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Temperature dependence of the energy GAP in GaAs
Volume 4 1A, number 3
25 September 1972
PHYSICS LE’ITERS
TEMPERATURE DEPENDENCE OF THE ENERGY GAP 1N GaAs K. SHINDO and K. HOSHINO Department of Physics, Faculty of Science. Tohoku University, Sendai, Japan
Received 17 July 1972 It is shown that electron-phonon self-energy effects are as important as the Debye-Walter screening of the pseudopotential in the temperature dependence of the band gap when the valence band is degenerate. Walter et al. [l] have calculated the temperature dependence of tire band gap in GaAs by taking the Debye-Wailer (D-W) screening of the pseudopotential into account. The purpose of this letter is to estimate the self-energy effect due to phonons (polaron effect) which were neglected in their treatment. We show that the temperature dependence of the band gap due to the polaron effect is as dominant as the effect of the D-W screening if the degeneracy of the valence band at I’@ = 0) is considered. The temperature dependence of the band gap at constant pressure is as follows,
(y$=(>)y+3Qvfg)T. where (Yis the coefficient of linear expansion. In this equation the first term on the right hand side is connected with the electron-phonon interaction and the second with the dilatation effect. At low temperature the latter is very small and negligible in GaAs. The term due to electron-phonon interactions consists of two effects; one is due to the D-W screening of the pseudopotential and the other due to the polaron effect. We first estimate the polaron effect in a simple model with non-degenerate holes and electrons having parabolic energy bands. For acoustic phonons we consider an isotropic solid with a simple deformation potential electron (or hole)-phonon coupling. For optical phonons the phenomenological Frohlich hamiltonian is used. In this hamiltonian the coupling constants depend only on the band masses. In a single band model we take the average of the contribution due to a heavy hole and a light hole. We estimate the difference (AX) between the self-energy at 80 K and at 0 K for each phonon. The experimental
value of AX is about-18 meV [2] . Walter et al. calculated AZ to be about-20 meV by considering only the D-W screening of the pseudopotential. In the simple model, where we neglect the degeneracy of the valence band, AZ due to acoustic phonons is-l .2 meV. The temperature dependence due to optical phonons is very small:.about-O.06 meV in spite of the large absolute value -12 meV. There are, however, many contributions from the transverse acoustic phonons to the self-energy, which are not included in the single band case, if we consider the degeneracy of the valence band. The orbital-strain Hamiltonian Hs at F can be written in conventional notation as [3], HS
=
-
--$d
a(eXr + Eyy t czz) - 3b [~~,,~,~
[(t; - 3 L2>cxy+ c.p]
cxy + c.p.1 .
The parameters (I, b and d are the deformation potentials. Pollak and Cardona [3] recently measured these quantities; uc - av =-4.0 eV, b =-4.0 eV, and d = -6.0 eV. We calculate the matrix elements of eq. (2) for phonon scattering by using the diagonalizing method of Ehrenreich and Overhauser [4; 51. In this calculation some models of the valence band are employed. It takes at least four constants to describe the valence bands near k = 0, three for the quadratic and one for the linear wave vector terms. These four have not been measured. In order to estimate the contributions from shear modes, the interaction (2) is evaluated for holes in a model isotropic material. Also, the linear wave vector terms are omitted and the two hole bands are considered parabolic with masses mn and ml.The resulting self243
Volume
41A, number
3
PHYSICS
energies are as follows, AZ, = -3.8 meV for the longi+ tudinal acoustic phonon, AZ, = -15 meV for the transverse acoustic phonons, and azl/aT =-4.78 X 10e4 (eV/K) at 80 K. Another effect due to the degeneracy is to set up piezoelectric coupling. This contribution can be easily estimated in zincblende crystals such as GaAs, and gives very small corrections to AZ of about -0.01 meV. We have approximately estimated the contribution to the electron-phonon self-energy due to the D-W screening of the pseudopotential, AC’. In this calculation we used Ashcroft’s potential screened by the dielectric function with a Hubbard exchange. The D-W screening factor is calculated by Vetelino et al. [6]. The lowest order self-energy in this case, AZ’/AZ, is of the order of low3 for each phonon. Thus the contribution to the self-energy due to the D-W screening effect is very small and negligible. The combined contribution of the self-energy effect (-18.8 meV) and the D-W screening effect (-20.0 meV) is about twice as much as the experimental value (-18.0 meV). Some improvements must, however, be made in the treatment of both effects. Walter et al. used the D-W factor of Ge in stead of
244
LETTERS
25 September
1972
GaAs. The D-W factor of GaAs, which is calculated by Vatelino et al. [6] , reduces the D-W screening effect overestimated by Walter et al. about 40%. We have adopted a model isotropic material. In the model the warping effect of the valence band is neglected and the lattice vibrations are assumed to be purely longitudinal and transverse. It is difficult to estimate these effects exactly for lack of the details of the phonon properties and the valence band parameters at present. Furthermore there is the uncertainty of deformation potentials. For instance, we find AZ about - 10 meV by using Blaslev’s data [7]. However, it is sure that the polaron effect is as important as the D-W screening effect. References Ill J.P. Walter, R.R.L.
PI [31 [41 [51 [61 [71
Zucca, M.L. Cohen and Y.R. Shen, Phys.Rev.Letters 24 (1970) 102. J.L. Shay, Phys.Rev. B4 (1971) 1385. F.H. Pollak and M. Cardona, Phys.Rev.172 (1968) 816. H. Ehrenreich and A.W. Overhauser, Phys.Rev. 104 (1956) 331. G.D. Mahan, J.Phys.Chem.Solids 26 (1965) 751. J.F. Vetelino, S.P. Gaur and S.S. Mitra, Phys.Rev. B5 (1972) 2360. I. Blashlev, Solid State Commun. 5 (1967) 315.
