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On the quantum limit behaviour of the magnetoresistance in non-parabolic semiconductors
Solid State Communications, Vol. 27, pp. 1401—1403. ©Pergamon Press Ltd. 1978. Printed in Great Britain.
0038—1098/78/0922—1401 $02.00/0
ON THE QUANTUM LIMIT BEHAVIOUR OF THE MAGNETORESISTANCE IN NON-PARABOLIC SEMICONDUCTORS J. Kossut” and J. Hajdu Institut für Theoretische Physik, der Universität zu Köln, West Germany (Received 14 June 1978 by B. Muhlschlegel) The quantum limit behaviour of the magnetoresistance in narrow-gap semiconductors is investigated. The non-parabolicity of the energy dispersion is taken into account in the density of states. Assuming constant level broadening, both the longitudinal and the transverse magnetoresistance obey asymptotic power laws. The exponents in these power laws are found to be larger than in the case of parabolic energy dispersion. 1. INTRODUCTION
for the peculiarities of the narrow-gap semiconductors
THE QUANTUM LIMIT behaviour of the magnetoresistaiice in semiconductors has been studied in several papers. Adams and Holstein [1] found for B -+00 an asymptotic power law
The results for the exponent a are collected in
~
[61.
Table 2 at the end of this note.
(1)
Table 1. Quantum limit behaviour of the Fermi energy.
where,in the transverse case, a = 5 fo’ short range impurity potentials and a = 3 for long range potentials. In the longitudinal case the corresponding exponents are a=2anda= I.Ifthedampingeffectoftheelectron-
Given is the power 13 ofthe asymptotic law CF Parabolic
Damping $3
r’=o —2
impurity scattering is taken into account by assuming constant level broadening, the exponents are a = 3 and a = 1 in the transverse case [2,31 and a = 1 and a =0 in the longitudinal case [2] for short range and long range potentials respectively. The impurity damping is not merely a small correction but it is essential to prevent the magnetoresistance from diverging at zero temperature. Adams [1] and obtained finite results by taking firstand the Holstein limit T -“0 then performing the integration with respect to the energy. Whereas
Energy dispersion
~
B~
Given by equation (4)
r#o —1
r=o
—5/2
r~#o —5/4
2. OUTLINE OF THE CALCULATIONS We consider in the following the two-band model of
—
a narrow-gap semiconductor [4]. The energy dispersion is taken to be 2k2]11/2 (2) ~ + 6g [Tlw(n+ ~)+ 6~ =
—
+
where X = (n,k~,k 2),e~iSthe energy gap, w = eB/mc, and m is the effective mass at the bottom of the conduction 2lc~2band. Assuming h2m CF + hw(n + ~) (3)
in the transverse case the self-consistent Bornapproximation confirms the results obtained in the constant damping approximation [2, 3]; rather substantial deviations occur in the longitudinal [21. the The aim of the present note is tocase investigate influence of the non-parabolicity of the energy dispersion on the exponent a which has been neglected in the previous works. In order to simplify the calculation we replace the wave functions by Landau functions. This approximation has been proved to be admissible for InSb-type semiconductors [4, 51- Our approach resembles the so-called “older scattering theory” in which the density of states has accounted *
—
—
‘~
the expression (2) can be approximated by —
e~
—
~-
(p,~ 1) + h2k~/2mp~ —
(4)
where p,~= (1 + 4hw(n + ~)/e~}”2.
On leave from the Institute of Physics, Polish Academy of Sciences, Warsaw, Poland.
For InSb at B = 1 T the condition (3) is satisfied for electron concentrations ~e ‘~ 1.8 x 1 0~m3 which is easily met in available samples.
1401
(5)
1402
MAGNETORESISTANCE IN NON-PARABOLIC SEMICONDUCTORS 2 In the most simple damping theoretical approxi~F (i + ~\ —~-~ ~ B~ mation the transverse conductivity is given by ~) 2
j
n 2 1e
=
de
~ A~(e)A~’(e)F~~’()(6)
(—. ~)
where n~is the impurity concentration,f the Fermi distribution function and A~(e) — Vo1k~
(~ ~~j2 + r2
k~
—
(7)
Fur
—
the density of states for fixed n. F~~’(e)A~(e) is the average squared diffusion velocity of the cyclotron center in the transition n n’. Since we approximate the wave function by Landau functions, the expression for F~~’(e) remains the same as in the case of parabolic energy dispersion [2, 3]. F~~’(e) depends on the ratio of the potential range r to the magnetic length R = -s/li/mw. For a degenerate electron gas in the quantum limit
Vol. 27, No. 12 (15)
and consequently AO(CF, F = 0) B512. (16) In order to simplify the calculation of A O(CF) we use the approxunate energy spectrum (4) which yields — (mp~)”2 A~(e) — (2irR)2h -~
~ ([(e
e~(p,~ — 1)/2)2 + F2]”2 + C — e — /p~ — 1)/2 + F2
—
Cg(p,~—
nIe2A~(eF)F~(eF)
F~(eF)~
1B’
r~R
I. B3
r ~R.
