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Theory of free-electron optical absorption in n-GaAs
Solid State Communications, Vol. 51, No. 3, pp. 179-183, 1984. Printed in Great Britain.
0038-1098/84 $3.00 + .00 Pergamon Press Ltd.
THEORY OF FREE-ELECTRON OPTICAL ABSORPTION IN n-GaAs Pawel Pfeffer* and Iza Gorczyca t International Centre for Theoretical Physics, Trieste, Italy and Wlodek Zawadzki Institute of Physics, Polish Academy of Sciences, 02668 Warsaw, Poland
(Received 3 November 1983 by M. Cardona) Free-electron optical absorption of Se-doped GaAs at room temperature is calculated and compared with existing experimental data. In addition to the standard features the theory takes into account a non-parabolic character of the conduction band, a short-range component of the Se donor potential and a plasmon generation in the presence of donors. Deformation potential constants for the conduction and the valence bands are calculated using the empirical pseudopotential method. A value of C = -- 15.7 eV is obtained for the conduction band and shown to agree with existing hydrostatic and uniaxial stress experiments. Assuming no impurity compensation and using no adjustable parameters we successfully describe the experimental free-electron absorption in GaAs as a function of photon wavelength for samples with various donor densities.
In an attempt to explain the observed free-electron absorption in GaAs we introduce into the theory the following new features: (1) a non-parabolic structure of the conduction band, both in the energies and the wave functions; (2) an independent calculation of the relevant deformation potential constants for acoustic phonon scattering;(3) short-range components of the donor potentials;(4) plasmon generation in the presence of donor potentials. In the theory of absorption due to one-electron excitations we use the Green function formalism. In the diagram development [5] we take the first three diagrams (second-order perturbation). As indicated previously [6], after summation over all combinations of photon and phonon absorption and emission, the expressions reduce to the second order perturbation theory for two weak interactions, involving a summation over all possible intermediate states. The electronic states are described using a three-level model of P6, Fa, P7 symmetry levels, resulting in a non-parabolic energywave vector dependence: h2k~/2m~ = e(1 + e/eg). We take the energy gap % = 1.51 eV, the spin-orbit A ---0.34 eV and m~ = 0.0653 mo at T = 300 K. The corresponding wave functions are S - P symmetry mixtures due to a finite value of the gap and spin mixtures due to the spin-orbit interaction [7]. The spin mixing introduces a possibility of optical and scattering spinflip transitions, which we include in the theory [8]. On the other hand, the intermediate states in the valence
1. INTRODUCTION EXPERIMENTALLY OBSERVED free-electron optical absorption in GaAs at room temperature has not been understood theoretically until now. Haga and Kimura [1] in the first serious effort to describe the freeelectron absorption in this material used a parabolic energy band model and took into account the electron scattering by optic phonons (polar interaction), acoustic phonons (deformation potential) and ionized impurities (screened Coulomb interaction). In order to explain the available experimental data of Spitzer and Whelan [2], Haga and Kimura had to assume unreasonably large compensation ratios in the investigated samples, which led them to suspect "the presence of some additional scatterers of types different from both lattice vibrations and ionized impurities"! Jensen [3], attributing all the absorption in GaAs at room temperature to polar optic phonon scattering, was able to account theoretically for only half of the observed absorption. Walukiewicz et al. [4], using a procedure similar to that of Haga and Kimura, also had to assume high compensation ratios in order to describe their absorption data. * Permanent address: Institute of Physics, Polish Academy of Sciences, 02668 Warsaw, Poland. t Permanent address: High Pressure Research Centre, Polish Academy of Sciences, 01142 Warsaw, Poland. 179
180
THEORY OF FREE-ELECTRON OPTICAL ABSORPTION IN n-GaAs
