Fermi Surface examples, elementary metals: v From www.phys.ufl.edu/fermisurface/periodic_table.html
Al
Si
La1-­‐xSrxCuO4
superconductor
(ARPES measurement)
Razzoli et al. New J. Physics 2010
ARPES: Angle-­‐resolved photoemission spectroscopy; Fermi surface & electron energy measurement technique (synchrotron source)
Recall:
kF = 3 3π 2 n
ε = 2 k 2 2m
2⎛m⎞
g (ε ) = 2 ⎜ 2 ⎟
π ⎝! ⎠
3/ 2
ε
Free electron result
⎛π 2 ⎞
1
c ≅ k BTg (ε F )⎜⎜ ⎟⎟ ≡ γT
V
⎝ 3 ⎠
Same as
(free electrons)
c≅
Sommerfeld
π 2 nk B ⎛ T
2
⎞
⎜ T ⎟
⎝ F⎠
Real metals with periodic potentials:
• Derived relationships containing g(ε) unchanged.
• However g(ε) will differ due to crystal-potential
Transport properties in metals:
Electrical conductivity: Classic relationship (Drude):
ne τ
j = −ne v = +
m
2
ne τ
σ=
m
2
• Electron wave-packets: in field, can show,
& see text ch. 13
“semiclassical model”
!
dk − e !
≈
E
dt
"
• Scattering; only allowed for electrons near Fermi surface.
• But as k changes, τ = collision time can define for all
electrons,
dn
n
= − ⇒ τ = lifetime
dt
τ
Fermi surface displaced.
• So classical formula may apply to Fermi gas situation;
scattering due to defects or phonons.
Not from regular crystal positions
Transport properties in metals:
Electrical conductivity: Classic relationship (Drude):
ne τ
j = −ne v = +
m
2
k F = 3 3π 2 n
ne τ
σ=
m
2
ε F = ! 2 k F 2 2m = ! 2 (3π 2 n) 2 / 3 2m
2⎛m⎞
g (ε F ) = 2 ⎜ 2 ⎟
π ⎝! ⎠
3/ 2
(
m 3π 2 n
εF =
! 2π 2
ne 2τ 2
e 2τ g (ε F )vF2 e 2τ
σ=
= g (ε F )ε F
=
m
3
m
3
)
1/ 3
(Expressed in terms of Fermi surface properties only)
• Classical formula often applied to Fermi gas situation;
scattering due to defects or phonons.
or other electrons, etc.
Transport properties:
Matthiessen’s rule: Scattering rates add.
1
τ
=
1
τ impurity
+
1
τ phonons
+ ...
• General rule; uncorrelated processes, metal resistivities add.
• Impurities always decrease mobility (increase resistivity)
• Alloy resistivity may be large & constant vs. T.
eτ
µ=
m
σ = neµ ≡ 1 / ρ
mobility
ρ = ρ impurity + ρ phonons + ...
With impurities
1
τ
∝ n phonon
High T,
Good metal
Transport properties:
Mobilities:
eτ
µ=
m
Measures scattering rate
• Mass also modified for states in crystal potential,
[example, graphene m* = 0]
µ ~ 100 cm V .s
RT
2
• GaAs,
µ ~ 1000 cm V .s
µ ~ 107 cm 2 V .s
• “world record”
• Copper,
2
Metal, n is constant, conductivity decrease vs. T.
Semiconductor n and σ increase as T increases.
Transport properties:
Mobilities:
eτ
µ=
m
Measures scattering rate
τ ~ 10 −13 s, ! ~ 10 −5 cm
µ ~ 100 cm V .s
RT
2
• GaAs,
µ ~ 1000 cm V .s
µ ~ 107 cm 2 V .s
• “world record”
• Copper,
2
quantum hall effect – systems;
electrons “ballistic”
“high electron
mobility transitor”
Transport properties:
1
Thermal conductivity, κ el = C!v
3
•
Works for degenerate
system too
• v (mean speed) is Fermi velocity for good metals.
• Specific heat we have seen:
C = γT
• Generally electron + phonon contributions to κ add.
κ el
2
κ el π ⎛ k B ⎞
=
⎜ ⎟T
σ
3 ⎝ e ⎠
2
Wiedemann-Franz law
(see ch. 1, different
result for classical
case)
⎛π 2 ⎞
1
c ≅ k BTg (ε F )⎜⎜ ⎟⎟ ≡ γT
V
⎝ 3 ⎠
g (ε F )vF2 e 2τ
σ=
3
Crystals:
Reading: Ch. 3, Ch. 4, Ch. 7.
Crystal = Bravais lattice + Basis.




R
=
n
a
+
n
a
+
n
a
Bravais lattice = repeated set of points,
1 1
2 2
3 3
!
di
Basis = location of atoms “decorating” the lattice
  
• Primitive Lattice vectors: a1 , a2 , a3
(not unique)
• Primitive cell. Space region, translated by all lattice vectors will fill space.
(not unique)
• Wigner Seitz primitive cell: space region closer to a given Lattice Point than
any other Lattice Point. Has the same point group symmetry as the lattice.
• Conventional cell: Larger than primitive cell, still tiles space, normally chosen
to show crystal symmetry. (Example: silicon conventional cell = cube; primitive
cell is that of the face-centered cubic [FCC] lattice, 4 times smaller.)
Symmetry classes
14 Bravais lattices
All Hexagonal
with
Space group
194
Including symmetry of point group
(lattice decoration),
230 “Space Group” Symmetry classes