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Effect of uniaxial stress on energy loss and scattering mechanism in p-type silicon
1. Phys. Chcm. Solids.
1972. Vol. 33. pp. I39- 143.
Pergamon
Press.
Printed
in Great
EFFECT OF UNIAXIAL STRESS SCATTERING MECHANISM Ludwig Boltzmann-lnstitut
Brilain
ON ENERGY LOSS IN p-TYPE SILICON
K. HESS fur Festkiirperphysik und lnstitut fur Angewandte A- 1060 Wien. Austria
(Received
10 December
1970;
in revisedform
14April
AND
Physik der Universitat.
1971)
Abstract-The conductivity by holes in p-type silicon and the warm carrier energy relaxation time are measured as a function of applied uniaxial stress at 90°K. The results are explained in terms of acoustic phonon scattering. The theoretical expressions allow one to determine the deformation potential constant b from measurements at stresses up to 3 x IO9dynlcm?. Furthermore, a negative differential resistivity at low temperatures and high stresses is obtained from theory. ZusammenfassungDie elektrische Leitfahigkeit und Energierelaxationszeit von p-Typ Silizium wurde in Abhangigkeit von uniaxialem Druck bei 90°K gemessen. Die Megergebnisse werden theoretisch mit Streuung an akustischen Phononen erkllrt. Aus dem Vergleich zwischen Theorie und Experiment erhalt man die Deformations-Potential-Konstante b. Weiters wird ein negativer differentieller Widerstand bei tiefen Temperaturen und hohen Driicken aus der Theorie erhalten.
1. INTRODUCTION
mation potential constant which is not completely known [ 18,191. Therefore with the same samples we have measured piezoresistance in a (100) direction and deduced the deformation potential constant.
relaxation time T, known as the time constant of energy transfer from the carriers to the lattice, has been studied in the elemental semiconductors germanium, silicon and tellurium [ l-61. The energy loss in p-type silicon was considered to be due to optical and acoustic phonon scattering [4]. Recently Asche[7] showed that the mobility variation with temperature below 150°K in p-type silicon can be explained by acoustic phonon scattering alone if non-parabolicity of the heavy hole band is taken into account. We report measurements of the warm hole 7, in p-type silicon at 90°K under applied uniaxial stress in the range of O-3 x lo9 dyn/ cm*. Further we present theoretical values of T, which are calculated for acoustic phonon scattering as the main energy loss mechanism. The good agreement between experimental and theoretical values shows that this scattering process does not only account for momentum randomization[7] but also for the energy loss in p-type silicon at 90°K. The contributions by light holes are neglected. These calculations require knowledge of the defor-
THE
ENERGY
2.THEORY
Pikus and Bir[ 10,l l] calculated the piezoresistance of p-type germanium and silicon for both very low and extremely high temperatures. Unfortunately, experimental data by Smith [ 121 in the low temperature range 77300°K were compared with the high temperature theoretical results. It has been shown by Asche [ 131 that only (p, - p,Jp, in the limit p + 0 was correctly calculated by Pikus and Bir. Here p is the pressure, and p,, the resistiving under applied stress p. Our calculation shows that the nonlinear variation of the piezoresistance with strain can be explained as a consequence of the population of the heavy hole band at low energy. The band structure of deformed p-type silicon is plotted schematically in Fig. 1, which shows that there is a distinct kink at a certain k-value which depends on the magnitude of the stress. The 139
K. HESS
and by m* above the kink. T,, is the momentum relaxation time for scattering by acoustical phonons given by: T,, = Cm$-3/2E-1~2
6
4
2
2 ARBITRARY
L
6 UNITS
Fig. I. Hole energy vs. k2 for the valence band structure of silicon and germanium under a compressive stress in N( 100) direction (full curves) and with no stress (dashed). Note the kink in the curve.
