Electronic Band Structure of
Solids
Introduction to Solid State Physics
http://www.physics.udel.edu/~bnikolic/teaching/phys624/phys624.html
1
What are quantum numbers?
 Quantum numbers label eigenenergies and eigenfunctions of a Hamiltonian
Sommerfeld:
Bloch:
and
k -vector ( k
is momentum)
k -vector ( k is the crystal momentum)
n (the band index).
•The Crystal Momentum is not the Momentum of a Bloch electron: the rate of change of an
electron momentum is given by the total forces on the electron, but the rate of change of
electronic crystal momentum is:
dr
1  n (k )
 v n (k )   nk vˆ  nk 
;
dt
k
d k
 e  E(r, t )  v n (k )  B(r, t ) 
dt
where forces are exerted only by the external fields, and not by the periodic field of the lattice.
PHYS 624: Electronic Band Structure of Solids
2
Semiclassical dynamics of Bloch electrons
•Bloch states have the property that their expectation values of r and k , follow
classical dynamics. The only change is that now  n (k ) (band structure) must be
used:
e
H classical   n  k  eA(r )   e (r ) 
B  Lk
2m
dp d ( k )
H dr
H



,
v

dt
dt
r dt
p
p
•A perfectly periodic ionic arrangement has zero resistance. Resistivity comes from
imperfections (example: a barrier induces a reflected and transmitted Bloch wave),
which control the mean-free path. This can be much larger than the lattice spacing.
•A fully occupied band does not contribute to the current since the electrons cannot be
promoted to other empty states with higher k . The current is induced by
rearrangement of states near the Fermi energy in a partially occupied band.
•Limits of validity:
eEa
n  const (no interband transitions)
2
 gap
(k )
eB
, c 
F
m

2
 gap
(k )
,  gap (k )   n (k )   n (k )
F
 gap ,  a
PHYS 624: Electronic Band Structure of Solids
3
What is the range of quantum numbers?
k
Sommerfeld:
runs through all of
k-space consistent with the Born-von
Karman periodic boundary
conditions:
 ( x  L, y, z )   ( x, y, z ) 
2

 ( x  L, y, z )   ( x, y  L, z )   k   nx , n y , nz   nx , n y , nz  0, 1, 2,
L

 ( x, y, z )   ( x, y, z  L) 
 ( x  L, y, z )   ( x, y, z )  0 


 ( x  L, y, z )   ( x, y  L, z )  0  k   nx , n y , nz   nx , n y , nz  1, 2,
L
 ( x, y, z )   ( x, y, z  L)  0 
Bloch: For each n,k runs through all wave vectors in a single
primitive cell of the reciprocal lattice consistent with the Bornvon Karman periodic boundary conditions; n runs through an
infinite set of discrete values.
PHYS 624: Electronic Band Structure of Solids
4
What are the energy levels?
Sommerfeld:
2
2
k
 (k ) 
2m
Bloch: For a given band index n,  n (k ) has
no simple explicit form. The only general
property is periodicity in the reciprocal space:
 n (k  G)   n (k )
PHYS 624: Electronic Band Structure of Solids
5
What is the velocity of electron?
Sommerfeld: The mean velocity of an electron in a
level with wave vector
is:
k
k 1 
v

m
k
Bloch: The mean velocity of an electron in a level with
band index
and wave vector
is:
n
Conductivity of a perfect crystal:
 
1  n (k )
v n (k ) 
k
k
NOTE: Quantum mechanical definition of a mean velocity

ˆ    dr  (r )
v  v
 (r )
mi
*
PHYS 624: Electronic Band Structure of Solids
6
What is the Wave function
Sommerfeld: The wave function of an electron with
wave vector
is:
ikr
k
k
1
 (r ) 
e
V
Bloch: The wave function of an electron with band index
and wave vector
is:
k
n
 k (r )  e nk (r )
ikr
where the function nk (r ) has no simple explicit form. The
only general property is its periodicity in the direct lattice
(i.e., real space):  (r  R)   (r)
nk
PHYS 624: Electronic Band Structure of Solids
nk
7
Sommerfeld vs. Bloch: Density of States
Sommerfeld → Bloch
D( ) 
2
dk     (k )   D( ) 
dk     (k ) 



