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Lattice vibration effects on electronic structure and photoemission
Solid State Communications, Vol. 88, No. 10, pp. 789-794, 1993. Printed in Great Britain.
0038-1098/93 $6.00 + .00 Pergamon Press Ltd
LATTICE VIBRATION EFFECTS ON ELECTRONIC STRUCTURE AND PHOTOEMISSION I. Delgadillo,* H. Gollisch and R. Feder Theoretische Festk6rperphysik, Universit/it Duisburg, 47048 Duisburg, FRG
(Received 27 August 1993, accepted 6 September 1993 by P.H. Dederichs) A theoretical approach to the electronic structure of solids at finite temperature has been developed on the basis of the adiabatic approximation. For any given temperature, correlated ion core displacement configurations on a large cluster are determined, which are consistent with experimental phonon dispersion relations. The electron Bloch spectral function, i.e. a generalized band structure, and photoemission intensities are obtained by a tight-binding recursion method for each configuration followed by a configurational average. Calculations for the 3d-band region of Cu yield the vibration-induced hole lifetime broadening (typically about 0.35 eV at 900 K) and real energy shifts (up to 4-0.1 eV). Due to a compensation by thermal lattice dilation, the d-band width changes only marginally with T. Comparison with experimental photoemission data from Cu(1 1 0) shows good agreement for the temperature dependence of peak height, line shape and energy shift. 1. INTRODUCTION TEMPERATURE effects in angle-resolved photoemission spectroscopy are - despite a fairly long history of study - not yet understood in a quantitatively satisfactory manner (cf. e.g. [1-5] and references therein). There have been mainly two types of theoretical approaches (see review in [4]): (a) a combination of a fight-binding initial bulk state with a plane-wave or augmented-plane-wave final state, as proposed by Shevchik [6], and (b) a one-step-model involving matrix elements between layer-KKR halfspace initial and final states (with the latter corresponding to time-reversed LEED states), as developed by Larsson and Pendry [7]. In both approaches the temperature dependence enters via Debye-Waller factors, i.e. correlations between the atomic vibrations are neglected. For the underlying electronic structure - the initial state in photoemission - more sophisticated many-body treatments of the electron-phonon system have been put forward [8, 9], which yield in particular the complex electron self-energy correction due to the electron-phonon interaction. This "thermal renormalization" of one-electron energies * Present address: Depto. de Fisica, Centro de Investigacion y de Estudios Avanzados del IPN, Apdo. Postal 14-740, 07000 Mexico, D.F.
in metals has been analyzed in more detail within the adiabatic approximation using second-order perturbation theory [10], but to our knowledge numerical results (well below the Fermi energy) have so far not been obtained. In this work, we investigate formally and numerically the temperature dependence - due to lattice vibrations and static lattice expansion - of the occupied electronic states and its implications for photoemission spectra. Within the adiabatic approximation, the one-electron Green function is evaluated in a tight-binding treatment of large clusters with static ion displacement configurations, which are consistent with experimental phonon spectra. Vibrational correlations are thus fully taken into account. Appropriate matrix elements of this Green function yield the total density of states, the Bloch spectral function and angle-resolved photoemission spectra. We have implemented this method numerically and applied it to the case of the 3d-band region of Cu. In particular, we obtain thermal shifts and broadenings of photoemission features in good agreement with experiment. 2. THEORY In the adiabatic approximation, the treatment of the electronic structure and of photoemission of a thermally vibrating crystalline system decomposes into two parts: firstly, the determination of static
789
790
LATTICE VIBRATION EFFECTS
"snapshots" of the ion core positions, and secondly, the calculation of the electronic properties for each of these ion core configurations followed by an average over them. At a given temperature T, in the harmonic approximation the displacement vectors ut of ion cores from their equilibrium lattice positions R ° are for crystals with one atom per unit cell - characterized by the well-known equal-time correlation function (cf. e.g. [11]) (u/. Ul,)r = N - - M Z exp (iq. [R° qj
Wqj
(1) where N is the number of ion cores in the normalization volume, M the ionic mass, w~- the dispersion relation of t h e j t h phonon mode, and hqj the average number of phonons in this mode
hqj = 1/{exp[hwqj/(ksT)]- 1}.
(2)
For infinite cubic crystals, the entire correlation information is given by "neighbour shell" correlation coefficients Q. defined as
Q~
(Ul'UI,(I,v))r for all l,
=
(3)
where l'(l, v) refers to the vth nearest neighbours of site R °, and Q0 = < u l ' u t > r is the mean square deviation. For a cluster (with periodic boundary conditions) consisting of N atoms and a given set of ut we define for a given set of displacements ut - equivalent order parameters -
P . = I/(NN.) Z u , . I
Zut,(t,~),
(4)
1'
where N. is the number of atoms in the vth neighbour shell. "Correct" displacement configurations {ut}, which are characterized by the correlation function given by equation (1) can then be obtained by finding the minimum of the quadratic form
F=
G,(Q.