e2ne —
[4
~ 2j
(9)
which is the non-parabolic counterpart of the usual parabolic cut-off energy C~= hw/2. Expression (18) and the more accurate cut-off energy [7] e~= ~o 2F, = ~ 1)/2 yield the same asymptotic behaviour
=
..!i+ 2
(18)
1/2
—
—
F~CO
(10)
0 CF
(11)
/
I. B’
r>>R.
In the quantum limit the Fermi energy ~F ~ determined by
J dAo(e). eF
(12)
The most natural choice of the cut-off energy is the minimal energy for which Ao(e, I’ = 0) X
—p00
~
~
with the quantum limit relaxation 2/h, time given by I /T~,(C~.)= A o(eF)D~(eF)R (B° r~R
=
—“
B514.
(19)
T
m
DOO(CF)
In the limit I’ 0~B the expressions (17) (for n = 0) and (13) coincide. This factalso indicates that the approximate energy spectrum (4) is suitable when studying the quantum limit behaviour. An obvious choice of the cut-off energy is now
(8)
The corresponding expression for the longitudinal conductivity is =
1)/2 (17)
—“
with
1/2
=
2 + (27rR)2
I li~ ~_eg[e2+eeteghw(n+~)1) ~2m
-1/2
(13)
3. RESULTS The results are collected in Tables power 1 and 2. shows the exponent in the asymptotic lawTable 1 xB~
(20)
~F6o
The impurity damping reduces the absolute value of $3 by a factor 1/2 both in the parabolic and in the nonparabolic case. On the other hand,non-parabolicity tends to increase the absolute value of $3 by a factor 5/4 both forf=Oandl’O. As Table 2 shows, the non-parabolicity tends to increase the exponent a. Since equation (4) can be viewed as a parabolic dispersion relation with an effective mass m(B) = pm, we may conclude that the increase of the powers a and $3 compared to their values in the parabolic case is mainly due to the magnetic field dependence of this mass. Acknowledgement We thank Dr. Yoshiyuki Ono for helpful discussions. —
is stifi real. Taking this value for ~ we obtain 1/2 Ihw 2 4 egLT+ (21rR)4ne2h2/8m~ —
In the limit B
-“
0o
.
(14)
Vol. 27, No. 12
MAGNETORESISTANCE IN NON-PARABOUC SEMICONDUCTORS
1403
Table 2. Quantum limit behaviour of the magnetoresistance. Given is the power a of the asymptotic law p Energy dispersion
Parabolic
a configuration
Damping
Given by (3)
(4)
a
Experiment
I’ = 0
F *0
F= 0
F
=
0
F*0
Range Transverse
Long Short
3 5
1 3
4 6
4 6
1.5 3.5
0.5_l.9a ~ ~
Longitudinal
Long Short
1 2
0 1
1.5 2.5
1.5 2.5
0.25 1.25
— —
a
Hg1 _~Cd~Te [8,9]
b
InSb [10,11]
REFERENCES 1.
ADAMS E.N.& HOLSTEIN T.D.,J. Phys. C7iem. Solids 10, 254 (1959).
2.
BERGERS D., diplomarbeit, University of Cologne (1976) (unpublished).
3.
BERGERS D. & HAJDU J., Solid State Commun. 20,683 (1976).
4.
ZAWADZKI W. & LAX B,Phys. Rev. Lett. 16, 1001(1966).
5.
LAX B., MAVROIDES J.G., ZEIGER HJ. & KEYES RJ.,Phys. Rev. 122,31(1961).
6. 7.
ZAWADZKI W.,Adv. Phys. 23,435 (1974). GOTZE W. & HAJDU J. (to be published inJ. Phys. C).
8. 9.
DORNHAUS R., NIMTZ G., SCHLABITZ W. & ZAPUNSKI P., Solid State Commun. 15,495 (1974). DORNHAUS R. & NIMTZ G., Solid State Commun. 22,41(1977).
10.
BECKMANN 0., HANAMURA E. & NEURINGER LJ.,Phys. Rev. Lett. 18,773 (1966).
11.