 2 
 2 
2
d
n
d
n B.Z .
PHYS 624: Electronic Band Structure of Solids
8
Bloch: van Hove singularities in the DOS
D( ) 
2
 2 
d
  dk     (k )  
n
n B.Z .
2
 2 
D( )d  
d
dS
n  dk   (k )
k n
Sn ( E )
1
dS k

 2 
d
d     k  n (k )  Δk   k  n (k ) k
D( )
PHYS 624: Electronic Band Structure of Solids
9
Bloch: van Hove singularities in the DOS
of Tight-Binding Hamiltonian
 n (k )  2t  cos(kx a)  cos(k y a)  cos(k z a)   k n (k )  2ta sin(k x a)  sin(k y a)  sin(k z a) 

k  (0, 0, 0) max


 
 
k  n (k )  0 for k    ,  ,   min
a
a
 a


  

 
 
k    , 0, 0  ,  0,  , 0  ,  0, 0,   saddle
a  
a
 a
 

1D
3D
Local DOS:  (r,  )    (r )  (    )
2

DOS: D( )   dr  (r,  )
PHYS 624: Electronic Band Structure of Solids
10
Sommerfeld vs. Bloch: Fermi surface
•Fermi energy    (T  0) represents the sharp occupancy
F
cut-off at T=0 for particles described by the Fermi-Dirac statitics.
•Fermi surface is the locus of points in reciprocal space where
F
k F
 (k )   F
No Fermi surface for insulators!
kF
Points of Fermi
“Surface” in 1D
PHYS 624: Electronic Band Structure of Solids
11
Sommerfeld vs. Bloch: Fermi surface in 3D
Sommerfeld: Fermi Sphere
Bloch: Sometimes sphere, but more likely anything else
For each partially filled band there will be a surface reciprocal space separating occupied
from the unoccupied levels → the set of all such surfaces is known as the Fermi surface
and represents the generalization to Bloch electrons of the free electron Fermi sphere.
The parts of the Fermi surface arising from individual partially filled bands are branches
of the Fermi surface: for each n solve the equation n (k )  F in
variable.

PHYS 624: Electronic Band Structure of Solids

k
12
Is there a Fermi energy of intrinsic Semiconductors?

•If F is defined as the energy separating the
highest occupied from the lowest unoccupied
level, then it is not uniquely specified in a solid
with an energy gap, since any energy in the gap
meets this test.
•People nevertheless speak of “the Fermi energy”
on an intrinsic semiconductor. What they mean is
the chemical potential, which is well defined at any
non-zero temperature. As T  0, the chemical
potential of a solid with an energy gap approaches
the energy of the middle of the gap and one
sometimes finds it asserted that this is the “Fermi
energy”. With either the correct of colloquial
definition,  n (k )   F does not have a solution in a
solid with a gap, which therefore has no Fermi
surface!
PHYS 624: Electronic Band Structure of Solids
13
DOS of real materials: Silicon, Aluminum, Silver
PHYS 624: Electronic Band Structure of Solids
14
Colloquial Semiconductor “Terminology” in Pictures
←PURE
DOPPED→
PHYS 624: Electronic Band Structure of Solids
15
Measuring DOS: Photoemission spectroscopy
Fermi Golden Rule: Probability per
unit time of an electron being ejected
is proportional to the DOS of occupied
electronic states times the probability
(Fermi function) that the state is
occupied:
1
I ( kin ) 
 ( kin )
 D( bin ) f ( bin )  D( kin     ) f ( kin     )
PHYS 624: Electronic Band Structure of Solids
16
Measuring DOS: Photoemission spectroscopy
Once the background is subtracted
off, the subtracted data is proportional
to electronic density of states
convolved with a Fermi functions.
We can also learn about DOS above the Fermi
surface using Inverse Photoemission where electron
beam is focused on the surface and the outgoing flux
of photons is measured.
PHYS 624: Electronic Band Structure of Solids
17
Fourier analysis of systems living on
periodic lattice
f (r )  f (r  R )  f (r )   fG eiGr
G
1
 iGr
iGr
fG 
d
r
e
f
(
r
);
e
dr  0


