients a and is w-d epen­ The energy di stribution of photoelectrons IS hence give n by dent. Such an approximation is however invalid at the Fermi edge, where, as Anderson [3] has pointed out, J(E) - vELAk·k.'(E- W)P': P"k" (14) we need an infinite nu mber of particle-hol e pairs to /;" ,1." represent INv*). 419 The edge problem has recently been treated by sev­ electron density of states and have indicated its relation eral people [4-6]. We choose here to give a brief ac­ to some measurable quan tities. We conclude from this count of Langreth's version [6] of Nozieres; and de that a theoretical investigation of this kind of experi- '), Dominicis' (ND) treatment of the problem in order to ments can not avoid the calculation of the one-electron show how the one·electron spectrum e nters the density of states of interacting electrons, but at the problem. The basic quantity needed for the evaluation same time due to a variety of effects there may be es­ of eq (18) is the correlation function sential modifications of the density of states as it ap­ pears in the actual experiment. Explicit results from ap- ' proximate calculations of the spectral function A and '" the density of states N(w} are discussed in sections 3 To evaluate that function ND study the multiple time and 4, and the actual comparison with experiment is function made in section 5.

(22) 3. One-Electron Spectrum of Conduction r Electrons The simple way, in which the core state operator en­ ters their model Hamiltonia n, causes the equation of The common picture of the one-electron spectra of motion for F to close onto itself and allows a solution as solids is obtained from energy band calculations, in a product, which the Schrodinger equation of independent elec­ trons in a periodic potential is solved. For the different Fkk'(T,T';t,t')=4>kk'(T,T';t,t') GcCt,t') (23) bands one obtains the electron energy as a function of wave-vector, E(k). The main contribution to the poten­ where Gc is the core electron Green function, tial seen by an electron is the average electrostatic field or the Hartree field. When going beyond the one-elec­ GcCt, t')= (N*IT(ac( t)at (t'))IN* (24) > tron approximation the next step would be to include dynamical effects of the interaction as well as exchange and 4> a valence electron Green function which obeys effects. Such effects can not be described by an ordina- j an equation with a transient potential from the core ry local potential. A generalized "potential," non-local hole. The x-ray spectrum is given by a convolution of in space and time, must be introduced. the core and valence electron spectra. Both spectra are Let us for simplicity study the case, where specific singular at the edge as a power law and so is also the effects of the periodicity are of minor importance, and resulting convolution. assume the distribution of conduction electrons to be The point in keeping the operator ac in eq (22) is that uniform. Due to the non-locality of the generalized " the time dependence of ak(t) then can be given by the "potential," its Fourier transform to momentum-energy same Hamiltonian as that used for IN*), while when space will show a dependence on both wave-vector k studying (N:J'IT(ait)ak,+)INj) , the time evolution of ak(t) is given by a Hamiltonian that does not include the dnd energy E, L (k, E). The quantity L is called the self­ potential from the core hole, and thus the usual many­ energy of the electron. Adding the self-energy to the average potential, we I body techniques do not apply. X-ray absorption can be ~ described in an analogous manner by the function F, have to solve the equation only replacing IN*) by IN). E= E(k) + L (k,E). (25) It should be mentioned that the ND treatment is based on a mod el Hamiltonia n that does not include in­ The self-energy includes all the interactions between teractions between the conduction electrons. This is the electron state considered and the system. This quite appropriate for their purposes of treating the edge necessarily includes dissipative effects, which lead to problem but is not sufficient for the spectrum far away the decay of the state. Consequently the self-energy from the edge. It is not clear if their treatment can be must be a complex quantity, thus extended to the more general case. We then have to resort to other methods, such as the approximation in (26) eq (19) or to a frontal attack on the dielectric function itself [7] . The spectral weight function defined in section 2 is re­ We have in this section given a definition of the one- lated to the self-energy according to the formula 1, sec- 420 tion 1 1 - 1m E(g,W) Plasmon peak

Electron -hole pairs I In the simple case that the self-energy I is independent --~~~------~~------w of energy, th e s pectral function for fix ed k will have a 2q [ Lorentzian shape and II determines the width of the line. When I de pends on energy there are no a priori FIGURE 1. Qualitative behavior of1m ,,-t (q,w)for smallq. restri ctions on the shape ofthe spectrum. Most of our present knowledge about the self-energy LTH approximation the plasmon is undamped for has been based on calculations, where vertex cor­ wave-numbers smaller than a critical value kc, whi ch at rections have been neglected , i.e. using the formulae metallic de nsities is of the same order of magnitude as [9-13] kp . L (k, w) = (2:)4 J eiWSv( k ') c 1 (k', w ')Co(k A great part of the lite rature about the electron gas is devoted to the search for an improved di electric fun c­ + k', w + w')dk'dw', (28) tion [15 - 20] , particularly because of the failure of th e LTH formula to describe interacti on effect at short wh ere di stances. The last word re main s to be said in this question, but it is inte resting to note the relative insen­ Go(k,w)=(w - E(k)+io sign (k-kp)) - I (29) sibility of the self-energy to the choice of dielectri c fun cti on [11 ,12,21]' An a pproxi mati on, whi ch has is the propagator for a non-interactin g electron, and proved to be most useful, is to take a plasmon·1ke s harp absorption for all k according to the formula

