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Effect of magnetic field on gunn effect in n-InSb
Solid State Communications,
Vol. 9, PP. 1877—1879, 1971. Pergamon Press.
Printed in Great Britain
EFFECT OF MAGNETIC FIELD ON GUNN EFFECT IN n—InSb U.P. Phadke and A.K. Chakravarti Department of Physics, Indian Institute of Technology, New Delhi—29, India
(Received 29 April 1971 by A.R. Verma)
The negative magnetic field coefficient of the threshold field for the Gunn effect in n—InSb at 77°K, which is apparently inconsistent with intervalley electron transfer mechanism, can be satisfactorily explained by purely geometric magnetoresistance effects.
THE GUNN effect stands for high frequency current oscillation, associated with travelling high field domains, in bulk semiconductors above a certain critical threshold field; this, as pointed 1 requires only the out by Ridley and Watkins, existence of a region of negative differential conductivity (ndc) in the I—V characteristics of these semiconductors. The physical mechanism underlying the ndc is the transfer of electrons from the high mobility conduction band to the low mobility ‘satellites’ at high electric fields. Since it is well known that the separation, A, between the conduction band and the upper lying valley decreases with pressure, the negative pressure coefficient of the threshold field for the Gunn effect in n—InSb2 lends support to the validity of this mechanism. This incidentally rules out the ndc due to the nonparabolic nature of the conduction band since it would lead to a positive pressure coefficient for the threshold field. It therefore appears to be well established that the ndc underlying the Gunn effect is due to the intervalley electron transfer mechanism. In a semiconductor with the band gap
and consequently the electron generation rate too slow to prevent the transfer of hot electrons to the upper minima if the observations are made on a short enough time scale (~10~sec). Now a magnetic field has a cooling effect on the carriers and should, therefore, cause an increase in the threshold field. This, in fact, has been observed in n—GaAs by Vorobiv; ~ the effect is quite small and is in confirmation with the Monte Carlo calculations of Boardman et al.4 However, very recently Muller and Ferry ~ have observed an appreciable reduction in the value of the threshold field with the application of a transverse magnetic field in the range 500 G to 3000 G for a 0.015 ~,< 0.015 x 0.135mm n—InSb sample with mobility p~= 5 x 105cm2/Vsec and carrier concentration n = 2 x 10 l4~~~ç3 at 77°K, the observation time was kept below 2.5 nsec to avoid the effects of carrier generation which is a much slower process as compared to the intervalley transfer; a longitudinal magnetic field over the same range gave only a slight decrease in the threshold field. Thus the effect of magnetic field apparently seems to be
E 9 < A (such as n—InSb E~ 0.22eV, A = 0.45eV at of 77°K), one would expect the occurrence the Gunn effect normally to be precluded by the impact ionization of electrons from the valence band into the conduction band. However, as was established by Smith et al.,2 the cross section for impact ionization should be too small
inconsistent with the electron 5 withtransfer a seriesmechof lucid anism. Muller andruled Ferry, arguments, have out most of the plausible explanations for the anomaly. Guetin and Hervouet ~ have observed a similar anomaly in n—GaAs (~ = 5300 cm2/Vsec) and attribute it partly to the geometric magnetoresistance at the
1877
1878
GUNN EFFECT IN n—InSb
contacts. Modifying their results, to take into account the variation of the mobility with the magnetic field, we obtain 2(1 — X(B))2]~2 (1)
E7(B) where E
=
ET(0)[1
+
Vol.9, No. 21
r
~
—
.
