2. F i f d El t
Fermi surfaces and Electron
dynamics
dynamics
¾Band structure calculations give E(k)
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
¾Form of Fermi surface is important
¾Fermi surface can be complicated due to
overlapping bands
overlapping bands.
3. Constructing Brillouin Zones
2D Square lattice. BZ constructed from the perpendicular
bisectors of the vectors joining a reciprocal lattice point
t i hb i l tti i t
to neighbouring lattice points 2π/a
1st B. Z.
2ndB. Z.
4. The Fermi Metals have a Fermi energy E
Surface
Metals have a Fermi energy, EF.
The Fermi Temperature,TF, is the
temperature at which kBTF = EF
temperature at which kBTF = EF.
All the free electron states within
a Fermi sphere in k-space are
a Fermi sphere in k space are
filled up to a Fermi wavevector,kF.
The surface of this sphere is
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
A Fermi surface still exists when the states are not free
electron states but it need not be a sphere.
5. Brillouin Zones and Fermi Surfaces
Empty Lattice model (limit of weak
Empty Lattice model (limit of weak
lattice potential):
States are Bloch states.Independent
E E2
States are Bloch states.Independent
states have k-vectors in first BZ.
No energy gaps at the BZ boundaries.
E1
kx π/a
−π/a 0
ky
[100]
kx = ky
E1
E2
y
E2
1
E1
1st B Z
k
−21/2π/a 0 21/2π/a
[110]
2st B. Z.
1st B. Z.
6. Fermi Contours in reduced Zone
E2
PLUS
PLUS
Parts of Fermi circle
moved into 1st BZ
1st B. Z. moved into 1st BZ
from 2nd BZ
2st B. Z.
. .
Reduced Zone scheme
Extended Zone scheme
8. E = -α –γ( Cos[kx x] - Cos[ky y]),
2D simple square
Lattice tight binding
Lattice tight binding
model.
Changing Fermi
Changing Fermi
Contour with
I i F i
Increasing Fermi
Energy.
http://dept.physics.upenn.edu/~mele/phys518/anims/Kronig/FermiSurf1.gif
9. BZs and Fermi Surfaces with gaps
E2
E1
E2
/ /
E1
Energy gaps make the Fermi contours
0
−π/a −π/a
kx
1st B Z Energy gaps make the Fermi contours
appear discontinuous at the BZ boundaries.
dE/dk = 0 at BZ boundaries. Fermi contour
2st B. Z.
1st B. Z.
perpendicular to BZ boundary.
10. BZs and Fermi Surfaces with gaps
E
ky
No gaps With gaps
E2
E1
E1
E2
E1
1st B Z
2st B. Z.
1st B. Z.
Energy gaps: Fermi contours appear discontinuous at the BZ boundaries.
dE/dk = 0 at BZ boundaries. Fermi contour perpendicular to BZ boundary.
11. Fermi Surfaces with gaps “Hole
Fermi Surfaces with gaps “Hole
like” orbits
like orbits
Periodic zone picture of
part of the Fermi contour
at energy E1.
1
On this part of the Fermi
contour electrons behave
lik iti l h d
like positively charged
“holes”. See later
12. Fermi Surfaces with gaps:
Fermi Surfaces with gaps:
“Electron like” orbits
Periodic zone picture of
Periodic zone picture of
part of the Fermi contour
at energy E2.
On this part of the Fermi
contour electrons behave
like negatively charged
like negatively charged
“electrons”. See later
13. Motion in a magnetic field
Motion in a magnetic field
Free electrons B
k
B
v
F ×
−
=
×
−
= )
/
( m
e
e =
The electrons move in circles in real space and in k-space.
Bloch electrons B
k
B
v
k
k ×
∇
−
=
×
−
= )
(
2
E
e
e
dt
d
=
=
In both cases the Lorentz force does not change the energy of the
electrons. The electrons move on contours of constant E.
y k
y ky
x kx
14. Electron and Hole orbits
dk
dk
dk
dk
B
)
k
(
E
e
dt
k
d
k
2
×
∇
−
=
=
Filled states are indicated in grey.
dE
dk
dk
dE
dt
dE
dk
dk
dE
dt
dE
dk dE
dk
dk
dE
dt
dk
dE
dt
dk
dt
dk
Bz
ky
ky dk
dt
dk
dk
dt dk
dt
dk
dk
Bz
Bz
ky
ky
(a) (b)
(a) (b)
kx
kx 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
15. Electron like orbits
Periodic zone
picture of Fermi
contour ( E ) near
contour ( E1 ) near
bottom of a band.
E1
Grad E
E
k /
/ 0
E1
kx π/a
−π/a 0
17. Tight binding simple cubic
d l F i S f
model:Fermi Surfaces
-α – γ(Cos[kx x] - Cos[ky y] - Cos[kz z]
Increasing Fermi Energy
h //h i b / l l /f i i l h l
http://home.cc.umanitoba.ca/~loly/fermiarticle.html
18. The Fermi Metals have a Fermi energy E
Surface
Metals have a Fermi energy, EF.
The Fermi Temperature,TF, is the
temperature at which kBTF = EF
temperature at which kBTF = EF.
All the free electron states within
a Fermi sphere in k-space are
a Fermi sphere in k space are
filled up to a Fermi wavevector,kF.
The surface of this sphere is
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
A Fermi surface still exists when the states are not free
electron states but it need not be a sphere.
