Download Fermi surfaces and metals

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
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
k F = 2π a
• For a divalent element
1
k F = 4π a
• For a trivalent element
k F = 6π 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, p.192)
Consider a wave packet with average location r and wave vector k,
G
then
⎧ G G 1 ∂ε (k )
n
G
⎪r (k ) =
=
k
∂
⎪
⎨ G
G G G
G
⎛
r
⎪ =k = q E + (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 << ε g
εg
εF
=ωc << ε g
εg
, =ωc ≅ 1.16 ⋅10−4 [ B / T ] eV
εF
May not be valid in small gap or heavily
doped semiconductors, but
“never close to being violated in a metal”
• 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 =ω << ε g
Bloch electron in an uniform electric field (Kittel, p.197)
G
G
G
G
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 τ
−eEτ / =
• Current density
1
j = ( − e)
V
∑v
k
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
G
G
G G
dk
v
G 1 ∂ε (k )
G
= −e × B, vk =
=
dt
c
= ∂k
G
G G
G G 1 d ε (k )
→ k ⋅ B = 0, k ⋅ vk =
=0
= dt
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 the 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?
G
eG G
=c G G G
G
=k = − r × B → r = − 2 B × k + r&
c
eB
G
=c ˆ G
G
G
→ 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⎞
G G ⎛
Why (q/c)A is momentum
dr
p
n
⋅
=
+
⎜
⎟h
v∫
2⎠
of field? See Kittel App. G.
⎝
G q G
G G
G
where p = pkin + p field = =k + A, q = −e
c
Gauge dependence prob?
e
e
G G
G G G
⋅
=
−
⋅
×
=
2
Φ
=
dr
k
dr
r
B
Not worse than the gauge
v∫
c v∫
c
dependence in qV.
e
G G e
dr ⋅ A = Φ
v
∫
c
c
1 ⎞ hc
⎛
⇒ Φn = ⎜ n + ⎟
2⎠ e
⎝
• the 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
Sn =
1 ⎞ 2π e
⎛
n
B
=
+
⎟
λB4 ⎜⎝
2 ⎠ =c
An
(Onsager, 1952)
B=0
• Energy of the orbit (for a spherical FS)
( =kn )
εn =
2
1⎞
⎛
= ⎜ n + ⎟ = ωc
2m
2⎠
⎝
Landau levels
• Degeneracy of the Landau level
(assuming spin degeneracy)
D=2
B≠0
2π eB / =c
( 2π / L )
2
Φ sample
2 BL2
=
=2
Φ0
hc / e
• Notice that the kz direction is not quantized
ε n,k
z
= 2 k z2
1⎞
⎛
= ⎜ n + ⎟ =ωc +
2⎠
2m
⎝
In the presence of B, the Fermi sphere
becomes a stack of cylinders.
Note:
• 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.
FS
E
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'= (n-1/2) 2πe/=c B' (B' > B)
⎛ 1 1 ⎞ 2π e
S⎜ − ⎟=
⎝ B B ' ⎠ =c
equal increment of
1/B reproduces
similar orbits
Detailed analysis (2D)
partially
filled
partially
filled
filled
filled
M =−
partially
filled
filled
∂E
∂B
Oscillation of the DOS at the Fermi energy (3D)
• Number of states are
proportional to areas of
cylinders in 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
Determination of FS
In the dHvA experiment of silver, the two
different periods of oscillation are due
two different extremal orbits
⎛ 1 1 ⎞ 2π e
Recall that Se ⎜ − ⎟ =
⎝ B B ' ⎠ =c
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
A111(belly)/A111(neck)=29