bands are neglected since the gap in GaAs is relatively large. 2. SCATTERING MODES. DEFORMATION POTENTIALS Due to the energy conservation containing the photon energy hw, scattering processes contributing to the free-carrier absorption involve higher values of momentum transfer q than those contributing to the direct-current mobility. This leads to an increased significance o f electron interactions with acoustic phonons and donors (short-range component), both being more effective at higher q values. The electron-acoustic phonon interaction is described using the deformation potential (dp) formalism. Within the three-level band model there exist five independent dp constants [9]. Their values are of considerable interest for various properties. The dp constant for 1'6 (conduction) edge in GaAs has been commonly assumed to be C ~ -- 7 eV (cf. [10, 11 ]). However, this value is not credible since an experimental pressure dependence of the energy gap gives C - - a = -- 8.5 eV (cf. Table 1), while the dielectric theory of the chemical bond gives C - - a = -- 8.2 eV [ 15 ] and pseudopotential calculations give C - - a = -- 9.9 eV [16]. (All the above values have been obtained taking the compressibility K = 1.34 x 10 -6 bar-l). Together with C ~ -- 7 eV it would lead to positive values of a in GaAs (the dp constant a determines a shift of the valence 1'8 edge under hydrostatic pressure), which contradicts all evidence concerning the I I I - V compounds. We have calculated the dp constants for GaAs using 'the empirical pseudopotential method and the rigid-ion model of the electron-phonon interaction [ 17]. In the standard notation the obtained values are: C = 15.7 eV, 10.3 eV, m --- 5.2 eV, 4.6 eV and the interband dp constant s ~ 0. This leads to a = (l + 2m)/3 = -- 6.9 eV, so that C - - a = -- 8.8 eV is in very good agreement with the experimental estimations quoted in Table 1. For the dp constants involved in splittings of the valence P8 edge under uniaxial stress, we calculate: b = (l - m)/3 = -- 1.7 eV and d = 2n/ x / 3 = -- 5.3, while the experimentally determined values are quoted in Table 1. Again the theory is in good agreement with the experimental data. (It can be seen that also purely experimental values of l, m and C - a cannot be reasonably reconciled with the value of C ~ -- 7 eV.) It should be mentioned that self-consistent OPW band calculations for GaAs give a shift of the conduction band edge due to varying lattice constant [37], which leads to C = -- 13.4 eV (using the above compressibility value). Our value of a agrees with the general prediction --
l
=
-
--
n
=
-
Vol. 5 l, No. 3
Table 1. Experimental values o f deformation potential constants in GaAs (in eV, all values should be taken with the minus sign) C-a 8.7 8.7 8.0
b ±0.4
8.93±0.9 8.38±0.8 8.14±0.8 8.07±0.8
d
Reference
2.1
-+0.1
6.5
-+0.3
1.7 2.0 2.0 2.0 1.75 4.1 1.76
+0.2 -+ 0.1 +0.2 +0.2 ± 0.3
4.4 5.4 6.0 5.3 5.55 6.0 4.59
±0.5 ± 0.3 +0.4 -+0.4 ± 1.1
-+ 0.1
1.66 + 0.1
± 0.25
4.52 +- 0.25
[12] [13] [14] [18] [19] [20] [21] [22] [23] [36] [36] [36] [36]
that the deformation potential of P8 valence level for a number o f I I I - V compounds is between -- 7 and eV [381. We conclude: (1) the deformation potential constant for I'6 conduction edge in GaAs is at least twice as large than the value commonly assumed until now and it is similar to that in InSb (cf. [6, 17, 24]); (2) the calculated dp constants correctly describe experimental behaviour o f GaAs under hydrostatic and uniaxial stress. The electron-donor interaction is taken in the form V = Vc + Vs, where Vc is the screened Coulomb potential and Vs is a short-range potential, depending on donor's chemical nature. There exist two matrix elements of the short-range part: A = (S[ VslS) and B = (XI VsLX), where S and X are s-like and p-like periodic band-edge functions. For Te donors in InSb pseudopotential calculations have given A = 0.7 x 10 -21 eV cm 3, a n d B = 0.95 A [25]. We make use of the fact that differences in the nucleus structure of the I n - S b - T e sequence are similar to those of the G a - A s Se sequence and calculate for the Se donors in GaAs: A = 0.7 x 10 -21 x ~2(GaAs)/~(InSb) eV cm 3, and B = A , where ~2(GaAs) = 4.52 x 10 -28 cm 3 and ~2(InSb) = 6.8 x 10 -23 cm 3 are volumes of the corresponding unit cells. Since Vc and Vs are due to the same impurity, one also deals with interference terms Vc Vs [26]. In our case, Vs has an opposite sign to Vc, so that the absorption due to the total Vc + Vs potential is lower than that due to Vc alone (cf. Fig. 2). For the electron-optic phonon (polar) coupling we use the Fr6hlich Hamiltonian with the effective (Callen) charge e* --- 0.20 e and the phonon energy h ~ L = 36.1 meV. The electron-electron scattering has not been included as an absorption mechanism since for a --
9
Vol. 51, No. 3
40 1
THEORY OF FREE-ELECTRON OPTICAL ABSORPTION IN n-GaAs
181
_- T=3OOK
T-3OOK ld G, (c n'T~)
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_]/ /~' /lit ~,'--~ .... ///?
