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On the oscillatory nature of the Debye screening length in degenerate semiconductors under magnetic quantization
PHYSICS LETTERS
Volume 53A, number 5
14 July 1975
ON THE OSCILLATORY NATURE OF THE DEBYE SCREENING LENGTH IN DEGENERATE SEMICONDUCTORS UNDER MAGNETIC QUANTIZATION A.N. CHAKRAVARTI and D. MUKHERJI Institute of Radio Physics and Electronics, University College of Science and Technology, 92 Acharya Rafblkz Chandm Road, Gdcutta 700009, India
Received 9 April 1975 The nature of the magnetic field dependence of the Debye screening length in degenerate semiconductors is shown to be oscillatory under the influence of magnetic quantization.
In recent years, the non-validity of the conventional Einstein relation for the diffusivity-mobility ratio of the carriers in semiconductors under conditions of degeneracy [l-4] has stimulated great interest in examining the basic parameters of degenerate semiconductors under various physical conditions. One such parameter is’the Debye screening length which is often used in calculations on semiconductor devices and is significantly influenced by the degree of degeneracy. Incidentally, the influence of magnetic quantization on the Debye screening length in degenerate semiconductors has not been reported in the literature. It would, therefore, be of much interest to determine this effect which is done in what follows for degenerate parabolic bands. The field of a charged impurity centre in a semiconductor modifies the average carrier density in its neighbourhood. Let nb be the density at a distance t from the centre and no the density for large values of r. The electrostatic potential 4 near the centre, assuming spherical symmetry, will then satisfy Poisson’s equation in the form -1 d2W_ --e(nb - n,)/e = e2 y dr2
2 $
L,,&O = (drl,/drlo)1/2
(3)
where n (= E&T) is the reduced Fermi energy and the subscripts H and 6 indicate the presence and absence of a quantizing magnetic field respectively. Further, for degenerate semiconductors having parabolic energy bands under the influence of magnetic quantization, we have the following relation [S] between QH and ~0: JImax c
$I=0
KQ@)
- ti - $ I l/2 = f [7@]
3/s,
7) B 0
(4)
where 8 = RoJkT, wc being the cyclotron frequency and $ is an integer or zero ($,, R oo/O). Eq. (4)
(1)
since nb = no - (dn,/d&-)
e@,e being the charge, e the static permittivity and EP the Fermi energy. The solution of eq. (1) may be written as
where l/L: = (e2/e) dn,/d&, Ld being the Debye screening length (the permittivity being expressed in rationalized m.k.s. units). We can, therefore, write
Fig. 1. Magnetic field dependence of the Debye screening length in degenerate semiconductors having parabolic energy bands under the influence of magnetic quantization.
403
Volume 53A. number 5
PHYSICS LETTERS
gives us
Hence, using eqs. (3)-(9, we have determined the plot of Lw/Ldo versus Q~/~Ias shown in tIg. 1. It is observed that the Debye screening length is an oscillatory function of the magnetic field in degenerate semiconductors having parabolic energy bands under the influence of magnetic quantization. The oscillatory dependence is due to the crossingover of the Fermi level by the sub-bands in steps resulting in a successive reduction in the number of occupied Landau levels as the magnetic tield is increased. For each coincidence of a Landau level with the Fermi level, there would be a discontinuity in the density-ofstates function resulting in a peak of the oscillations. Thus, the peaks should occur whenever the Fermi energy is a multiple of the energy separation between two successive Landau levels, i.e. when the equality
404
14 July 1975
EF = (J/ + f ) em/m* is satisfied, and hence the period T of the oscillations when plotted against the reciprocal magnetic field will be given by r = kz?a/m*EF.It is, therefore, noted that the origin of the oscillations in the Debye length is the same as that of the Shubnikov-de Haas oscillations. The authors are indebted to Professor J.N. Bhar for his keen interest in the work.
References [l] P.T. Landsberg, Proc. Roy. Sot. 213A (1952) 226. [2] F.A. Lindholm and R.W. Ayers, Roe. IEEE 56 (1968) 371. [3] A.N. Chakravarti and D.P. Parui, Phys. Lett. 43A (1973) [4] ?h. Chakravarti and B.R. Nag, Intemat. J. Electron. 37 (1974) 281. [S] J.S. Blakemore, Semiconductor statistics (Pergamon Press, London, 1962) p. 91.