is the bare Coul omb potential. The constant kfo' means · (31) the F ermi momentum, and 0 is a positive infinitesimal. To make connection with another approximation, we note that the Hartree-Fock approximation corresponds where Wp is the cl assical plasma frequency a nd w(k ) th e ) to the choice E = 1 in eq (28). As it stands, eq (28) ex­ resonance frequency at wave numbe r k. The frequency presses the lowest order coupling to the density fluctua­ w(k) reduces to Wp when I k l~O and is proportional to tions of the conduction electrons, described by the Ikl 2 for large Ikl. This formula correctl y re presents the wave -vector- and frequency-dependent dielectric func­ small and large w-limits and giv es a quite reasonable in­ tion E(k,w). This function contains information about terpolation for intermediate w values [1, 1]. With the static screening, given by E(k,O), but also important this choice one can perform the frequency integration .., effects of the dynamic behavior of the density fluctua­ in eq (28) and obtain tions. The typical s mall-k-behavior of the spectral func­ tion for this kind of excitations, -1m c l(k,w), is in­ (k w) = __1_ Jd3q v(q)no(k q) di cated in fi gure 1. The clearcut classification of the ex­ ~ + L.. ' (27T)3 E(q, E(u+q) -w) citations into electron-h ole pairs and plasmons is characteristi c for th e Linearized Time dependent Har- + Wp2 Jd3q v(q) 1 (32) ::> tree (LTH) or Random Phase Approximation [14]. For (27T)3 2w(q) w-E(h+q)-w(q) s mall wave-l engths the electron-hole pair excitations become of increasin g importance, while in the k~ 0- The first term in eq (32) is a screened exchange poten­ limit the plas mon exhaus ts completely the s um rule for tial, no(k) being the momentum di stribution for inde· 1m c l, i. e. it is the only excitatio n. The latter property pendent electrons, and the second term describes the ~ follow s from general arguments about charge conserva· correlation hole around the electron [12,22J . ti on and translation invariance [8, p. 288], and should Because of the plasma resonance in the di electric be valid for any u seful approximation for E(k,w). In the function there will be a rapid variation in the real part 421

378- 324 0 - 70 - 8 ------but a broad resonance with a sharp onset at w = E(k) +

L 2 3 Wp. ~ Figure 3 illustrates how the parabolic quasi particle >- 0 C> dispersion law is accompanied by a second branch of n:: w z -2 the spectrum, the plasmaron. In fi gure 4 the momen­ ~ lL. tum distribution function for the interacting electrons jj -4 is given. Vl The characteristic structure due to plasmon effects <{ 0.6 kr 1.0 kr 1.4 k, will modify the density of states. This is demonstrated [0.6 I schematically in figure 5. The conduction band will C> t Z·0.56 Z= 0.63 retain approximately its parabolic shape. At an energy ~ 0.4 bO.58 Wp below the Fermi edge there is the onset of the -' Z·027 ;;, 0.2 second band due to the plasmaron states. This band is tJ w (\ terminated at low energies with a rather distinct edge. ~ 0.0 For unoccupied states there is an extra contribution to -2 0 2 -2 0 2 -2 0 2 the density of states starting at an energy Wp above the ENERGY W/Wp ENERGY W/Wp ENERGY W/Wp Fermi level. Figure 6 gives a survey of the density of FIGURE 2. The seLf-energy ~ and the co rresponding spectraL function state curves at four different values of the electron gas A for fs = 5 [21]. The crossings be tween the Re ~ curves and the parameter rs (rs = 2 corresponding roughly to the elec­ straight Lines give the soLutions to the Dyson equation. The numbers at the peaks indicate the strengths of the Lines. tron density of AI, and rs= 4 to that of Na). So far we have discussed the gross effects due to the l coupling between electrons and plasmons. These are of ~ at the frequency w = E(k) + Wp for electrons and at quite well re presented by the plasmon-pole approxima­ W = E(k) - Wp for holes. This behavior of the electron tion for the dielectric function in eq (31). Electron-hole gas seems to have been first noticed and discussed by pair excitations are however not included, to re present Hedin et al. [23J and has been studied in great detail by Lundqvist [13,21J. Typical curves for ~ obtained with this dielectric function are shown in the upper half of figure 2. In the lower part of figure 2 we have given 4 the results for A(k,w) calculated from eq (27). For k = kf' there is only one strong peak in the spec· 3 tral function, corresponding to the usual quasi particle.