theory by Pikus and Bir[ 10-l 11 for the interpretation of piezoresistance measurements in the temperature range 77-300X, assumes that all carriers have energies above the kink. To get an estimate of the contribution of the carriers below the kink, we average the momentum relaxation time over the whole energy range. For the sake of simplicity the band structure above the kink is assumed to be undeformed. Since the contribution to piezoresistance of the carriers above the kink is small for reasonable values of the deformation potential constant this assumption seems to be justified. However, this contribution is taken into account through a correction factor F. F is given by: F = pA/pL, where pA/pj, is the ratio of specific resistivity with and without pressure obtained from the transport theory of Pikus and Bir[ 111. If acoustical phonon scattering is assumed to be the dominant scattering mechanism and stress and field are applied in the ( 100) direction, p,, is given by:
rnt is the density of states mass, denoted by m,* below the kink and by m* above the kink in the e(k) relation. C is an energy independent constant. The brackets in equation (1) denote the average over all energies. This average has to be taken separately from 0 to l I and from l 1 to w. This gives for l/p,: ” de exp (-•/kT,)e+-$. de exp (- E/~T~)E} Here N is the normalisation N = (k7’,,)5/*m*““[1
(3).
factor:
+ (eJkT,,)“‘*y*(5/2,
4k~h)(l-S1)1.
(4)
In these equations n is the carrier concentration, T,, the heavy hole temperature, l 1 the energy which corresponds to the kink in the e(k) curve, T the lattice temperature, S, = (m,T/m*)3’* and y* (ct, b) is a tabulated function[ 141. e1 is obtained from the e(k) relation [ll]: El = b(GT-
G,) (m*/2B04,,)
+ 1) I
(5)
where b is the deformation potential constant, components of the strain tensor and B, is a characteristic parameter of the warped surfaces. Equations (1) and (2) yield: l ii are the diagonal
PO -=PII
UP (+o
= FL1 -
(4kTh)2y*(2,
1 - (EJ~T,,)~‘*~* Here m, is the effective mass in ( 100) direction, which is denoted by m,, below the kink
(2)
l JkT,r)(l -&IF)1 O/2, ~,/kT,t) (1 --s,) (6)
where S, = (m*/m,,,)
EFFECT
OF UNIAXIAL
The entire pressure dependence is contained in the stress dependence of l 1 and F. It is well known that this type of equation can describe a negative differential resistance (NDR)[IS, 161. Ridley and Watkins[l6] have shown that in the case of scattering by acoustic phonons a NDR should appear if the ratio of masses S1*j3 is approximately 0,05. pJpO is plotted vs. hole temperature Th in Fig. 2 to
30
90
I50
210
270
^ ., -n
330
E, 9 h2q2/8m*.
s,+h’#/lm.
de[{ (IV,+ 1) exp
de[{(N,+
X
1) exp
(--Au/q)}
I r,+h'O'/8mZ
x exp (-- 4kTdl
(8)
where u is the sound velocity, lnCP the mean free path of the carriers below the kink and I,, the mean free path above the kink. In the case of m,* = m* and I,,,, = I,,, equation (6) reduces to the well known formula for the change of the phonon density with time [ 181. In equation (6) N,, is the effective density of states in the deformed crystal, which is derived from the assumption that the carrier density is not changed when stress is applied to the crystal: lo’& l 1/* exp (-e/k
Th)
+ m*3/2 I O&E&2 exp (- l /k T,,) El > = ( 1/NC)m*3’2 I mde l 1’* exp (-e/k 0
T,,)
(9)
N, is the effective density of states in the undeformed crystal. The total energy loss (de/ at), is given by [ 181:
(($)) x i-q
= (!,$r’
1-w
s” (%)/xp
,EJkTh) & (;+T)
L&
(7)
where q is the acoustic wave vector. Now the time derivative of the phonon density (aN,/ ar), can be written as:
I
+ 2, i;‘“N ac CP
00
390
point out the strong dependence of pP on the hole temperature T,,. A NDR was found experimentally with highly deformed p-type germanium [ 171 at low temperature. To calculate the energy loss we use a similar procedure. The change of energy loss due to the deformation is neglected for carriers above the kink in the e(k) curve. Furthermore, we assume:
141
x exp (- e/kT,,)]
(l/N,,)(m,*“”
Th
Fig. 2. pP/pO vs. hole temperature. S, = 2.55, S, = 0,21, p = 1O’Odyn/cm2.
