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Theoretical Aspects OP Electron Damping
84
THEORETICAL ASPECTS OF ELECTRON DAMPING
I. Bartoe
Institut d'Electronique, USTHB, Alger + +permanent address: Institute of Physics, Czech.Acad.Sci, Prague
The surface sensitivity of LEED and of electron spectroscopies of surfaces is caused by the interaction among electrons, which is particularly strong for electrons with energies of the order of tens till hundreds of elec tronvolts. In LEED, where only the elastically back scattered electrons are studied, their trajectory through a crystal cannot be too long as otherwise an inelastic collision takes place which removes the electron from its elastic channel. Thus only electrons propagating, not too deep under the surface can contribute to the detected signal. In photoemission, for the same reasons, only electrons excited from atoms close enough to the surface can reach the surface without any loss of their energy. Inelastic collisions of electrons, being responsible for the loss of investigated electron from the elastic channel, cause the effective absorption or damping of the electron flux. This central factor of surface sensitivity is usually treated in a highly simplified manner however: by adding an imaginary constant contribution to the real potential to obtain the optical potential. This is in a marked contrast to the real part of the potential in which crystalline periodicity is fully respected. The simplified description appeared suffioient for surface crystallography by LEED and ..... for determining of electron dispersion relations E(k) by angular resolved photoemission. For subtler effects, and, in particular at very low energies the damping should be treated more carefUlly. Complicated many-electron problem can be reduced to the oneelectron problem if only the elastic chennel of the incident/excited electron is investigated [1] • The effective potential, so called optical potential, is no more real; its imaginary component is responsible for the electron absorption (damping) in the elastic channel. This can be illustrated on a simple example of the electron diffraction, described by a Hamiltonian with just a constant pure imaginary optical potencial Voi [1] • The eleotron wave function is exponentially damped, and the uncertainly re-
lation between energy and time leads to a broadening: any structure the r/v protile has to be broader than Voi• Density ot electron states in crystals. Let us begin the investigation with the simplest optical potential, i.e. with real periodic potential complemented by a constant imaginary part representing homogeneous electron damping in the crystal. In perfect crystals without damping, electron states characterised by wave vectors k are stationary. This is no more true when imaginary component is added to the potential. The damping ot the electron wave tunction in time leads to the energy broadening ot the electron state. Which is best represented by partial densities ot states in the k-space: discrete d-tunctions ot the undamped. crystal become broadened when imaginary component is introduced into the crystal potential [2] • The optical potential with just a constant imaginary part represents electron damping which is homogeneous throughout the crystal. This is only the zeroth approximation because valence electrons, which are mainly responsible tor inelastio oollisions ot low energy eleotrons, are in tact distributed non-unitormly throughout the crystal. Valence electrons are distributed with the periodicity ot the orystal lattioe; thus, also the imaginary part of the opti.. cal potential has this periodicity and can be expanded into the Fourier series. Let us investigate the consequences ot the nonhomogenity of the damping on a simple onedimensional model with the Hamiltonian H. (1) H • T + (Vor + i Voi) + (Vgr + iVgi) cos gx. The non-homogeneoWi damping, introduced by Vgi = 0, brings assymetry into the protile of the partial density of states (Fig.1). The real and imaginary parts of the optical potential can, in lim! ting cases, be either in phase (Vgi Vgr > 0) or out-ofphase (Vgi Vgr < 0). The real part of the potential determines the space localization of electrons: bonding states below the gap are concentrated predominantly in the regions olose to the potential minima whereas antibonding states (above the gap) have maimal ampl1tudes at potential maxima. The minima of the imaginary component of the potential, giVing effective inorease