For the other k-values in the fi gure, we find three solu· '-. n. tions of the Dyson equation 3 2 I "­ W=E(k)+Lf{(k,w). (33) :3 01 One of them, however, falls at w = E(k) ± Wp , where the 0:: w damping is very strong, and therefore this solution is ef. Z w fectively suppressed. Of the two remaining solutions, O+-----~~------one corresponds to the usual quasi particle, i.e. a bare electron s urrounded by a cloud of virtual plasmons and electron· hole excitations. For hole states, i.e. for k < kF , -1 a new state appears which has an energy lower than I I that corresponding to a hole plus a plasmon, i.e. e(k)­ Wp. This result from a low order treatment corresponds '1 to a coherent state of hole-plasmon pairs, and may be thought of as holes coupled to real plasmons. This cou­ pled state, which has been called a plasmaron, has a o 1 large oscillator strength, and thus gives an essential MOMENTUM contribution to the sum rule for the one-electron spec· FIGURE 3. The spectraL weight function A( k,w) given by LeveLcurves trum. For electron states, k > kF , there is no sharp state indicating the vaLue offiwpA, at rs = 4 (fiwp = 0.435 Ry) [13]. 422 particle energy in the form

E(k) = E(k) + V(k), (34)

where V(k) can be interpreted as an effective exchange L. and correlation potential. In fi gure 9 suc h a potential is Vi iJ shown. It has a remarkably weak k-dependence for E moderate wave vectors, its value lying roughly halfway :J C between the Slater [25] and "2/3 Slater" values

FIGU RE 4. The momenllLm. distribution Junction n(k) at r.,= 2,3,4 and from eq (28) is practically the same as the Hartree value 5 [13] . E(kF)' a result which is in accord with the experimental findings. For large momenta (k ?: kl" + kc) there is a charac· One-elec tron teristic k-d e pendence of V(k), whi ch mi ght influe nce properties of electrons involved in photoemi ssion and )

Plasmaron edge

+ Energv w p Plasmon Fermi edge

FIGU RE 5. Density oj states Jar conduction electrons.

-4 -3 -2 -I o them we mu st turn to the LTH dielectric function or ENERGY <1<, hi gher approximations. These excitations are responsi· FIGURE 6. The density oj slates Jar the val ues oJthe electron gas ) ble for the broadening of the quasi particle peak and parameter r, = 2,3,4,5. The dashed curve is the result oJthe one· electron theo ry, and the vertical broken line indicates the Fermi level the Auger tail at the bottom of the main band in figures [13]. 5 and 6 as well as the broad spectral we ight contours in figure 3. 1.5,------, Figure 7 shows ~J(k,E(k)) calculated from eq (28) and is an indication of the high damping rate for quasi parti· cles with k greater than k,.. + kc, i.e., in the region where they can decay into plasmons [10,24]. However, as il· lus trated by the typical spectral forms for the quasi par·

ticle peak in figure 8, the quasi particle peak is always ] di stingui shable from the s pectral bac kground, its width 0.5 f(k) is s maller than its excitation energy E(k) -t-L. ,/,,_---1------From eq (2 8) one can also draw information about the exchange and correlation "potential" for the quasi par· " ticles. The inadequacy of the Hartree·Fock approxima· I J + 5 tion in a metal is well· known, and in band calculations k/kf - the effe cts of exchange and correlation are commonly FIGURE 7. The imaginary part oj the quasi·particle selfenergy 1m ~(k,E(k))/EF as a Junction oj the momentum k oj the electron at simulated by usi ng a local potential, such as the Slater diJJerent metallic densities (EF = 0.921,0.409,0.230, and 0.147 Ry at [25] or the Caspar [26,27] expression s. After solving f s = 2, 3, 4 and 5, respectively) . The dashed curve is Quinn's result at the Dyson equation one can write the resulting quasi· f s = 2 [10,13]. 423 electron-electron interaction , as derived from eq (28), 8.0 are small [11,12], e.g. ome/m = -.01 for Al and ome/m = .06 for N a, and the properties of the quasi particles close to the Fermi surface are dominated by the elec· tron-phonon interaction. The effects of the ele'ctron-phonon interaction on the quasi-particle dispersion la w follow in a straightforward way using perturbation theory in the Brillouin-Wigner fo rm, thus