X
STRESS
(-hulq)
x I:pq($$)
q3 (10)
where @N&r),,,, is the time rate of change of the density N, produced by the holes in a band of density of states mass m,*. @N&t),,,. is the corresponding quantity, with mass m*. In equation ( 12) the quantity A is defined as
-N,}
A = N,,/N,.
(11)
K. HESS
142
In the warm electron region T,, is approximately equal to the lattice temperature T. With this assumption we obtain: 0.8
+ exp (-
l JkT,J
I
.
(12) 0.6
3. EXPERIMENTAL
The apparatus for the determination of the energy relaxation time T, of warm carriers from the phase dependent harmonic mixing signal has been described previously[S]. The arrangement for applying uniaxial stress to the sample via a piston is shown in Fig. 3. The sample is mounted in a ridged waveguide.
-P 0.4 I x
b
103
2
dYn/&
Fig. 4. Comparison between measured and calculated values of p,,/p, vs. p for p-type silicon at 90°K. The directions of applied stress and electric field are (100). The deformation potential constant b is assumed to be 2 eV. Curve (a) calculated from equation (4). Curve (b) according to Pikus and Bir.
Table 1. Constants used in the calculation m,,, and m,T are taken.from Ref. [91 wtl = 0~196n1, tn: = 0.24
Fig. 3. Experimental arrangement for the measurements of the energy relaxation time 7, and the piezoresistance.
Filamentary samples have been prepared from p-type silicon having a resistivity of 320 R-cm and a mobility of 700 cm*/vs. at room temperature. At 90°K the resistivity is 25 R-cm and the mobility 9000 cm*/vs. Typical dimensions are 0,08 X 0,07 X 2,0 cm”. Aluminium contacts were alloyed to the sample in a hydrogen atmosphere. The piezoresistance was measured with a Wheatstone bridge. 4. RESULTS
Figure 4 shows a comparison of measured and calculated values of pI,,pO vs. p for p-type silicon at 90°K. The calculated pJpO curve (a) fits the experimental results in the whole pressure range investigated better than curve
m,,
ttl* = 0,49m,, B, = I.1
b= 2 eV F = I.03
(b) calculated according to Pikus and Bir. The constants used in the calculation are listed in Table 1. The nonlinear behaviour of the piezoresistance cannot be explained in terms of the theory of Pikus and Bir. Theoretical values deduced from their theory are included in Fig. 4 (curve b) for comparison. Measured values of the ratio of energy relaxation times are plotted vs. stress in Fig. 5. For pressures above 2 x IO9 dyn/cm* the theoretical curve is indicated. Because of the limitation expressed by equation (5) the calculations would not be valid at lower pressures. 5. CONCLUSIONS
It is shown that the dependence of the energy relaxation time on uniaxial stress in the warm carrier region at 90°K can be des-
EFFECT
OF UNIAXIAL
STRESS
143
conductivity of holes in p-Si is almost negligible if no stress is applied. Our work shows that this conclusion may be extended to include piezoresistance and the energy loss mechanism in the warm carrier range with and without applied stress. Acknowledgement-The author wishes to thank Prof. Dr. K. Seeger and Dr. H. Kahlert for the continuous guidance during the course of this work and Dr. G. Bauer for carefully reading the manuscript.
,I 0
-P 4
I x IO9
2
dYnh2
3
Fig. 5. Comparison between measured and calculated values of T,~/T,” for p-type silicon at 90°K. The direction of stress and electric field is ( 100); b = 2 eV. The error bars indicate the experimental error in rcD.
