04.11.2011
Materialwissenschaften I
WiSe 11/12
A brief introduction to band
structure and electron
transport in materials
Priv.-Doz. Dr. Bert Nickel
[email protected]
Literatur
Hunklinger: Solid state physics
R. Tilley: Understanding solids
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Todays questions
• what is the electronic structure of a solid ?
• how do electrons behave in a solid ?
• what is the origin of metallic
semiconducting, and insulating behaviour?
Band structures
(e) free electron (fcc)
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key terms
•
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•
•
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plane wave, standing wave, wave vector k
Bragg condition, reduced Brillouin zone
dispersion relation w(k)
group velocity vg = dw/dk
effective mass of an electron
Band gap
Bloch oscillation
Band structure, Pauli principle
Fermi energy, Fermi wave length
Waves: definition of terms
wave vector k :
(2p/lX, 2p/lY, 2p/lZ)
plane wave :
E(x > 0,t) = E0 exp i(wt - k.r)
phase velocity :
v=w/k
two interfering plane waves :
E = 2 E0 exp i(w0t - k0.r) cos [(dwt - dk.r)/2]
group velocity :
vg = dw/dk  group velocity
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E (Amplitude)
Waves: definition of terms
t=0
1
t
0
-1
0
1
2
3
x in units of wave length l
4
dispersion relation:
free electron in a constant field
Energy
Energy (k) =
 2k2/(2m)

Wave vector (k)
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two counter propagating waves form a standing wave
E (Amplitude)
E = 2 exp i (wt) cos(k.r)
t=0
1
0
-1
t = T/2
0
1
2
3
x in units of wave length l
4
1
the two
solutions
0
-1
0
E (Amplitude)
E (Amplitude)
n1 n2
1
2
3
n
n
x 1 in2 units of wave length l
k = Pp/a , P = 1, 2, 3, ...

Bragg condition
4
1
0
-1
0
1
2
3
x in units of wave length l
4
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Energy
electron in an ideal crystal
Energy gap
-2
-1
0
1
k in units of (p/a)
2
E in units of E
Brillouin zone
5
4
3
2
1
0
-2
-1
0
1
k in units of (p/a0)
2
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reduced zone scheme
extended zone scheme
repeated zone scheme
Energy
Band structure and Pauli‘s principle
EF
Empty band
Filled band
-2
-1
0
1
k in units of (p/a)
2
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Energy
Effect of an applied field on
electrons in a crystal
EF
-2
-1
0
1
k in units of (p/a)
2
Energy
example: Na (3s1) crystal
EF = 8.13 eV
-2
a = 0.423 nm
-0.877
0.877
-1
0
1
k in units of kBRAGG
2
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Band structures
(e) free electron (fcc)
Energie in eV
Art
Material
0K
300 K
C (als Diamant)
indirekt
5,4
5,46–6,4
Si
indirekt
1,17
1,12
Ge
indirekt
0,75
0,67
Se
direkt
1,74
indirekt
2,36
IV-IV-Verbindungen
SiC 3C
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III-V-Verbindungen
InP
direkt
1,42
1,27
InAs
direkt
0,43
0,355
InSb
direkt
0,23
0,17
InN
direkt
0,7
InxGa1-xN
direkt
0,7–3,37
GaN
direkt
3,37
GaP 3C
indirekt
2,26
GaSb
direkt
0,81
0,69
GaAs
direkt
1,52
1,43
AlxGa1-xAs
x<0,4
,x>0,4 direkt
indirekt
1,42–2,16
II-VI-Verbindungen
TiO2
ZnO
direkt
ZnS
ZnSe
3,03
3,2
3,436
3,37
3,56
direkt
2,70
CdS
2,42
CdSe
1,74
CdTe
1,45
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