{]O5 + fp J (35 ) E(k) - E(p)+w(k - p) . \ 0,, 0 In this equation gk- p is the matrix element for the elec­ (]O5 tron-phonon coupling and w(k-p) the phonon frequen­ c y. 0 , 0 5 The qualitative effect of the phonons is to flatten out the dispersion curve in the immediate neighborhood of the Fermi surface, and this gives rise to the enhance­ If . ment of the density of states (fig. 10), or, equivalently, 0 of the thermal effective mass and one obtains from eq CJ/Cr­ (35) FIGU RE 8. Typical spectralformsfor the quasi·particle peak of the mp,, = m 0 + '\) spectral weight function in diff erent momentum regions. The scaLes are different/or the three curves, the energy w is measured/rom the Fermi LeveLf.t , and the curves are drawn/or f s = 4 [24]. (36) ,

LEED experiments. It is not clear, howe ver, how much where No(E f ) is the density of states without electron­ of this structure that may be observed due to the short­ phonon interaction, Ip i = I kl = kF and the integration lifetime of the electrons at these energies. extends over the full solid angle. We want to stress again that the di scussion we have Ash croft and Wilkins [ 32J first calculated the cor­ given of the one-electron spectrum is based on the as­ rections for Na, Al and Pb using eq (36), and several sumption that vertex corrections are small. As similar calculations have been published over the last di scussed in the next section recent work by Langreth [29J shows that vertex corrections in the core electron problem can have a quite large effect on the form of satellite structures, while their effect on the quasi parti­ cle properties seems to be small. Preliminary investi ga­ -1 I- - tions by one of us (L.H.) show similar strong vertex ef­ "2/3 -Slater" fects on the conduction band satellite. The details of the plas maron structure should thus not be taken very ----- seriously. -2 / The quasi particle states close to the Fermi surface Slater are of particular interest due to their importance for --- thermal and trans port properties [30J. To study these problems the quasi particle density of states or density -3 of levels rather than the on e-electron density of states o 2 3 is of importance. Because of interactions the quasi-par­ ticle dispersion law is distorted, corresponding to the FIGU RE 9. Exchange-correLation potentiaL/or an eLectron gas at f s = well-known mass enhancement at the Fermi surface. 4 co mpared with the SLater and the Gtispcir(2/3 SLater) approximations The corrections to the free-electron mass m due to the [28]. 424 electron depends only weakly on th e state of th e outer electrons. The energy levels on the oth er hand are shifted by an appreciable amount compared to the cor­ -- N responding atomic levels, typically of the order of 5 to bs 10 e V, which is large on the scale of valence electron energies but is a small relative change in the energy of a core electron. The core shifts can be measured accu­ rately by the method of x-ray photoemission spectrosco­ -4 -2 o 2 4 pyas well as by x-ray absorption and inelastic scatter­ ing of fast electrons. FIGURE 10. Density of qltasi·particle levels for Na at T = oaK [31]. The shift of the quasi particle energy of a core elec­ few years. The most accurate values of A are those tron comes partly from changes in the average Cou­ deduced from tunneling data by McMillan and Rowell lomb field, partly from polarization effects. The [33] . For exa mple, the two methods give the values A Coulomb shift is due to the different valence charge dis­ = 0.49 and 0.38, respectively, for AI; A = 1.05 and 1.3, tribution relative to that in a free atom. It generally respectively, for Pb. results in a decrease of the binding energy. The It is obvious that the enhancement varies with tem­ poLarization shift co mes from th e relaxati on of the perature and that no enh ancement is left at hi gh tem­ valence charge di stribution around th e hole created peratures, where the phonon system behaves like a when we remove the electron. The valence electrons flu ctuating classical medium_ are drawn in towards the positive hole in the ion. This Similarly to the case of electron-electron interaction effect decreases the binding energy by half of the the electron-phonon interacti on gives a characteristi c change in the Coulomb potential calculated at the core structure to the spectral function_ Engelsberg and site, i. e. precisely the amount obtained if we calculate Schrieffer [34] pointed out th at in th e neighborhood of the self-energy of th e hole usin g electrostatics. The the F ermi surface the quasi particle picture will no shift in the core energy thus contains information about longer apply_ They calculated the spectral function for the valence electron distribution and polarizability, an Einstein model and for a Debye spectrum and ob­ measured with the core electron as a probe. Theoreti cal tained a spectral function with a very complex struc­ calculations for simple metals (Li , Na, K, AI) are in very ture in the region close to the Fermi s urface_ For sodi­ good agreement with the experime ntall y observed um no dramatic effects occur because of the rather peaks in the XPS spectra [35] . weak electron-phonon interaction [ 31]. In analogy with the strong effects of the interacti on We conclude this section by noting that inclusion of between particles and plasmons previously di scussed, dynamical and exchange effects, in particular con­ there is a corresponding coupling between a hole in the sideration of the electron-electron interaction tells us core and the density flu ctuations of the co nduction why the one-electron theory works so well in explaining electrons. This leads to a strong structure in th e core many gross features of metals and what limitations it electron spectrum [36]. has. However , at leas t in a low-order treatment there We assume for simplicity that we can neglect the are also ne w structures introduced by the interaction, spatial extension of the core electron wave function. schematically characterized as due to the resonant Calculation of the self-energy to lowest order in the coupling between electrons and plasmons. In section 5 screened interacti on gives the formula we will discuss possibilities to observe this structure. L (E) = - (2~)4 J d 3qdwv(q) [E - 1 (q, w) -1] 4. One-Electron Spectrum of Core Electrons 1 (37) '7 The preceding discussion has emphasized the strong W-E+En+io ' effects of the electron-plas mon coupling on the spec­ trum of conduction electrons. Similar strong effects where En is the core quasi-particle energy. Reme mber­ occur in the spectra of core electron s. For simplicity we ing that c 1 (q, w) has a strong resonance in the plasmon limit the di scussion to simple metals with small cores, regime, we see that after integration over the frequen­ I so that the core electrons can be physically distin­ cy, the self-energy will show a resonance behavior in ~ gui shed from the valence electrons and be well local- the energy region E = En - Wp • This rapid variation will ized to a particular ion. The wave function of a core give rise to two solutions of the Dyson equation. The 425 cle state and approximately the same strength corresponds to high excitations of the conduc­ tion electrons as described by the broad satel­ lite structure. 0.8 As discussed in section 2 the spectral function has the form (cf. eq (16»