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102_-, :
"
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,,X/,..,";,;:,' 'ic
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I I t AI
IVl t ~ )X (~m)
Fig. 1. The refractive index at room temperature vs photon wavelengths for various n-GaAs samples (cf. Table 1), as calculated from the reflectivity data [2]. The dashed line for sample 2 is interpolated. Note the change of scale in the upper part of the figure. parabolic energy band its contribution vanishes identically [27] and the non-parabolicity of the band in GaAs is not very pronounced. We include screening of all the electron interactions with lattice vibrations and lattice imperfections by movable electrons, also of those due to short range potentials of acoustic phonons [28] and donors [29]. In the last two cases the screening effects are weak. The dynamic dielectric function e(q, co) of the electron gas is approximated by the static function for small q values e(q, O) = 1 + (qXoo)-2 for phonons and e(q, 0) = 1 + (qXo) -2 for impurities, where ko and k.. are the respective screening lengths, calculated using eo = 12.9 and eo, = 10.9 (cf. [8, 30]). 3. PLASMONS Until now we have considered the one-electron excitations. Photons cannot be absorbed generating directly plasmons [31 ]. However, as shown by Mycielski and Mycielski [32], if the symmetry of the system is broken by impurities or defects, the plasmon generation becomes possible and it can constitute an important mechanism of optical absorption for hco > hcop [6]. We include this mechanism using the prescription given in [32], which is not based on the second-order perturbation theory for two perturbations.
toP
/ ./I,
cc,.://
lO' _--
.://
/l/CC' I
1
I
l lJlJl
5
,,,I
I
10 20z (tim)
Fig. 2. Experimental optical absorption of Se-doped GaAs sample with n = 5.4 x 10 TM cm -3 vs photon wavelength [2] and the theory for the free-electron range (solid line). Various contributions to the total absorption are also indicated: AC = acoustic, OP = optical, CCe = Coulomb, CCt = Coulomb, plus short-range, PL = plasmon. Average squares of the plasma frequency co~ = (4rr2N/ e oo) ( 1/m *), where 1/m * = (1/h 2k) (de/dk), and of the velocity (v2), where v = (I/h) (de/dk), appearing in the description of this mechanism are calculated for the nonparabolic band (cf. [33]). Strictly speaking, one can exactly separate the plasma excitation from the oneelectron excitations only in absence of the Landau damping, that is in the range 0 ~ q/k F ~ 0.8, where q and kF are the plasmon and the Fermi wavevectors, respectively. This leads to a very restrictive criterion for the validity of the plasmon generation theory: cop co <~ 1.25 cop. A somewhat less stringent condition is q ~ qse, where q~e = 2rr/Xse is the screening wavevector of the electron gas. This leads to a validity criterion cop ~< co ~< 2cop. However, as follows from the random-phase approximation for the dynamic dielectric function (cf. [34]), also in the presence of not too strong Landau damping: q/kF > 0.8, one can differentiate to some extent between the one-electron and the collective excitations. 4. RESULTS AND DISCUSSION The absorption constant is inversely proportional to the refractive index N, which can be determined
182
THEORY OF FREE-ELECTRON OPTICAL ABSORPTION IN n-GaAs
Table 2. GaAs sample characteristics (1/m*)o p denotes average values o f the inverse effective mass for the nonparabolic band (el [33]) are photon wavelengths corresponding to respective plasma frequencies Sample
Donor
n ( c m -3 )
mS(l/m*)
Xp(/am)
2 3 4 5 6
Se Se Se Te Se
1.3 x 1017 4.9 x 1017 1.09 x l0 is 1.12x1018 5.4 x 1018
0.92 0.91 0.90 0.89 0.82
81.6 42.2 28.5 28.2 13.4
10" T~
o( (crn-1 }
v
- -
v v
v eq _
x
~vv v
×
10' n
¢..
