Page 211 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Transport Theory
We know from the discussion about Bloch oscillations that for real band
structures the velocity vectors in (6.38) have to be replaced by
⁄
v = ∂ E ∂ p . This leads to more complicated integrals for the transport
coefficients. In the same way a momentum–dependent relaxation time τ
must be kept in the integrand. Thus we must be aware of the fact that we
are in the regime of a harmonic approximation with constant relaxation
time. Let us rewrite the two equations (6.35) as

∇T i
j = – N ∇η – N -------- = – -- (6.41a)
n 11 12 e
T
∇T
j = – N ∇η – N -------- (6.41b)
Q 21 22 T
i
where is the electron current density.

Transport Two special cases help us to understand the significance of the transport
Coefficients coefficients:
and their
Significance 1. In the absence of a thermal gradient and for a homogeneous material
(∇µ = 0 ) we can easily verify that N is proportional to the elec-
11
σ
trical conductivity , exactly
2
Electrical σσ σ σ= q N 11 (6.42)
Conductivity
From (6.11) we know that the electrical current density for electrons
–
may be written as q j = – qv n = σE . Thus the electron drift
n d
velocity is given by v = – µ E and the constant of proportionality is
e
d
called the electron mobility
σ
Carrier µ = ------ (6.43)
e
Mobility nq
σ
We are already well familiar with the coefficient and now under-
stand the microscopic origin from which it comes. The absence of a

208 Semiconductors for Micro and Nanosystem Technology