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Scattering mechanisms in p-type GaSb in the temperature range 30–300°K
J. Phys. Chem. Solids, 1973, Vol. 34, pp. 2231-2234. Pergamon Press. Printed in Greal Britain
SCATTERING MECHANISMS IN p-TYPE GaSb IN THE TEMPERATURE RANGE 3 0 - 3 0 0 ~ D. BARJON, A. RAYMOND, B. PISTOULET and J. L. ROBERT Centre d'Etudes d'Electronique des Solides, associe au C.N.R.S., Universite des Sciences et Techniques du Langnedoc, 34060 Montpellier, Cedex, France (Received 27 July 1972; in revised form 3 M a y 1973)
Abstract--In a former paper[l], we have shown that the magnetoresistance coefficient in p-type GaSb(l+~) remains close to 1 at 77~ and that the mobilities ratio remains equal to 6 in the temperature range 77-300~ We show from these results that between 30 and 300~ the predominant scattering is a mixed scattering by lattice vibrations and ionized impurities. Interband scattering is the predominant process for light holes, while heavy holes undergo intraband scattering. In this temperature range, this mechanism accounts for the mobility variation, a result which had not been found so far on p -type GaSb. for = a ( k T ) 3/2
1. I N T R O D U C T I O N
IN A FORMERpaper [1], we have shown that the magnetoresistance coefficient
E is the energy and " a " is a constant. For ionized impurities scattering (3):
1 + ~: - ('r3}(r) of p-type GaSh remains almost equal to 1 at 77~ and that the mobilities ratio remains constant and close to 6 in the temperature range 77-300~ In this paper we intend to deduce from these results the scattering m e c h a n i s m involved as well as the mobility variations of each type of carriers between 30 and 300~ and the variation of the Hall coefficient r = (r=}/(r)2; r is found to remain nearly constant and equal to 1 in this temperature range. We show that the variations of the mobility vs temperature may be fully explained between 30 and 300~ by a mixed scattering of lattice vibrations and ionized impurities, and that optical mode scattering is ineffective in this temperature range.
[ E ~':=
=
rot = b ( k T ) 3/=
b is a slowly varying function of the energy, which can be considered in first approximation as a constant when we use the method of Conwell and Weisskopf[4]. We suppose that the sample is partly compensated and that the number of ionized centers is constant at low temperature. If both scattering mechanisms are assumed to be independant, the relaxation time is given by:
1
I
1
T
TL
TI
- =--+-
whence
2. V A L U E S O F r A N D 1 + ~ I N T H E C A S E OF MIXED SCATTERING BY LATTICE VIBRATIONS AND IONIZED IMPURITIES
ToLX 312
r-x~+~
For lattice scattering, the expression of the relaxation time is written (2):
with
[ E ~-t/2
x = ~ -E~ 2231
and
/ 3 = ttie o I _ b(kT)_3
(I)
2232
D. BARJON
et al.
The average value of the relaxation time is written:
"
{
1.5
~ x 3/2 exp ( -
x)~'(x) dx
f x 3n exp ( - x ) dx =
4
f/
x3/21"(x)exp ( - x ) dx. I 0"5
The expressions of r and 1 + ~ are then:
~ X9/2exp r=
3N/H -o m
(-- X) dx
4 [fo|
(x2+/3) 3
(1 + ~) vs/3//3 + 1.
d x . ~'~ x 3 exp ( - x ) dx J0 (x2+/3)
[ fff x9/2exp (- x) dx] 2 (x2 + /3) 2
The variations of these two functions vs /3/(/3+1) are given on Figs. 1 and 2. L e t ' s notice in particular that for/3 = 0 (only lattice scattering) r = 1,18 and 1 + ~ = 1,275. On the other hand, for/3/(/3 + 1) = 1 (only ionized impurities scattering) r = 1,93 and 1 + ~ = 1,57.