25 September 1972
PHYSICS LE’ITERS
TEMPERATURE DEPENDENCE OF THE ENERGY GAP 1N GaAs K. SHINDO and K. HOSHINO Department of Physics, Faculty of Science. Tohoku University, Sendai, Japan
Received 17 July 1972 It is shown that electron-phonon self-energy effects are as important as the Debye-Walter screening of the pseudopotential in the temperature dependence of the band gap when the valence band is degenerate. Walter et al. [l] have calculated the temperature dependence of tire band gap in GaAs by taking the Debye-Wailer (D-W) screening of the pseudopotential into account. The purpose of this letter is to estimate the self-energy effect due to phonons (polaron effect) which were neglected in their treatment. We show that the temperature dependence of the band gap due to the polaron effect is as dominant as the effect of the D-W screening if the degeneracy of the valence band at I’@ = 0) is considered. The temperature dependence of the band gap at constant pressure is as follows,
(y$=(>)y+3Qvfg)T. where (Yis the coefficient of linear expansion. In this equation the first term on the right hand side is connected with the electron-phonon interaction and the second with the dilatation effect. At low temperature the latter is very small and negligible in GaAs. The term due to electron-phonon interactions consists of two effects; one is due to the D-W screening of the pseudopotential and the other due to the polaron effect. We first estimate the polaron effect in a simple model with non-degenerate holes and electrons having parabolic energy bands. For acoustic phonons we consider an isotropic solid with a simple deformation potential electron (or hole)-phonon coupling. For optical phonons the phenomenological Frohlich hamiltonian is used. In this hamiltonian the coupling constants depend only on the band masses. In a single band model we take the average of the contribution due to a heavy hole and a light hole. We estimate the difference (AX) between the self-energy at 80 K and at 0 K for each phonon. The experimental
value of AX is about-18 meV [2] . Walter et al. calculated AZ to be about-20 meV by considering only the D-W screening of the pseudopotential. In the simple model, where we neglect the degeneracy of the valence band, AZ due to acoustic phonons is-l .2 meV. The temperature dependence due to optical phonons is very small:.about-O.06 meV in spite of the large absolute value -12 meV. There are, however, many contributions from the transverse acoustic phonons to the self-energy, which are not included in the single band case, if we consider the degeneracy of the valence band. The orbital-strain Hamiltonian Hs at F can be written in conventional notation as [3], HS
=
-
--$d
a(eXr + Eyy t czz) - 3b [~~,,~,~
[(t; - 3 L2>cxy+ c.p]
cxy + c.p.1 .
The parameters (I, b and d are the deformation potentials. Pollak and Cardona [3] recently measured these quantities; uc - av =-4.0 eV, b =-4.0 eV, and d = -6.0 eV. We calculate the matrix elements of eq. (2) for phonon scattering by using the diagonalizing method of Ehrenreich and Overhauser [4; 51. In this calculation some models of the valence band are employed. It takes at least four constants to describe the valence bands near k = 0, three for the quadratic and one for the linear wave vector terms. These four have not been measured. In order to estimate the contributions from shear modes, the interaction (2) is evaluated for holes in a model isotropic material. Also, the linear wave vector terms are omitted and the two hole bands are considered parabolic with masses mn and ml.The resulting self243
Volume
41A, number
3
PHYSICS
energies are as follows, AZ, = -3.8 meV for the longi+ tudinal acoustic phonon, AZ, = -15 meV for the transverse acoustic phonons, and azl/aT =-4.78 X 10e4 (eV/K) at 80 K. Another effect due to the degeneracy is to set up piezoelectric coupling. This contribution can be easily estimated in zincblende crystals such as GaAs, and gives very small corrections to AZ of about -0.01 meV. We have approximately estimated the contribution to the electron-phonon self-energy due to the D-W screening of the pseudopotential, AC’. In this calculation we used Ashcroft’s potential screened by the dielectric function with a Hubbard exchange. The D-W screening factor is calculated by Vetelino et al. [6]. The lowest order self-energy in this case, AZ’/AZ, is of the order of low3 for each phonon. Thus the contribution to the self-energy due to the D-W screening effect is very small and negligible. The combined contribution of the self-energy effect (-18.8 meV) and the D-W screening effect (-20.0 meV) is about twice as much as the experimental value (-18.0 meV). Some improvements must, however, be made in the treatment of both effects. Walter et al. used the D-W factor of Ge in stead of
244
LETTERS
25 September
1972
GaAs. The D-W factor of GaAs, which is calculated by Vatelino et al. [6] , reduces the D-W screening effect overestimated by Walter et al. about 40%. We have adopted a model isotropic material. In the model the warping effect of the valence band is neglected and the lattice vibrations are assumed to be purely longitudinal and transverse. It is difficult to estimate these effects exactly for lack of the details of the phonon properties and the valence band parameters at present. Furthermore there is the uncertainty of deformation potentials. For instance, we find AZ about - 10 meV by using Blaslev’s data [7]. However, it is sure that the polaron effect is as important as the D-W screening effect. References Ill J.P. Walter, R.R.L.
PI [31 [41 [51 [61 [71
Zucca, M.L. Cohen and Y.R. Shen, Phys.Rev.Letters 24 (1970) 102. J.L. Shay, Phys.Rev. B4 (1971) 1385. F.H. Pollak and M. Cardona, Phys.Rev.172 (1968) 816. H. Ehrenreich and A.W. Overhauser, Phys.Rev. 104 (1956) 331. G.D. Mahan, J.Phys.Chem.Solids 26 (1965) 751. J.F. Vetelino, S.P. Gaur and S.S. Mitra, Phys.Rev. B5 (1972) 2360. I. Blashlev, Solid State Commun. 5 (1967) 315.