8.3 Low-Field Transport
229
(a)
300
200
100
0
10
18
10
19
10
20
10
21
10
22
Electron mobilty (cm /Vs)
2
Donor concentration (cm )
-3
Si
(b)
n-Si
5
6
4
3
2
1
10 20
10 21
Impurity custering Z
Donor concentration (cm )
-3
Fig. 8.3 a Electron mobility in highly doped silicon. Experimental data (symbols) from various sources and modeling
with ionized impurity scattering with (solid line) and without (dashed line) considering impurity clustering. b Effective
impurity cluster charge Z D . Adapted from [722]
8.3.4 Deformation Potential Scattering
Acoustic phonons with small wavevector, i.e. a wavelength large compared to the unit cell, can have
TA or LA character. The TA phonons represent a shear wave (with zero divergence), the LA phonons
are a compression wave (with zero rotation). The LA is a plane wave of displacement δR parallel to
the k-vector q,
δR = A sin (q · R − ωt) .
(8.18)
The strain tensor is given by
i j =
1
2
q i A j + q j A i
cos (q R − ωt) .
(8.19)
It has a diagonal form i j = q i A j for q and ω → 0. Therefore, the LA phonon creates an oscillatory
volume dilatation (and compression) with amplitude q · A. This volume modulation affects the position
of the band edges. For the conduction-band edge the energy change is related to the volume change by
the hydrostatic deformation potential E ac.def. = V ∂ E C /∂V . Since the modulation is small compared
to the energy of the charge carriers, it is mostly an elastic scattering process. The Hamilton operator
for the LA scattering is
ˆ
H = E ac.def. (q · A) .
(8.20)
The size of the LA amplitude is given by the number of phonons in the mode that is given by the Bose–
Einstein distribution, N ph ( = [exp(
kT
)]
−1 . The mobility due to acoustic deformation potential
scattering is found to be
μ ac.def. =
2
√
2π e
4 c l
3 m ∗5/2 E
2
ac.def.
(kT )
−3/2
,
(8.21)
where c l = ρc
LA
s , ρ being the density and c s being the sound velocity. The scattering time increases
like τ ∝ E
−1/2 with the kinetic energy [714].
The acoustical deformation potential scattering is important at high temperatures. It is dominating
in nonpolar semiconductors (Ge, Si) at high temperatures (typically at and above room temperature).
229
(a)
300
200
100
0
10
18
10
19
10
20
10
21
10
22
Electron mobilty (cm /Vs)
2
Donor concentration (cm )
-3
Si
(b)
n-Si
5
6
4
3
2
1
10 20
10 21
Impurity custering Z
Donor concentration (cm )
-3
Fig. 8.3 a Electron mobility in highly doped silicon. Experimental data (symbols) from various sources and modeling
with ionized impurity scattering with (solid line) and without (dashed line) considering impurity clustering. b Effective
impurity cluster charge Z D . Adapted from [722]
8.3.4 Deformation Potential Scattering
Acoustic phonons with small wavevector, i.e. a wavelength large compared to the unit cell, can have
TA or LA character. The TA phonons represent a shear wave (with zero divergence), the LA phonons
are a compression wave (with zero rotation). The LA is a plane wave of displacement δR parallel to
the k-vector q,
δR = A sin (q · R − ωt) .
(8.18)
The strain tensor is given by
i j =
1
2
q i A j + q j A i
cos (q R − ωt) .
(8.19)
It has a diagonal form i j = q i A j for q and ω → 0. Therefore, the LA phonon creates an oscillatory
volume dilatation (and compression) with amplitude q · A. This volume modulation affects the position
of the band edges. For the conduction-band edge the energy change is related to the volume change by
the hydrostatic deformation potential E ac.def. = V ∂ E C /∂V . Since the modulation is small compared
to the energy of the charge carriers, it is mostly an elastic scattering process. The Hamilton operator
for the LA scattering is
ˆ
H = E ac.def. (q · A) .
(8.20)
The size of the LA amplitude is given by the number of phonons in the mode that is given by the Bose–
Einstein distribution, N ph ( = [exp(
kT
)]
−1 . The mobility due to acoustic deformation potential
scattering is found to be
μ ac.def. =
2
√
2π e
4 c l
3 m ∗5/2 E
2
ac.def.
(kT )
−3/2
,
(8.21)
where c l = ρc
LA
s , ρ being the density and c s being the sound velocity. The scattering time increases
like τ ∝ E
−1/2 with the kinetic energy [714].
The acoustical deformation potential scattering is important at high temperatures. It is dominating
in nonpolar semiconductors (Ge, Si) at high temperatures (typically at and above room temperature).