86
~g, --002
-10
Fig. 1: Partial densities of states of the nearly free electron model at the Brillouin Zone boundary. The symmetrical two-peak profile for homogeneous damping (V dotoi=O.1; ted line) becomes asymmetrical when inhomogeneous damping is added (Vgi = 0)
COPPER EXITED ELECTRON LIFETIMES leVI
° Knapp et al. (1979)
_0
• This experiment
........ 0
OL_ _---'5
10
T
Y
~
....-0
-'-
L-_---I
15 20 ELECTRON ENERGV leV)
Fig. 2: Experimentally determined electron lifetimes (eV) for copper. Open circles are obtained by tuning the photon frequency [3], full circles by variation of the polar emission angle [4J. The energy is measured from Ep •
87
of damping, thus affect mostly one of the two types of electron states; the same is true about maxima which give effective decrease of damping. In Fig. 1 , in the out-of-phase case, partial density of states is sharpened for bonding states whereas it is broadened for the anti-bonding ones. Angular resolved photoemission profiles. Broadening of electron energy bands in crystals influences profiles of energy distribution curves in the angular resolved photoemission. While without damping (sharp bands) the spectrum should ideally consist of discrete ~-functions for monochromatic exciting radiation (direct vertical transitions), with increasing damping these lines have to become broadened. As in the process of optical excitation electron-hole pairs are created, both, finite lifetime of a hole and of an electron, have to determine jointly the broadening of the peaks in the profiles. Apart, from the two lifetimes, also the slopes of the two participating bands intervene. The halfwidths of the resulting Lorentzians ~ can be expressed by a simple formula in terms of these four quanti ties [3,4] • _ If the electron structure of a crystal is known, still depends on the two lifetimes: ~ of a hole and ~ of an electron. ~can be determined, utilizing transitions from occupied electron states close to the Fermi level (aJ,( = 0 for electrons at EF in perfect crystals). Analysis of the excited electron band broadening can be performed by turning the frequency of the exciting radiation (synchrotron source) [3] ; alternatively, the angle of incidence of the radiation can be turned at a fixed energy of the radiation [4] • The analysis of peak widths in the angular resolved photoemission thus provides the energy dependence of the imaginary part of the optical potential (Fig.2). Some discrepancies observed mainly in the vicinity of the Brillouin zone. indicate that non-homogeneities of electron damping should be taken into account. For the holes below~, the analysis is made easy if initial bands studied are flat in the direction perpendicular to the surface of a crystal; then becomes equal to ~. This
?
f
88 0.50
a
b 0.40
0.30
a: 0.20
0.10
0.0
9.60
9.80
10.00
E
10.20 ('10")
10.40
10.60
Fig. 3: Reflectivity from the semiinfin1te NFE crystal with the potential (1) fors homogeneous damping (Vol=0.1, Vgi=O, curve u) and lnhomoheneous damping: (a) in-phase (Vgi=0.005), and (b) out-of-phase (Vgi = 0.005)
16 12 19
21 E reV1 23
25
Fig.4s Intensity profiles for the OO-beam, specularly reflected from Cu(III), for polar angles of incidence:(a) = 0° (normal incidence), (b) .. 10(C) = 2° , and azimuthal angle = 0°. Damping along the surface reduced by 25 % from its value in bulk (Voi • 1.75 eV).