=6
-
'k ux
_
x
xX
3
10°-
I
1
, //I,
5
II,Jll,,,I 10
t
30
%(/zm) Fig. 3. Experimental optical absorption of Te- and Sedoped GaAs samples with various free-electron concentrations [2] vs photon wavelength and the theory for the free-electron range. Solid lines = total theoretical absorption, dashed lines = the same with plasmon contribution excluded. directly from reflectivity experiments in the low-loss range of optical wavelengths. Figure 1 shows N as a function o f X for different GaAs samples, calculated from the reflectivity data of Spitzer and Whelan [3]. In Table 1 we quote the main sample characteristics. The dashed line in Fig. 1 is interpolated between those
Vol. 51, No. 3
for lower and higher electron concentrations, since the reflectivity data for the sample have not been published. In Fig. 2 we show the experimental optical absorption of the sample 6 and the theory for the free-electron range of photon wavelengths. Various modes contributing to the total absorption are also indicated. The theory, assuming no impurity compensation and containing no adjustable parameters, gives an excellent fit to the data. All the four new theoretical features mentioned above are necessary for this agreement. The plasmon generation is an important absorption mechanism and it is clear why the previous authors, by not taking it into account, had to assume high compensation ratios in order to explain the experimental data. A sharp decrease of the theoretical absorption above X ~ 13.4/lm is due to the increase of N above the plasma edge (cf. Fig. 1) and to a disappearance of plasmon generation at X > Xp. Clearly, both features are directly related. Above the plasma edge the sample is strongly reflecting and absorption measurements become difficult, but a verification of the maximum predicted by the theory does not seem impossible. Fig. 3 shows comparison between the experimental data for various GaAs samples and our theory for the long-wavelength range of free-electron absorption. Again, we take the number of donors equal to the number of free electrons, i.e. we assume no compensation. Since for decreasing electron densities a separation of one-electron and plasma excitations becomes progressively doubtful, we indicate the theoretical absorption both with and without the plasmon contribution. It can be seen that the theory successfully accounts for the data without any adjustable parameters. We do not describe the data for sample 5 since we do not have sufficient information on the short-range potential Vs of Te donors in GaAs. It can be seen that sample 4, exhibits lower a values. This illustrates directly the importance of short-range electron-donor interactions for the freeelectron absorption. The theory contains three points which, in our opinion, require further elaboration. First, the threelevel band model, used here for GaAs, is not quite adequate for this material. However, since the momentum matrix elements with higher conduction bands are considerably smaller than the ones we have included (cf. [35]), it is believed that a more complex treatment would not modify much the results. The second, more universal problem is that of screening effects in the presence of two perturbations, which we have included only approximately. Finally, it is not clear how to deal with the electron gas in which single-particle and collective excitations are not well separable. More experimental data are needed in order to investigate practical importance of these problems for the free-electron absorption.
Vol. 51, No. 3
THEORY OF FREE-ELECTRON OPTICAL ABSORPTION IN n-GaAs
Acknowledgements - Two of the authors (P.P. and I.G.) would like to thank Professor Abdus Salam, the International Atomic Energy Agency and UNESCO for hospitality at the International Centre for Theoretical Physics, Trieste.
17. 18. 19. 20.
REFERENCES 1. 2.
3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
E. Haga & H. Kimura, J. Phys. Soc. Japan 19, 658 (1964). W.G.Spitzer & Y.M. Whelan, Phys. Rev. 114, 59 (1959). A. Jensen, Ann. Phys. 80, 284 (1973). W. Walukiewicz, J. Lagowski, L. Jastnebski, M. Lichtensteiger & H.C. Gatos, J. Appl. Phys. 50, 899 (1973). E.N. Foo & N. Tzoar, Phys. Rev. 187,1000 (1969). P. Pfeffer & W. Zawadzki, Proc. 16th Intern. Conf. Phys. Semicond. (Edited by M. Averous), Phys. I17B-I18B, 425 (1983). W. Zawadzki, Handbook on Semiconductors (Edited by W. Paul), Vol. 1, p. 689. North Holland (1982). W. Zawadzki & W. Szymanska, Phys. Status Solidi (b) 45,415 (1971). W. Szymanska, P. Boguslawki & W. Zawadzki, Phys. Status Solidi (b) 65,641 (1974). K. Fletcher & P.N. Butcher, J. Phys. C5, 212 (1972). D.L. Rode,Phys. Rev. B2, 1012 (1970). J. Feinlib, S. Groves, W. Paul & R. Zallen, Phys. Rev. 131, 2070 (1963). F.H. Pollak, M. Cardona & K.L. Shaklee, Phys. Ret,. Lett. 16, 942 (1966). G.A.N. Connell,High Temp. High Pres. 1,77 (1969). D.L. Camphausen, G.A.N. Connell & W. Paul, Phys. Rev. Lett. 26,184(1971). Y.F. Tsay, S.S. Mitra & B. Bendow, Phys. Rev. BI0, 1476 (1974).
21. 22. 23. 24. 25. 26. 27. 28. 29. 30.
31. 32. 33. 34. 35. 36. 37. 38.