V pcell pcell
pcell
 (k )   (k  G )   (k )   R eiRk
dk
R  V pcell
  2 
3
e
BZ
Born-von Karman:
3
f (r )   f k e , k  
ikr
k
e
ikR
e
ikR
R
k
i 1
mi
bi ;
Ni
fk 
R
 iRk
 (k )
f (r)  f (r  Ni ai ) 
1
Vcrystal

crystal
dr e ikr f (r );

eikr dr  0
crystal
 N  k ,0 
3

 R   ni ai , 0  ni  N i , N  N1 N 2 N 3 ; k  BZ
 N  R ,0 
i 1

PHYS 624: Electronic Band Structure of Solids
18
Fouirer analysis of Schrödinger equation
U (r )  U (r  R)  U (r )  U G eiGr , G  R  2 m
G
2


2
ˆ
H  (r)   
  U (r)   (r)   (r),  (r)   Ck eikr
k
 2m

2 2
k
ikr
i ( k  +G)r
ikr
C
e

C
U
e


C
e



k
k G
k
2
m
k
k G
k
2 2




k
ikr
k   k  G :  e 
   Ck  U G Ck-G   0, for all r
k
G

 2 m

 2k 2 
   Ck  U G Ck-G  0, for all k   k   (k )

G
 2m

Potential acts to couple Ckwith its reciprocal space translation Ck  G
and the
problem decouples into N independent problems for each k within the first BZ.
PHYS 624: Electronic Band Structure of Solids
19
Fourier analysis, Bloch theorem, and
its corollaries
 k (r )   Ck G e
i ( k G ) r
G

 iGr  ikr
   Ck G e  e
G 

 (r )   (r  R)

 Hˆ  k G (r )   (k  G ) k G (r )
1.  k G (r )   k (r )  
ˆ

 H  k (r )   (k  G ) k (r )   (k )   (k  G )
 2

2
2.  
   ik   U (r)  nk (r)   n (k )nk (r)
 2m

2


2
3.  
   ik   U (r)  n,k (r)   n (k )n,k (r)
 2m

•Each zone n is indexed by a
vector and, therefore, has as
many energy levels as there
are distinct k vector values
within the Brillouin zone, i.e.:
4. n*,k (r )  n , k (r ) ,  n (k )   n (k ) (Kramers theorem)
PHYS 624: Electronic Band Structure of Solids
k
N  N1 N2 N3
20
“Free” Bloch electrons?
2
•Really free electrons → Sommerfeld
degenerate eigenvalues.
k2
continuous spectrum with infinitely
 n (k ) 
2m
•  n (k )= n (k + G) does not mean that two electrons with wave vectors k and k  G
have the same energy, but that any reciprocal lattice point can serve as the origin of n (k. )

•In the case of an infinitesimally
small periodic potential there is
periodicity, but not a real
potential. The  n (k )function
than is practically the same as in
the case of free electrons, but
starting at every point in
reciprocal space.
Bloch electrons in the limit
U  0 : electron moving through an empty lattice!
PHYS 624: Electronic Band Structure of Solids
21
Schrödinger equation for “free” Bloch
electrons
2

 0
2
0
0
 2m    ik    (k )  k (r)  0  (k )  2m  k  G 


1 iGr
0
k (r ) 
e
Counting of Quantum States:
V
2
2
G
Extended Zone Scheme: Fix
(i.e., the BZ) and then count
region corresponding to that zone.
Reduced Zone Scheme: Fix
equivalent states in all BZ.
k
k
vectors within the
in any zone and then, by changing
PHYS 624: Electronic Band Structure of Solids
G
, count all
22
“Free” Bloch electrons at BZ boundary

e
iGx / 2
e
 iGx / 2

cos
x
a
PHYS 624: Electronic Band Structure of Solids

e
iGx / 2
e
 iGx / 2

sin
x
a
23
“Free” Bloch electrons at BZ boundary
•Second order perturbation theory, in crystalline potential, for the reduced
2
zone scheme:
 k (G )   (G)  G  k U G  k  
0
k
H
r G  k   (r )e
0
nk
ikr
Hk U G k
 k0 (G )   k0 (H)
 ...  O(U 3 )
1 i (G k)r

e
V
1
G  k U H  k  U G  ei (  H+G+G) dr  U G G,G-H  U G-H
V G
G
1
G  k U H  k   U (r )dr
2
V
U G
second order correction   0
0