,
— -
—
~B
1d field in the pres7
~
/
0 ,
is the magnetoresistance. As X(B) vs. B data is not available for this n—InSb sample, we have
/
i5~
, 7
calculated X(B) from equation (1) for different calues of B using the results of Muller and Ferry5 for E~(B).The results, along with the experimental results of Bate et al.’ for a n—lnSb sample with n = 1.7 x 10’~cm3 at 77°K, for the sake of comparison are plotted in Fig. 1. The magnetoresistance X(B) was found to obey an empirical relation X(B) c~B 0.075 in the region B = 500— 3000 G. Keeping in view the large variations in the value of the magneto-resistance8 due to the inhomogeneities in the sample, the pulling rate of the crystal, surface currents etc. and the essentially qualitative nature of the theory, our results for the variation of the magneto-resistance with the magnetic field, which, in turn, governs the threshold field, are in satisfactory agreement with the experimental resuits of Bate et al.’ In fact a rigorous theory would give
ET(B)
=
ET(0) cosO
(2)
where 0, the Hall angle, should be calculated from appropriate averages over a suitable distribution function. However circumverting the calculation of 0 we deduce a self consistent verification of the essential idea behind the above theory. For a sample with small length to width ratio (1/cu << 1), the Hall field is approximately zero everywhere in the bulk and hence proceeding along the same lines as Guetin and Hervouet 6 it can be shown that
[E~(B)L,~,
=
E~(0)sec 0
Eliminating the Hall angle 0 between equations (2) and (3) we obtain
[E~(B)I 1,~
[E~(B)~j1~,
1
=
2 [ET(0)1 (4)
So the experimental verification of the relation (4) which involves the measurement of threshold fields in presence of a transverse
/
~ 70.
FIG. 1. Shows the variation of the magnetoresistance X(B) with the magnetic field B. The solid curve denotes our results calculated from the experimental results of Muller and Ferry ~ for 0.015 x 0.015 x 0.35nmm n—InSb sample witha carrier concentration = 2 x 10 ‘4cm~ and mobility ,u 5cm2/Vsec at 77°K; the 0 =is5 reproduced x 10 dashed curve from the experimental results of Bate et al.7 on a n—InSb sample with n 1.7 x 10 4cm~ and Hall mobility = 6.2 x 10~cm2, Vsec at 77°K. magnetic field for two limiting dimensions of the sample, i/cu >> 1 and 1/cu << 1 (using Corbino disk geometry), and the threshold field in the absence of magnetic field, s.hould avoid the cumbersome calculations of the distribution function in n—InSb at 77°K.9 The slight reduction in the threshold field due to the application of a longitudinal magnetic field can be attributed to the anisotropy of the electron-lattice scattering and a relation identical to equation (4) exists for this case also. Finally, we conclude that effect of magnetic field on the Gunn effect in n—InSb at 77°K is consistent with the intervalley electron transfer mechanism and the negative magnetic field coefficient of the threshold field for Gunn effect may be explained by purely geometric magnetoresistance effects keeping in mind the variation of the mobility with magnetic field.
Vol. 9, No. 21
GUNN EFFECT IN n—InSb
Acknowledgements — The present investigation was supported by Council of Scientific and Industrial Research, India and Environmental
1879
Scientific Services Administration, Boulder, Colorado, U.S.A.
REFERENCES 1.
RIDLEY B.K. and WATKINS T.B.,Proc. Phys. Soc. 78, 293 (1961).
2.
SMITH J.E., NATHAN M.I., McGRODDY J.C., POROWSKI S.A. and PAUL W., App. Phys. Lett. 15, 242 (1969).
3.
VOROBIV V.N., Soy. Phys. Semicond. 4, 800 (1970).
4.
BOARDMAN A.D., FAWCETT W. and RUCH J.G.,
5.
MULLER E. and FERRY D.K., Solid State Commun. 8, 855 (1970).
6.
GUETIN P. and HERVOUET C., Proc. IEEE, 56, 1957 (1968).
7.
BATE R., WILLARDSON R.K. and BEET A.C., J. Phys. Chem. Solids, 9, 119 (1959).
8.
WEISS H., J. appl. Phys. 32, 2064 (1961).
9.
FAWCETT W. and RUCH J.G., Appi. Phys. Lett. 15, 368 (1969).
Phys. Status. Solidi (a) (Jan. 1971 issue).
Der negative Magnetfeldkoeffizient des Schwellenrelds für den Gunn schen Effekt in n—InSb bei 77°K, der mit der Wirkungsweise des intervalley Elektronenffbergangs anscheinend unvereinbar ist, kann durch rein geomet rische Magnetores istenzeffekte befriedigend erkl~rtwerden.
