- Email: [email protected]
The energy relaxation due to electron-two short wave length transverse acoustic phonons in semiconductors
Volume 54A, number 5
PHYSICS LETFERS
6 October 1975
THE ENERGY RELAXATION DUE TO ELECTRON-TWO SHORT WAVE LENGTH TRANSVERSE ACOUSTIC PHONONS IN SEMICONDUCTORS S.S. KUBAKADDI and B.S. KRISHNAMURTHY Department of Physics, Karnatak Universiti’, Dharwar, Karnataka, India Received 18 August 1975 The energy relaxation time, due to the interaction of electrons with two short wave length acoustic phonons, is calculated in nondegenerate semiconductors and compared with an experiment in n-type lnSb at low temperatures.
Recent experiments [1, 2] have established that the carrier two-phonon interaction is an important scattering mechanism in semiconductors. In a two-phonon process all phonons can take part unlike the case of one-phonon process. In the latter process the energy and momentum conservation restricts the available phonons to a small fraction of the Brillouin zone. In the present note we wish to examin the contribution of the two short wave length transverse acoustic phonons (2 SWTA) to the carrier energy relaxation process. Following Ganguly and Ngai [3] we write the matrix element for 2 SWTA phonon process (1) Vq,q hD(q, q)/p~2a2(wqwq’)1/2 where D(q, q’) is the 2SWTA phonon effective deformation potential, Wq is the phonon frequency of wave vector q, a is the lattice constant and p is the mass density of the crystal. It should be noted that the deformation potential concept invoked in obtaining eq. (1) requires that Q = q + q’ lies in the long wave length region. In what follows we shall calculate the average energy loss rate due to 2SWTA phonons (both emitted and both absorbed) following the kinetic method assuming a displaced Maxwellian distribution for electrons at electron temperature Te and a simple parabolic band model. We get for the average energy loss rate I ~3
(,~) D
2
(w
+
w’)312
q
m*3/2
~7/2~1/2 (7/2)1/2 exp(7/2)K
~‘
1(7/2)NN’[l
—
exp(y0
—
y)].
(2)
In eq. (2) y = h (w + w’)/kB Te, “~= ~ (u.’ + w’)/k~T, N and N’ are the equilibrium phonon distribution at lattice temperature T and qm is the zone edge value of q. Further, we have assumed for simplicity that = w and wq’ = w’ are independent of the wave vectors. In fact, it is known that, in InSb the transverse acoustic branch of the lattice dispersion is relatively flat [4] in short wave length region. Now we obtain from (2) the electron energy relaxation time, using r~ = (2/3 kB)(dP/dTC), 512h1I2 m*3/2 1 1 / D ~2 ~~ +~ “ -~q~ 2 T2 exp(—y)(y/2)1/2 exp(7/2)K 1 (3) 1(y/2){l +~r~+ [K0(y/2)/K1ey/2)]} q ap in for hw/kBT> 1. In eq. (3) K 1(y/2) and K0(y/2) are the modified Bessel’s functions. Now in order to compare the above theory with an experiment we shall consider an n-type InSb. In fact, the experimental results of the electron power loss as a function of electron temperature, in the range of 1 .5 to 30 K, have been presented by many authors [5—8] It has been suggested that [1, 6, 8] the two zone edge acoustic phonon (2 ZETA) scattering may be responsible for the energy relaxation in the temperature of 10 to 22 K. In fig. 1 we have plotted dp/d Te as a function of T~for various independent processes such as the acoustic phonon (both deformation and piezoelectric) [9], the polar optical phonon [10] and 2 ZETA phonons as well as their combinations using the following parameters: D = 620 eV [11] a = 6.4789 A, h(w + w’)/kB = 124 K (for TA phonons at X point) [12], the deformation potential for the one phonon scattering E1 = 7.2 eV, the piezoelectric constant 13 = 3 X I o~ .