0.6 <{. Q. 3 and shows a powerlaw singularity at the Fermi edge. o. The first order theory correctly predicts a singularity but of a somewhat different form, namely (wln 2w) - t. The core electron spectrum can be obtained exactly if we use a simple model Hamiltonian 0.2 H = w+a+ aa+ L gq(b q + b ~ Q) + L wqbtbq, (39) Q Q Here a+ is the creation operator for a core electron of energy E and bq + the creation operator for a plasmon of -3 -1 o energy Wq. The exact solution of this problem has been given by Langreth [29], and we now give a brief ac­ FIGURE 11. Quasi-particle peak in the spectral function for a core electron and the associate plasmon satellite structure for different count of his work, which also shows the close resem­ densities of the conduction electrons, measured by the electron gas blance of the problem to the Miissbauer or the impurity­ parameter r" [36]. phonon problem. The self-energy l in this model is given by the sum second solution, however, gives a quite broad peak in of all diagrams where the core electron is dressed with the spectral function. The results for different densities plasmons (fig. 12). For the true Hamiltonian including of the electron gas are illustrated in figure 11. The spec­ all electron-electron interactions we have the same set trum is measured from the shifted quasi particle ener­ of diagrams, where the plasmon propagator is replaced gy, i.e. the zero of energy corresponds to complete by a screened interaction v(q) E- t (q,w). Except for the relaxation of the electrons around the hole. first diagram the bare Coulomb potential gives no con­ We summarize the characteristic features of the core tribution (the hole propagates in only one direction) and spectrum we can thus replace E- t by (E - t - 1). By choosing the ) a. A large polarization shift of the core quasi parti­ dielectric function in eq (31) and the coupling I cle level. b. The shape of the spectrum is independent of gl = v( q)wll(2wq), (40) the core level considered. This results from neglecting the actual size of the core wave the plasmon diagrams and the screened interaction dia­ function. grams become the same. The core Green function c. A pronounced satellite structure, which starts Gc(t) = - i( T(a(t) a+(0))) (41) at W = - Wp and has a broad peak. d. An extended tail on the low energy side of the can be written as (cf. eq (38» quasi-particle peak. This tail is due to the coupling between the hole and screened elec­ tron-hole pair excitations, and corresponds to states of the whole system involving two holes, where 10) is the plasmon vacuum state and fI is the one in the core and one in the conduction band Hamiltonian for the plasmon in the presence of the core plus one excited electron above the Fermi sea. e. There is an appreciable reduction of the spec­ tral strength of the quasi particle. Approxi­ mately half of the spectral strength cor­ ~=2~1 +2~1"'" responds to excitations close to the quasi parti- FIGURE 12. Diagramsfor the core electron selfenergy. 426 hole, AC(W)j H= 2:gq(bq+ b ~ q) + 2: wqbtbq (43) q II Quasi- I Satellites particle ~ A canonical tran sformation shifts the zero point of the I plasmon vibrations, thus I I I (44) I I / ____ " W ~ where " /wp

-2 -1 a q q

(45) F,GU RE 13. Comparison of the first order (dotted line) and th e exact result (full curve)for the core spectralfunctionfrom the model ) We may hence write Hamiltoni(tn in eq (3 9). fn this model the quasi-particle peak is a /l. function. The dash.ed curve indicates a more realisticfonn of that (0 I eif1t I0) = (0 Ie - AeiHoteA I° )e - i~