89
is, e.g. the case of excitations from occupied bands ot surface states (energy depending on k l l only). Intensity profiles in LEED. For the desoription of intensities of low energy electron beams, diffraoted back from crystal surfaces, dynamioal theory of diffraction is needed. One form, particularly suitable for incorporation of the damping effects is based on surface Green functions [5] • Reflectivity of a simple onedimensional nearly free electron model is total for energies corresponding to the gap. When homogeneous damping is introduced, the flat-topped I/V profiles get eroded into well-known bell-shaped curves [5] • The effect ot the inhomogeneous damping i8 closely related to that discussed above in the context of densities of electron states. The effective damping increase in the lower part of the gap is accompanied by its decrease in the upper part and vice verse. In the latter case (out-of-phase), the reflectivity is increased at lower energies and decreased at higher energies in the gap (Fig.J). In a more realistic situation tor Cu(100) similar effeots are encountered when inhomogeni ty is introduoed by changing the imaginary part of the optical potential within the spheres. The profiles get modified: the eftective energy position is shifted and the peakwidths are slightly changed. The changes are small because of presence of two counter-directed effeots in the gap, however [6] . The role of damping inhomogeni ty may be bigger i t pronounced structures in I/V profiles are present close outside the gap (Bragg peak). In the reflectivity from Cu(III) at around 20 eV two band structure features intenene: one above and the other just below the Bragg peak; they are revealed in I/V calculations with negligible damping [7J • Calculations, phenomenologically encorporating the damping inhomogeni ty, provide profiles with a sharp structure below the Bragg peak and no struoture above it (Fig.4) in agreement with experimental data. This resembles the DE results in the out-of-phase case; the valenoe electron density, being reduced at atomic nuclei, should then provide attenuated effective damping there. In this way, damping inhomogenity can
90
give rise to the existence of sharp structures in the I/V profiles; the structures are sharper than predicated from the uncertainly relations from the mean value of the optical potential VOte Conclusion. Damping of excited electron states in crystals, complemen~ ting dispersion relations E(k), can be obtained from profiles in LEED and angular resolved photoemission. Inhomogenity of damping leads to interesting nonnegligible effects, in the vicinity of Bragg peaks. References
[7] [8]
Pendry J.B.: in LEED-Surface Structures of Solids, J~SMF Fraha 1972, p. 305 Bartos I.: Phys.stat.sol.(6) 1£g (1984) K 159 Knapp J.A., Himpsel F.J., Eastman D.E.:Phys.Rev.B19(1979) 4952 Grepstad J.K., Slagsvold B.J., Bartos I.: J.Phys. F 12(1982) 1679 VelickY B., Bartos I.: in LEED-Surface Structures of Solids, J~SMF Praha 1972, p. 423 Bartos r., Koukal J.: in The Structure of Surfaces, Springer 1985, p.113 16 Bosse J.e., Lopez J.: Surf.Sci. 162 (1985) 953 Bartos I., Koukal J.: Surf. Sci , 183 (1987) 21
THEORETICAL ASPECTS OF ELECTRON DAMPING
I. Bartoe
Institut d'Electronique, USTHB, Alger + +permanent address: Institute of Physics, Czech.Acad.Sci, Prague
The surface sensitivity of LEED and of electron spectroscopies of surfaces is caused by the interaction among electrons, which is particularly strong for electrons with energies of the order of tens till hundreds of elec tronvolts. In LEED, where only the elastically back scattered electrons are studied, their trajectory through a crystal cannot be too long as otherwise an inelastic collision takes place which removes the electron from its elastic channel. Thus only electrons propagating, not too deep under the surface can contribute to the detected signal. In photoemission, for the same reasons, only electrons excited from atoms close enough to the surface can reach the surface without any loss of their energy. Inelastic collisions of electrons, being responsible for the loss of investigated electron from the elastic channel, cause the effective absorption or damping of the electron flux. This central factor of surface sensitivity is usually treated in a highly simplified manner however: by adding an imaginary constant contribution to the real potential to obtain the optical potential. This is in a marked contrast to the real part of the potential in which crystalline periodicity is fully respected. The simplified description appeared suffioient for surface crystallography by LEED and ..... for determining of electron dispersion relations E(k) by angular resolved photoemission. For subtler effects, and, in particular at very low energies the damping should be treated more carefUlly. Complicated many-electron problem can be reduced to the oneelectron problem if only the elastic chennel of the incident/excited electron is investigated [1] • The effective potential, so called optical potential, is no more real; its imaginary component is responsible for the electron absorption (damping) in the elastic channel. This can be illustrated on a simple example of the electron diffraction, described by a Hamiltonian with just a constant pure imaginary optical potencial Voi [1] • The eleotron wave function is exponentially damped, and the uncertainly re-