Z
-
p.)2
(5)
//
subject to the condition E ut = 0. The c, are suitable weight coefficients. Starting from some random configuration {at} with P0 = Q0, a "correct" configuration {ut} is determined iteratively by a molecular-dynamics method. We now proceed to calculations - for nonmagnetic elemental d-band metals - the electronic structure and photoemission intensities. In a nonrelativistic tight-binding approximation including s, p
Vol. 88, No. 10
and d bands we require the Hamiltonian matrix elements nilmi,l, m, between orbitals lm and I'm' on sites R i = R ° :~ ui. To obtain these matrix elements, we first determine the corresponding matrix elements n~ilm,i,i,m, for the equilibrium configuration {R ° } by means of a tight-binding fit to a band structure obtained from first principles by a self-consistent LMTO calculation. The modification of the interatomic matrix elements due to the displacements u i is then approximated via a parametrization (cf. [12]) involving a dependence on the interatomic distances d as d -5, d - 3 / 2 and d -2 for the (d, d), (d, sp) and (sp, sp) matrix elements, respectively. For a given displacement configuration, the total density of states and the Bloch spectral function are subsequently obtained from the appropriate diagonal Green function matrix elements, which are evaluated by means of Haydock's recursion scheme [13]. Finally, these quantities are averaged over a set of displacement configurations. For photoemission, we employ a bulk interband transition model. Following [14], the photocurrent in the direction k is obtained as the diagonal element of the initial-state Green function with respect to the state A. plk), where A is the magnetic vector potential of the photon field, p the momentum operator and [k) a bulk final state. Strictly, the latter should also be calculated for the individual displacement configurations. But since its damping is dominated by electron-electron processes, we choose to ignore at present its modification by phonon processes. For the numerical application of the above method to Cu, we make the following specifications. The phonon dispersion relations w,d, which are required as input for the correlation function [equation (1)], are obtained from the usual harmonic-approximation dynamical matrix after modelling the force constant matrix empirically from elastic constants [15]. Anharmonic effects are approximately taken into account in the "quasi-harmonic approximation" as follows. For a given T, the set {R °} is taken as the equilibrium configuration of the thermally expanded lattice at T. Phonon spectra and thence displacement configurations are calculated on the basis of elastic constants appropriate for this T. The cubic clusters are taken to consist of 4 x 83 = 2048 atoms. The number of configurations found sufficient for convergence is typically five. Hole lifetime effects are easily incorporated in the recursion calculations via an imaginary self-energy part, which leads to a Lorentzian broadening of the spectra. Physically, the hole lifetime is determined by electron-hole scattering and by electron-phonon scattering. Since the latter is already inherent in our adiabatic-approximation approach, we explicitly
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Fig. 1. Bloch spectral function in the d-band region (with energies given relative to EF) of Cu for k along F - X with relative values as indicated in the individual panels, for temperatures T = 0 (thin solid lines) and T = 900K with (thick solid lines) and without (dashed lines) thermal lattice expansion. All spectra include a (hole lifetime) imaginary self-energy part of 0.02eV. The T = 0 curves have been reduced by a factor 10 to facilitate comparison. The dotted lines are spectra simply obtained by convoluting the T = 0 spectra by a Gaussian of 0.35 eV F W H M . The inset shows the corresponding usual T = 0 band structure. model only the former by an imaginary self-energy part, which we for simplicity assume - in the d-band range of Cu - as a constant between 0.02 and 0.04 eV. 3. RESULTS AND DISCUSSION As a typical example, we show in Fig. 1 the Bloch spectral function ImG(E,k) for Cu with k along F - X. For T = 0, the spectral function exhibits sharp peaks (broadened by a small hole life-time of 0.02 eV
791
due to Auger-type processes), which are seen to reflect the conventional (non-relativistic) band structure shown in the inset. In particular, at F the two spectral function peaks near -3.1 eV and -2.4 eV correspond to the points 1-'25, and 1"12, respectively, in the band structure. At X, the four peaks between -5.5 eV and -1.SeV correspond to the points Xl, X2,, X2 and Xs. For T -- 900 K, the peaks of the spectral function (bold solid lines in Fig. 1) are seen to be broadened, asymmetrical in shape and displaced in energy. Since we assumed only 0.02eV for the electron-hole interaction contribution to the hole lifetime, the overwhelming contribution to the observed broadening of about 0.35 eV can be ascribed to the electron-phonon interaction, i.e. technically speaking to the corresponding imaginary electron self-energy part. The energy shift and the shape asymmetry are emphasized by comparing with the (dotted) curves obtained by naively representing thermal effects by convoluting the T - - 0 spectra with a Gaussian of 0.35eV F W H M . The energy shift is produced jointly by thermal expansion of the equilibrium lattice constant and by the lattice vibrations. To isolate the effect of the latter, we show in Fig. 1 also spectra obtained with T = 900 K lattice vibrations and the T = 0 lattice constant (dashed lines). The amount of the energy shift of the these spectral function peaks is very small for FEy, i.e. near the centre of the d-bands. For lower/ higher energies, it becomes increasingly larger towards negative/positive values. This broadening of the d-band region with temperature becomes plausible if one regards the spectral function as arising from an incoherent superposition of contributions from regions with lattice constants reduced and enhanced relative to T = 0 equilibrium value. Due to the d -5 dependence of the d-band interatomic matrix elements (see Section 2), the Hamiltonian becomes biassed in favour of reduced lattice constant configurations, i.e. in favour of a widening of the d-band. The amount of the vibration-induced energy shift is seen to be about -0.15eV for XI, and about 0.1 eV for 2"5. A particularly large shift (of about 0.25 eV) is found for the sp-F~ke A 1 band, which for T = 0 is at E = -1.75 eV for k = 0.7. The temperature dependence of the spectral peak broadening and of the energy shifts for selected d-band points together with the change of the d-band width at X is shown in more detail in Fig. 2. Thermal lattice expansion by itself is seen to reduce the d-band width, as one would qualitatively expect. From 300 K to 900 K, the d-band gets narrower by about 0.15 eV. As can be seen from the two upper panels of Fig. 2, the corresponding energy shift is strongest for the bottom of the d-band (Xl point) and almost
792
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Fig. 2. Thermal effects on the 3d-bands of Cu: Increase of peak widths (FWHM) at selected points (top panel) and shifts in energy (relative to the T = 300 K values) as functions of T for topmost Xpoint (Ks) (second panel), lowest X-point (X0 (third panel), A 2, at k = 0.5 (fourth panel) and d-band width (bottom panel). In panels two to five the symbols connected by dashed, by thin solid and by thick solid lines represent the effects of static thermal lattice expansion, lattice vibrations, and both together, respectively.