Moreover, in order that r and 1 + ~ should remain nearly equal to 1, /3/(/3 + 1) must be included b e t w e e n 0,1 and 0,8. 3. STUDY OF THE MOBILITY VARIATION VS THE TEMPERATURE
The expression of the mobility is: e(~! met
where mc~ is the conduction mass of the considered carrier and (z,) =
--
t.5--
t 0
I
dx]2
f~ X6 exp ( - x )
2
k
Fig. 2. Variation of the magnetoresistance coefficient
(x 2+/3)2
(x 2+/3) l+~=Jo
#/J7+l
J
05
,B/J+I Fig. 1. Variation of r =
vs/3//3 + 1.
. / x3/2 \ a(kT) -sn \x--y-~l.
The variation of (x3~21(x2+/3))vs/3/(/3 + 1) is given on Fig. 3. If the value of the ratio/3/(/3 + 1) is fixed at a given temperature, it is possible to determine the mobility variation vs the temperature. On Fig. 4 we see that the best fit is obtained when we choose /3/(/3 + 1)= 0,83 at 77~ When the t e m p e r a t u r e varies f r o m 77 to 300~ this ratio decreases f r o m 0,83 to 0,076 and so at a high t e m p e r a t u r e one tends to have a mixed scattering in which the lattice scattering is preponderant. In this t e m p e r a t u r e range, 1 + ~ and r remain equal to 1 with a 10 per cent accuracy. Moreover, the fact that the mobilities ratio remains c o n s t a n t [ I - 5 ] can be accounted for
SCATTERING MECHANISMS IN p-TYPE GaSb
A X e4~X
0.~
V
I
O,5
I
X 312
Fig. 3. Variation of the expression < x - ~ > v s
/3 /3+]"
2233
laxation time is the same for both types of holes. In the case of ionized impurities scattering, Argyres and Adams[7], and Herring[8] have shown that the probability of a collision is also proportional to the number of possible final states. Taking into account the great density of ionized impurities, the preponderant scattering process for the light carriers is an interband scattering, as it has been proved by Brown and Bray [9] for p - t y p e Germanium. Therefore, the relaxation times, of both types of carriers can be considered as equal and so, in the case of a mixed scattering, the values of fl are the same for both types of carriers in the temperature range 30-300~
0,9
4. DISCUSSION
F\ 0.~
E
2
+
The scattering process above considered accounts perfectly for the results obtained between 30 and 300~ However, above the D e b y e temperature (OD= 280~ for GaSb [10]) the mixed scattering by lattice vibrations and optical mode [11] allows us to account for the observed results. Indeed, in the case of optical mode ( T > 0 D ) the relaxation time is given by[12]: [ E y/~
o.i
I
I
I
I
io
2o
~o
50
T,
I
I
z5 too
I
I
Top= c(kT) '/2.
2oo 3oo
~
Fig. 4. Light holes mobility vs temperature: + Experimental points, ~Theoretical points for /3 /3+1= 0.83 at 77~ ----Theoretical points for ~+1 = 0.85 at 77~ by the considered scattering process. Brooks [6] has shown for Germanium and Silicon, that the lattice scattering is an interband scattering for light holes and an intraband scattering for heavy holes. Therefore, the probability of a collision being proportional to the number of possible final states, the re-
By using relation (1) we obtain: TOLx312
T=X2+/3, with /3' = a ( k T ) 2. The variations of r and 1 + ~ are plotted on curves 1 and 2. The coefficients r and 1 + ~ remain nearly equal to 1 w h e n / 3 ' / ( f i ' + 1) is included between 0,8 and 0,1. In this hypothesis,
2234
D. BARJON et al.
the mobilities ratio remains once more equal to the reverse of the conduction masses ratio. However, such a mixed scattering can't account for the variations of carriers mobility as soon as the temperature is lower than the Debye temperature. In the temperature range 4,2-20~ the mixed scattering by optical mode (T <~ 0o) and ionized impurities seems probable. Indeed, in this case Ehrenreich [13, 14] has shown that in GaAs the carrier mobility is proportion to T ~ : such a variation of the mobility is observed on GaSb in the considered temperature range. In short, at 77~ this study shows that the magnetoresistance coefficient (1 + ~) as well as r are very close to 1. In the temperature range 77-300~ the mixed scattering by lattice vibrations and ionized impurities shows that r remains alsmost equal to 1. Moreover, this scattering process allows us to compute the mobility variations in the temperature range 30-300~ a result which had not been found so far on p-type GaSb.
