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Anomalous temperature dependence of the electrical resistivity of heavily doped n-type germanium below 1 K
Volume 40A, number 2
PHYSICS LETTERS
3 July 1972
ANOMALOUS TEMPERATURE DEPENDENCE OF THE ELECTRICAL RESISTIVITY OF HEAVILY DOPED n-TYPE GERMANIUM BELOW 1 K S. GONDA Electrotechnical Laboratory, Tanashi, Tokyo, Japan Received 25 April 1972
It is found that the resistivity of heavily doped, one-valley n-Ge decreases with decreasing temperature below 1 K contrary to the behavior above 1 K. This anomaly cannot be explained by the mechanisms proposed up to the present.
In first approximation, temperature-independent resistivity is expected at low temperatures in heavily doped semiconductors. However, even at temperatures lower than one-hundredth of the degeneracy temperature TD, a small but conspicuous temperature dependence of the resistivity is observed [1 3]. In n-type germanium the resistivity versus temperature curve has a positive gradient with a negative derivative neai I K, or log T dependence in a narrow temperature range [1, 3]. On the other hand, in germanium stressed uniaxially in the [111] direction so that all electrons are in one valley, the resistivity increases with decreasing temperature above 1 K as shown by Katz [4]. These behaviors in As-doped germanium near 1 K have been attributed to intervalley electron-electron scattering in the four-valley case [5] or screened Coulomb scattering by ionized impurities in the onevalley case [6]. Does the resistivity of stressed germaniurn continue to increase as the temperature decreases below 1 K? This note presents the answer obtained experimentally, that is, the temperature dependence of the resistivity of n-type germanium with and without a sufficiently large stress below I K. Samples were bridge-type, Sb- or As-doped germanium with carrier concentrations from 1017 to 1018 cm The degree of compensation was less than 10 percent. The cross section was about 0.45 X 0.45 mm2 ~.
and the length was 6mm. Temperatures below 1 K were attained using an adiabatic demagnetization cryostat [7]. A uniaxial compressional stress was applied in the [111] directions of the sample held in a clamp jig by means of screwing a piston at room ternperature. The clamp jig was attached to the heat link
which was a group of 1500 copper wires embedded in the cooling salt and cooled down to 0.1 K. The magnitude of the stress applied to the sample was checked at 4.2 K by comparing piezo-resistance with the value preliminary measured by the ordinary pushrod method. The resistivity was measured for current flowing parallel to the direction of stress. Fig. 1 shows the typical temperature dependence of the resistivity of n-type germanium doped with Lero-stress resistivity decreases l.2X l0’8Sbcm with lowering temperature, as shown in fig. 1(a), and temperature variation tends to the cease. One-valley resistivity of the sample, upon which a uniaxial stress of more than I 0~dyne/cm2 was applied in the [Ill] direction, increases with lowering temperature down to about 1.5 K, as shown in fig. 1(b). It should be emphasized at temperatures below 1.5K the resistivity decreases with lowering temperature, and the temperature dependence can be approximately described by log T. The same tendency as this case is observed in the sample doped with arsenic. Phonon scattering, resonance scattering [8], dipole scattering [9] and s.d scattering [3] were discussed as the origin of the temperature dependence near 1 K. These were found, however, to be inconsistent with experimental data [4, 10]. If the collision time by impurity scattering is assumed to decrease for E ~ 2E 0, and to increase obeying the usual ionized impurity scattering formula for E> 3E0 with increasing electron energy E, where E0 is Fermi energy, the temperature dependence of unstressed and stressed germanium can be well explained above 1 K [11]. But this phenornenological theory also can not explain the ~.
Volume 40A, number 2
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PHYSICS LETTERS
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3 July 1972
Coulomb scattering has been proposed [6]. According 2 to the calculation the expression ~p/p 0 — y(T/TD) was derived for temperatures well below TD, where p 0 is the resistivity at T = OK, and y is a function of the number of valley v. For v = 1, y is always negative for degenerate doping. Below about I K, however, these results are in disagreement with the present data. Therefore this anomalous temperature dependence seems to require another mechanism different from the one proposed up to the present. The details of experi-
406 a 4.04
mental data and discussion will be reported in a
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4.02
TD
86K
forth-coming paper.
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The author would like to express his sincere thanks
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to Prof. W. Sasaki for suggesting this investigation and
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to Drs. N. Mikoshiba and C. Yamanouchi for their stimulating discussions.