Vol. 88, No. 10
negligible at the top (Ks point). This shift of the dband centre towards higher energy is due to a readjustment of the Fermi energy, which is determined by the (also) narrowed sp-band. As already discussed above and evident from Fig. 2, lattice vibrations by themselves widen the d-band, i.e. they tend to compensate the narrowing due to lattice expansion. Since both effects are not strictly linear in T, their combination (filled circles the bottom panel of Fig. 2) can result in both narrowing (relative to the T = 300K value) (most notable for T = 0 and T = 600 K) and widening (above 650 K). Our results compare - in trends and orders of magnitude - favourably with an experimental determination of the temperature dependence of the Cu dbands between 300 and 675 K by means of photoemission [1]. For the increase of the peak widths (FWHM), i.e. twice the increase of the imaginary selfenergy part, a temperature-broadening coefficient of about 10-4eVK -1 is reported in [1]. Although the broadening is actually not linear in T, it is interesting to note that this agrees well with our result for X5 between 300 and 450 K (see top panel of Fig. 2), but is by a factor of about 2 smaller for higher T up to 675 K. On the other hand, the top panel of Fig. 4 of [1] suggests an increase of the FWHM of very roughly 100meV from 300 to 675K, which matches quite well our finding of 85 meV. The energy of the upper d-band was found in [1] to shift upwards by 45-65 meV from 300 to 675 K, while we find 64eV. For the change of the d-band width, a reduction of 19 4- 7.5 meV is obtained in [1] near the L point. Since our d-band width is taken at X, we cannot strictly compare, but our reduction by 7 meV for 600 K and expansion by 11 meV for 675 K are not far off. If we define the positions of the asymmetrically broadened peaks not by their maxima but by their centres of gravity, we get instead -11 and 0meV, which is closer to experiment. As is evident from the bottom panel of Fig. 2, these near-zero values arise from the compensation of the about 100meV d-band narrowing due to lattice dilation by about 100 meV widening due to lattice vibrations. To make contact with a recent high-resolution temperature-dependent photoemission experiment [5,16], we have calculated normal photoemission intensities from Cu(1 10) for h w = 16eV radiation linearly polarized with the magnetic vector potential A parallel to the [1-10] direction. In this geometry, nonrelativistic dipole selection rules (cf. e.g. [17] p. 205 and references therein) require the initial state (of the perfectly ordered lattice) to be of symmetry t y p e s )"~4 and S 4 for k along £(E)K and U(S)X, respectively. For the photon energy hLv = 16eV, the
LATTICE VIBRATION EFFECTS
Vol. 88, No. 10
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Energy (eV) Fig. 3. Normal photoemission from Cu(1 10) by 16eV photons at temperatures (in K) as indicated. Top panel: theoretical spectra obtained for fully correlated atomic vibrations (thick lines) and for uncorrelated vibrations (thin lines), without imaginary self-energy part V/and without Gaussian broadening. Central panel: theoretical spectra for correlated vibrations including an electron-hole contribution V,. = -0.04 eV and a Gaussian broadening of 0.09 eV FWHM. Bottom panel: experimental spectra by WiLrtenberg et al. [5] (shifted in energy such as to match the T = 155K peak position, which is not specified in [5], with theory and earlier data). initial state suitable for a direct bulk interband transition is of type $4 with k close to the X2 point. As is shown by the calculated T = 0 results (the dotted vertical line in the top panel of Fig. 3 and including the hole lifetime plus an "experimentalresolution" Gaussian broadening of 0.09 eV (according to [5]) - the dotted spectrum in the central panel), the binding energy of this initial state is about -1.75eV. While the experimental data [5] do not give its binding energy, earlier photoemission data [18] place this state at about -2.2 eV. This difference is - in addition to a small spin-orbit correction - mostly due to the well-
793