,
389
Volume 54A, number 5
PHYSICS LETTERS /0
/0
6 October 1975
/
_______________________________________________________
d
e
/ ~ /~
Fig. 1. Plot of dP/dTe as a function of electron temperature Tc for n-type InSb. [5] arc shown with the fol3 x , n = 2.2 x 1014 c1n3 andExperimental ~, n = 1.1 X points lOis cin3, Theoretical curves for lowing carrier concentrations: •, n = 5 X 1013 cm the following scattering mechanisms: (a) acoustic + polar optical + 2 ZETA, (b) acoustic + 2 ZETA, (c) 2 ZETA, (d) acoustic + polar optical, (e) polar optical and (f) acoustic.
dyn1/2 cm_l,m*
3,OD 278 K, e 0,p = 5.82 g cm 0 17.18 and e~= 15.68. It is interesting to note (10 to 20 K) that there is a good agreement between the curve a (combination of all one phonon and 2 ZETA phonon processes) and the experimental points of [5] However, the curve c indicates that in the temperature range of 12 to 18 K the power loss due to 2 ZETA phonons scattering is dominant over the one phonon scattering mechanisms and the sudden increase in the observed power loss is accounted for in this temperature range. It may be noted that the kinetic method and the perturbation theory is questionable as pointed out [7, 8] in the example considered for experimental comparison. However, since no quantum treatment exists and the criterion for the validity of the method is weakly violated as it is in the cross over region, the applicability of the method is assumed in the present investigation. It should be realized that even with the approximate model the dominance of the two phonon scattering in the temperature range 12 to 18 K is appreciable. One of the authors (SSK) is pleased to acknowledge CS. IR New Delhi, India for award of fellowship during the period of this investigation. =
0.0139 m
.
[1] [2] [3] [4] [5] ]6] [7] [8] [9] [10] [11] [12]
390
R.A. Stradling and R.A. Wood, J. Phys. C. Solid State Physics 3 (1970) 2425. K.L. Ngai and E.J. Johnson, Phys. Rev. Lett. 29 (1972) 1607. A.K. Ganguly and K.L. Ngai, Phys. Rev. B 8(1973) 5654. DL. Price, J.M. Rowe and R.M. Milklow, Phys. Rev. B 3 (1971) 1268. J.R. Sandercock, Solid State Commun. 7 (1969) 721. J.J. Whalen and CR. Westgatc, J. AppI. Phys. 43 (1972) 1965. J.P. Maneval and A. Zylbersztein and H.F. Budd, Phys. Rev. Lett. 23 (1969) 848. 11. Kahlert and C. Bauer, Phys. Rev. B 7 (1973) 2670. Sh.M. Kogan, Soy. Phys. Solid State 4 (1963) 1813. R. Stratton, Proc. Roy. Soc. 246 (1957) 406. R. Kaplan and K.L. Ngai, Comm. Solid State Phys. 6(1974)17. D.L. Stierwalt, J. Phys. Soc. Japan. Suppl. 21(1966) 58.
PHYSICS LETFERS
6 October 1975
THE ENERGY RELAXATION DUE TO ELECTRON-TWO SHORT WAVE LENGTH TRANSVERSE ACOUSTIC PHONONS IN SEMICONDUCTORS S.S. KUBAKADDI and B.S. KRISHNAMURTHY Department of Physics, Karnatak Universiti’, Dharwar, Karnataka, India Received 18 August 1975 The energy relaxation time, due to the interaction of electrons with two short wave length acoustic phonons, is calculated in nondegenerate semiconductors and compared with an experiment in n-type lnSb at low temperatures.
Recent experiments [1, 2] have established that the carrier two-phonon interaction is an important scattering mechanism in semiconductors. In a two-phonon process all phonons can take part unlike the case of one-phonon process. In the latter process the energy and momentum conservation restricts the available phonons to a small fraction of the Brillouin zone. In the present note we wish to examin the contribution of the two short wave length transverse acoustic phonons (2 SWTA) to the carrier energy relaxation process. Following Ganguly and Ngai [3] we write the matrix element for 2 SWTA phonon process (1) Vq,q hD(q, q)/p~2a2(wqwq’)1/2 where D(q, q’) is the 2SWTA phonon effective deformation potential, Wq is the phonon frequency of wave vector q, a is the lattice constant and p is the mass density of the crystal. It should be noted that the deformation potential concept invoked in obtaining eq. (1) requires that Q = q + q’ lies in the long wave length region. In what follows we shall calculate the average energy loss rate due to 2SWTA phonons (both emitted and both absorbed) following the kinetic method assuming a displaced Maxwellian distribution for electrons at electron temperature Te and a simple parabolic band model. We get for the average energy loss rate I ~3
(,~) D
2
(w
+
w’)312
q
m*3/2
~7/2~1/2 (7/2)1/2 exp(7/2)K
~‘
1(7/2)NN’[l
—
exp(y0
—
y)].
