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Fermi surfaces and metals
• construction of Fermi surface
• semiclassical electron dynamics
• de Haas-van Alphen effect
• experimental determination of Fermi surface
(the Sec on “Calculation of energy bands” will be skipped)
Dept of Phys
M.C. Chang
• Higher Brillouin zones
(for square lattice)
• Reduced zone scheme
3
2
1
2
2
2
3
3
3
3
3
3
Every Brillouin zone
has the same area
3
• At zone boundary, k points to the plane bi-secting the G
G
vector, thus satisfying the Laue condition
k  Gˆ 
2
• Bragg reflection at zone boundaries produce energy gaps (Peierls, 1930)
Beyond the 1st
Brillouin zone
BCC crystal
FCC crystal
Fermi surface for (2D) empty lattice
3
2
• For a monovalent element,
the Fermi wave vector
kF  2 a
• For a divalent element
1
kF  4 a
• For a trivalent element
kF  6 a
• Distortion due to lattice potential
1st BZ
2nd BZ
A larger Fermi "sphere " (empty lattice)
• Extended zone scheme
• Reduced zone scheme
• Periodic zone scheme
Again if we turn on the
lattice potential, then
the corners will be
smoothed.
Fermi surface of alkali metals (monovalent, BCC lattice)
kF = (3π2n)1/3
n = 2/a3
→ kF = (3/4π)1/3(2π/a)
ΓN=(2π/a)[(1/2)2+(1/2)2]1/2
∴ kF = 0.877 ΓN
Fermi spheres of alkali metals
4π/a
Percent deviation of k from the
free electron value (1st octant)
Fermi surface of noble metals (monovalent, FCC lattice)
Band structure
(empty lattice)
kF = (3π2n)1/3,
n = 4/a3
→ kF = (3/2π)1/3(2π/a)
ΓL= ___
kF = ___ ΓL
Fermi surface
(a cross-section)
Fermi surfaces of noble metals
Periodic zone scheme
Fermi surface of Al (trivalent, FCC lattice)
1st BZ
• Empty lattice approximation
2nd BZ
• Actual Fermi surface
Fermi surfaces and metals
• construction of Fermi surface
• semiclassical electron dynamics
• de Haas-van Alphen effect
• experimental determination of Fermi surface
important
Semiclassical electron dynamics (Kittel, Chap 8, p.192)
Consider a wave packet with average location r and wave vector k, then

1  n (k )
r
(
k
)


k


 k  q  E  r (k )  B 



c



Derivation
neglected here
• Notice that E is the external field, which does not include the
lattice field. The effect of lattice is hidden in εn(k) !
Range of validity
• This looks like the usual Lorentz force eq. But It is valid only when
Interband transitions can be neglected. (One band approximation)
eEa   g
g
F
c   g
g
,
F
May not be valid in small gap or heavily
doped semiconductors, but
“never close to being violated in a metal”
c  1.16 104  B / T  eV
• E and B can be non-uniform in space, but they have to be much
smoother than the lattice potential.
• E and B can be oscillating in time, but with the condition    g
Bloch electron in an uniform electric field (Kittel, p.197)
dk
 eE  k (t )  eEt
dt
• Energy dispersion (periodic zone scheme, 1D)
ε(k)
k
π/a
-π/a
v(k)
k
• In a DC electric field, the electrons decelerate and reverse
its motion at the BZ boundary.
• A DC bias produces an AC current ! (called Bloch oscillation)
• Partially filled band without scattering
E
• Partially filled band with scattering time 
eE / 
• Current density
1
j  (  e)
vk