known fact (cf. [19,20]) that self-consistent LMTOcalculations using the standard Hedin-Lundqvist exchange-correlation approximation yield C u d bands about 0.4eV ("self-energy correction") too high in energy compared to photoemission. This correction is, however, found to have practically no influence on the thermal lattice vibration effects, to which we now turn our attention. With increasing temperature, the calculated photoemission spectra (top panel of Fig. 3) change along the lines already encountered for the Bloch spectral function (see Fig. 1, especially around the X 2 point): asymmetric broadening and shift in energy corresponding to the imaginary and real parts of the electron-phonon self-energy - together with a reduction in height. In particular, we obtain peak widths (FWHM) of 0.085eV and 0.193eV at T = 432K and 668K, respectively, i.e. imaginary self-energy parts of 0.0425 eV and 0.0965 eV. To see the influence of correlations between the atomic vibrations, we have in addition calculated corresponding spectra (thin lines in the top panel of Fig. 3) ignoring these correlations, i.e. taking into account only the on-site order parameter P0 [cf. equation (4) ]. The peaks for T = 432 K and T = 668 K are seen to be about 40% broader and slightly shifted towards higher energy, as is qualitatively plausible for reduced order. The temperature behaviour of the (correlatedvibration) spectra is preserved if an electron-hole interaction contribution to the hole lifetime and the experimental energy resolution are taken into account (central part of Fig. 3). Comparison with the experimental spectra measured at T = 155 K and T = 668 K [5] (bottom panel of Fig. 3) shows very good agreement, except for some deviation in the low-energy tails, which experimentally include some inelastic background, and a small extra peak around -1.42 eV in the T--- 155 K data, which is produced by spin-orbit coupling (which we neglected in the present calculations). 4. CONCLUSION The quantitative validity of our calculations of the temperature dependence of the electronic d-band structure of Cu has been demonstrated by the level of agreement reached with experimental photoemission data. The electron-phonon contribution to the electron self-energy has been found to be non-linear in temperature and to depend substantially both on energy and k. Its imaginary part attains values up to 0.2eV at 900K. Its real part, i.e. energy shifts of spectral peaks, reaches up to +0.1 eV near the top and up to -0.1 eV near the bottom of the d-band. The
794
LATTICE VIBRATION EFFECTS
corresponding widening of the d-band is, however, effectively compensated by thermal lattice expansion. Vibrations and expansion jointly lead to an almost uniform shift of the d-bands (typically about 70 meV relative to the T -- 0 positions for T = 600 K) towards the Fermi energy. Our results suggest interesting consequences for systems with only partially filled d-bands, i.e. 3d transition metals and especially 3d-ferromagnets. In particular, at elevated temperatures a sizeable electron-phonon contribution to the imaginary part of the electronic self-energy may render the quasiparticle concept invalid in the vicinity of the Fermi energy. Acknowledgements - This work was funded by the German DAAD and by the Mexican CONACyT.
7. 8. 9. 10. 11.
12. 13.
14. 15.
REFERENCES 1. 2. 3.
4. 5. 6.
J.A. Knapp, F.J. Himpsel, A.R. Williams and D.E. Eastman, Phys. Rev. B19, 2844 (1979). R. Courths & S. Hiifner, Physics Reports 112, 53 (1984). R.C. White, C.S. Fadley, M.~ Sagurton, P. Roubin, D. Chandesris, J. Lecante, C. Guillot & Z. Hussain, Phys. Rev. B35, 1147 (1987). P.O. Nilsson, L. Ilver, H.I. Starnberg & D.S.L. Law, J. Phys. C.M. 1, 6159 (1989). J. Wiirtenberg, W. Kuch, S. Langsdorf, E. Dietz & U. Gerhardt, Physics Scripta 41, 634 (1990). N.J. Shevchik, Phys. Rev. B16, 3428 (1976); 20, 3020 (1978).
16. 17. 18. 19. 20.
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C.G. Larsson & J.B. Pendry, J. Phys. C14, 3089 (1981). E.G. Maksimov, Sov. Phys. JETP 42, 1138 (1976). P.B. Allen, Phys. Rev. B18, 5217 (1978). P.B. Allen & V. Heine, J. Phys. C9, 2305 (1976). A.A. Maradudin, Chapt. 1 in Dynamical Properties of Solids, Vol. 1 (Edited by G.K. Horton & A.A. Maradudin), North Holland, Amsterdam (1974). W.A. Harrison, Electronic Structure and the Properties of Solids, Freeman and Co., San Francisco (1980). R. Haydock, in Solid State Physics: Advances in Research and Applications (Edited by F. Seitz, D. Turnbull and H. Ehrenreich), Vol. 35, Academic Press, New York (1980). R. McLean & R. Haydock, J. Phys. C10, 1929 (1977). Landolt-Boernstein, Numerical Data and Functional Relationships in Science and Technology, New Series III/2 (Editor in Chief: K.H. Hellwege), Springer, Berlin, Heidelberg and New York (1969). J. Wiirtenberg, Dissertation, Universit~it Frankfurt (1987). R. Feder, in Polarized Electrons in Surface Physics (Edited by R. Feder), World Scientific, Singapore (1985). E. Dietz & F.J. Himpsel, Solid State Commun. 30, 235 (1979). R. Courths, H. Wern, U. Hau, B. Cord, V. Bachelier & S. Hiifner, J. Phys. F14, 1559 (1984). S.V. Halilov, H. Gollisch, E. Tamura & R. Feder, J. Phys. CM 5, 4711 (1993).
