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Magnetoresistance in the extreme quantum limit in n-type InSb
PHYSICS
Volume 24A, number 1
LETTERS
F’(K) decreased faster in this retion than F(K) and if S ’ f 0 (in [4] S ’ = 3 was used) in contra&c tion with [5]. This indicates that the magnetic electron density perturbation extends to the first neighbour iron atoms, in qualitative agreement with the change in the total magnetization per Mn impurity atom. The authors are indebted to Professor for many helpful discussions.
1. T.Wolfram
2. 3. 4. 5. 6.
L. Pgl
2 January 1967
and J.Callawav, _ Phvs. _ Rev. 130 (1963) 2207. T.Wolfram and W.Hall, Phys. Rev. 143 (1966) 284. Ju.A. Isjumov and M. V. Medvedjev, Zh. Eksperim. i Teor. Fiz. 48 (1966) 574. V. Jaccarino, L.R.Walker and G.K.Wertheim, Phvs. Rev. Letters 13 (19641 752. M.F.Collins and G.G.iow, Proc. Phys. Sot. 86 (1965) 535. A.Arrott and J.E. Noakes, Iron and its dilute solid solutions (Interscience Publishers Inc., New York, 1963) p. 81.
*****
MAGNETORESISTANCE LIMIT
IN THE EXTREME IN n-TYPE InSb
QUANTUM
M. A. KINCH* Clarendon Laboratory,
Oxford, England
Received 30 November 1966
The magnetoresistance of n-type InSb has been investigated in the temperature region 10’ - 77’K under “extreme quantum limitn.transportconditions. &- is found to vary as approximately BIT-~. This is attributed to micro-inhomogeneity effects.
In conditions of magnetic field B such that tiwc >> kT, and WcTm >> 1, where wc = eB/m* (MKS) is the cyclotron frequency and Tm the momentum relaxation time, for a nondegenerate semiconductor, all electrons are accommodated in the lowest (n = 0) Landau level, and the “extreme quantum limit” transport theory must be employed. The properties of n-type InSb at low temperatures are such that the above conditions are satisfied at relatively small values of B, give; by fiwc x 1.5 x BX lo:25 joule >>kT, or B X x T- >> 102. wc~m 2 1 for B 2 250 G at temperatures below 77OK. Measurements on this material should offer a simple means of testing the various theories of quantum transport [I]. Sladek [2] has performed such experiments in the region 50° - 1lOoK on a polycrystalline specimen with n - 3 X 1014 cm-3, and finds a linear dependence of magnetoresistance PT on magnetic field, together with an approximate temperature variation of TWoa4. He concludes that piezoelectric scattering provides the best explanation of his data. However, the fit is only semiquantitative and is somewhat unsatisfactory in that recent mobility results at these temperatures [3,4] indicate that momentum
scattering is due primarily to ionized impurities, acoustical mode deformation potential and polar optical mode phonons. We have extended these magnetoresistance studies down to the hydrogen temperature range. Two advantages are gained by this: a) no ambiguity exists with regard to the dominant momentum loss mechanism, the mobility p a T1.1 and is typical of ionized impurity scattering; and b) larger values of the parameter Fiw,/kT are obtained, thus obviating the necessity of considering Landau levels higher than n = 0. The theoretical predictions of Adams and Holstein [l] for ionized impurity scattering in this case give for the transverse resistivity PTm a B’T-Q. The results obtained for sample InSb 12, a single crystal with n - 8 x lO+I3 cm-3, are shown in figs. 1 and 2, from which it is apparent that pT m BIT-I. This dependence is typical of all specimens measured, some of which were etched and other mechanically polished. It is interesting to note that where our measurements $ Present address: Texas Instruments Incorporated, Dallas, Texas.
23
PHYSICS
Volume 24A, number 1
121 ohm
LETTERS
10 OK
cm
IO-
2 January 1967
ohm. cm
f a-
r,
1394
1‘
OK
J?