Vol. 9, PP. 1877—1879, 1971. Pergamon Press.
Printed in Great Britain
EFFECT OF MAGNETIC FIELD ON GUNN EFFECT IN n—InSb U.P. Phadke and A.K. Chakravarti Department of Physics, Indian Institute of Technology, New Delhi—29, India
(Received 29 April 1971 by A.R. Verma)
The negative magnetic field coefficient of the threshold field for the Gunn effect in n—InSb at 77°K, which is apparently inconsistent with intervalley electron transfer mechanism, can be satisfactorily explained by purely geometric magnetoresistance effects.
THE GUNN effect stands for high frequency current oscillation, associated with travelling high field domains, in bulk semiconductors above a certain critical threshold field; this, as pointed 1 requires only the out by Ridley and Watkins, existence of a region of negative differential conductivity (ndc) in the I—V characteristics of these semiconductors. The physical mechanism underlying the ndc is the transfer of electrons from the high mobility conduction band to the low mobility ‘satellites’ at high electric fields. Since it is well known that the separation, A, between the conduction band and the upper lying valley decreases with pressure, the negative pressure coefficient of the threshold field for the Gunn effect in n—InSb2 lends support to the validity of this mechanism. This incidentally rules out the ndc due to the nonparabolic nature of the conduction band since it would lead to a positive pressure coefficient for the threshold field. It therefore appears to be well established that the ndc underlying the Gunn effect is due to the intervalley electron transfer mechanism. In a semiconductor with the band gap
and consequently the electron generation rate too slow to prevent the transfer of hot electrons to the upper minima if the observations are made on a short enough time scale (~10~sec). Now a magnetic field has a cooling effect on the carriers and should, therefore, cause an increase in the threshold field. This, in fact, has been observed in n—GaAs by Vorobiv; ~ the effect is quite small and is in confirmation with the Monte Carlo calculations of Boardman et al.4 However, very recently Muller and Ferry ~ have observed an appreciable reduction in the value of the threshold field with the application of a transverse magnetic field in the range 500 G to 3000 G for a 0.015 ~,< 0.015 x 0.135mm n—InSb sample with mobility p~= 5 x 105cm2/Vsec and carrier concentration n = 2 x 10 l4~~~ç3 at 77°K, the observation time was kept below 2.5 nsec to avoid the effects of carrier generation which is a much slower process as compared to the intervalley transfer; a longitudinal magnetic field over the same range gave only a slight decrease in the threshold field. Thus the effect of magnetic field apparently seems to be
E 9 < A (such as n—InSb E~ 0.22eV, A = 0.45eV at of 77°K), one would expect the occurrence the Gunn effect normally to be precluded by the impact ionization of electrons from the valence band into the conduction band. However, as was established by Smith et al.,2 the cross section for impact ionization should be too small
inconsistent with the electron 5 withtransfer a seriesmechof lucid anism. Muller andruled Ferry, arguments, have out most of the plausible explanations for the anomaly. Guetin and Hervouet ~ have observed a similar anomaly in n—GaAs (~ = 5300 cm2/Vsec) and attribute it partly to the geometric magnetoresistance at the
1877
1878
GUNN EFFECT IN n—InSb
contacts. Modifying their results, to take into account the variation of the mobility with the magnetic field, we obtain 2(1 — X(B))2]~2 (1)
E7(B) where E
=
ET(0)[1
+
Vol.9, No. 21
r
~
—
.