0038-1098/93 $6.00 + .00 Pergamon Press Ltd
LATTICE VIBRATION EFFECTS ON ELECTRONIC STRUCTURE AND PHOTOEMISSION I. Delgadillo,* H. Gollisch and R. Feder Theoretische Festk6rperphysik, Universit/it Duisburg, 47048 Duisburg, FRG
(Received 27 August 1993, accepted 6 September 1993 by P.H. Dederichs) A theoretical approach to the electronic structure of solids at finite temperature has been developed on the basis of the adiabatic approximation. For any given temperature, correlated ion core displacement configurations on a large cluster are determined, which are consistent with experimental phonon dispersion relations. The electron Bloch spectral function, i.e. a generalized band structure, and photoemission intensities are obtained by a tight-binding recursion method for each configuration followed by a configurational average. Calculations for the 3d-band region of Cu yield the vibration-induced hole lifetime broadening (typically about 0.35 eV at 900 K) and real energy shifts (up to 4-0.1 eV). Due to a compensation by thermal lattice dilation, the d-band width changes only marginally with T. Comparison with experimental photoemission data from Cu(1 1 0) shows good agreement for the temperature dependence of peak height, line shape and energy shift. 1. INTRODUCTION TEMPERATURE effects in angle-resolved photoemission spectroscopy are - despite a fairly long history of study - not yet understood in a quantitatively satisfactory manner (cf. e.g. [1-5] and references therein). There have been mainly two types of theoretical approaches (see review in [4]): (a) a combination of a fight-binding initial bulk state with a plane-wave or augmented-plane-wave final state, as proposed by Shevchik [6], and (b) a one-step-model involving matrix elements between layer-KKR halfspace initial and final states (with the latter corresponding to time-reversed LEED states), as developed by Larsson and Pendry [7]. In both approaches the temperature dependence enters via Debye-Waller factors, i.e. correlations between the atomic vibrations are neglected. For the underlying electronic structure - the initial state in photoemission - more sophisticated many-body treatments of the electron-phonon system have been put forward [8, 9], which yield in particular the complex electron self-energy correction due to the electron-phonon interaction. This "thermal renormalization" of one-electron energies * Present address: Depto. de Fisica, Centro de Investigacion y de Estudios Avanzados del IPN, Apdo. Postal 14-740, 07000 Mexico, D.F.
in metals has been analyzed in more detail within the adiabatic approximation using second-order perturbation theory [10], but to our knowledge numerical results (well below the Fermi energy) have so far not been obtained. In this work, we investigate formally and numerically the temperature dependence - due to lattice vibrations and static lattice expansion - of the occupied electronic states and its implications for photoemission spectra. Within the adiabatic approximation, the one-electron Green function is evaluated in a tight-binding treatment of large clusters with static ion displacement configurations, which are consistent with experimental phonon spectra. Vibrational correlations are thus fully taken into account. Appropriate matrix elements of this Green function yield the total density of states, the Bloch spectral function and angle-resolved photoemission spectra. We have implemented this method numerically and applied it to the case of the 3d-band region of Cu. In particular, we obtain thermal shifts and broadenings of photoemission features in good agreement with experiment. 2. THEORY In the adiabatic approximation, the treatment of the electronic structure and of photoemission of a thermally vibrating crystalline system decomposes into two parts: firstly, the determination of static
789
790
LATTICE VIBRATION EFFECTS
"snapshots" of the ion core positions, and secondly, the calculation of the electronic properties for each of these ion core configurations followed by an average over them. At a given temperature T, in the harmonic approximation the displacement vectors ut of ion cores from their equilibrium lattice positions R ° are for crystals with one atom per unit cell - characterized by the well-known equal-time correlation function (cf. e.g. [11]) (u/. Ul,)r = N - - M Z exp (iq. [R° qj
Wqj
(1) where N is the number of ion cores in the normalization volume, M the ionic mass, w~- the dispersion relation of t h e j t h phonon mode, and hqj the average number of phonons in this mode
hqj = 1/{exp[hwqj/(ksT)]- 1}.
(2)
For infinite cubic crystals, the entire correlation information is given by "neighbour shell" correlation coefficients Q. defined as
Q~
(Ul'UI,(I,v))r for all l,
=
(3)
where l'(l, v) refers to the vth nearest neighbours of site R °, and Q0 = < u l ' u t > r is the mean square deviation. For a cluster (with periodic boundary conditions) consisting of N atoms and a given set of ut we define for a given set of displacements ut - equivalent order parameters -
P . = I/(NN.) Z u , . I
Zut,(t,~),
(4)
1'
where N. is the number of atoms in the vth neighbour shell. "Correct" displacement configurations {ut}, which are characterized by the correlation function given by equation (1) can then be obtained by finding the minimum of the quadratic form
F=
G,(Q.