V k filled states
• Why the oscillation is not observed in ordinary crystals?
To complete a cycle (a is the lattice constant),
eET/ = 2π/a → T=h/eEa
For E=104 V/cm, and a=1 A, T=10-10 sec.
But electron collisions take only about 10-14 sec.
∴ a Bloch electron cannot get to the zone boundary without de-phasing.
To observe it, one needs
• a stronger E field → but only up to about 106 V/cm (for semicond)
• a larger a → use superlattice (eg. a = 100 A)
• reduce collision time → use crystals with high quality
(Mendez et al, PRL, 1988)
• Bloch oscillators generate THz microwave:
frequency ～ 1012~13,
wave length λ ～ 0.01 mm - 0.1mm
(Waschke et al, PRL, 1993)
important
Bloch electron in an uniform magnetic field
1  (k )
k
1 d  (k )
 k  B  0, k  vk 
0
dt
dk
v
  e  B,
dt
c
vk 
Therefore, 1. Change of k is perpendicular to the B field,
k|| does not change
and 2. ε(k) is a constant of motion
This determines uniquely an electron orbit on the FS:
B
• For a spherical FS, it just gives
the usual cyclotron orbit.
• For a connected FS, there
might be open orbits.
Cyclotron orbit in real space
The above analysis gives us the orbit in k-space.
What about the orbit in r-space?
e
c
k   r  B  r   2 Bk  r
c
eB
c ˆ
 r (t )  r (0)  
B  [k (t )  k (0)]
eB
r-orbit
k-orbit
⊙
• r-orbit is rotated by 90 degrees from the k-orbit and
scaled by c/eB ≡ λB2
• magnetic length λB = 256 A at B = 1 Tesla
Fermi surfaces and metals
• construction of Fermi surface
• semiclassical electron dynamics
• de Haas-van Alphen effect
• experimental determination of Fermi surface
De Haas-van Alphen effect (1930)
Silver
In a high magnetic field, the magnetization
of a crystal oscillates as the magnetic field
increases.
Similar oscillations are observed in other
physical quantities, such as
• magnetoresistivity
Resistance
in Ga
(Shubnikov-de Haas effect, 1930)
• specific heat
• sound attenuation
… etc
Basically, they are all due to the quantization of electron
energy levels in a magnetic field (Landau levels, 1930)
Quantization of the cyclotron orbits
• In the discussion earlier, the radius of the cyclotron orbit can be varied
continuously, but due to their wave nature, the orbits are in fact quantized.
• Bohr-Sommerfeld quantization rule (Onsager, 1952)
for a closed cyclotron orbit,
1

dr

p

n


h

2

Why (q/c)A is momentum
of field? See Kittel App. G.
q
A, q  e
c
Gauge dependence prob?
e
e
dr

k


dr

r

B

2

Not worse than the gauge

c
c
dependence in qV.
e
e
dr

A



c
c

1  hc
1


 n   n   ,
An  n   n   2B2
2 e
B 
2

where p  pkin  p field  k 
also
• the flux through an r -orbit is quantized in units of
flux quantum: hc/e≡Φ0=4.14·10-7 gauss．cm2
Mansuripur’s Paradox Kirk T. McDonald
important
• Since a k-orbit (circling an area S) is closely related
to a r-orbit (circling an area A), the orbits in k-space
are also quantized (Onsager, 1952)
Sn 
1  2 e
1  2



n

B

n




B4 
2 c
2  B2

An
B=0
• Energy of the orbit (for a spherical FS)

n 
kn  
1
  n   c Landau levels
2m
2

2
• Degeneracy of the Landau level
D2
spin
2 eB / c
 2 / L 
B≠0
2

2 BL2

 2 sample
hc / e
0
Highly
degenerate
• Notice that, in 3D, the kz direction is not quantized
  n,kz
2 2
kz
1

  n   c 
2
2m

In the presence of B, the Fermi sphere
becomes a stack of cylinders.
Remarks:
• Fermi energy ～ 1 eV, cyclotron energy ～ 0.1 meV (for B = 1 T)
∴ the number of cylinders usually ～ 10000
need low T and high B to observe the fine structure.
• Radius of cylinders  B , so they expand as we increase B.
The orbits are pushed out of the FS one by one.
Cyclotron orbits
FS
E
Landau levels
larger level separation,
and larger degeneracy
(both  B)
EF
B
• Successive B’s that produce orbits with the same area:
Sn = (n+1/2) 2πe/c B
Sn-1'= (n-1/2) 2πe/c B' (B' > B)
 1 1  2 e
S  
c
 B B'
equal increment of
1/B reproduces
similar orbits
Quantitative details (2D)
partially
filled
N=50
filled
partially
filled
filled
DB
partially
filled
filled
M 
E
B
Oscillation of the DOS at Fermi energy (3D)
• Number of states in dε are
proportional to areas of
cylinders in an energy shell.
Two extremal
orbits
• The number of states at EF are
highly enhanced when there are
extremal orbits on the FS.
• There are extremal orbits at
regular interval of 1/B.
• This oscillation in 1/B can be
detected in any physical quantity
that depends on the DOS
.
In the dHvA experiment of silver, the two
different periods of oscillation are due
two different extremal orbits.
Determination of FS
 1 1  2 e
Recall that Se    
c
 B B'
Therefore, from the two periods we can
determine the ratio between the sizes of
the "neck" and the "belly“.
A111(belly)/A111(neck)=27
A111(belly)/A111(neck)=51
B
A111(belly)/A111(neck)=29