102 CHAPTER 3. CHARGE TRANSPORT IN MATERIALS
band;∞andsare the high frequency and static dielectric constants of the semiconductor,
whileois the free space dielectric constant)
W(k)=e^2 m∗^1 /^2 ωo
4 π√
2
(
o
∞−
o
s)
1+2αE′
γ^1 /^2 (E)Fo(E, E′)×
{
n(ωo) absorption
n(ωo)+1 emission}
. (3.3.18)
where
E′ = E+ωofor absorption
= E−ωofor emission
γ(E)=E(1 +αE)Fo(E, E′)=C−^1(
Aln∣
∣∣
∣
γ^1 /^2 (E)+γ^1 /^2 (E′)
γ^1 /^2 (E)−γ^1 /^2 (E′)∣
∣∣
∣+B
)
A =[2(1+αE)(1 +αE′)+α{γ(E)+γ(E′)}]^2
B = − 2 αγ^1 /^2 (E)γ^1 /^2 (E′)
= ×[4(1 +αE)(1 +αE′)+α{γ(E)+γ(E′)}]a
C =4(1+αE)(1 +αE′)(1 + 2αE)(1 + 2αE′)It is important to examine typical values of scattering rates from these processes. The values for
GaAs are shown in figure 3.9.Notethatthephononemissionprocesscanstartonlyafterthe
electronhasenergyequaltothephononenergy. Optical phonon scattering is the most important
scattering mechanism for high-field or high-temperature transport of electrons. The emission rate
is stronger than the absorption rate by the raten(ω 0 p+1)ton(ω 0 ). Optical phonon emission is
the dominant mechanism for electrons to lose energy they gain from the electric field.
Ionized impurity scattering
An important scattering mechanism is due to ionized dopants. The scattering potential is
Coulombic in nature, except that the potential is suppressed by screening effects due to free
carriers. The screening is due to the presence of the other free electrons or holes, which form
a cloud around the ion. There are several models for the ionized impurity scattering potential.
A good approximation for the potential seen by electrons in a semiconductor is given by the
screened Coulombic potential
V(r)=e^2
e−λr
r(3.3.19)
where
λ^2 =ne^2
kBT