Z
-
p.)2
(5)
//
subject to the condition E ut = 0. The c, are suitable weight coefficients. Starting from some random configuration {at} with P0 = Q0, a "correct" configuration {ut} is determined iteratively by a molecular-dynamics method. We now proceed to calculations - for nonmagnetic elemental d-band metals - the electronic structure and photoemission intensities. In a nonrelativistic tight-binding approximation including s, p
Vol. 88, No. 10
and d bands we require the Hamiltonian matrix elements nilmi,l, m, between orbitals lm and I'm' on sites R i = R ° :~ ui. To obtain these matrix elements, we first determine the corresponding matrix elements n~ilm,i,i,m, for the equilibrium configuration {R ° } by means of a tight-binding fit to a band structure obtained from first principles by a self-consistent LMTO calculation. The modification of the interatomic matrix elements due to the displacements u i is then approximated via a parametrization (cf. [12]) involving a dependence on the interatomic distances d as d -5, d - 3 / 2 and d -2 for the (d, d), (d, sp) and (sp, sp) matrix elements, respectively. For a given displacement configuration, the total density of states and the Bloch spectral function are subsequently obtained from the appropriate diagonal Green function matrix elements, which are evaluated by means of Haydock's recursion scheme [13]. Finally, these quantities are averaged over a set of displacement configurations. For photoemission, we employ a bulk interband transition model. Following [14], the photocurrent in the direction k is obtained as the diagonal element of the initial-state Green function with respect to the state A. plk), where A is the magnetic vector potential of the photon field, p the momentum operator and [k) a bulk final state. Strictly, the latter should also be calculated for the individual displacement configurations. But since its damping is dominated by electron-electron processes, we choose to ignore at present its modification by phonon processes. For the numerical application of the above method to Cu, we make the following specifications. The phonon dispersion relations w,d, which are required as input for the correlation function [equation (1)], are obtained from the usual harmonic-approximation dynamical matrix after modelling the force constant matrix empirically from elastic constants [15]. Anharmonic effects are approximately taken into account in the "quasi-harmonic approximation" as follows. For a given T, the set {R °} is taken as the equilibrium configuration of the thermally expanded lattice at T. Phonon spectra and thence displacement configurations are calculated on the basis of elastic constants appropriate for this T. The cubic clusters are taken to consist of 4 x 83 = 2048 atoms. The number of configurations found sufficient for convergence is typically five. Hole lifetime effects are easily incorporated in the recursion calculations via an imaginary self-energy part, which leads to a Lorentzian broadening of the spectra. Physically, the hole lifetime is determined by electron-hole scattering and by electron-phonon scattering. Since the latter is already inherent in our adiabatic-approximation approach, we explicitly
LATTICE VIBRATION EFFECTS
Vol. 88, No. 10 I
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Fig. 1. Bloch spectral function in the d-band region (with energies given relative to EF) of Cu for k along F - X with relative values as indicated in the individual panels, for temperatures T = 0 (thin solid lines) and T = 900K with (thick solid lines) and without (dashed lines) thermal lattice expansion. All spectra include a (hole lifetime) imaginary self-energy part of 0.02eV. The T = 0 curves have been reduced by a factor 10 to facilitate comparison. The dotted lines are spectra simply obtained by convoluting the T = 0 spectra by a Gaussian of 0.35 eV F W H M . The inset shows the corresponding usual T = 0 band structure. model only the former by an imaginary self-energy part, which we for simplicity assume - in the d-band range of Cu - as a constant between 0.02 and 0.04 eV. 3. RESULTS AND DISCUSSION As a typical example, we show in Fig. 1 the Bloch spectral function ImG(E,k) for Cu with k along F - X. For T = 0, the spectral function exhibits sharp peaks (broadened by a small hole life-time of 0.02 eV
791
due to Auger-type processes), which are seen to reflect the conventional (non-relativistic) band structure shown in the inset. In particular, at F the two spectral function peaks near -3.1 eV and -2.4 eV correspond to the points 1-'25, and 1"12, respectively, in the band structure. At X, the four peaks between -5.5 eV and -1.SeV correspond to the points Xl, X2,, X2 and Xs. For T -- 900 K, the peaks of the spectral function (bold solid lines in Fig. 1) are seen to be broadened, asymmetrical in shape and displaced in energy. Since we assumed only 0.02eV for the electron-hole interaction contribution to the hole lifetime, the overwhelming contribution to the observed broadening of about 0.35 eV can be ascribed to the electron-phonon interaction, i.e. technically speaking to the corresponding imaginary electron self-energy part. The energy shift and the shape asymmetry are emphasized by comparing with the (dotted) curves obtained by naively representing thermal effects by convoluting the T - - 0 spectra with a Gaussian of 0.35eV F W H M . The energy shift is produced jointly by thermal expansion of the equilibrium lattice constant and by the