(
G
)


G k
k (G + G )
PHYS 624: Electronic Band Structure of Solids
24
“Free” Bloch electrons at BZ boundary
•For perturbation theory to work, matrix elements of crystal potential have to
be smaller than the level spacing of unperturbed electron → Does not hold at
the BZ boundary!
2
2
k 2   G + k     G2  2Gk 
 2m
2m 
0 at BZ boundary  Laue diffraction!
 k0 (G  0)   k0 (G) 
2
   eikr   ei (G+k)r
2
(k )   (k )   4 U G 

G'
G'
G'

0

0
k    G 0 (k )   G 0 (k  )  U G ;  G 0 (k )   G 0 (k  )  U G
2
2
2


G 0
1 0
(k )   G 0 (k )   G0  (k ) 
2
PHYS 624: Electronic Band Structure of Solids

0
G 0
0
G
2
25
Extended vs. Reduced vs. Repeated
Zone Scheme
•In 1D model, there is always a gap at the Brillouin zone boundaries, even for an
arbitrarily weak potential.
•In higher dimension, where the Brillouin zone boundary is a line (in 2D) or a surface
(in 3D), rather than just two points as here, appearance of an energy gap depends on
the strength of the periodic potential compared with the width of the unperturbed band.
PHYS 624: Electronic Band Structure of Solids
26
Fermi surface in 2D for free Sommerfeld electrons
4 2
 2 
2
b 
, Scell  a 2
 
Scell
 a 
2
S BZ
S state
S BZ
4 2
2 2



, N  N1 N 2
2 N 2 NScell Scrystall
S Fermi  N e S state
2 2 S BZ
 2 
 kF 
a
2
S Fermi  2 N e S state  S BZ  k F  1.128
S Fermi  zN e S state
Sb

 0.798
2
a

a
S BZ
z
, for crystal with z-valence atoms
2
PHYS 624: Electronic Band Structure of Solids
27
Fermi surface in 2D for “free” Bloch electrons
•There are empty states in the first BZ and occupied
states in the second BZ.
•This is a general feature in 2D and 3D: Because of
the band overlap, solid can be metallic even when if
it has two electrons per unit cell.
PHYS 624: Electronic Band Structure of Solids
28
Fermi surface is orthogonal to the BZ boundary
 (k ) 
 (k )   (k  G )

2
0
k  (k ) 
0
 0 (k )   0 (k  G) 
4
1 
 2
1
k  0 (k )  k  0 (k  G ) 

2
2
0
2
 UG
2
(k )   0 (k  G )  k  0 (k )  k  0 (k  G ) 

0
(k )   (k  G ) 
0
4
2
 UG
2
at the BZ boundary :  0 (k )   0 (k  G)
k

k  (k )   k + k + G    + G   G  0
2m
m2

2
PHYS 624: Electronic Band Structure of Solids
2
29
Tight-binding approximation
→Tight Binding approach is completely opposite to “free” Bloch electron: Ignore core
electron dynamics and treat only valence orbitals localized in ionic core potential.
There is another way to generate band gaps in the electronic DOS → they naturally
emerge when perturbing around the atomic limit. As we bring more atoms together or
bring the atoms in the lattice closer together, bands form from mixing of the orbital
states. If the band broadening is small enough, gaps remain between the bands.
PHYS 624: Electronic Band Structure of Solids
30
Constructing Bloch functions from
atomic orbitals
PHYS 624: Electronic Band Structure of Solids
31
From localized orbitals to wave functions overlap
PHYS 624: Electronic Band Structure of Solids
32
Tight-binding method for single s-band
→Tight Binding approach is completely opposite to “free” Bloch electron: Ignore core
electron dynamics and treat only valence orbitals localized in ionic core potential.
Notation: x i   (x  Ri )
PHYS 624: Electronic Band Structure of Solids
33
One-dimensional case
→Assuming that only nearest neighbor orbitals overlap:
PHYS 624: Electronic Band Structure of Solids
34
One-dimensional examples:
s-orbital band vs. p-orbital band
tp  0
ts  0
ts  t p
 kl ( x) 