20kG
20 OK 10 kG
l-
4&g&
\, 10
0
4
Fig. 1. PT-H,
8
H-k&s
16
20
Fig. 2. pT -T,
at various lattice temperatures.
overlapped those of Sladek similar results were obtained. It is not possible to account for this dependence by appealing to existing theory on the various scattering mechanisms. Indeed it seems unlikely that any one scattering mechanism, as implied by the B dependence, can be responsible for the results over the whole range 10’ - lOOoK. The more likely explanation is that these results are due to the difficulties which are inherent in this tupe of measurement. In these high mobility semiconductors these are: a) geometry effects, b) contact effects, c) inhomogeneity effects and d) surface effects. Sladek reproduced his results with a specimen of length/width ratio - 23 and deduced that a) was unimportant. Because our results, where they overlap, are similar we will neglect a). d) was thought to be negligible because similar results were obtained with etched and mechanically polished surfaces. The effect of contacts is also thought to be minimal in that the magnetoresistance displayed by specimens with first four and then six contacts showed very little change. It should be stated, however, that if the contacts were on a face which was perpendicular to B then a noticeable increase inpT was observed, as would be expected because of the wholesale shorting of the Hall voltage. We are thus left with c). The treatment of inhomogeneities by Herring [5] yields for the classical case of large B transverse magnetoresistance saturation
*****
24
T----1
OK
100
at two values of transverse magnetic
Pyy(B)Ieff
field.
=
Pyy(B)+EJ
[w;‘” ]
where X is - unity and ( > denotes a spatial average. This expression should apply equally well in this case in that the predicted pyy(B) is independent of B, although it is only strictly valid if the correction term is <
References 1. E. N.Adams and T.H.Holstein,
J. Phys. Chem. Solids 10 (1959) 254. 2. R.J.Sladek, J. Phys. Chem. Solids 16 (1960) 1. 3. 4. 5. 6.
I.M.Tsidilkovski, Phys. Stat. Sol. 8 (1965) 253. M.A.Kinch, Brit. J. Appl. Phys. 17 (1966) 1257. C.Herring, J. Appl. Phys. 31 (1960) 1939. I. L.Drichko and I.V.Mochan, Sov. Phys. Solid State 7 (1966) 2634.
Volume 24A, number 1
LETTERS
F’(K) decreased faster in this retion than F(K) and if S ’ f 0 (in [4] S ’ = 3 was used) in contra&c tion with [5]. This indicates that the magnetic electron density perturbation extends to the first neighbour iron atoms, in qualitative agreement with the change in the total magnetization per Mn impurity atom. The authors are indebted to Professor for many helpful discussions.
1. T.Wolfram
2. 3. 4. 5. 6.
L. Pgl
2 January 1967
and J.Callawav, _ Phvs. _ Rev. 130 (1963) 2207. T.Wolfram and W.Hall, Phys. Rev. 143 (1966) 284. Ju.A. Isjumov and M. V. Medvedjev, Zh. Eksperim. i Teor. Fiz. 48 (1966) 574. V. Jaccarino, L.R.Walker and G.K.Wertheim, Phvs. Rev. Letters 13 (19641 752. M.F.Collins and G.G.iow, Proc. Phys. Sot. 86 (1965) 535. A.Arrott and J.E. Noakes, Iron and its dilute solid solutions (Interscience Publishers Inc., New York, 1963) p. 81.
*****
MAGNETORESISTANCE LIMIT
IN THE EXTREME IN n-TYPE InSb
QUANTUM
M. A. KINCH* Clarendon Laboratory,
Oxford, England
Received 30 November 1966
The magnetoresistance of n-type InSb has been investigated in the temperature region 10’ - 77’K under “extreme quantum limitn.transportconditions. &- is found to vary as approximately BIT-~. This is attributed to micro-inhomogeneity effects.