lation between energy and time leads to a broadening: any structure the r/v protile has to be broader than Voi• Density ot electron states in crystals. Let us begin the investigation with the simplest optical potential, i.e. with real periodic potential complemented by a constant imaginary part representing homogeneous electron damping in the crystal. In perfect crystals without damping, electron states characterised by wave vectors k are stationary. This is no more true when imaginary component is added to the potential. The damping ot the electron wave tunction in time leads to the energy broadening ot the electron state. Which is best represented by partial densities ot states in the k-space: discrete d-tunctions ot the undamped. crystal become broadened when imaginary component is introduced into the crystal potential [2] • The optical potential with just a constant imaginary part represents electron damping which is homogeneous throughout the crystal. This is only the zeroth approximation because valence electrons, which are mainly responsible tor inelastio oollisions ot low energy eleotrons, are in tact distributed non-unitormly throughout the crystal. Valence electrons are distributed with the periodicity ot the orystal lattioe; thus, also the imaginary part of the opti.. cal potential has this periodicity and can be expanded into the Fourier series. Let us investigate the consequences ot the nonhomogenity of the damping on a simple onedimensional model with the Hamiltonian H. (1) H • T + (Vor + i Voi) + (Vgr + iVgi) cos gx. The non-homogeneoWi damping, introduced by Vgi = 0, brings assymetry into the protile of the partial density of states (Fig.1). The real and imaginary parts of the optical potential can, in lim! ting cases, be either in phase (Vgi Vgr > 0) or out-ofphase (Vgi Vgr < 0). The real part of the potential determines the space localization of electrons: bonding states below the gap are concentrated predominantly in the regions olose to the potential minima whereas antibonding states (above the gap) have maimal ampl1tudes at potential maxima. The minima of the imaginary component of the potential, giVing effective inorease
86
~g, --002
-10
Fig. 1: Partial densities of states of the nearly free electron model at the Brillouin Zone boundary. The symmetrical two-peak profile for homogeneous damping (V dotoi=O.1; ted line) becomes asymmetrical when inhomogeneous damping is added (Vgi = 0)
COPPER EXITED ELECTRON LIFETIMES leVI
° Knapp et al. (1979)
_0
• This experiment
........ 0
OL_ _---'5
10
T
Y
~
....-0
-'-
L-_---I
15 20 ELECTRON ENERGV leV)
Fig. 2: Experimentally determined electron lifetimes (eV) for copper. Open circles are obtained by tuning the photon frequency [3], full circles by variation of the polar emission angle [4J. The energy is measured from Ep •
87
of damping, thus affect mostly one of the two types of electron states; the same is true about maxima which give effective decrease of damping. In Fig. 1 , in the out-of-phase case, partial density of states is sharpened for bonding states whereas it is broadened for the anti-bonding ones. Angular resolved photoemission profiles. Broadening of electron energy bands in crystals influences profiles of energy distribution curves in the angular resolved photoemission. While without damping (sharp bands) the spectrum should ideally consist of discrete ~-functions for monochromatic exciting radiation (direct vertical transitions), with increasing damping these lines have to become broadened. As in the process of optical excitation electron-hole pairs are created, both, finite lifetime of a hole and of an electron, have to determine jointly the broadening of the peaks in the profiles. Apart, from the two lifetimes, also the slopes of the two participating bands intervene. The halfwidths of the resulting Lorentzians ~ can be expressed by a simple formula in terms of these four quanti ties [3,4] • _ If the electron structure of a crystal is known, still depends on the two lifetimes: ~ of a hole and ~ of an electron. ~can be determined, utilizing transitions from occupied electron states close to the Fermi level (aJ,( = 0 for electrons at EF in perfect crystals). Analysis of the excited electron band broadening can be performed by turning the frequency of the exciting radiation (synchrotron source) [3] ; alternatively, the angle of incidence of the radiation can be turned at a fixed energy of the radiation [4] • The analysis of peak widths in the angular resolved photoemission thus provides the energy dependence of the imaginary part of the optical potential (Fig.2). Some discrepancies observed mainly in the vicinity of the Brillouin zone. indicate that non-homogeneities of electron damping should be taken into account. For the holes below~, the analysis is made easy if initial bands studied are flat in the direction perpendicular to the surface of a crystal; then becomes equal to ~. This
?