,
— -
—
~B
1d field in the pres7
~
/
0 ,
is the magnetoresistance. As X(B) vs. B data is not available for this n—InSb sample, we have
/
i5~
, 7
calculated X(B) from equation (1) for different calues of B using the results of Muller and Ferry5 for E~(B).The results, along with the experimental results of Bate et al.’ for a n—lnSb sample with n = 1.7 x 10’~cm3 at 77°K, for the sake of comparison are plotted in Fig. 1. The magnetoresistance X(B) was found to obey an empirical relation X(B) c~B 0.075 in the region B = 500— 3000 G. Keeping in view the large variations in the value of the magneto-resistance8 due to the inhomogeneities in the sample, the pulling rate of the crystal, surface currents etc. and the essentially qualitative nature of the theory, our results for the variation of the magneto-resistance with the magnetic field, which, in turn, governs the threshold field, are in satisfactory agreement with the experimental resuits of Bate et al.’ In fact a rigorous theory would give
ET(B)
=
ET(0) cosO
(2)
where 0, the Hall angle, should be calculated from appropriate averages over a suitable distribution function. However circumverting the calculation of 0 we deduce a self consistent verification of the essential idea behind the above theory. For a sample with small length to width ratio (1/cu << 1), the Hall field is approximately zero everywhere in the bulk and hence proceeding along the same lines as Guetin and Hervouet 6 it can be shown that
[E~(B)L,~,
=
E~(0)sec 0
Eliminating the Hall angle 0 between equations (2) and (3) we obtain
[E~(B)I 1,~
[E~(B)~j1~,
1
=
2 [ET(0)1 (4)
So the experimental verification of the relation (4) which involves the measurement of threshold fields in presence of a transverse
/
~ 70.
FIG. 1. Shows the variation of the magnetoresistance X(B) with the magnetic field B. The solid curve denotes our results calculated from the experimental results of Muller and Ferry ~ for 0.015 x 0.015 x 0.35nmm n—InSb sample witha carrier concentration = 2 x 10 ‘4cm~ and mobility ,u 5cm2/Vsec at 77°K; the 0 =is5 reproduced x 10 dashed curve from the experimental results of Bate et al.7 on a n—InSb sample with n 1.7 x 10 4cm~ and Hall mobility = 6.2 x 10~cm2, Vsec at 77°K. magnetic field for two limiting dimensions of the sample, i/cu >> 1 and 1/cu << 1 (using Corbino disk geometry), and the threshold field in the absence of magnetic field, s.hould avoid the cumbersome calculations of the distribution function in n—InSb at 77°K.9 The slight reduction in the threshold field due to the application of a longitudinal magnetic field can be attributed to the anisotropy of the electron-lattice scattering and a relation identical to equation (4) exists for this case also. Finally, we conclude that effect of magnetic field on the Gunn effect in n—InSb at 77°K is consistent with the intervalley electron transfer mechanism and the negative magnetic field coefficient of the threshold field for Gunn effect may be explained by purely geometric magnetoresistance effects keeping in mind the variation of the mobility with magnetic field.
Vol. 9, No. 21
GUNN EFFECT IN n—InSb
Acknowledgements — The present investigation was supported by Council of Scientific and Industrial Research, India and Environmental
1879
Scientific Services Administration, Boulder, Colorado, U.S.A.
REFERENCES 1.
RIDLEY B.K. and WATKINS T.B.,Proc. Phys. Soc. 78, 293 (1961).
2.
SMITH J.E., NATHAN M.I., McGRODDY J.C., POROWSKI S.A. and PAUL W., App. Phys. Lett. 15, 242 (1969).
3.
VOROBIV V.N., Soy. Phys. Semicond. 4, 800 (1970).
4.
BOARDMAN A.D., FAWCETT W. and RUCH J.G.,
5.
MULLER E. and FERRY D.K., Solid State Commun. 8, 855 (1970).
6.
GUETIN P. and HERVOUET C., Proc. IEEE, 56, 1957 (1968).
7.
BATE R., WILLARDSON R.K. and BEET A.C., J. Phys. Chem. Solids, 9, 119 (1959).
8.
WEISS H., J. appl. Phys. 32, 2064 (1961).
9.
FAWCETT W. and RUCH J.G., Appi. Phys. Lett. 15, 368 (1969).
Phys. Status. Solidi (a) (Jan. 1971 issue).
Der negative Magnetfeldkoeffizient des Schwellenrelds für den Gunn schen Effekt in n—InSb bei 77°K, der mit der Wirkungsweise des intervalley Elektronenffbergangs anscheinend unvereinbar ist, kann durch rein geomet rische Magnetores istenzeffekte befriedigend erkl~rtwerden.