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394
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References
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[1] W. Sasaki and R. Dc Bruyn Ouboter, Physica 27(1961)
1 I
I)
0.1
1 T
(
877.
1.1
0 K)
hg. 1. Temperature dependence ot the resistIvity 3, (a) without ol n-type stress, ~b) with a uniaxial compressional germanium doped with 1.2x 1018stress Sb cnfin the [111] direction.
temperature dependence below 1K [121. As the origin of the (T/TD)2 dependence above 1 K in As-doped germanium, intervalley electron-electron scattering has been proposed [5], but at least the positive temperature coefficient in the resistivity versus temperature curve for one-valley germanium below 1K can not be understood by this scattering, because intervalley scattering can not take place in the one-valley case. As the origin of negative temperature coefficient of one-valley germanium above 1 K screened
[2] W. Sasaki, S. Gonda and M. Inouc, J. Phys. Soc. Japan 18(1963)914. [3] (1965) W. Sasaki 1092. and M. Nakamura, J. Phys. Soc. Japan 20 [4] M.J. Katz, Phys. Rev. 140 (1965) A 1323. [5] M.J. Katz, S.H. Koenig and A.A. Lopez, Phys. Rev. Letters 15(1965) 828. [6] T. MeeksandJ.B. Krieger, Phys. Rev. 185 (1969) 1068. [7] S. Gonda, Cryophysics cryoengineering Bull. Intern. Institute ofand Retrigeration, 1970,(Suppl. Annex 2) ~, 205. [8] D. Long, J.D. Zook, P.W. Chapman and ON. Tutte, Solid State Communications 2 (1964) 191. [9] S.H. Koenig, Proc. Intern. Conf. on Physics of semi 1962,Japan p 5 20 (1965) 2293. [10] conductors, W. Sasaki,J. Exeter Phys. Soc. [11] T. Kurosawa and W. Sasaki, J. Phys. Soc. Japan 31 (1971) 953. [12] T. Kurosawa, private communication.
PHYSICS LETTERS
3 July 1972
ANOMALOUS TEMPERATURE DEPENDENCE OF THE ELECTRICAL RESISTIVITY OF HEAVILY DOPED n-TYPE GERMANIUM BELOW 1 K S. GONDA Electrotechnical Laboratory, Tanashi, Tokyo, Japan Received 25 April 1972
It is found that the resistivity of heavily doped, one-valley n-Ge decreases with decreasing temperature below 1 K contrary to the behavior above 1 K. This anomaly cannot be explained by the mechanisms proposed up to the present.
In first approximation, temperature-independent resistivity is expected at low temperatures in heavily doped semiconductors. However, even at temperatures lower than one-hundredth of the degeneracy temperature TD, a small but conspicuous temperature dependence of the resistivity is observed [1 3]. In n-type germanium the resistivity versus temperature curve has a positive gradient with a negative derivative neai I K, or log T dependence in a narrow temperature range [1, 3]. On the other hand, in germanium stressed uniaxially in the [111] direction so that all electrons are in one valley, the resistivity increases with decreasing temperature above 1 K as shown by Katz [4]. These behaviors in As-doped germanium near 1 K have been attributed to intervalley electron-electron scattering in the four-valley case [5] or screened Coulomb scattering by ionized impurities in the onevalley case [6]. Does the resistivity of stressed germaniurn continue to increase as the temperature decreases below 1 K? This note presents the answer obtained experimentally, that is, the temperature dependence of the resistivity of n-type germanium with and without a sufficiently large stress below I K. Samples were bridge-type, Sb- or As-doped germanium with carrier concentrations from 1017 to 1018 cm The degree of compensation was less than 10 percent. The cross section was about 0.45 X 0.45 mm2 ~.