f
88 0.50
a
b 0.40
0.30
a: 0.20
0.10
0.0
9.60
9.80
10.00
E
10.20 ('10")
10.40
10.60
Fig. 3: Reflectivity from the semiinfin1te NFE crystal with the potential (1) fors homogeneous damping (Vol=0.1, Vgi=O, curve u) and lnhomoheneous damping: (a) in-phase (Vgi=0.005), and (b) out-of-phase (Vgi = 0.005)
16 12 19
21 E reV1 23
25
Fig.4s Intensity profiles for the OO-beam, specularly reflected from Cu(III), for polar angles of incidence:(a) = 0° (normal incidence), (b) .. 10(C) = 2° , and azimuthal angle = 0°. Damping along the surface reduced by 25 % from its value in bulk (Voi • 1.75 eV).
89
is, e.g. the case of excitations from occupied bands ot surface states (energy depending on k l l only). Intensity profiles in LEED. For the desoription of intensities of low energy electron beams, diffraoted back from crystal surfaces, dynamioal theory of diffraction is needed. One form, particularly suitable for incorporation of the damping effects is based on surface Green functions [5] • Reflectivity of a simple onedimensional nearly free electron model is total for energies corresponding to the gap. When homogeneous damping is introduced, the flat-topped I/V profiles get eroded into well-known bell-shaped curves [5] • The effect ot the inhomogeneous damping i8 closely related to that discussed above in the context of densities of electron states. The effective damping increase in the lower part of the gap is accompanied by its decrease in the upper part and vice verse. In the latter case (out-of-phase), the reflectivity is increased at lower energies and decreased at higher energies in the gap (Fig.J). In a more realistic situation tor Cu(100) similar effeots are encountered when inhomogeni ty is introduoed by changing the imaginary part of the optical potential within the spheres. The profiles get modified: the eftective energy position is shifted and the peakwidths are slightly changed. The changes are small because of presence of two counter-directed effeots in the gap, however [6] . The role of damping inhomogeni ty may be bigger i t pronounced structures in I/V profiles are present close outside the gap (Bragg peak). In the reflectivity from Cu(III) at around 20 eV two band structure features intenene: one above and the other just below the Bragg peak; they are revealed in I/V calculations with negligible damping [7J • Calculations, phenomenologically encorporating the damping inhomogeni ty, provide profiles with a sharp structure below the Bragg peak and no struoture above it (Fig.4) in agreement with experimental data. This resembles the DE results in the out-of-phase case; the valenoe electron density, being reduced at atomic nuclei, should then provide attenuated effective damping there. In this way, damping inhomogenity can
90
give rise to the existence of sharp structures in the I/V profiles; the structures are sharper than predicated from the uncertainly relations from the mean value of the optical potential VOte Conclusion. Damping of excited electron states in crystals, complemen~ ting dispersion relations E(k), can be obtained from profiles in LEED and angular resolved photoemission. Inhomogenity of damping leads to interesting nonnegligible effects, in the vicinity of Bragg peaks. References
[7] [8]
Pendry J.B.: in LEED-Surface Structures of Solids, J~SMF Fraha 1972, p. 305 Bartos I.: Phys.stat.sol.(6) 1£g (1984) K 159 Knapp J.A., Himpsel F.J., Eastman D.E.:Phys.Rev.B19(1979) 4952 Grepstad J.K., Slagsvold B.J., Bartos I.: J.Phys. F 12(1982) 1679 VelickY B., Bartos I.: in LEED-Surface Structures of Solids, J~SMF Praha 1972, p. 423 Bartos r., Koukal J.: in The Structure of Surfaces, Springer 1985, p.113 16 Bosse J.e., Lopez J.: Surf.Sci. 162 (1985) 953 Bartos I., Koukal J.: Surf. Sci , 183 (1987) 21