REFERENCES 1. ROBERT J. L., PISTOULET B., BARJON D. and RAYMOND A., J. Phys. Chem. Solids 34, 2221 (1973). 2. BEER A. C., Galvanomagnetic effects in semiconductor, Suppl. 4 to Solid State Physics, p. 111. Academic Press, New York (1963). 3. BROOKS H., Phys. Rev. 83, 879 (1951). 4. BEER A. C., Galvanomagnetic effects in semiconductors, Suppl. 4 to Solid State Physics, p. 113. Academic Press, New York (1963). 5. LEPETRE T. and ROBERT J. L., C.R. Acad. Sci. Paris, B325, 275 (1972). 6. BROOKS H., Advances in Electronics and Electron Physics (Edited by L. Marton), Vol. 7, p. 152. Academic Press, New York (1955). 7. ARGYRES P. N. and ADAMS E. N., Phys. Rev. 104, 900 (1956). 8. HERRING C., Bell Syst. Tech J. 34, 237 (1955). 9. BROWN D. M. and BRAY R., Phys. Rev. 127, 1593 (1962). 10. SIROTA N. N., Semiconductors and Semimetals (Edited by R. K. Willardson and A. C. Beer), Vol. 4, p. 130. Academic Press, New York (1968). 11. WILEY J. D. and DIDOMENICO M., Phys. Rev. B2, 427 (1970). 12. TSIDIL'KOVSKII I. M., Thermomagnetic Effects in Semiconductors. Infosearch, London. 13. EHRENREICH H., J. Phys. Chem. Solids 8, 130 (1959). 14. EHRENREICH H,, Phys. Rev. 120, 1951 (1960).
SCATTERING MECHANISMS IN p-TYPE GaSb IN THE TEMPERATURE RANGE 3 0 - 3 0 0 ~ D. BARJON, A. RAYMOND, B. PISTOULET and J. L. ROBERT Centre d'Etudes d'Electronique des Solides, associe au C.N.R.S., Universite des Sciences et Techniques du Langnedoc, 34060 Montpellier, Cedex, France (Received 27 July 1972; in revised form 3 M a y 1973)
Abstract--In a former paper[l], we have shown that the magnetoresistance coefficient in p-type GaSb(l+~) remains close to 1 at 77~ and that the mobilities ratio remains equal to 6 in the temperature range 77-300~ We show from these results that between 30 and 300~ the predominant scattering is a mixed scattering by lattice vibrations and ionized impurities. Interband scattering is the predominant process for light holes, while heavy holes undergo intraband scattering. In this temperature range, this mechanism accounts for the mobility variation, a result which had not been found so far on p -type GaSb. for = a ( k T ) 3/2
1. I N T R O D U C T I O N
IN A FORMERpaper [1], we have shown that the magnetoresistance coefficient
E is the energy and " a " is a constant. For ionized impurities scattering (3):
1 + ~: - ('r3}(r) of p-type GaSh remains almost equal to 1 at 77~ and that the mobilities ratio remains constant and close to 6 in the temperature range 77-300~ In this paper we intend to deduce from these results the scattering m e c h a n i s m involved as well as the mobility variations of each type of carriers between 30 and 300~ and the variation of the Hall coefficient r = (r=}/(r)2; r is found to remain nearly constant and equal to 1 in this temperature range. We show that the variations of the mobility vs temperature may be fully explained between 30 and 300~ by a mixed scattering of lattice vibrations and ionized impurities, and that optical mode scattering is ineffective in this temperature range.