183
I. Gorczyca, Phys. Status Solidi (b) 108,205 (1981). I. Balslev, Solid State Commun. 5, 315 (1967). R.N. Bhargava & M.I. Nathan, Phys. Rev, 161, 695 (1967). F.H. Pollak & M. Cardona, Phys. Rev. 172,816 (1968). I. Balslev, Phys. Rev. 177, 1173 (1969). A. Gavini & M. Cardona, Phys. Rev. B1,672 (1970). P.Y. Yu, M. Cardona & F.H. Pollak, Phys. Rev. B3, 340 (1971). W. Zawadzki & J. Szymanska, Phys. Chem. Sol. 32, 1151 (1971). I. Gorczyca, Phys. Status Solidi (b) 103,529 (1981). E. Litwin-Staszewska, S. Porowski & A.S. Filipchenko, Phys. Status Solidi (b ) 48,525 (1971). M. Suffczynski, Phys. Rev. 140, A147 (1965). P. Boguslawski& J. Mycielski,J. Phys. CIO, 2143 (1977). P. Boguslawski& J. Mycielski,J. Phys. C13, 1019 (1980). G.E. Stillman, C.M. Wolfe & Y.O. Dimmock, Semiconductors and Semimetals (Edited by R.K. Willardson & A.C. Beer), Vol. 12, p. 169. Academic Press (1977). J. Blinowski & A. Mycielski,Phys. Lett. A50, 88 (1977). J. Mycielski & A. Mycielski, Phys. Rev. BI8, 1859 (1978). W. Zawadzki, Adv. Phys. 23,435 (1974). G.D. Mahan, Many-PartielePhysics, p. 431. Plenum Press, New York (1981). C. Hermann & C. Weisbuch, Phys. Rev. BI5,823 (1977). M. Chandrasekhar & F.H. Pollak, Phys. Rev. B15, 2127(1977). T.C. Collins, D.J. Stukel & R.N. Euwema, Phys. Rev. B1,724 (1970). J.A. Verg6s, D. G16tzel, M. Cardona & O.K. Andersen, Phys. Status Solidi (b) 113, 519 (1982).
0038-1098/84 $3.00 + .00 Pergamon Press Ltd.
THEORY OF FREE-ELECTRON OPTICAL ABSORPTION IN n-GaAs Pawel Pfeffer* and Iza Gorczyca t International Centre for Theoretical Physics, Trieste, Italy and Wlodek Zawadzki Institute of Physics, Polish Academy of Sciences, 02668 Warsaw, Poland
(Received 3 November 1983 by M. Cardona) Free-electron optical absorption of Se-doped GaAs at room temperature is calculated and compared with existing experimental data. In addition to the standard features the theory takes into account a non-parabolic character of the conduction band, a short-range component of the Se donor potential and a plasmon generation in the presence of donors. Deformation potential constants for the conduction and the valence bands are calculated using the empirical pseudopotential method. A value of C = -- 15.7 eV is obtained for the conduction band and shown to agree with existing hydrostatic and uniaxial stress experiments. Assuming no impurity compensation and using no adjustable parameters we successfully describe the experimental free-electron absorption in GaAs as a function of photon wavelength for samples with various donor densities.
In an attempt to explain the observed free-electron absorption in GaAs we introduce into the theory the following new features: (1) a non-parabolic structure of the conduction band, both in the energies and the wave functions; (2) an independent calculation of the relevant deformation potential constants for acoustic phonon scattering;(3) short-range components of the donor potentials;(4) plasmon generation in the presence of donor potentials. In the theory of absorption due to one-electron excitations we use the Green function formalism. In the diagram development [5] we take the first three diagrams (second-order perturbation). As indicated previously [6], after summation over all combinations of photon and phonon absorption and emission, the expressions reduce to the second order perturbation theory for two weak interactions, involving a summation over all possible intermediate states. The electronic states are described using a three-level model of P6, Fa, P7 symmetry levels, resulting in a non-parabolic energywave vector dependence: h2k~/2m~ = e(1 + e/eg). We take the energy gap % = 1.51 eV, the spin-orbit A ---0.34 eV and m~ = 0.0653 mo at T = 300 K. The corresponding wave functions are S - P symmetry mixtures due to a finite value of the gap and spin mixtures due to the spin-orbit interaction [7]. The spin mixing introduces a possibility of optical and scattering spinflip transitions, which we include in the theory [8]. On the other hand, the intermediate states in the valence
