Some Transport Coefficients in Heavily Doped N-Type Silicon
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Date
1999-06
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Addis Ababa University
Abstract
The Boltzmann transport equation has been solved applying the relaxation tinIe
approxinIation in the presence of weak electric and magnetic fields to obtain a general
expression for the anisotropic PaIt of the distIibution fimction of the degenerate fi·ee
electron gas in heavily doped n-type silicon. The isotropic palt of the distIibution fimction
was assumed to be the Fenni-Dirac distIibution fimction and the constant energy sUliaces in
k-space are considered to be ellipsoidal. The energy band distOltion was taken into
consideration by using the Slotboom's approximation for the general expression of the
quantum density of states in heavily doped semiconductors given by Kane.
Employing the results of the Boltzmann transpOlt equation expression have been derived
for the electron dlift mobility, the Hall coefficient, and the coefficient of transverse
magnetoresistance valid for both the parabolic and non-parabolic density of states assuming
ionized impmity scattering to be the dominant scatteIing mechanism.
Finally, a nmneIical calculation for the nOlmalized Felmi energy, electron dlift mobility,
Hall coefficient, and coefficient of transverse magnetoresistance using the parabolic and
the non-parabolic density of states in the impmity concentration range fi-om Ixl018 to
2xl020 cm·3 at 3QooK reavils that taking the band tail states in the quantum density of
states reduces the magnitudes of these coefficients by as much as 37%, 59%, 54%, and
24 %, respectively.
silicon using Mahan's [15J treatment to ca.!culate many body shifts in both the conduction
and valance bands and the non-parabolic density of states suggested by Slot boom [12J and
from their calculations they obtained excellent agreement between the theoretical and the
experimental values for both optical and electrical band gap narrowing in heavily doped
n-type silicon for various doping concentrations.
Sharma [16J has calculated the diffusion mobility ratio, i.e., D / Ji in a heavily doped
n-type silicon using the non-parabolic density of states and has compared these results
with those obtained by using the parabolic density of states. His calculations have shown
that the difference between the two ca.!culations can be as high as 20% : Moreover, from
his calculations he also concluded that any serious calculation of the transport coefficients
in heavily doped materials must incorporate the effect of band tails.
In light of these excellent agreements between theory and experiment, as confirmed
by Lee and Fossum [14J, one expects that the band tails arising due to the randomness of
impurities should significantly modify the transport properties of highly doped materials.
Therefore, it seems useful to calculate theoretically the various transport coefficients in
heavily doped semiconductors taking into account the band tail density of states. However,
to the best of our knowledge no theoretical calculation of any other transport coefficients
in heavily doped silicon has been reported in the literature which take into account the
band tails due to the random impurity distribution.
In this research wok, an attempt will be made to illustrate that taking the band tails
into account can significantly affect the calculation of the transport coefficients in heavily
doped n-type silicon having many-valley type energy band structure with ellipsoidal constant
energy surfaces. In this thesis, we shall calculate the three most important transport
coefficients, i.e., the drift mobility, the Hall coefficient, and the coefficient of magnetoresistance
in heavily doped n-type silicon taking into account the anisotropy of the crystal
structure, the degeneracy of the free carriers, and the distortion of the energy band structure.
The impurities are considered to be donors and contribute a single electron to the
conduction band. We also assumed that at 3000 J(, all the impurities are ionized, and thus
the electron concentration in the conduction band is equal to the donor concentration Nd •
In section (2.1) we discuss the energy band structure of lightly and heavily doped
n-type silicon, and give the explicit expressions of the parabolic and the non-parabolic
2
densi ty of states.
In section (2.2) we first formula.te the Boltzma.n transport equation using the relaxation
time approximation in the presence of dc electric and magnetic fields and then by
linearizing this general equation, we solve it to obtain an expression for the anisotropic
part of the electron distribution function.
In section (2.3), we derive a general expression of the electron current density J in
terms of the conductivity coefficients using the anisotropic part of the distribution function
derived in section (2.2). Next we evaluate the three conductivity coefficients assuming
ellipsoidal constant energy surfaces. Finally, assuming a transverse orientation for the
magnetic field and the current, it is seen that most of the components of these tensors
will vanish and we are left only with a few of them. As a result of this assumption, the
general expression for the current density will reduce to a simplified expression and thus
we get an expression for the current density which is the modified form of that obtained
by F. Seitz [1].
In chapter 3, we derive a general expression for the drift mobility, the Hall coefficient,
and the coefficient of magnetoresistance using the parabolic as well as the non-parabolic
density of states.
Finally, in chapter 4, we use the general expressions derived in chapter 3 to compute
the drift mobility, the Hall coefficient, and the coefficient of magnetoresistance numerically
in n-type silicon in the range of impurity concentration from 1 X 1018 to 2 X 1020 cm-3
at 3000 K. We also plot the normalized drift mobility, Hall coefficient, and coefficient
of magnetoresistance as a function of nn. At the end we present a comparison of our
results obtained by using the non-parabolic density of states with the corresponding values
calculated by using the parabolic density of states
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Keywords
Heavily Doped N-Type Silicon