and the length was 6mm. Temperatures below 1 K were attained using an adiabatic demagnetization cryostat [7]. A uniaxial compressional stress was applied in the [111] directions of the sample held in a clamp jig by means of screwing a piston at room ternperature. The clamp jig was attached to the heat link
which was a group of 1500 copper wires embedded in the cooling salt and cooled down to 0.1 K. The magnitude of the stress applied to the sample was checked at 4.2 K by comparing piezo-resistance with the value preliminary measured by the ordinary pushrod method. The resistivity was measured for current flowing parallel to the direction of stress. Fig. 1 shows the typical temperature dependence of the resistivity of n-type germanium doped with Lero-stress resistivity decreases l.2X l0’8Sbcm with lowering temperature, as shown in fig. 1(a), and temperature variation tends to the cease. One-valley resistivity of the sample, upon which a uniaxial stress of more than I 0~dyne/cm2 was applied in the [Ill] direction, increases with lowering temperature down to about 1.5 K, as shown in fig. 1(b). It should be emphasized at temperatures below 1.5K the resistivity decreases with lowering temperature, and the temperature dependence can be approximately described by log T. The same tendency as this case is observed in the sample doped with arsenic. Phonon scattering, resonance scattering [8], dipole scattering [9] and s.d scattering [3] were discussed as the origin of the temperature dependence near 1 K. These were found, however, to be inconsistent with experimental data [4, 10]. If the collision time by impurity scattering is assumed to decrease for E ~ 2E 0, and to increase obeying the usual ionized impurity scattering formula for E> 3E0 with increasing electron energy E, where E0 is Fermi energy, the temperature dependence of unstressed and stressed germanium can be well explained above 1 K [11]. But this phenornenological theory also can not explain the ~.
Volume 40A, number 2
L
E22.e -~
F F
E 227 22.6
II I
L
22.9
PHYSICS LETTERS
H
b 0
0000
o
~
0
J
~,,
~
~
I
I
0
i’i~
I I
I I
0.1 I
I ~
1 ~I I
10
3 July 1972
Coulomb scattering has been proposed [6]. According 2 to the calculation the expression ~p/p 0 — y(T/TD) was derived for temperatures well below TD, where p 0 is the resistivity at T = OK, and y is a function of the number of valley v. For v = 1, y is always negative for degenerate doping. Below about I K, however, these results are in disagreement with the present data. Therefore this anomalous temperature dependence seems to require another mechanism different from the one proposed up to the present. The details of experi-
406 a 4.04
mental data and discussion will be reported in a
0 0
4.02
TD
86K
forth-coming paper.
00
The author would like to express his sincere thanks
4.00 E
39~
0
to Prof. W. Sasaki for suggesting this investigation and
0
to Drs. N. Mikoshiba and C. Yamanouchi for their stimulating discussions.
0
3.96 0
394
0 0
3.92
Ii’6
~0
References
000
390
o00 I
[1] W. Sasaki and R. Dc Bruyn Ouboter, Physica 27(1961)
1 I
I)
0.1
1 T
(
877.
1.1
0 K)
hg. 1. Temperature dependence ot the resistIvity 3, (a) without ol n-type stress, ~b) with a uniaxial compressional germanium doped with 1.2x 1018stress Sb cnfin the [111] direction.
temperature dependence below 1K [121. As the origin of the (T/TD)2 dependence above 1 K in As-doped germanium, intervalley electron-electron scattering has been proposed [5], but at least the positive temperature coefficient in the resistivity versus temperature curve for one-valley germanium below 1K can not be understood by this scattering, because intervalley scattering can not take place in the one-valley case. As the origin of negative temperature coefficient of one-valley germanium above 1 K screened
[2] W. Sasaki, S. Gonda and M. Inouc, J. Phys. Soc. Japan 18(1963)914. [3] (1965) W. Sasaki 1092. and M. Nakamura, J. Phys. Soc. Japan 20 [4] M.J. Katz, Phys. Rev. 140 (1965) A 1323. [5] M.J. Katz, S.H. Koenig and A.A. Lopez, Phys. Rev. Letters 15(1965) 828. [6] T. MeeksandJ.B. Krieger, Phys. Rev. 185 (1969) 1068. [7] S. Gonda, Cryophysics cryoengineering Bull. Intern. Institute ofand Retrigeration, 1970,(Suppl. Annex 2) ~, 205. [8] D. Long, J.D. Zook, P.W. Chapman and ON. Tutte, Solid State Communications 2 (1964) 191. [9] S.H. Koenig, Proc. Intern. Conf. on Physics of semi 1962,Japan p 5 20 (1965) 2293. [10] conductors, W. Sasaki,J. Exeter Phys. Soc. [11] T. Kurosawa and W. Sasaki, J. Phys. Soc. Japan 31 (1971) 953. [12] T. Kurosawa, private communication.