1. INTRODUCTION EXPERIMENTALLY OBSERVED free-electron optical absorption in GaAs at room temperature has not been understood theoretically until now. Haga and Kimura [1] in the first serious effort to describe the freeelectron absorption in this material used a parabolic energy band model and took into account the electron scattering by optic phonons (polar interaction), acoustic phonons (deformation potential) and ionized impurities (screened Coulomb interaction). In order to explain the available experimental data of Spitzer and Whelan [2], Haga and Kimura had to assume unreasonably large compensation ratios in the investigated samples, which led them to suspect "the presence of some additional scatterers of types different from both lattice vibrations and ionized impurities"! Jensen [3], attributing all the absorption in GaAs at room temperature to polar optic phonon scattering, was able to account theoretically for only half of the observed absorption. Walukiewicz et al. [4], using a procedure similar to that of Haga and Kimura, also had to assume high compensation ratios in order to describe their absorption data. * Permanent address: Institute of Physics, Polish Academy of Sciences, 02668 Warsaw, Poland. t Permanent address: High Pressure Research Centre, Polish Academy of Sciences, 01142 Warsaw, Poland. 179
180
THEORY OF FREE-ELECTRON OPTICAL ABSORPTION IN n-GaAs
bands are neglected since the gap in GaAs is relatively large. 2. SCATTERING MODES. DEFORMATION POTENTIALS Due to the energy conservation containing the photon energy hw, scattering processes contributing to the free-carrier absorption involve higher values of momentum transfer q than those contributing to the direct-current mobility. This leads to an increased significance o f electron interactions with acoustic phonons and donors (short-range component), both being more effective at higher q values. The electron-acoustic phonon interaction is described using the deformation potential (dp) formalism. Within the three-level band model there exist five independent dp constants [9]. Their values are of considerable interest for various properties. The dp constant for 1'6 (conduction) edge in GaAs has been commonly assumed to be C ~ -- 7 eV (cf. [10, 11 ]). However, this value is not credible since an experimental pressure dependence of the energy gap gives C - - a = -- 8.5 eV (cf. Table 1), while the dielectric theory of the chemical bond gives C - - a = -- 8.2 eV [ 15 ] and pseudopotential calculations give C - - a = -- 9.9 eV [16]. (All the above values have been obtained taking the compressibility K = 1.34 x 10 -6 bar-l). Together with C ~ -- 7 eV it would lead to positive values of a in GaAs (the dp constant a determines a shift of the valence 1'8 edge under hydrostatic pressure), which contradicts all evidence concerning the I I I - V compounds. We have calculated the dp constants for GaAs using 'the empirical pseudopotential method and the rigid-ion model of the electron-phonon interaction [ 17]. In the standard notation the obtained values are: C = 15.7 eV, 10.3 eV, m --- 5.2 eV, 4.6 eV and the interband dp constant s ~ 0. This leads to a = (l + 2m)/3 = -- 6.9 eV, so that C - - a = -- 8.8 eV is in very good agreement with the experimental estimations quoted in Table 1. For the dp constants involved in splittings of the valence P8 edge under uniaxial stress, we calculate: b = (l - m)/3 = -- 1.7 eV and d = 2n/ x / 3 = -- 5.3, while the experimentally determined values are quoted in Table 1. Again the theory is in good agreement with the experimental data. (It can be seen that also purely experimental values of l, m and C - a cannot be reasonably reconciled with the value of C ~ -- 7 eV.) It should be mentioned that self-consistent OPW band calculations for GaAs give a shift of the conduction band edge due to varying lattice constant [37], which leads to C = -- 13.4 eV (using the above compressibility value). Our value of a agrees with the general prediction --