In conditions of magnetic field B such that tiwc >> kT, and WcTm >> 1, where wc = eB/m* (MKS) is the cyclotron frequency and Tm the momentum relaxation time, for a nondegenerate semiconductor, all electrons are accommodated in the lowest (n = 0) Landau level, and the “extreme quantum limit” transport theory must be employed. The properties of n-type InSb at low temperatures are such that the above conditions are satisfied at relatively small values of B, give; by fiwc x 1.5 x BX lo:25 joule >>kT, or B X x T- >> 102. wc~m 2 1 for B 2 250 G at temperatures below 77OK. Measurements on this material should offer a simple means of testing the various theories of quantum transport [I]. Sladek [2] has performed such experiments in the region 50° - 1lOoK on a polycrystalline specimen with n - 3 X 1014 cm-3, and finds a linear dependence of magnetoresistance PT on magnetic field, together with an approximate temperature variation of TWoa4. He concludes that piezoelectric scattering provides the best explanation of his data. However, the fit is only semiquantitative and is somewhat unsatisfactory in that recent mobility results at these temperatures [3,4] indicate that momentum
scattering is due primarily to ionized impurities, acoustical mode deformation potential and polar optical mode phonons. We have extended these magnetoresistance studies down to the hydrogen temperature range. Two advantages are gained by this: a) no ambiguity exists with regard to the dominant momentum loss mechanism, the mobility p a T1.1 and is typical of ionized impurity scattering; and b) larger values of the parameter Fiw,/kT are obtained, thus obviating the necessity of considering Landau levels higher than n = 0. The theoretical predictions of Adams and Holstein [l] for ionized impurity scattering in this case give for the transverse resistivity PTm a B’T-Q. The results obtained for sample InSb 12, a single crystal with n - 8 x lO+I3 cm-3, are shown in figs. 1 and 2, from which it is apparent that pT m BIT-I. This dependence is typical of all specimens measured, some of which were etched and other mechanically polished. It is interesting to note that where our measurements $ Present address: Texas Instruments Incorporated, Dallas, Texas.
23
PHYSICS
Volume 24A, number 1
121 ohm
LETTERS
10 OK
cm
IO-
2 January 1967
ohm. cm
f a-
r,
1394
1‘
OK
J?
20kG
20 OK 10 kG
l-
4&g&
\, 10
0
4
Fig. 1. PT-H,
8
H-k&s
16
20
Fig. 2. pT -T,
at various lattice temperatures.
overlapped those of Sladek similar results were obtained. It is not possible to account for this dependence by appealing to existing theory on the various scattering mechanisms. Indeed it seems unlikely that any one scattering mechanism, as implied by the B dependence, can be responsible for the results over the whole range 10’ - lOOoK. The more likely explanation is that these results are due to the difficulties which are inherent in this tupe of measurement. In these high mobility semiconductors these are: a) geometry effects, b) contact effects, c) inhomogeneity effects and d) surface effects. Sladek reproduced his results with a specimen of length/width ratio - 23 and deduced that a) was unimportant. Because our results, where they overlap, are similar we will neglect a). d) was thought to be negligible because similar results were obtained with etched and mechanically polished surfaces. The effect of contacts is also thought to be minimal in that the magnetoresistance displayed by specimens with first four and then six contacts showed very little change. It should be stated, however, that if the contacts were on a face which was perpendicular to B then a noticeable increase inpT was observed, as would be expected because of the wholesale shorting of the Hall voltage. We are thus left with c). The treatment of inhomogeneities by Herring [5] yields for the classical case of large B transverse magnetoresistance saturation
*****
24
T----1
OK
100
at two values of transverse magnetic
Pyy(B)Ieff
field.
=
Pyy(B)+EJ
[w;‘” ]
where X is - unity and ( > denotes a spatial average. This expression should apply equally well in this case in that the predicted pyy(B) is independent of B, although it is only strictly valid if the correction term is <
References 1. E. N.Adams and T.H.Holstein,
J. Phys. Chem. Solids 10 (1959) 254. 2. R.J.Sladek, J. Phys. Chem. Solids 16 (1960) 1. 3. 4. 5. 6.
I.M.Tsidilkovski, Phys. Stat. Sol. 8 (1965) 253. M.A.Kinch, Brit. J. Appl. Phys. 17 (1966) 1257. C.Herring, J. Appl. Phys. 31 (1960) 1939. I. L.Drichko and I.V.Mochan, Sov. Phys. Solid State 7 (1966) 2634.