Page 247 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Interacting Subsystems
1
1
------------------- =
k
ω
ω
ω – 1 pv ---- 1 + ---- pv + … (7.22)
k
Thus we have for the dielectric function
2 pv ∂f
1
e
k
2 ∫
(
,
ε p ω) ≈ 1 + --------------- --------- 1 + ---- pv + … 0 d k (7.23)
3
k
p 4π 3 ω ω ∂ E k()
Ω k
in which the first term in the power series vanishes due to
(
E k() = E – k) . Now let us assume parabolic bands with
2 2
E k() = — k ⁄ ( 2m∗ ) which gives us a second order term of the form
2
—
i i ∑
—
—
2 2
( pv ) pv( ) =  ∑ ---- p k   ---- p k  = ---------- p k (7.24)
k k  m   m j j  2
i j 3m
which, inserted into (7.23), results in a dielectric function in the long
wavelength limit p → 0
2 2 ∂f
e —
2 2∫
(
,
ε 0 ω) = 1 + ---------------------- k 4 0 dk (7.25)
2
3m ω π ∂ E k()
Ω k
This we write another way to read
ω 2 pl
εω() = 1 – -------- (7.26)
ω 2
where we defined the plasma frequency by
2 2 ∂f
4 
e —
0
2 2∫
ω 2 pl = ---------------- k – E k()   dk (7.27)

3m π ∂
Ω k
The dielectric function vanishes at the plasma frequency and thus the
effective potential shows a singularity at the plasma frequency since
Ψ eff () Ψ ext () ε ω() . Let us calculate the plasma frequency for a
⁄
ω
ω =
(
degenerate Fermi gas, for which f 0 = Θ E – E ) is the distribution
F
k
244 Semiconductors for Micro and Nanosystem Technology