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3.225 Quiz 2007
e=1.602x10-19 C
mo=9.11x10-31 kg
c=2.998x108 m/sec
εo=8.854x10-12 F/m
kb=1.38x10-23 J/K
h=6.626x10-34 J-sec
ħ=1.054x10-34 J-sec
A=6.022x1023 mole-1
1. 1­D Silicon
Silicon has a valence 4, a=0.5428nm
(a) Using the Drude model, calculate the estimated conductivity along the 1-D wire
n=
Z 4
ne2 τ
= , σ=
→ calculate using τ = 10-14 s
a a
m0
(b) What is the plasma frequency?
ωp =
ne 2
m 0ε 0
c) Now let us have electron waves instead of electron particles in our 1-D Si; determine
kF and EF
nπ
4
2π
, n = ⇒ k F 1−d =
2
a
a
2
2
2 2
ℏ kF
2π ℏ
=
= * 2
2m
ma
k F 1−d =
EF 1−d
1
(d) Derive the density of states for the 1-D Si for free electron waves
g (E) =
1-d
dN dk 1
2kL
2mE
, N=
, k=
dk dE L
π
ℏ
⇒ g( E ) =
1-d
2m 1
πℏ E
(e) If the band gap of interest is (ћ2π2/(2m0a2)), draw the E vs. band structure for
nearly free electron silicon
2
2. 2­D Semiconductor
(a) Derive the density of states for a 2-D semiconductor, g(E)=m*/(πћ2)
dN dk 1
k 2 L2
2m*E
, N=
, k=
dk dE L
2π
ℏ
*
m
2-d
⇒ g (E) = 2
πℏ
g (E)
2-d
=
(b) Assume that Eg = 1.1eV for this intrinsic 2-D semiconductor (i.e. same as for 3-D
Si). Calculate n at room temperature and compare to 3-D Si, where n = 1010 cm-3 for
intrinsic material. Assume m*=m0.
Number of e- per area in conduction band = n (as 2-d semi conductor)
∞
n=
∫ f ( E ) g ( E )dE, f ( E ) ∼
2-d
EC
g 2-d ( E ) =
⇒ n=
m*
, ⇒ n=
πℏ
∞
Eg
 E-E F 
exp  and E C = E g (as intrinsic)
 , EF =
2
 k bT 
2-d
∫ f ( E ) g ( E )dE =
EC
∞
Eg 
 E
m*
exp  +
 dE
∫
πℏ Eg
 k b T 2k b T 
 Eg 
m* k b T
exp  
πℏ
 2k b T 
n2-d < n3-d as unlike g3-d(E), g2-d
(E) is not an increasing function of E (in fact it is a
constant).
3. Dielectric Properties
Describe ε(ω) for 1-D Si in the context of the sources of polarization.
Just electronic polarization…draw ε(ω) vs ω. Show ωoe etc.
3