[ E ~':=
=
rot = b ( k T ) 3/=
b is a slowly varying function of the energy, which can be considered in first approximation as a constant when we use the method of Conwell and Weisskopf[4]. We suppose that the sample is partly compensated and that the number of ionized centers is constant at low temperature. If both scattering mechanisms are assumed to be independant, the relaxation time is given by:
1
I
1
T
TL
TI
- =--+-
whence
2. V A L U E S O F r A N D 1 + ~ I N T H E C A S E OF MIXED SCATTERING BY LATTICE VIBRATIONS AND IONIZED IMPURITIES
ToLX 312
r-x~+~
For lattice scattering, the expression of the relaxation time is written (2):
with
[ E ~-t/2
x = ~ -E~ 2231
and
/ 3 = ttie o I _ b(kT)_3
(I)
2232
D. BARJON
et al.
The average value of the relaxation time is written:
"
{
1.5
~ x 3/2 exp ( -
x)~'(x) dx
f x 3n exp ( - x ) dx =
4
f/
x3/21"(x)exp ( - x ) dx. I 0"5
The expressions of r and 1 + ~ are then:
~ X9/2exp r=
3N/H -o m
(-- X) dx
4 [fo|
(x2+/3) 3
(1 + ~) vs/3//3 + 1.
d x . ~'~ x 3 exp ( - x ) dx J0 (x2+/3)
[ fff x9/2exp (- x) dx] 2 (x2 + /3) 2
The variations of these two functions vs /3/(/3+1) are given on Figs. 1 and 2. L e t ' s notice in particular that for/3 = 0 (only lattice scattering) r = 1,18 and 1 + ~ = 1,275. On the other hand, for/3/(/3 + 1) = 1 (only ionized impurities scattering) r = 1,93 and 1 + ~ = 1,57.
Moreover, in order that r and 1 + ~ should remain nearly equal to 1, /3/(/3 + 1) must be included b e t w e e n 0,1 and 0,8. 3. STUDY OF THE MOBILITY VARIATION VS THE TEMPERATURE
The expression of the mobility is: e(~! met
where mc~ is the conduction mass of the considered carrier and (z,) =
--
t.5--
t 0
I
dx]2
f~ X6 exp ( - x )
2
k
Fig. 2. Variation of the magnetoresistance coefficient
(x 2+/3)2
(x 2+/3) l+~=Jo
#/J7+l
J
05
,B/J+I Fig. 1. Variation of r =
vs/3//3 + 1.
. / x3/2 \ a(kT) -sn \x--y-~l.
The variation of (x3~21(x2+/3))vs/3/(/3 + 1) is given on Fig. 3. If the value of the ratio/3/(/3 + 1) is fixed at a given temperature, it is possible to determine the mobility variation vs the temperature. On Fig. 4 we see that the best fit is obtained when we choose /3/(/3 + 1)= 0,83 at 77~ When the t e m p e r a t u r e varies f r o m 77 to 300~ this ratio decreases f r o m 0,83 to 0,076 and so at a high t e m p e r a t u r e one tends to have a mixed scattering in which the lattice scattering is preponderant. In this t e m p e r a t u r e range, 1 + ~ and r remain equal to 1 with a 10 per cent accuracy. Moreover, the fact that the mobilities ratio remains c o n s t a n t [ I - 5 ] can be accounted for
SCATTERING MECHANISMS IN p-TYPE GaSb
A X e4~X
0.~
V
I
O,5
I
X 312
Fig. 3. Variation of the expression < x - ~ > v s
/3 /3+]"
2233
laxation time is the same for both types of holes. In the case of ionized impurities scattering, Argyres and Adams[7], and Herring[8] have shown that the probability of a collision is also proportional to the number of possible final states. Taking into account the great density of ionized impurities, the preponderant scattering process for the light carriers is an interband scattering, as it has been proved by Brown and Bray [9] for p - t y p e Germanium. Therefore, the relaxation times, of both types of carriers can be considered as equal and so, in the case of a mixed scattering, the values of fl are the same for both types of carriers in the temperature range 30-300~
0,9
4. DISCUSSION
F\ 0.~
E
2
+
The scattering process above considered accounts perfectly for the results obtained between 30 and 300~ However, above the D e b y e temperature (OD= 280~ for GaSb [10]) the mixed scattering by lattice vibrations and optical mode [11] allows us to account for the observed results. Indeed, in the case of optical mode ( T > 0 D ) the relaxation time is given by[12]: [ E y/~
o.i
I
I
I
I
io
2o
~o
50
T,
I
I
z5 too
I
I
Top= c(kT) '/2.
