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Lattice mobility in n-germanium at low temperatures
PHYSICS
Volume 26A, number 2
LATTICE
MOBILITY
IN
LETTERS
n-GERMANIUM
16 December 1967
AT
LOW
TEMPERATURES
K. BAUMANN, P. KOCEVAR and M. KRIECHBAUM Institut fir theoretische Physik der Universittit Graz, Graz, Austria Received 8 November 1967
The lattice mobility /.l~ of hot electrons is calculated for a sample of n-germanium at 4.0°K. Assuming 5 X lo14 donors per cm3 and a cross-dimension of 0.1 cm, we find /.&L0~ Ti0e9, Te being the electron temperature. _ Conwell [l] has given a theory of the disturbance of the acoustic phonon ‘distribution caused by hot electrons in a many-valley semiconductor at low temperatures. The phonon generation rate is calculated using the electron-phonon scattering matrix elements of Herring and Vogt [2]. For the decrease of the phonon number only boundary scattering is taken into account. Thus the phonon relaxation time is given by L/U,, where L is the smallest cross-dimension of the sample, and u, are the sound velocities. The phonon distribution derived by Conwell may be combined with the expression for the electron relaxation time tensor as given in [2]. In this way we have calculated numerically the electron mobility in n-Ge for six selected electron temperatures between 30°K and 150’K. The other parameters are chosen in accordance with an experiment performed by Seeger [3]. Especially, the sample temperature has been taken as 4.2oK, the carrier concentration as 5 x 1014/cm3 , and L = 0.117 cm. The other constants are the same as in [l]. The electric field vector is assumed to point into the [loo] direction. The field strength belonging to each electron temperature is found from energy balance. Our results are shown in fig. 1. It can be seen that the mobility ,u versus electron temperature T, and field strength E has practically a pure power behaviour , viz.
p cc .,Oag,
and
~1cc E-o.75 .
ELr$TRIC
FIEL,;
(V/cm)
1
EL&&
TEMPE~~URE
loK
)
Fig. 1. The circles represent the points for which the computation was done.
simple parabolic conduction band. Thus we see that the use of spheroidal energy valleys is essential for obtaining the correct field dependence of the mobility. We are grateful to K.Seeger for much valuable advice. Thanks are also due to the Mathematisches Institut der Technischen Hochschule Wien for making available their computer.
(1)
It should be mentioned
tive reasoning
derives
@ 0~ Teoe5,
that Conwell by qualitadifferent power laws, viz.
and
/.~a E-Oa5 .
References
(1’)
The reason for this discrepancy lies in the fact that the dependence of ip (defined in formula (31) of ref. 1 on Te is not negligible. The results (l’), have already been found by Paranjape [4] using a 62
1. 2. 3. 4.
E. M. Conwell, Phys. Rev. 135 (1964) A814. C.Herring and E.Vogt, Phys. Rev. 101 (1956) 944. K.Seeger, Z. Phys. 182 (1965) 510. V.V.Paranjape, Proc. Phys. Sot. (London) 80 (1962) 971.
Volume 26A, number 2
LATTICE
MOBILITY
IN
LETTERS
n-GERMANIUM
16 December 1967
AT
LOW
TEMPERATURES
K. BAUMANN, P. KOCEVAR and M. KRIECHBAUM Institut fir theoretische Physik der Universittit Graz, Graz, Austria Received 8 November 1967
The lattice mobility /.l~ of hot electrons is calculated for a sample of n-germanium at 4.0°K. Assuming 5 X lo14 donors per cm3 and a cross-dimension of 0.1 cm, we find /.&L0~ Ti0e9, Te being the electron temperature. _ Conwell [l] has given a theory of the disturbance of the acoustic phonon ‘distribution caused by hot electrons in a many-valley semiconductor at low temperatures. The phonon generation rate is calculated using the electron-phonon scattering matrix elements of Herring and Vogt [2]. For the decrease of the phonon number only boundary scattering is taken into account. Thus the phonon relaxation time is given by L/U,, where L is the smallest cross-dimension of the sample, and u, are the sound velocities. The phonon distribution derived by Conwell may be combined with the expression for the electron relaxation time tensor as given in [2]. In this way we have calculated numerically the electron mobility in n-Ge for six selected electron temperatures between 30°K and 150’K. The other parameters are chosen in accordance with an experiment performed by Seeger [3]. Especially, the sample temperature has been taken as 4.2oK, the carrier concentration as 5 x 1014/cm3 , and L = 0.117 cm. The other constants are the same as in [l]. The electric field vector is assumed to point into the [loo] direction. The field strength belonging to each electron temperature is found from energy balance. Our results are shown in fig. 1. It can be seen that the mobility ,u versus electron temperature T, and field strength E has practically a pure power behaviour , viz.
p cc .,Oag,
and
~1cc E-o.75 .
ELr$TRIC
FIEL,;
(V/cm)
1
EL&&
TEMPE~~URE
loK
)
Fig. 1. The circles represent the points for which the computation was done.
simple parabolic conduction band. Thus we see that the use of spheroidal energy valleys is essential for obtaining the correct field dependence of the mobility. We are grateful to K.Seeger for much valuable advice. Thanks are also due to the Mathematisches Institut der Technischen Hochschule Wien for making available their computer.
(1)
It should be mentioned
tive reasoning
derives
@ 0~ Teoe5,
that Conwell by qualitadifferent power laws, viz.
and
/.~a E-Oa5 .
References
(1’)
The reason for this discrepancy lies in the fact that the dependence of ip (defined in formula (31) of ref. 1 on Te is not negligible. The results (l’), have already been found by Paranjape [4] using a 62
1. 2. 3. 4.
E. M. Conwell, Phys. Rev. 135 (1964) A814. C.Herring and E.Vogt, Phys. Rev. 101 (1956) 944. K.Seeger, Z. Phys. 182 (1965) 510. V.V.Paranjape, Proc. Phys. Sot. (London) 80 (1962) 971.