cribed consistently with the same model as was used to explain the piezoresistance. This model interprets the nonlinear behaviour of the piezoresistance and energy relaxation time in a satisfactory way by taking into account of the contribution of the carriers below the kink in the e(k) curve. It allows one to determine deformation potential constants from measurements at stresses up to 3 X lo” dyn/cm’. It should be emphasized that acoustical phonon scattering was the only scattering mechanism involved in the present calculations. Asche[7] showed that the contribution of optical phonon scattering to the
REFERENCES 1. GIBSON A. E., GRANVILLE J. W. and PAIGE E.G.S.,J.Phys.Che~n.So/idsl9.198~1961). 2. MORGAN T. N. and KELLY C. E.. Phys. Reu. 137.A 1573 (1965). 3. SEEGER K.. Z. Phys. 172.68 ( 1963). 4. HESS K. and SEEGER K.. Z. Phys. 218.43 I (19691. 5. KAHLERT H.. HESS K. and SEEGER K.. Solid Sfotc Commun. 7, I 149 ( 1969). 6. HOLM-KENNEDY J. W. and CHAMPLIN K. S.. Appl. Phys. Letf. 15. 382 (19691. 7. ASCHE M., VON BORZESKOWSKI J. and SARBEI 0. G.. Phys. Stntus Solidi 38.357 (1970). 8. RISKAER 8.. Phy. Rev. 152,845 (1966). 9. HENSEL J. C. and FEHER G. H., Phys. Rev. 129, 1041 (1963). 10. PIKUS G. E. and BIR G. L.. Fii. fuerd. Tela 1, 1642 (19691. 11. PIKUS G. E. and BIR G. L.. Soviet Phys. solid State 1, 12 ( 1960). 12. SMITH G. S.. Phys. Rev. 94,42 (1954). 13. ASCHE M.. SARBEI 0. G. and VASETZKI V. M., Phps. Status Solidi 18. 749 (1966). 14. ABRAMOVlTZ M. and STEGUN J. A., Handbook of Marhetmricul Fmctions. pp. 261 Dover, New York (1965). 15. OMAR M. A., Phvs. Rev. 171,925 (1968). 16. RIDLEY B. K. and WATKINS T. D.. Proc. Phps. Sot. Land. 78,293 ( I96 1). 17. KASTAL’SKII A. A. and RYVKIN S. M.. Souief Phps. Semicond. 1,523 (1967). 18. CONWELL E. M., Higli Field Trtrnsporr i,t Semiconductors. Academic Press, New York and London (1967).
1972. Vol. 33. pp. I39- 143.
Pergamon
Press.
Printed
in Great
EFFECT OF UNIAXIAL STRESS SCATTERING MECHANISM Ludwig Boltzmann-lnstitut
Brilain
ON ENERGY LOSS IN p-TYPE SILICON
K. HESS fur Festkiirperphysik und lnstitut fur Angewandte A- 1060 Wien. Austria
(Received
10 December
1970;
in revisedform
14April
AND
Physik der Universitat.
1971)
Abstract-The conductivity by holes in p-type silicon and the warm carrier energy relaxation time are measured as a function of applied uniaxial stress at 90°K. The results are explained in terms of acoustic phonon scattering. The theoretical expressions allow one to determine the deformation potential constant b from measurements at stresses up to 3 x IO9dynlcm?. Furthermore, a negative differential resistivity at low temperatures and high stresses is obtained from theory. ZusammenfassungDie elektrische Leitfahigkeit und Energierelaxationszeit von p-Typ Silizium wurde in Abhangigkeit von uniaxialem Druck bei 90°K gemessen. Die Megergebnisse werden theoretisch mit Streuung an akustischen Phononen erkllrt. Aus dem Vergleich zwischen Theorie und Experiment erhalt man die Deformations-Potential-Konstante b. Weiters wird ein negativer differentieller Widerstand bei tiefen Temperaturen und hohen Driicken aus der Theorie erhalten.