+ 2\ 2: nfq~O(W-E - LlE+ Wq+Wq') + ... } (50) 5. Qualitative Discussion of Some Experiments • qq' To co mpare the first order results (the first diagram in We shall discuss some differe nt types of experiments fi g. 12) with the e xact solution we have used the simple utilizing the connections with the one-electron spec­ ~ di spersion law W (' = Wp + q2 (in units of the Fermi e nergy trum discussed in section 2 and the results from the ap­ and the F ermi momentum), which allows an analytic proximate calculations re ported in sections 3 and 4. We solution. The res ult for the electron density of sodi um can really not put forward much more than guesses metal (rs= 4) is given in fi gure 13. about where many-body effects may possibly occur. Since the dielectric fun ction in eq (31) contains no The difficulty is that the predictions by the one-electron particle-hole pairs, th e quasi particle peak is a o-func- approximation have seldom been worked out in enough 7 tion. A more reali stic shape of the peak is indicated in detail to give reliable level densities and matrix ele­ the fi gure. The exact solution has structure also at ments, and this knowledge is required both to evaluate - 3wp, -4wp, etc, which is not shown in the figure_ the many-body effects per se as well as to find out how This s tructure, however, has weak edges and carries much of the experimental structure that is accounted only a few percent of the oscill ator strength. for by the one-electron approximation. Also the experi­ Comparing the results of the first order calculation mental data are sometimes not as accurate and reLable 1 and the exact solution we note: as one would need. Thus s urface conditions are often 1. The energy s hift AE is exactly the same. not under good enough control, bac kground effects are 427 poorly known and disturbing secondary effects are not limits the approximate validity to the light metallic ele­ carefully analyzed and subtracted. ments such as Li, Be, Na, Mg, Al and K and excludes The discussion will neglect the effects of final state e.g. all transition metals. interactions between the electron and hole. This means Through the recent work by Nozieres and de that the treatment is a simple extension of the usual Dominicis and others [ 4·6J the possibility of a singular theory for interband transitions in which the density of structure at the Fermi edge seems to be well states for electrons and holes in band theory is replaced established. The magnitude of this Fermi edge peak by the corresponding quantities including many-elec· and the influence of this effect on the intensity at ener· tron interactions as illustrated in fi gures 6, 11, and 13. gies outside the immediate vicinity of the edge is how· Although certainly of restricted validity it seems that ever so far unsettled, althou gh important progress has the predictions of such an approach are worthwhile to been made [38]. As regards the main band it is clear summarize. that the presence of the core hole will give an enhance· ment of the intensity [IJ. The ac~ual magnitude of this 5.1. X-Ray Photoemission (XPS) enhancement factor and its variation over the main band is so far not well·known. To obtain the one·elec· This experiment has a fairly c1earcut relation to the tron density of states in the main band from experi· one· electron spectrum. Ideally the energy distribution ments seems to require b oth careful calculations of of photoelectrons will be given by eq (I S). However, this dipole matrix elements and better estimates of the equation is valid only if the photoelectrons leave the enhancement factor. solid without being scattered. Structure due to satel· Below the main band the edge for plasmon produc· lites in the density of states will thus be mixe d with tion is well establi shed experimentally [1,39] . The in· structure due to energy losses. tensity of the satellite structure is strongly affected by The losses to volume excitations are proportional to intricate cancellation mechanisms due to the presence the sample thickness, while the intensity of the satellite of the core hole and the theoretical predictions are un· structUFe in the one·electron spectrum has a definite certain. The plas maron edge (cf. figs. 5 and 6) has been relation to the intensity of the quasi particle peak. Thus searched for in AI, but the experiment was not conclu· by varying the sample thickness one should be able to sive [40]. As discussed in section 3, it is not clear separate the two kinds. of processes. whether this edge is a feature of an exact solution of the Theory predicts an asymmetric form of the core· problem or just a result of the low order treatment. A quasi particle peak [4,5,6,29,36,37] and satellite struc· clear experimental confirmation or di smissal would be ture starting at Wp below the main peak. According to of great aid for the further study of these many· body ef­ the Langreth model solution there s hould be two satel· fects. lites also >xith ~n asymmetric line shape (cf. fi g. 13), while the first order theory predicts the satellite in 5.3. X-Ray Absorption fi gure 11. There should be a satenite structure in the conduction halil.d, as well. Even if the exact shape of Consideration of these experiments requires a treat­ this structure might differ from the results of the low· ment of final state interactions but in the absence of a order theory discussed in section 3, the total intensity detailed theory for these w e shall here take a simple of the satellite band should be appreciable. point of vi ew and treat the s tructure in the one-electron 1 An experimental verification of the many-body struc· spectra as additional levels or groups of levels in a one­ ture would be of great aid for the further development electron scheme. A similar discussion of plasmon ef­ of the theory. As regards the position of the core levels fects in x-ray absorption in metals was given by Ferrell there seems to be a good agreement between theory [41] . Due to the presence of satellite structure in both and experime nts but further experimental and theoreti· core and conduction band spectra there should be a cal work on this problem would help to clarify how characteristic structure above the threshold [36] . point defects polarize and distort their surroundings. Accurate x-ray absorption spectra for simple metals have recently [42] bee n obtained, which show an edge 5.2. Soft X-Ray Emission anomaly very similar to what one may expect from the Mahan exciton effect [4-6]. The fine structure of the We consider only the simplest possible case where absorption coeffi cient for the LII,l1l transition in mag­ we can assume complete relaxation of the Fermi gas nesium is shown in fi gure 14 [43]. Immediately above around the hole before the emission takes place. This the edge there is more detailed structure which 428 density_ The position of the plasmaron peak should be given better at higher densities and the di screpancy with experiment at lower densities is in no way alarm­ ) ing. More serious is the fact that Langreth's calculation L1IESCA) Mg has shown the higher order effects to be strong. The strong peak could of course also be due to many-body effects involving final-state interactions.