(2)
In eq. (2) y = h (w + w’)/kB Te, “~= ~ (u.’ + w’)/k~T, N and N’ are the equilibrium phonon distribution at lattice temperature T and qm is the zone edge value of q. Further, we have assumed for simplicity that = w and wq’ = w’ are independent of the wave vectors. In fact, it is known that, in InSb the transverse acoustic branch of the lattice dispersion is relatively flat [4] in short wave length region. Now we obtain from (2) the electron energy relaxation time, using r~ = (2/3 kB)(dP/dTC), 512h1I2 m*3/2 1 1 / D ~2 ~~ +~ “ -~q~ 2 T2 exp(—y)(y/2)1/2 exp(7/2)K 1 (3) 1(y/2){l +~r~+ [K0(y/2)/K1ey/2)]} q ap in for hw/kBT> 1. In eq. (3) K 1(y/2) and K0(y/2) are the modified Bessel’s functions. Now in order to compare the above theory with an experiment we shall consider an n-type InSb. In fact, the experimental results of the electron power loss as a function of electron temperature, in the range of 1 .5 to 30 K, have been presented by many authors [5—8] It has been suggested that [1, 6, 8] the two zone edge acoustic phonon (2 ZETA) scattering may be responsible for the energy relaxation in the temperature of 10 to 22 K. In fig. 1 we have plotted dp/d Te as a function of T~for various independent processes such as the acoustic phonon (both deformation and piezoelectric) [9], the polar optical phonon [10] and 2 ZETA phonons as well as their combinations using the following parameters: D = 620 eV [11] a = 6.4789 A, h(w + w’)/kB = 124 K (for TA phonons at X point) [12], the deformation potential for the one phonon scattering E1 = 7.2 eV, the piezoelectric constant 13 = 3 X I o~ .
,
389
Volume 54A, number 5
PHYSICS LETTERS /0
/0
6 October 1975
/
_______________________________________________________
d
e
/ ~ /~
Fig. 1. Plot of dP/dTe as a function of electron temperature Tc for n-type InSb. [5] arc shown with the fol3 x , n = 2.2 x 1014 c1n3 andExperimental ~, n = 1.1 X points lOis cin3, Theoretical curves for lowing carrier concentrations: •, n = 5 X 1013 cm the following scattering mechanisms: (a) acoustic + polar optical + 2 ZETA, (b) acoustic + 2 ZETA, (c) 2 ZETA, (d) acoustic + polar optical, (e) polar optical and (f) acoustic.
dyn1/2 cm_l,m*
3,OD 278 K, e 0,p = 5.82 g cm 0 17.18 and e~= 15.68. It is interesting to note (10 to 20 K) that there is a good agreement between the curve a (combination of all one phonon and 2 ZETA phonon processes) and the experimental points of [5] However, the curve c indicates that in the temperature range of 12 to 18 K the power loss due to 2 ZETA phonons scattering is dominant over the one phonon scattering mechanisms and the sudden increase in the observed power loss is accounted for in this temperature range. It may be noted that the kinetic method and the perturbation theory is questionable as pointed out [7, 8] in the example considered for experimental comparison. However, since no quantum treatment exists and the criterion for the validity of the method is weakly violated as it is in the cross over region, the applicability of the method is assumed in the present investigation. It should be realized that even with the approximate model the dominance of the two phonon scattering in the temperature range 12 to 18 K is appreciable. One of the authors (SSK) is pleased to acknowledge CS. IR New Delhi, India for award of fellowship during the period of this investigation. =
0.0139 m
.
[1] [2] [3] [4] [5] ]6] [7] [8] [9] [10] [11] [12]
390
R.A. Stradling and R.A. Wood, J. Phys. C. Solid State Physics 3 (1970) 2425. K.L. Ngai and E.J. Johnson, Phys. Rev. Lett. 29 (1972) 1607. A.K. Ganguly and K.L. Ngai, Phys. Rev. B 8(1973) 5654. DL. Price, J.M. Rowe and R.M. Milklow, Phys. Rev. B 3 (1971) 1268. J.R. Sandercock, Solid State Commun. 7 (1969) 721. J.J. Whalen and CR. Westgatc, J. AppI. Phys. 43 (1972) 1965. J.P. Maneval and A. Zylbersztein and H.F. Budd, Phys. Rev. Lett. 23 (1969) 848. 11. Kahlert and C. Bauer, Phys. Rev. B 7 (1973) 2670. Sh.M. Kogan, Soy. Phys. Solid State 4 (1963) 1813. R. Stratton, Proc. Roy. Soc. 246 (1957) 406. R. Kaplan and K.L. Ngai, Comm. Solid State Phys. 6(1974)17. D.L. Stierwalt, J. Phys. Soc. Japan. Suppl. 21(1966) 58.