1. INTRODUCTION
mation potential constant which is not completely known [ 18,191. Therefore with the same samples we have measured piezoresistance in a (100) direction and deduced the deformation potential constant.
relaxation time T, known as the time constant of energy transfer from the carriers to the lattice, has been studied in the elemental semiconductors germanium, silicon and tellurium [ l-61. The energy loss in p-type silicon was considered to be due to optical and acoustic phonon scattering [4]. Recently Asche[7] showed that the mobility variation with temperature below 150°K in p-type silicon can be explained by acoustic phonon scattering alone if non-parabolicity of the heavy hole band is taken into account. We report measurements of the warm hole 7, in p-type silicon at 90°K under applied uniaxial stress in the range of O-3 x lo9 dyn/ cm*. Further we present theoretical values of T, which are calculated for acoustic phonon scattering as the main energy loss mechanism. The good agreement between experimental and theoretical values shows that this scattering process does not only account for momentum randomization[7] but also for the energy loss in p-type silicon at 90°K. The contributions by light holes are neglected. These calculations require knowledge of the defor-
THE
ENERGY
2.THEORY
Pikus and Bir[ 10,l l] calculated the piezoresistance of p-type germanium and silicon for both very low and extremely high temperatures. Unfortunately, experimental data by Smith [ 121 in the low temperature range 77300°K were compared with the high temperature theoretical results. It has been shown by Asche [ 131 that only (p, - p,Jp, in the limit p + 0 was correctly calculated by Pikus and Bir. Here p is the pressure, and p,, the resistiving under applied stress p. Our calculation shows that the nonlinear variation of the piezoresistance with strain can be explained as a consequence of the population of the heavy hole band at low energy. The band structure of deformed p-type silicon is plotted schematically in Fig. 1, which shows that there is a distinct kink at a certain k-value which depends on the magnitude of the stress. The 139
K. HESS
and by m* above the kink. T,, is the momentum relaxation time for scattering by acoustical phonons given by: T,, = Cm$-3/2E-1~2
6
4
2
2 ARBITRARY
L
6 UNITS
Fig. I. Hole energy vs. k2 for the valence band structure of silicon and germanium under a compressive stress in N( 100) direction (full curves) and with no stress (dashed). Note the kink in the curve.
theory by Pikus and Bir[ 10-l 11 for the interpretation of piezoresistance measurements in the temperature range 77-300X, assumes that all carriers have energies above the kink. To get an estimate of the contribution of the carriers below the kink, we average the momentum relaxation time over the whole energy range. For the sake of simplicity the band structure above the kink is assumed to be undeformed. Since the contribution to piezoresistance of the carriers above the kink is small for reasonable values of the deformation potential constant this assumption seems to be justified. However, this contribution is taken into account through a correction factor F. F is given by: F = pA/pL, where pA/pj, is the ratio of specific resistivity with and without pressure obtained from the transport theory of Pikus and Bir[ 111. If acoustical phonon scattering is assumed to be the dominant scattering mechanism and stress and field are applied in the ( 100) direction, p,, is given by:
rnt is the density of states mass, denoted by m,* below the kink and by m* above the kink in the e(k) relation. C is an energy independent constant. The brackets in equation (1) denote the average over all energies. This average has to be taken separately from 0 to l I and from l 1 to w. This gives for l/p,: ” de exp (-•/kT,)e+-$. de exp (- E/~T~)E} Here N is the normalisation N = (k7’,,)5/*m*““[1
(3).
factor:
+ (eJkT,,)“‘*y*(5/2,
4k~h)(l-S1)1.
(4)
In these equations n is the carrier concentration, T,, the heavy hole temperature, l 1 the energy which corresponds to the kink in the e(k) curve, T the lattice temperature, S, = (m,T/m*)3’* and y* (ct, b) is a tabulated function[ 141. e1 is obtained from the e(k) relation [ll]: El = b(GT-
G,) (m*/2B04,,)
+ 1) I
(5)
where b is the deformation potential constant, components of the strain tensor and B, is a characteristic parameter of the warped surfaces. Equations (1) and (2) yield: l ii are the diagonal
PO -=PII
UP (+o
= FL1 -
(4kTh)2y*(2,
1 - (EJ~T,,)~‘*~* Here m, is the effective mass in ( 100) direction, which is denoted by m,, below the kink
(2)
l JkT,r)(l -&IF)1 O/2, ~,/kT,t) (1 --s,) (6)
where S, = (m*/m,,,)
EFFECT
OF UNIAXIAL
The entire pressure dependence is contained in the stress dependence of l 1 and F. It is well known that this type of equation can describe a negative differential resistance (NDR)[IS, 161. Ridley and Watkins[l6] have shown that in the case of scattering by acoustic phonons a NDR should appear if the ratio of masses S1*j3 is approximately 0,05. pJpO is plotted vs. hole temperature Th in Fig. 2 to
30
90
I50
210
270
^ ., -n
330
E, 9 h2q2/8m*.