l
=
-
--
n
=
-
Vol. 5 l, No. 3
Table 1. Experimental values o f deformation potential constants in GaAs (in eV, all values should be taken with the minus sign) C-a 8.7 8.7 8.0
b ±0.4
8.93±0.9 8.38±0.8 8.14±0.8 8.07±0.8
d
Reference
2.1
-+0.1
6.5
-+0.3
1.7 2.0 2.0 2.0 1.75 4.1 1.76
+0.2 -+ 0.1 +0.2 +0.2 ± 0.3
4.4 5.4 6.0 5.3 5.55 6.0 4.59
±0.5 ± 0.3 +0.4 -+0.4 ± 1.1
-+ 0.1
1.66 + 0.1
± 0.25
4.52 +- 0.25
[12] [13] [14] [18] [19] [20] [21] [22] [23] [36] [36] [36] [36]
that the deformation potential of P8 valence level for a number o f I I I - V compounds is between -- 7 and eV [381. We conclude: (1) the deformation potential constant for I'6 conduction edge in GaAs is at least twice as large than the value commonly assumed until now and it is similar to that in InSb (cf. [6, 17, 24]); (2) the calculated dp constants correctly describe experimental behaviour o f GaAs under hydrostatic and uniaxial stress. The electron-donor interaction is taken in the form V = Vc + Vs, where Vc is the screened Coulomb potential and Vs is a short-range potential, depending on donor's chemical nature. There exist two matrix elements of the short-range part: A = (S[ VslS) and B = (XI VsLX), where S and X are s-like and p-like periodic band-edge functions. For Te donors in InSb pseudopotential calculations have given A = 0.7 x 10 -21 eV cm 3, a n d B = 0.95 A [25]. We make use of the fact that differences in the nucleus structure of the I n - S b - T e sequence are similar to those of the G a - A s Se sequence and calculate for the Se donors in GaAs: A = 0.7 x 10 -21 x ~2(GaAs)/~(InSb) eV cm 3, and B = A , where ~2(GaAs) = 4.52 x 10 -28 cm 3 and ~2(InSb) = 6.8 x 10 -23 cm 3 are volumes of the corresponding unit cells. Since Vc and Vs are due to the same impurity, one also deals with interference terms Vc Vs [26]. In our case, Vs has an opposite sign to Vc, so that the absorption due to the total Vc + Vs potential is lower than that due to Vc alone (cf. Fig. 2). For the electron-optic phonon (polar) coupling we use the Fr6hlich Hamiltonian with the effective (Callen) charge e* --- 0.20 e and the phonon energy h ~ L = 36.1 meV. The electron-electron scattering has not been included as an absorption mechanism since for a --
9
Vol. 51, No. 3
40 1
THEORY OF FREE-ELECTRON OPTICAL ABSORPTION IN n-GaAs
181
_- T=3OOK
T-3OOK ld G, (c n'T~)
r \
_]/ /~' /lit ~,'--~ .... ///?
~ ~2
l?/" 'i,
102_-, :
"
d I "!,i
,,X/,..,";,;:,' 'ic
:- \
I I t AI
IVl t ~ )X (~m)
Fig. 1. The refractive index at room temperature vs photon wavelengths for various n-GaAs samples (cf. Table 1), as calculated from the reflectivity data [2]. The dashed line for sample 2 is interpolated. Note the change of scale in the upper part of the figure. parabolic energy band its contribution vanishes identically [27] and the non-parabolicity of the band in GaAs is not very pronounced. We include screening of all the electron interactions with lattice vibrations and lattice imperfections by movable electrons, also of those due to short range potentials of acoustic phonons [28] and donors [29]. In the last two cases the screening effects are weak. The dynamic dielectric function e(q, co) of the electron gas is approximated by the static function for small q values e(q, O) = 1 + (qXoo)-2 for phonons and e(q, 0) = 1 + (qXo) -2 for impurities, where ko and k.. are the respective screening lengths, calculated using eo = 12.9 and eo, = 10.9 (cf. [8, 30]). 3. PLASMONS Until now we have considered the one-electron excitations. Photons cannot be absorbed generating directly plasmons [31 ]. However, as shown by Mycielski and Mycielski [32], if the symmetry of the system is broken by impurities or defects, the plasmon generation becomes possible and it can constitute an important mechanism of optical absorption for hco > hcop [6]. We include this mechanism using the prescription given in [32], which is not based on the second-order perturbation theory for two perturbations.
