Solid State Theory

Physics 545

Fermi Surfaces
 F
Fermi
i surfaces
f and
d Electron
El t
dynamics
¾Band structure calculations give E(k)
¾E(k) determines the dynamics of the electrons
¾It is E(k) at the Fermi Surface that is important
¾Form of Fermi surface is important
¾Fermi surface can be complicated due to
overlapping bands.
bands
 Constructing Brillouin Zones
2D Square lattice. BZ constructed from the perpendicular
bisectors of the vectors joining a reciprocal lattice point
t neighbouring
to i hb i lattice
l tti points
i t 2π/a

1st B. Z.

2ndB. Z.
 The Fermi Metals have a Fermi energy,
energy EF.
Surface The Fermi Temperature,TF, is the
temperature at which kBTF = EF.
All the free electron states within
a Fermi sphere in k k-space
space are
filled up to a Fermi wavevector,kF.
The surface of this sphere is
called the Fermi surface.
On the Fermi surface the free
electrons have a Fermi velocity vF
= hkF/me.

A Fermi surface still exists when the states are not free
electron states but it need not be a sphere.
 Brillouin Zones and Fermi Surfaces
Empty Lattice model (limit of weak
lattice potential):
E E2
States are Bloch states.Independent
states have k-vectors in first BZ. E1
No energy gaps at the BZ boundaries.

ky −π/a 0 kx π/a
[100]
E2
E1 kx = ky
E2

E1

1st B.
B Z.
Z
2st B. Z. −21/2π/a 0 k [110] 21/2π/a
 Fermi Contours in reduced Zone
E2

PLUS

Parts of Fermi circle

1st B.. Z.. moved into 1st BZ
2st B. Z. from 2nd BZ

Extended Zone scheme Reduced Zone scheme

 Fermi Contours
i periodic
in i di Zone
Z
E2

1st B.
B Z.
Z
2st B. Z.
 E = -α –γ( Cos[kx x] - Cos[ky y]),

2D simple square
Lattice tight binding
model.
Changing Fermi
Contour with
I
Increasing
i Fermi
F i
Energy.

http://dept.physics.upenn.edu/~mele/phys518/anims/Kronig/FermiSurf1.gif
 BZs and Fermi Surfaces with gaps

E2

E1
E2

E1
−π/a
/ 0 −π/a
/
kx

1st B.
B Z.
Z Energy gaps make the Fermi contours
appear discontinuous at the BZ boundaries.
2st B. Z.
dE/dk = 0 at BZ boundaries. Fermi contour
perpendicular to BZ boundary.
 BZs and Fermi Surfaces with gaps
No gaps With gaps
ky E2
E2
E1 E1

1st B.
B Z.
Z
2st B. Z.

Energy gaps: Fermi contours appear discontinuous at the BZ boundaries.

dE/dk = 0 at BZ boundaries. Fermi contour perpendicular to BZ boundary.
Fermi Surfaces with gaps “Hole
like” orbits
like
Periodic zone picture of
part of the Fermi contour
at energy E1.
On this part of the Fermi
contour electrons behave
lik positively
like iti l charged
h d
“holes”. See later
 Fermi Surfaces with gaps:
“Electron like” orbits
Periodic zone picture of
part of the Fermi contour
at energy E2.
On this part of the Fermi
contour electrons behave
like negatively charged
“electrons”. See later
 Motion in a magnetic field
Free electrons F = −ev × B = −(e= / m)k × B

The electrons move in circles in real space and in k-space.

dk e e
Bloch electrons = − v × B = − 2 ∇ k E (k ) × B
dt = =
In both cases the Lorentz force does not change the energy of the
electrons. The electrons move on contours of constant E.
y ky

x kx
 Electron and Hole orbits
dk e Filled states are indicated in grey.
= − 2 ∇ k E(k ) × B
dt = dk
dt
dE
dk dE
dk
ky dt dk
ky
Bz

(a) (b)

kx kx

(a) Electron like orbit centred on k = 0. Electrons move anti-clockwise.

(b) Hole like orbit. Electrons move clockwise as if they have positive
charge
 Electron like orbits

Periodic zone
picture of Fermi
contour ( E1 ) near
bottom of a band.

E1

E Grad E

E1
−π/a
/ 0 kx π/a
/
 Hole like
orbits
Periodic zone
picture of the
Fermi contour at
the top of a band
Grad E
E2
E

E2

−π/a
π/a 0 kx π/a
/
 Tight binding simple cubic
model:Fermi
d lF i Surfaces
S f
-α – γ(Cos[kx x] - Cos[ky y] - Cos[kz z]

Increasing Fermi Energy

h //h
http://home.cc.umanitoba.ca/~loly/fermiarticle.html
i b / l l /f i i l h l
 The Fermi Metals have a Fermi energy,
energy EF.
Surface The Fermi Temperature,TF, is the
temperature at which kBTF = EF.
All the free electron states within
a Fermi sphere in k k-space
space are
filled up to a Fermi wavevector,kF.
The surface of this sphere is
called the Fermi surface.
On the Fermi surface the free
electrons have a Fermi velocity vF
= hkF/me.

A Fermi surface still exists when the states are not free
electron states but it need not be a sphere.
 Sodium Copper

http://www.phys.ufl.e
d /f
du/fermisurface/http
i f /h

Strontium
Lead
Palladium
Tungsten
g
Yttrium
Y
Thorium
 Rhenium
Re