Page 132 - Electrical Properties of Materials

114 The band theory of solids

A similar formula applies in the z-direction. So, oddly enough, the effect-
ive mass may be quite different in different directions. Physically, this means
that the same electric field applied in different directions will cause varying
If you are fond of mathematics you amounts of acceleration. This is bad enough, but something even worse may
may think of the effective mass (or happen. There can be a term like
rather of its reciprocal) as a tensor
quantity, but if you dislike tensors 2 ∂ E –1
2
just regard the electron in a crystal m ∗ xy = . (7.46)
as an extremely whimsical particle ∂k x ∂k y
which, in response to an electric With our simple model, m ∗ turns out to be infinitely large, but it is worth
field in the (say) z-direction, may xy
noting that in general an electric field applied in the x-direction may accelerate
move in a different direction.
an electron in the y-direction. As far as I know there are no electronic devices
making use of this effect; if you want to invent something quickly, bear this
possibility in mind.

7.6 The effective number of free electrons
Let us now leave the fanciful world of three dimensions and return to the
∗ Do not mistake this for the rate of mathematically simpler one-dimensional case. In a manner rather similar to
change of electric current under station- the derivation of effective mass we can derive a formula for the number of
ary conditions. For the steady state to ap- electrons available for conduction. According to eqn (7.40),
ply one must take collisions into account
as well. 2
dv g 1 ∂ E
= eE . (7.47)
2
dt ∂k 2
We have here the formula for the acceleration of an electron. But we have
not only one electron, we have lots of electrons. Every available state may be
† filled by an electron; so the total effect of accelerating all the electrons may be
We have already done it twice be-
fore, but since the density of states obtained by a summation over all the occupied states. We wish to sum dv g /dt
is a rather difficult concept (making for all electrons. Multiplying by the electron charge,
something continuous having previously
stressed that it must be discrete), and
since this is a slightly different situation, d (ev g )
we shall do the derivation again. Re- dt
member, we are in one dimension, and
we are interested in the number of states is nothing else but the rate of change of electric current that flows initially
in momentum space in an interval dp x . ∗
According to eqn (6.31), when an electric field is applied. Thus,
h
p x = n x , dI d
2 = (ev g )
dt dt
where n x is an integer. So for unit length 2
there is exactly one state and for a 1 2 ∂ E
length dp x the number of states, dn x ,is = 2 e E ∂k 2 , (7.48)
(2/h)dp x , which is equal to (1/π)dk x .
We have to divide by 2 because only pos-
itive values of n are permitted and have or, going over to integration,
to multiply by 2 because of the two pos-
2
sible values of spin. Thus, the number of dI 1 1 d E
2
states in a dk x interval remains = e E dk, (7.49)
dt 2 π dk 2
1
dk x .
π †
where the density of states in the range dk is dk/π.