1 Chapter Six Free Electron Fermi Gas What determines if the crystal will be a metal, an insulator, or a semiconductor ? Band structures of solids filled states empty states Conduction band partially filled filled states empty states E g E g >>k B T empty states filled states E g Metal semiconductor Insulator Conduction electrons are available at high T or by doping No conduction electrons E Valence band filled / Conduction band empty E g <k B T Conduction electrons are available
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1
Chapter Six Free Electron Fermi Gas
What determines if the crystal will be a metal, an insulator, or a semiconductor ?
Band structures of solids
filledstates
empty states
Conduction bandpartially filled
filledstates
empty states
Eg
Eg>>kBT
empty states
filledstates
Eg
Metal semiconductor Insulator
Conduction electrons are available
at high T or by doping
No conduction electrons
E
Valence band filled / Conduction band emptyEg<kBT
Conduction electrons are available
2
Basic idea : pushing atoms together to form a crystal
free atoms crystalsmolecules
discrete energy levels splitting of levels band of states
Low energy levels remain discrete and localized on atoms. Core states
High energy levels split to form bands of closely energy levels that can extend through the crystal
valence and conduction bands
3
Free electron model – treat conduction electrons as free particles
Circular motion in both real and k- spaces in free electron model
εk ∝r
ky
kz
kx
B
= constant
Electron at εF moves in orbits along the Fermi surface sphere.
True for all Fermi surfaces, not only for free electrons.
For transport properties, important factor is ωcτ , phase change of electron between two successive scattering events.
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Hall Effect
xnevxjJ
zBB
d==
=r
r
- - - - - -
+ + + + + +E
FB
e-, vdFE
Magnetic field
current density
zy
x
e-, vdx
y
z FBB
In general,
ne1
BjE
Rx
yH −=≡
Hall coefficient
Hall effect revealsdensity and sign of charge carriers.
yjBRynejBE
)y(njBEeBvqEqF
H≡=
−+=×+=
r
rrrvr
( )thickness×==I
VjE
ρ y
x
yH
Hall resistivity [ Ωm ]
Transverse
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0.91Cs
1.11Rb
1.01K
1.01Na
0.81Li
valenceMetal expH
theorH /RR
Alkali metals : OK
1.51Au
1.21Ag
1.41Cu Noble metals :numerically incorrect
-0.33Al
-0.83Zn
-1.22Cd
-0.22Be Higher-valent metals : wrong sign
one hole
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Thermal conductivity
TH TL∇T
jU
dxdTκjU −= the energy transmitted
across unit area per unit timeThe flux of the thermal energy
0E
U
Tjκ
=∇
−≡ r κ : thermal conductivity coefficient
( )( )TκE TLJ
TLE σJ
TU
T
∇−+=
∇−+=rrr
rrr thermal electric current densityElectric current density
Heat current density
In a open-circuit heat measurement,
TκTL
TκTLTLJ
TLE 0J
2T
TTu
T
∇
−=
∇−
∇=
∇=→=
r
rr
rrr
σ
σ
σ
σ
2T
0J
U*
TLκ
Tjκ
−=
∇−≡
=
r
In fact, the 2nd term, LT, is very small in most metals and semiconductors.
Hence, κ* = κ
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Heat current from phonon – previous chapter
τCv31Cv
31κ 2
gg == l
2FF
F
BB
2e
mv21ε
εTknkπ
21C
=
=Tτ
mnkπ
31κ
2B2
e =
In pure metal, the electronic contribution is dominant at all Ts.
In impure metals or disordered materials,τ is reduced by collisions with impurities, and the phonon contribution may be comparable with the electronic contribution.
Apply to free electrons
Ratio of Thermal to Electrical Conductivity2
B2
ek
3πL
=LTT
ek
3π
τ/mne3m / Tτnkπκ 2
B2
2
2B
2e ≡
==
σLorenz number
Wiedemann-Franz law Lth = 2.45 × 10-8 Watt-Ω/K2
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A temperature-independent Lorenz number depends on the relaxation processes for electrical and thermal conductivity being the same – which is not true at all temperatures.