s,+h’#/lm.
de[{ (IV,+ 1) exp
de[{(N,+
X
1) exp
(--Au/q)}
I r,+h'O'/8mZ
x exp (-- 4kTdl
(8)
where u is the sound velocity, lnCP the mean free path of the carriers below the kink and I,, the mean free path above the kink. In the case of m,* = m* and I,,,, = I,,, equation (6) reduces to the well known formula for the change of the phonon density with time [ 181. In equation (6) N,, is the effective density of states in the deformed crystal, which is derived from the assumption that the carrier density is not changed when stress is applied to the crystal: lo’& l 1/* exp (-e/k
Th)
+ m*3/2 I O&E&2 exp (- l /k T,,) El > = ( 1/NC)m*3’2 I mde l 1’* exp (-e/k 0
T,,)
(9)
N, is the effective density of states in the undeformed crystal. The total energy loss (de/ at), is given by [ 181:
(($)) x i-q
= (!,$r’
1-w
s” (%)/xp
,EJkTh) & (;+T)
L&
(7)
where q is the acoustic wave vector. Now the time derivative of the phonon density (aN,/ ar), can be written as:
I
+ 2, i;‘“N ac CP
00
390
point out the strong dependence of pP on the hole temperature T,,. A NDR was found experimentally with highly deformed p-type germanium [ 171 at low temperature. To calculate the energy loss we use a similar procedure. The change of energy loss due to the deformation is neglected for carriers above the kink in the e(k) curve. Furthermore, we assume:
141
x exp (- e/kT,,)]
(l/N,,)(m,*“”
Th
Fig. 2. pP/pO vs. hole temperature. S, = 2.55, S, = 0,21, p = 1O’Odyn/cm2.
X
STRESS
(-hulq)
x I:pq($$)
q3 (10)
where @N&r),,,, is the time rate of change of the density N, produced by the holes in a band of density of states mass m,*. @N&t),,,. is the corresponding quantity, with mass m*. In equation ( 12) the quantity A is defined as
-N,}
A = N,,/N,.
(11)
K. HESS
142
In the warm electron region T,, is approximately equal to the lattice temperature T. With this assumption we obtain: 0.8
+ exp (-
l JkT,J
I
.
(12) 0.6
3. EXPERIMENTAL
The apparatus for the determination of the energy relaxation time T, of warm carriers from the phase dependent harmonic mixing signal has been described previously[S]. The arrangement for applying uniaxial stress to the sample via a piston is shown in Fig. 3. The sample is mounted in a ridged waveguide.
-P 0.4 I x
b
103
2
dYn/&
Fig. 4. Comparison between measured and calculated values of p,,/p, vs. p for p-type silicon at 90°K. The directions of applied stress and electric field are (100). The deformation potential constant b is assumed to be 2 eV. Curve (a) calculated from equation (4). Curve (b) according to Pikus and Bir.
Table 1. Constants used in the calculation m,,, and m,T are taken.from Ref. [91 wtl = 0~196n1, tn: = 0.24
Fig. 3. Experimental arrangement for the measurements of the energy relaxation time 7, and the piezoresistance.