Volume 53A, number 5
14 July 1975
ON THE OSCILLATORY NATURE OF THE DEBYE SCREENING LENGTH IN DEGENERATE SEMICONDUCTORS UNDER MAGNETIC QUANTIZATION A.N. CHAKRAVARTI and D. MUKHERJI Institute of Radio Physics and Electronics, University College of Science and Technology, 92 Acharya Rafblkz Chandm Road, Gdcutta 700009, India
Received 9 April 1975 The nature of the magnetic field dependence of the Debye screening length in degenerate semiconductors is shown to be oscillatory under the influence of magnetic quantization.
In recent years, the non-validity of the conventional Einstein relation for the diffusivity-mobility ratio of the carriers in semiconductors under conditions of degeneracy [l-4] has stimulated great interest in examining the basic parameters of degenerate semiconductors under various physical conditions. One such parameter is’the Debye screening length which is often used in calculations on semiconductor devices and is significantly influenced by the degree of degeneracy. Incidentally, the influence of magnetic quantization on the Debye screening length in degenerate semiconductors has not been reported in the literature. It would, therefore, be of much interest to determine this effect which is done in what follows for degenerate parabolic bands. The field of a charged impurity centre in a semiconductor modifies the average carrier density in its neighbourhood. Let nb be the density at a distance t from the centre and no the density for large values of r. The electrostatic potential 4 near the centre, assuming spherical symmetry, will then satisfy Poisson’s equation in the form -1 d2W_ --e(nb - n,)/e = e2 y dr2
2 $
L,,&O = (drl,/drlo)1/2
(3)
where n (= E&T) is the reduced Fermi energy and the subscripts H and 6 indicate the presence and absence of a quantizing magnetic field respectively. Further, for degenerate semiconductors having parabolic energy bands under the influence of magnetic quantization, we have the following relation [S] between QH and ~0: JImax c
$I=0
KQ@)
- ti - $ I l/2 = f [7@]
3/s,
7) B 0
(4)
where 8 = RoJkT, wc being the cyclotron frequency and $ is an integer or zero ($,, R oo/O). Eq. (4)
(1)
since nb = no - (dn,/d&-)
e@,e being the charge, e the static permittivity and EP the Fermi energy. The solution of eq. (1) may be written as
where l/L: = (e2/e) dn,/d&, Ld being the Debye screening length (the permittivity being expressed in rationalized m.k.s. units). We can, therefore, write
Fig. 1. Magnetic field dependence of the Debye screening length in degenerate semiconductors having parabolic energy bands under the influence of magnetic quantization.
403
Volume 53A. number 5
PHYSICS LETTERS
gives us
Hence, using eqs. (3)-(9, we have determined the plot of Lw/Ldo versus Q~/~Ias shown in tIg. 1. It is observed that the Debye screening length is an oscillatory function of the magnetic field in degenerate semiconductors having parabolic energy bands under the influence of magnetic quantization. The oscillatory dependence is due to the crossingover of the Fermi level by the sub-bands in steps resulting in a successive reduction in the number of occupied Landau levels as the magnetic tield is increased. For each coincidence of a Landau level with the Fermi level, there would be a discontinuity in the density-ofstates function resulting in a peak of the oscillations. Thus, the peaks should occur whenever the Fermi energy is a multiple of the energy separation between two successive Landau levels, i.e. when the equality
404
14 July 1975
EF = (J/ + f ) em/m* is satisfied, and hence the period T of the oscillations when plotted against the reciprocal magnetic field will be given by r = kz?a/m*EF.It is, therefore, noted that the origin of the oscillations in the Debye length is the same as that of the Shubnikov-de Haas oscillations. The authors are indebted to Professor J.N. Bhar for his keen interest in the work.
References [l] P.T. Landsberg, Proc. Roy. Sot. 213A (1952) 226. [2] F.A. Lindholm and R.W. Ayers, Roe. IEEE 56 (1968) 371. [3] A.N. Chakravarti and D.P. Parui, Phys. Lett. 43A (1973) [4] ?h. Chakravarti and B.R. Nag, Intemat. J. Electron. 37 (1974) 281. [S] J.S. Blakemore, Semiconductor statistics (Pergamon Press, London, 1962) p. 91.