toP
/ ./I,
cc,.://
lO' _--
.://
/l/CC' I
1
I
l lJlJl
5
,,,I
I
10 20z (tim)
Fig. 2. Experimental optical absorption of Se-doped GaAs sample with n = 5.4 x 10 TM cm -3 vs photon wavelength [2] and the theory for the free-electron range (solid line). Various contributions to the total absorption are also indicated: AC = acoustic, OP = optical, CCe = Coulomb, CCt = Coulomb, plus short-range, PL = plasmon. Average squares of the plasma frequency co~ = (4rr2N/ e oo) ( 1/m *), where 1/m * = (1/h 2k) (de/dk), and of the velocity (v2), where v = (I/h) (de/dk), appearing in the description of this mechanism are calculated for the nonparabolic band (cf. [33]). Strictly speaking, one can exactly separate the plasma excitation from the oneelectron excitations only in absence of the Landau damping, that is in the range 0 ~ q/k F ~ 0.8, where q and kF are the plasmon and the Fermi wavevectors, respectively. This leads to a very restrictive criterion for the validity of the plasmon generation theory: cop co <~ 1.25 cop. A somewhat less stringent condition is q ~ qse, where q~e = 2rr/Xse is the screening wavevector of the electron gas. This leads to a validity criterion cop ~< co ~< 2cop. However, as follows from the random-phase approximation for the dynamic dielectric function (cf. [34]), also in the presence of not too strong Landau damping: q/kF > 0.8, one can differentiate to some extent between the one-electron and the collective excitations. 4. RESULTS AND DISCUSSION The absorption constant is inversely proportional to the refractive index N, which can be determined
182
THEORY OF FREE-ELECTRON OPTICAL ABSORPTION IN n-GaAs
Table 2. GaAs sample characteristics (1/m*)o p denotes average values o f the inverse effective mass for the nonparabolic band (el [33]) are photon wavelengths corresponding to respective plasma frequencies Sample
Donor
n ( c m -3 )
mS(l/m*)
Xp(/am)
2 3 4 5 6
Se Se Se Te Se
1.3 x 1017 4.9 x 1017 1.09 x l0 is 1.12x1018 5.4 x 1018
0.92 0.91 0.90 0.89 0.82
81.6 42.2 28.5 28.2 13.4
10" T~
o( (crn-1 }
v
- -
v v
v eq _
x
~vv v
×
10' n
¢..
=6
-
'k ux
_
x
xX
3
10°-
I
1
, //I,
5
II,Jll,,,I 10
t
30
%(/zm) Fig. 3. Experimental optical absorption of Te- and Sedoped GaAs samples with various free-electron concentrations [2] vs photon wavelength and the theory for the free-electron range. Solid lines = total theoretical absorption, dashed lines = the same with plasmon contribution excluded. directly from reflectivity experiments in the low-loss range of optical wavelengths. Figure 1 shows N as a function o f X for different GaAs samples, calculated from the reflectivity data of Spitzer and Whelan [3]. In Table 1 we quote the main sample characteristics. The dashed line in Fig. 1 is interpolated between those
Vol. 51, No. 3
for lower and higher electron concentrations, since the reflectivity data for the sample have not been published. In Fig. 2 we show the experimental optical absorption of the sample 6 and the theory for the free-electron range of photon wavelengths. Various modes contributing to the total absorption are also indicated. The theory, assuming no impurity compensation and containing no adjustable parameters, gives an excellent fit to the data. All the four new theoretical features mentioned above are necessary for this agreement. The plasmon generation is an important absorption mechanism and it is clear why the previous authors, by not taking it into account, had to assume high compensation ratios in order to explain the experimental data. A sharp decrease of the theoretical absorption above X ~ 13.4/lm is due to the increase of N above the plasma edge (cf. Fig. 1) and to a disappearance of plasmon generation at X > Xp. Clearly, both features are directly related. Above the plasma edge the sample is strongly reflecting and absorption measurements become difficult, but a verification of the maximum predicted by the theory does not seem impossible. Fig. 3 shows comparison between the experimental data for various GaAs samples and our theory for the long-wavelength range of free-electron absorption. Again, we take the number of donors equal to the number of free electrons, i.e. we assume no compensation. Since for decreasing electron densities a separation of one-electron and plasma excitations becomes progressively doubtful, we indicate the theoretical absorption both with and without the plasmon contribution. It can be seen that the theory successfully accounts for the data without any adjustable parameters. We do not describe the data for sample 5 since we do not have sufficient information on the short-range potential Vs of Te donors in GaAs. It can be seen that sample 4, exhibits lower a values. This illustrates directly the importance of short-range electron-donor interactions for the freeelectron absorption. The theory contains three points which, in our opinion, require further elaboration. First, the threelevel band model, used here for GaAs, is not quite adequate for this material. However, since the momentum matrix elements with higher conduction bands are considerably smaller than the ones we have included (cf. [35]), it is believed that a more complex treatment would not modify much the results. The second, more universal problem is that of screening effects in the presence of two perturbations, which we have included only approximately. Finally, it is not clear how to deal with the electron gas in which single-particle and collective excitations are not well separable. More experimental data are needed in order to investigate practical importance of these problems for the free-electron absorption.
Vol. 51, No. 3
THEORY OF FREE-ELECTRON OPTICAL ABSORPTION IN n-GaAs
Acknowledgements - Two of the authors (P.P. and I.G.) would like to thank Professor Abdus Salam, the International Atomic Energy Agency and UNESCO for hospitality at the International Centre for Theoretical Physics, Trieste.
17. 18. 19. 20.
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