lattice vibrations. To isolate the effect of the latter, we show in Fig. 1 also spectra obtained with T = 900 K lattice vibrations and the T = 0 lattice constant (dashed lines). The amount of the energy shift of the these spectral function peaks is very small for FEy, i.e. near the centre of the d-bands. For lower/ higher energies, it becomes increasingly larger towards negative/positive values. This broadening of the d-band region with temperature becomes plausible if one regards the spectral function as arising from an incoherent superposition of contributions from regions with lattice constants reduced and enhanced relative to T = 0 equilibrium value. Due to the d -5 dependence of the d-band interatomic matrix elements (see Section 2), the Hamiltonian becomes biassed in favour of reduced lattice constant configurations, i.e. in favour of a widening of the d-band. The amount of the vibration-induced energy shift is seen to be about -0.15eV for XI, and about 0.1 eV for 2"5. A particularly large shift (of about 0.25 eV) is found for the sp-F~ke A 1 band, which for T = 0 is at E = -1.75 eV for k = 0.7. The temperature dependence of the spectral peak broadening and of the energy shifts for selected d-band points together with the change of the d-band width at X is shown in more detail in Fig. 2. Thermal lattice expansion by itself is seen to reduce the d-band width, as one would qualitatively expect. From 300 K to 900 K, the d-band gets narrower by about 0.15 eV. As can be seen from the two upper panels of Fig. 2, the corresponding energy shift is strongest for the bottom of the d-band (Xl point) and almost
792
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Vol. 88, No. 10
negligible at the top (Ks point). This shift of the dband centre towards higher energy is due to a readjustment of the Fermi energy, which is determined by the (also) narrowed sp-band. As already discussed above and evident from Fig. 2, lattice vibrations by themselves widen the d-band, i.e. they tend to compensate the narrowing due to lattice expansion. Since both effects are not strictly linear in T, their combination (filled circles the bottom panel of Fig. 2) can result in both narrowing (relative to the T = 300K value) (most notable for T = 0 and T = 600 K) and widening (above 650 K). Our results compare - in trends and orders of magnitude - favourably with an experimental determination of the temperature dependence of the Cu dbands between 300 and 675 K by means of photoemission [1]. For the increase of the peak widths (FWHM), i.e. twice the increase of the imaginary selfenergy part, a temperature-broadening coefficient of about 10-4eVK -1 is reported in [1]. Although the broadening is actually not linear in T, it is interesting to note that this agrees well with our result for X5 between 300 and 450 K (see top panel of Fig. 2), but is by a factor of about 2 smaller for higher T up to 675 K. On the other hand, the top panel of Fig. 4 of [1] suggests an increase of the FWHM of very roughly 100meV from 300 to 675K, which matches quite well our finding of 85 meV. The energy of the upper d-band was found in [1] to shift upwards by 45-65 meV from 300 to 675 K, while we find 64eV. For the change of the d-band width, a reduction of 19 4- 7.5 meV is obtained in [1] near the L point. Since our d-band width is taken at X, we cannot strictly compare, but our reduction by 7 meV for 600 K and expansion by 11 meV for 675 K are not far off. If we define the positions of the asymmetrically broadened peaks not by their maxima but by their centres of gravity, we get instead -11 and 0meV, which is closer to experiment. As is evident from the bottom panel of Fig. 2, these near-zero values arise from the compensation of the about 100meV d-band narrowing due to lattice dilation by about 100 meV widening due to lattice vibrations. To make contact with a recent high-resolution temperature-dependent photoemission experiment [5,16], we have calculated normal photoemission intensities from Cu(1 10) for h w = 16eV radiation linearly polarized with the magnetic vector potential A parallel to the [1-10] direction. In this geometry, nonrelativistic dipole selection rules (cf. e.g. [17] p. 205 and references therein) require the initial state (of the perfectly ordered lattice) to be of symmetry t y p e s )"~4 and S 4 for k along £(E)K and U(S)X, respectively. For the photon energy hLv = 16eV, the
LATTICE VIBRATION EFFECTS
Vol. 88, No. 10
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Energy (eV) Fig. 3. Normal photoemission from Cu(1 10) by 16eV photons at temperatures (in K) as indicated. Top panel: theoretical spectra obtained for fully correlated atomic vibrations (thick lines) and for uncorrelated vibrations (thin lines), without imaginary self-energy part V/and without Gaussian broadening. Central panel: theoretical spectra for correlated vibrations including an electron-hole contribution V,. = -0.04 eV and a Gaussian broadening of 0.09 eV FWHM. Bottom panel: experimental spectra by WiLrtenberg et al. [5] (shifted in energy such as to match the T = 155K peak position, which is not specified in [5], with theory and earlier data). initial state suitable for a direct bulk interband transition is of type $4 with k close to the X2 point. As is shown by the calculated T = 0 results (the dotted vertical line in the top panel of Fig. 3 and including the hole lifetime plus an "experimentalresolution" Gaussian broadening of 0.09 eV (according to [5]) - the dotted spectrum in the central panel), the binding energy of this initial state is about -1.75eV. While the experimental data [5] do not give its binding energy, earlier photoemission data [18] place this state at about -2.2 eV. This difference is - in addition to a small spin-orbit correction - mostly due to the well-