2oo 3oo
~
Fig. 4. Light holes mobility vs temperature: + Experimental points, ~Theoretical points for /3 /3+1= 0.83 at 77~ ----Theoretical points for ~+1 = 0.85 at 77~ by the considered scattering process. Brooks [6] has shown for Germanium and Silicon, that the lattice scattering is an interband scattering for light holes and an intraband scattering for heavy holes. Therefore, the probability of a collision being proportional to the number of possible final states, the re-
By using relation (1) we obtain: TOLx312
T=X2+/3, with /3' = a ( k T ) 2. The variations of r and 1 + ~ are plotted on curves 1 and 2. The coefficients r and 1 + ~ remain nearly equal to 1 w h e n / 3 ' / ( f i ' + 1) is included between 0,8 and 0,1. In this hypothesis,
2234
D. BARJON et al.
the mobilities ratio remains once more equal to the reverse of the conduction masses ratio. However, such a mixed scattering can't account for the variations of carriers mobility as soon as the temperature is lower than the Debye temperature. In the temperature range 4,2-20~ the mixed scattering by optical mode (T <~ 0o) and ionized impurities seems probable. Indeed, in this case Ehrenreich [13, 14] has shown that in GaAs the carrier mobility is proportion to T ~ : such a variation of the mobility is observed on GaSb in the considered temperature range. In short, at 77~ this study shows that the magnetoresistance coefficient (1 + ~) as well as r are very close to 1. In the temperature range 77-300~ the mixed scattering by lattice vibrations and ionized impurities shows that r remains alsmost equal to 1. Moreover, this scattering process allows us to compute the mobility variations in the temperature range 30-300~ a result which had not been found so far on p-type GaSb.
REFERENCES 1. ROBERT J. L., PISTOULET B., BARJON D. and RAYMOND A., J. Phys. Chem. Solids 34, 2221 (1973). 2. BEER A. C., Galvanomagnetic effects in semiconductor, Suppl. 4 to Solid State Physics, p. 111. Academic Press, New York (1963). 3. BROOKS H., Phys. Rev. 83, 879 (1951). 4. BEER A. C., Galvanomagnetic effects in semiconductors, Suppl. 4 to Solid State Physics, p. 113. Academic Press, New York (1963). 5. LEPETRE T. and ROBERT J. L., C.R. Acad. Sci. Paris, B325, 275 (1972). 6. BROOKS H., Advances in Electronics and Electron Physics (Edited by L. Marton), Vol. 7, p. 152. Academic Press, New York (1955). 7. ARGYRES P. N. and ADAMS E. N., Phys. Rev. 104, 900 (1956). 8. HERRING C., Bell Syst. Tech J. 34, 237 (1955). 9. BROWN D. M. and BRAY R., Phys. Rev. 127, 1593 (1962). 10. SIROTA N. N., Semiconductors and Semimetals (Edited by R. K. Willardson and A. C. Beer), Vol. 4, p. 130. Academic Press, New York (1968). 11. WILEY J. D. and DIDOMENICO M., Phys. Rev. B2, 427 (1970). 12. TSIDIL'KOVSKII I. M., Thermomagnetic Effects in Semiconductors. Infosearch, London. 13. EHRENREICH H., J. Phys. Chem. Solids 8, 130 (1959). 14. EHRENREICH H,, Phys. Rev. 120, 1951 (1960).