AMIRKHANOV K1iJ. & BASHIROV R.S., Soy. Phys. Semicond. 1,558 (1967).
0038—1098/78/0922—1401 $02.00/0
ON THE QUANTUM LIMIT BEHAVIOUR OF THE MAGNETORESISTANCE IN NON-PARABOLIC SEMICONDUCTORS J. Kossut” and J. Hajdu Institut für Theoretische Physik, der Universität zu Köln, West Germany (Received 14 June 1978 by B. Muhlschlegel) The quantum limit behaviour of the magnetoresistance in narrow-gap semiconductors is investigated. The non-parabolicity of the energy dispersion is taken into account in the density of states. Assuming constant level broadening, both the longitudinal and the transverse magnetoresistance obey asymptotic power laws. The exponents in these power laws are found to be larger than in the case of parabolic energy dispersion. 1. INTRODUCTION
for the peculiarities of the narrow-gap semiconductors
THE QUANTUM LIMIT behaviour of the magnetoresistaiice in semiconductors has been studied in several papers. Adams and Holstein [1] found for B -+00 an asymptotic power law
The results for the exponent a are collected in
~
[61.
Table 2 at the end of this note.
(1)
Table 1. Quantum limit behaviour of the Fermi energy.
where,in the transverse case, a = 5 fo’ short range impurity potentials and a = 3 for long range potentials. In the longitudinal case the corresponding exponents are a=2anda= I.Ifthedampingeffectoftheelectron-
Given is the power 13 ofthe asymptotic law CF Parabolic
Damping $3
r’=o —2
impurity scattering is taken into account by assuming constant level broadening, the exponents are a = 3 and a = 1 in the transverse case [2,31 and a = 1 and a =0 in the longitudinal case [2] for short range and long range potentials respectively. The impurity damping is not merely a small correction but it is essential to prevent the magnetoresistance from diverging at zero temperature. Adams [1] and obtained finite results by taking firstand the Holstein limit T -“0 then performing the integration with respect to the energy. Whereas
Energy dispersion
~
B~
Given by equation (4)
r#o —1
r=o
—5/2
r~#o —5/4
2. OUTLINE OF THE CALCULATIONS We consider in the following the two-band model of
—
a narrow-gap semiconductor [4]. The energy dispersion is taken to be 2k2]11/2 (2) ~ + 6g [Tlw(n+ ~)+ 6~ =
—
+
where X = (n,k~,k 2),e~iSthe energy gap, w = eB/mc, and m is the effective mass at the bottom of the conduction 2lc~2band. Assuming h2m CF + hw(n + ~) (3)
in the transverse case the self-consistent Bornapproximation confirms the results obtained in the constant damping approximation [2, 3]; rather substantial deviations occur in the longitudinal [21. the The aim of the present note is tocase investigate influence of the non-parabolicity of the energy dispersion on the exponent a which has been neglected in the previous works. In order to simplify the calculation we replace the wave functions by Landau functions. This approximation has been proved to be admissible for InSb-type semiconductors [4, 51- Our approach resembles the so-called “older scattering theory” in which the density of states has accounted *
—
—
‘~
the expression (2) can be approximated by —
e~
—
~-
(p,~ 1) + h2k~/2mp~ —
(4)
where p,~= (1 + 4hw(n + ~)/e~}”2.
On leave from the Institute of Physics, Polish Academy of Sciences, Warsaw, Poland.
For InSb at B = 1 T the condition (3) is satisfied for electron concentrations ~e ‘~ 1.8 x 1 0~m3 which is easily met in available samples.
1401
(5)
1402
MAGNETORESISTANCE IN NON-PARABOLIC SEMICONDUCTORS 2 In the most simple damping theoretical approxi~F (i + ~\ —~-~ ~ B~ mation the transverse conductivity is given by ~) 2
j
n 2 1e
=
de
~ A~(e)A~’(e)F~~’()(6)
(—. ~)
where n~is the impurity concentration,f the Fermi distribution function and A~(e) — Vo1k~
(~ ~~j2 + r2
k~
—
(7)
Fur
—
the density of states for fixed n. F~~’(e)A~(e) is the average squared diffusion velocity of the cyclotron center in the transition n n’. Since we approximate the wave function by Landau functions, the expression for F~~’(e) remains the same as in the case of parabolic energy dispersion [2, 3]. F~~’(e) depends on the ratio of the potential range r to the magnetic length R = -s/li/mw. For a degenerate electron gas in the quantum limit
Vol. 27, No. 12 (15)
and consequently AO(CF, F = 0) B512. (16) In order to simplify the calculation of A O(CF) we use the approxunate energy spectrum (4) which yields — (mp~)”2 A~(e) — (2irR)2h -~
~ ([(e
e~(p,~ — 1)/2)2 + F2]”2 + C — e — /p~ — 1)/2 + F2
—
Cg(p,~—
nIe2A~(eF)F~(eF)
F~(eF)~
1B’
r~R
I. B3
r ~R.