5.4_ Photoemission in the Ultraviolet ) The basic m echanisms of the photoemission process are still not well understood. It may be regarded in one extreme as a pure surface effect and in another as a > pure volume effect. In the latter case we have to ac­ count for the very important inelastic scattering effects of the outgoing photoelectrons. If the surface effects are not too strong, and if we can sort out the inelasti­ cally scattered electrons, the pnotoemission results in the ultraviolet s hould reflect structure due to the satel­ lite band of the conduction electrons_ There is so me 80 85 90 95 leV) hint of suc h a s tructure in the recent results for cesium 70 90 100 110 120 1 0 leV) [46J (fig. 15) at an e nergy of about 4 eV below the Fermi edge. However, the structure could also be due to a loss FIGURE 14. The/ine st ructure of the absorption coefficientfor the LII .", transition in m.agnesium [43]. of two s urface plasmons [46J.

possibly co uld be due to.ordinary band structure effects in the final state. At still higher energies comes a stron g peak [43-45] _ • We will argue that this peak may be a many-body ef­ llw=I02eV fect. In table 1 we give results for the posi tions of the peaks measured from the threshold [43J , and compare o with the position of the core plasmaron peak di scussed E (; c: )- in section 3 (cf. fig. 11). There is good agreement for c: .2 I aluminum but not for the other metals. z o The importan t point, however, is that Epeak/ Wp is a i= ::J very smooth function of the electron density_ ActuaUy m n: f-­ its value is closely 0_8 Ts in all cases. This indicates that (f) the peak is associated with properties of the electron o >­ gas rather than being due to oscillator strength effects. t?n: w Z The latter would be connected with the properties of w the ion core and there is then no obvious reason to ex­ z o n: pect a regular variation with the conduction electron f-­ u W ...J W f2 TABLE 1. Position of the strong peak in soft x-ray absorption and o :r: comparison with the location of the satellite peak (€pm) in 0.. figure J 1.

Element rs Epeak, eV Epeak/Wp o 234 567 8 9 KINETIC ENERGY leV) AI...... 2.07 24 1.6 1.7 Mg...... 2.66 22 2.1 1.8 FIGURE 15. Photoelectron energy distribu.tion curves at hw = 10.2 eV Na...... 4.00 18 3.2 2.1 for Na, K, Rb and Cs [46]. The horizo ntal bars indicate the values of t the s urface (full curve) and volume (dashed) plas mons. 429 5.5. Optical Absorption in the Ultraviolet one-dimensional momentum distribution [53J. From recent measurements of the Compton profiles of Li, Na J The description of optical properties of metals in and Al the linear momentum distribution has been terms of Drude (see e.g. refs 47 ,48) and interband con­ deri ved [54J . The result for Li is shown in figure 16. ~,

