108 CHAPTER 3. CHARGE TRANSPORT IN MATERIALS
Example 3.3The mobility of electrons in pure GaAs at 300 K is 8500 cm^2 /V·s. Calculate
the relaxation time. If the GaAs sample is doped at Nd=10^17 cm−^3 , the mobility
decreases to 5000 cm^2 /V·s. Calculate the relaxation time due to ionized impurity
scattering.
The relaxation time is related to the mobility by
τsc(1) =
m∗μ
e
=
(0. 067 × 0. 91 × 10 −^30 kg)(8500× 10 −^4 m^2 /V·s)
1. 6 × 10 −^19 C
=3. 24 × 10 −^13 s
If the ionized impurities are present, the time is
τsc(2)=
m∗μ
e
=1. 9 × 10 −^13 s
The total scattering rate is the sum of individual scattering rates. Since the scattering rate
is inverse of scattering time we find that (this is called Mathieson’s rule) the
impurity-related timeτsc(imp)is given by
1
τsc(2)
=
1
τsc(1)
+
1
τsc(imp)
which gives
τsc(imp)=4. 6 × 10 −^13 s
Example 3.4The mobility of electrons in pure silicon at 300 K is 1500 cm^2 /Vs. Calculate
the time between scattering events using the conductivity effective mass.
The conductivity mass for indirect semiconductors, such as Si, is given by (see Appendix
C)
m∗σ =3
(
2
m∗t
+
1
m∗
)− 1
=3
(
2
0. 19 mo
+
1
0. 98 mo
)− 1
=0. 26 mo
The scattering time is then
τsc =
μm∗σ
e
=
(0. 26 × 0. 91 × 10 −^30 )(1500× 10 −^4 )
1. 6 × 10 −^19
=2. 2 × 10 −^13 s
Example 3.5Consider two semiconductor samples, one Si and one GaAs. Both materials
are dopedn-type atNd=10^17 cm−^3. Assume 50 % of the donors are ionized at 300 K.
Calculate the conductivity of the samples. Compare this conductivity to the conductivity
of undoped samples.