e2ne —
[4
~ 2j
(9)
which is the non-parabolic counterpart of the usual parabolic cut-off energy C~= hw/2. Expression (18) and the more accurate cut-off energy [7] e~= ~o 2F, = ~ 1)/2 yield the same asymptotic behaviour
=
..!i+ 2
(18)
1/2
—
—
F~CO
(10)
0 CF
(11)
/
I. B’
r>>R.
In the quantum limit the Fermi energy ~F ~ determined by
J dAo(e). eF
(12)
The most natural choice of the cut-off energy is the minimal energy for which Ao(e, I’ = 0) X
—p00
~
~
with the quantum limit relaxation 2/h, time given by I /T~,(C~.)= A o(eF)D~(eF)R (B° r~R
=
—“
B514.
(19)
T
m
DOO(CF)
In the limit I’ 0~B the expressions (17) (for n = 0) and (13) coincide. This factalso indicates that the approximate energy spectrum (4) is suitable when studying the quantum limit behaviour. An obvious choice of the cut-off energy is now
(8)
The corresponding expression for the longitudinal conductivity is =
1)/2 (17)
—“
with
1/2
=
2 + (27rR)2
I li~ ~_eg[e2+eeteghw(n+~)1) ~2m
-1/2
(13)
3. RESULTS The results are collected in Tables power 1 and 2. shows the exponent in the asymptotic lawTable 1 xB~
(20)
~F6o
The impurity damping reduces the absolute value of $3 by a factor 1/2 both in the parabolic and in the nonparabolic case. On the other hand,non-parabolicity tends to increase the absolute value of $3 by a factor 5/4 both forf=Oandl’O. As Table 2 shows, the non-parabolicity tends to increase the exponent a. Since equation (4) can be viewed as a parabolic dispersion relation with an effective mass m(B) = pm, we may conclude that the increase of the powers a and $3 compared to their values in the parabolic case is mainly due to the magnetic field dependence of this mass. Acknowledgement We thank Dr. Yoshiyuki Ono for helpful discussions. —
is stifi real. Taking this value for ~ we obtain 1/2 Ihw 2 4 egLT+ (21rR)4ne2h2/8m~ —
In the limit B
-“
0o
.
(14)
Vol. 27, No. 12
MAGNETORESISTANCE IN NON-PARABOUC SEMICONDUCTORS
1403
Table 2. Quantum limit behaviour of the magnetoresistance. Given is the power a of the asymptotic law p Energy dispersion
Parabolic
a configuration
Damping
Given by (3)
(4)
a
Experiment
I’ = 0
F *0
F= 0
F
=
0
F*0
Range Transverse
Long Short
3 5
1 3
4 6
4 6
1.5 3.5
0.5_l.9a ~ ~
Longitudinal
Long Short
1 2
0 1
1.5 2.5
1.5 2.5
0.25 1.25
— —
a
Hg1 _~Cd~Te [8,9]
b
InSb [10,11]
REFERENCES 1.
ADAMS E.N.& HOLSTEIN T.D.,J. Phys. C7iem. Solids 10, 254 (1959).
2.
BERGERS D., diplomarbeit, University of Cologne (1976) (unpublished).
3.
BERGERS D. & HAJDU J., Solid State Commun. 20,683 (1976).
4.
ZAWADZKI W. & LAX B,Phys. Rev. Lett. 16, 1001(1966).
5.
LAX B., MAVROIDES J.G., ZEIGER HJ. & KEYES RJ.,Phys. Rev. 122,31(1961).
6. 7.
ZAWADZKI W.,Adv. Phys. 23,435 (1974). GOTZE W. & HAJDU J. (to be published inJ. Phys. C).
8. 9.
DORNHAUS R., NIMTZ G., SCHLABITZ W. & ZAPUNSKI P., Solid State Commun. 15,495 (1974). DORNHAUS R. & NIMTZ G., Solid State Commun. 22,41(1977).
10.
BECKMANN 0., HANAMURA E. & NEURINGER LJ.,Phys. Rev. Lett. 18,773 (1966).
11.
AMIRKHANOV K1iJ. & BASHIROV R.S., Soy. Phys. Semicond. 1,558 (1967).