tributions is often in qualitative agreement with experi­ In section 3 approximate values of the momentum I ment. To go beyond that description we need to know distribution for an electron gas have been shown (fig. 4). the dielectric fun ction including final state interactions_ The corresponding linear momentum distribution The many-body effects may show up as changes in in­ shows a too small reduction compared to the free-elec­ tensity and as new structures_ Attempts to account for tron case to reproduce the experimental results for Li the former have been made e.g. by Mahan [49J, who (fig. 16) and AI, while for Nathe electron gas curve falls -C considered the contributions from virtually exchanged almost entirely within the experimental region. plasmons. Absorption caused by electron gas effects As band structure effects could be expected to be has been considered by Hopfield [50J. He observed more important in Li and Al than in Na, this kind of cor­ that while the electron gas by itself cannot absorb rection has also been calculated using the OPW radiation the effect of a weak perturbing potential from method [55J. As shown in figure 16 these effects phonons or disorder is enough to provide the necessary reduce the discrepancy, even if a quantitative agree­ momentum conservation for the absorption process and ment has not been obtained. th us allow the plasmon resonances of the electron gas This kind of experiment provides a way of illuminat­ to show up. ing another aspect of the distribution of electrons in A straightforward way to extend the one-electron metals and provides a useful way of checking theoreti­ joint density of states expression is to make a convolu­ cal models including many-body interactions. tion of the spectral weights of occupied and unoccupied y states. This has been done for semiconductors by Bar­ 6. Concluding Remarks dasis and Hone [51J who in addition considered vertex corrections. They obtained improved agreement with This paper has presented a discussion of the possible experiment. Calculations for metals by the convolution nature of many-electron effects on the density of states. approximation indicate the existence of a plasmon-in­ It is based on a study of the one-electron spectrum in­ duced structure at photon energies above En + Wp [52J , cluding interactions and points out the existence of where En is the interband threshold energy. There are characteristic satellite structure in the density of states • some experimental indications of structure beyond the of electrons and holes in simple metals. Considering ordinary interband absorption in this energy region. the joint density of states of electrons and holes as in the theory of interband transitions certain predictions 5.6. The Compton Effect about possible effects in x-ray and optical spectra can be made. All this material is however only qualitative X-ray scattering from an electron gas in the regime J and tentative. The structure in the one-particle spec- i of large momentum transfer is a direct measure of the trum has been calculated in low order and considerable Ifp) (0.") changes may result by including higher-order effects. Further, the convolution of the electron and hole spec- trum implies neglecting the final state interaction (iJ between the electron and the hole. The final state in- , teractions may partly cancel out the structure in the "\., electron and hole spectrum, and it leads to charac­ teristic new effects such as edge singularities, and of course also gives an overall distortion of the spectrum. With all these reservations, however, the discussion 2 points out the existence of a number of possible in­ teresting effects which offer a challenge for further stu­ dy. In assessing the possibility of pursuing this ap­ proach to obtain quantitative theoretical results one has to consider critically the present state of the art with re­ p (!4,u) gard to ordinary band theory. Indeed, rather little has FIGURE 16. The linear momentum distributionfor Li [54,55]. yet been done in a quantitative way to calculate spectra 430 especiaUy with regard to oscillator strengths. Such [26] Gaspar, R., Acta Ph ys. Hung. 3, 263 (1954). ,. more detail ed knowledge from energy band calcula· [27] Kohn , W. , and Sham, L. J. , Ph ys. Re v. 140, A11 33 (1965). tions also forms a necessary prerequisite for making [28] Hedin, L. , Lundqvist, B. l. , and Lundqvisl, S. (10 be publi s hed). [29] Langreth , D. C. (to be published). ) qua ntitative state me nts about many·electron effects. [30] e.g., Pines, D. , and Noz ieres, P. , "The Th eory of Quantum Liquids," Vol. I, (Benjamin , New York , 1966). 7. References [31] Grimvall, G., J. Phys. Che m. Solid s 29, 1221 (1968); Phys. Kon· dens. Materie 6, 15 (1967). [1] For a general re vi ew, see L. Hedin and S. Lundqvist, Solid [32] Ashcroft, N. W., and Wilkins, J. W. , Ph ys . Leit ers 14, 285 State Phys ics 23, (F. Seitz, D: Turnbull and H. Ehrenreich, (1965). Editors (Academic Press, New York, 1969). [33] McMillan, W. L., and Rowell, J. M., in "Superconductivity," ) [2] Hedin , L. , in "Soft X·Ray Spectra and the Electronic Structure R. D. Parks, Editor (Dekker, Inc., New York, 1969). of Metals and Mate rials," D. Fabian, Editor (Academi c Press, [34] Engelsberg, 5., and Schrieffer, J. R. , Phys. Rev. 131, 993 New York, 1968). (1963). [3] And erso n, P. W., Phys. Rev. Letters 18,1049 (1967). [35] Hedin , L., Ark. Fys. 30, 231 (1965). [4] e.g., Mahan, C. D. , Ph ys. Rev. 163,612 (1967): Bergersen, B., [36] Lundqvist, B. I., Phys. Kondens. Materie 9,236 (1969). > and Brouers, F. , J. Phys. Chem. 2, 65 1 (1969); Mizu no, Y. , and [37] Doniach, S., a nd Sunji c, M. (to be published). Ishikawa, K., J. f'hys. Soc. 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