793
known fact (cf. [19,20]) that self-consistent LMTOcalculations using the standard Hedin-Lundqvist exchange-correlation approximation yield C u d bands about 0.4eV ("self-energy correction") too high in energy compared to photoemission. This correction is, however, found to have practically no influence on the thermal lattice vibration effects, to which we now turn our attention. With increasing temperature, the calculated photoemission spectra (top panel of Fig. 3) change along the lines already encountered for the Bloch spectral function (see Fig. 1, especially around the X 2 point): asymmetric broadening and shift in energy corresponding to the imaginary and real parts of the electron-phonon self-energy - together with a reduction in height. In particular, we obtain peak widths (FWHM) of 0.085eV and 0.193eV at T = 432K and 668K, respectively, i.e. imaginary self-energy parts of 0.0425 eV and 0.0965 eV. To see the influence of correlations between the atomic vibrations, we have in addition calculated corresponding spectra (thin lines in the top panel of Fig. 3) ignoring these correlations, i.e. taking into account only the on-site order parameter P0 [cf. equation (4) ]. The peaks for T = 432 K and T = 668 K are seen to be about 40% broader and slightly shifted towards higher energy, as is qualitatively plausible for reduced order. The temperature behaviour of the (correlatedvibration) spectra is preserved if an electron-hole interaction contribution to the hole lifetime and the experimental energy resolution are taken into account (central part of Fig. 3). Comparison with the experimental spectra measured at T = 155 K and T = 668 K [5] (bottom panel of Fig. 3) shows very good agreement, except for some deviation in the low-energy tails, which experimentally include some inelastic background, and a small extra peak around -1.42 eV in the T--- 155 K data, which is produced by spin-orbit coupling (which we neglected in the present calculations). 4. CONCLUSION The quantitative validity of our calculations of the temperature dependence of the electronic d-band structure of Cu has been demonstrated by the level of agreement reached with experimental photoemission data. The electron-phonon contribution to the electron self-energy has been found to be non-linear in temperature and to depend substantially both on energy and k. Its imaginary part attains values up to 0.2eV at 900K. Its real part, i.e. energy shifts of spectral peaks, reaches up to +0.1 eV near the top and up to -0.1 eV near the bottom of the d-band. The
794
LATTICE VIBRATION EFFECTS
corresponding widening of the d-band is, however, effectively compensated by thermal lattice expansion. Vibrations and expansion jointly lead to an almost uniform shift of the d-bands (typically about 70 meV relative to the T -- 0 positions for T = 600 K) towards the Fermi energy. Our results suggest interesting consequences for systems with only partially filled d-bands, i.e. 3d transition metals and especially 3d-ferromagnets. In particular, at elevated temperatures a sizeable electron-phonon contribution to the imaginary part of the electronic self-energy may render the quasiparticle concept invalid in the vicinity of the Fermi energy. Acknowledgements - This work was funded by the German DAAD and by the Mexican CONACyT.
7. 8. 9. 10. 11.
12. 13.
14. 15.
REFERENCES 1. 2. 3.
4. 5. 6.
J.A. Knapp, F.J. Himpsel, A.R. Williams and D.E. Eastman, Phys. Rev. B19, 2844 (1979). R. Courths & S. Hiifner, Physics Reports 112, 53 (1984). R.C. White, C.S. Fadley, M.~ Sagurton, P. Roubin, D. Chandesris, J. Lecante, C. Guillot & Z. Hussain, Phys. Rev. B35, 1147 (1987). P.O. Nilsson, L. Ilver, H.I. Starnberg & D.S.L. Law, J. Phys. C.M. 1, 6159 (1989). J. Wiirtenberg, W. Kuch, S. Langsdorf, E. Dietz & U. Gerhardt, Physics Scripta 41, 634 (1990). N.J. Shevchik, Phys. Rev. B16, 3428 (1976); 20, 3020 (1978).
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C.G. Larsson & J.B. Pendry, J. Phys. C14, 3089 (1981). E.G. Maksimov, Sov. Phys. JETP 42, 1138 (1976). P.B. Allen, Phys. Rev. B18, 5217 (1978). P.B. Allen & V. Heine, J. Phys. C9, 2305 (1976). A.A. Maradudin, Chapt. 1 in Dynamical Properties of Solids, Vol. 1 (Edited by G.K. Horton & A.A. Maradudin), North Holland, Amsterdam (1974). W.A. Harrison, Electronic Structure and the Properties of Solids, Freeman and Co., San Francisco (1980). R. Haydock, in Solid State Physics: Advances in Research and Applications (Edited by F. Seitz, D. Turnbull and H. Ehrenreich), Vol. 35, Academic Press, New York (1980). R. McLean & R. Haydock, J. Phys. C10, 1929 (1977). Landolt-Boernstein, Numerical Data and Functional Relationships in Science and Technology, New Series III/2 (Editor in Chief: K.H. Hellwege), Springer, Berlin, Heidelberg and New York (1969). J. Wiirtenberg, Dissertation, Universit~it Frankfurt (1987). R. Feder, in Polarized Electrons in Surface Physics (Edited by R. Feder), World Scientific, Singapore (1985). E. Dietz & F.J. Himpsel, Solid State Commun. 30, 235 (1979). R. Courths, H. Wern, U. Hau, B. Cord, V. Bachelier & S. Hiifner, J. Phys. F14, 1559 (1984). S.V. Halilov, H. Gollisch, E. Tamura & R. Feder, J. Phys. CM 5, 4711 (1993).