Filamentary samples have been prepared from p-type silicon having a resistivity of 320 R-cm and a mobility of 700 cm*/vs. at room temperature. At 90°K the resistivity is 25 R-cm and the mobility 9000 cm*/vs. Typical dimensions are 0,08 X 0,07 X 2,0 cm”. Aluminium contacts were alloyed to the sample in a hydrogen atmosphere. The piezoresistance was measured with a Wheatstone bridge. 4. RESULTS
Figure 4 shows a comparison of measured and calculated values of pI,,pO vs. p for p-type silicon at 90°K. The calculated pJpO curve (a) fits the experimental results in the whole pressure range investigated better than curve
m,,
ttl* = 0,49m,, B, = I.1
b= 2 eV F = I.03
(b) calculated according to Pikus and Bir. The constants used in the calculation are listed in Table 1. The nonlinear behaviour of the piezoresistance cannot be explained in terms of the theory of Pikus and Bir. Theoretical values deduced from their theory are included in Fig. 4 (curve b) for comparison. Measured values of the ratio of energy relaxation times are plotted vs. stress in Fig. 5. For pressures above 2 x IO9 dyn/cm* the theoretical curve is indicated. Because of the limitation expressed by equation (5) the calculations would not be valid at lower pressures. 5. CONCLUSIONS
It is shown that the dependence of the energy relaxation time on uniaxial stress in the warm carrier region at 90°K can be des-
EFFECT
OF UNIAXIAL
STRESS
143
conductivity of holes in p-Si is almost negligible if no stress is applied. Our work shows that this conclusion may be extended to include piezoresistance and the energy loss mechanism in the warm carrier range with and without applied stress. Acknowledgement-The author wishes to thank Prof. Dr. K. Seeger and Dr. H. Kahlert for the continuous guidance during the course of this work and Dr. G. Bauer for carefully reading the manuscript.
,I 0
-P 4
I x IO9
2
dYnh2
3
Fig. 5. Comparison between measured and calculated values of T,~/T,” for p-type silicon at 90°K. The direction of stress and electric field is ( 100); b = 2 eV. The error bars indicate the experimental error in rcD.
cribed consistently with the same model as was used to explain the piezoresistance. This model interprets the nonlinear behaviour of the piezoresistance and energy relaxation time in a satisfactory way by taking into account of the contribution of the carriers below the kink in the e(k) curve. It allows one to determine deformation potential constants from measurements at stresses up to 3 X lo” dyn/cm’. It should be emphasized that acoustical phonon scattering was the only scattering mechanism involved in the present calculations. Asche[7] showed that the contribution of optical phonon scattering to the
REFERENCES 1. GIBSON A. E., GRANVILLE J. W. and PAIGE E.G.S.,J.Phys.Che~n.So/idsl9.198~1961). 2. MORGAN T. N. and KELLY C. E.. Phys. Reu. 137.A 1573 (1965). 3. SEEGER K.. Z. Phys. 172.68 ( 1963). 4. HESS K. and SEEGER K.. Z. Phys. 218.43 I (19691. 5. KAHLERT H.. HESS K. and SEEGER K.. Solid Sfotc Commun. 7, I 149 ( 1969). 6. HOLM-KENNEDY J. W. and CHAMPLIN K. S.. Appl. Phys. Letf. 15. 382 (19691. 7. ASCHE M., VON BORZESKOWSKI J. and SARBEI 0. G.. Phys. Stntus Solidi 38.357 (1970). 8. RISKAER 8.. Phy. Rev. 152,845 (1966). 9. HENSEL J. C. and FEHER G. H., Phys. Rev. 129, 1041 (1963). 10. PIKUS G. E. and BIR G. L.. Fii. fuerd. Tela 1, 1642 (19691. 11. PIKUS G. E. and BIR G. L.. Soviet Phys. solid State 1, 12 ( 1960). 12. SMITH G. S.. Phys. Rev. 94,42 (1954). 13. ASCHE M.. SARBEI 0. G. and VASETZKI V. M., Phps. Status Solidi 18. 749 (1966). 14. ABRAMOVlTZ M. and STEGUN J. A., Handbook of Marhetmricul Fmctions. pp. 261 Dover, New York (1965). 15. OMAR M. A., Phvs. Rev. 171,925 (1968). 16. RIDLEY B. K. and WATKINS T. D.. Proc. Phps. Sot. Land. 78,293 ( I96 1). 17. KASTAL’SKII A. A. and RYVKIN S. M.. Souief Phps. Semicond. 1,523 (1967). 18. CONWELL E. M., Higli Field Trtrnsporr i,t Semiconductors. Academic Press, New York and London (1967).