 e   x  na 
ikx
n 
l
 Re   kl ( x) 
bandwidth: W  4dtl  2 ztl
PHYS 624: Electronic Band Structure of Solids
35
Wannier Functions
→It would be advantageous to have at our disposal localized wave functions with vanishing
overlap i j   ij : Construct Wannier functions as a Fourier transform of Bloch wave
functions!
PHYS 624: Electronic Band Structure of Solids
36
Wannier functions as orthormal basis set
Ri  a
Ri  a
1D example: decay as power law,
so it is not completely localized!
PHYS 624: Electronic Band Structure of Solids
37
Band theory of Graphite and Carbon Nanotubes
(works also for MgB2 ): Application of TBH method
•Graphite is a 2D network made of 3D carbon
atoms. It is very stable material (highest melting
temperature known, more stable than diamond).
It peels easily in layers (remember pencils?).
•A single free standing layer would be hard to
peel off, but if it could be done, no doubt it would
be quite stable except at the edges – carbon
nanotubes are just this, layers of graphite which
solve the edge problem by curling into closed
cylinders.
•CNT come in ‘’single-walled” and “multi-walled”
forms, with quantized circumference of many
sizes, and with quantized helical pitch of many
types.
Lattice structure of graphite layer: There are two carbon atoms per cell,
designated as the A and B sublattices. The vector
connects the two
sublattices and is not a translation vector. Primitive translation vectors are
. PHYS 624: Electronic Band Structure of Solids

a, b
38
2
Chemistry of Graphite: sp hybridization, covalent
bonds, and all of that
1 A, B
2 A, B
s 
px
3
3
1 A, B 1 A, B 1 A, B

s
px 
py
3
6
2
1 A, B 1 A, B 1 A, B

s 
px 
py
3
6
2
4A, B  pzA, B
1A, B 
2A, B
3A, B

b ( onding )
i
1 A
1 A
B
a ( ntibonding )
 i  i  ,  i
 i  iB 
2
2
PHYS 624: Electronic Band Structure of Solids
39
Truncating the basis to a single  orbital per atom
s
p ,p
•The atomic
orbitals as well as the
atomic carbon
functions form strong
x y
bonding orbitals which are doubly occupied
and lie below the Fermi energy. They also
form strongly antibonding orbitals which are
high up and empty.
•This leaves space on energy axis near the
Fermi level for
orbitals (they point
perpendicular to the direction of the bond
between them)


•The
orbitals form two bands, one
bonding band lower in energy which is
doubly occupied, and one antibonding band
higher in energy which is unoccupied.
•These two bands are not separated by a
gap, but have tendency to overlap by a
small amount leading to a “semimetal”.
PHYS 624: Electronic Band Structure of Solids
Eigenstates of translation operator:
kA 
1
N
ikm
e
 A (r  m)
m
1
ikm
e
B (r  m )

N m
B (r )   A (r  τ )
kB 
Bloch eigenstates:
kn   kA   kB
   1
2
2
40
Diagonalize 2 x 2 Hamiltonian
 kA Hˆ kA
kA Hˆ kB 
H (k )  

 kB Hˆ kA
ˆ
kB H kB 

1
kA Hˆ kA   eik ( m-m)  dr A (r  m) Hˆ  A (r  m)
N mm
kB Hˆ kB   drB (r ) Hˆ B (r )  kA Hˆ kA   dr A (r ) Hˆ  A (r )
kA Hˆ kB   dr A (r ) Hˆ B (r )  e  ika  dr A (r ) Hˆ B (r  a)  e  ikb  dr A (r ) Hˆ B (r  b)
t
  (k )  t 1  eika  eikb
  (k )  t 3  2cos(ka)  2cos(kb)  2cos(k (a  b)) 
1/ 2
PHYS 624: Electronic Band Structure of Solids
41
Band structure plotting: Irreducible BZ
 (r )   R | Ta   (r )   (R | Ta  r )   kn ( R 1r  R 1a)
n
Rk
n
k
R | Ta  r  Rr  a, R | Ta 
PHYS 624: Electronic Band Structure of Solids
1
n
k
1
 R |  R a
1
1
1
42
Graphite band structure in pictures
 (k )
•Plot
for some special directions in reciprocal space: there are three
directions of special symmetry which outline the “irreducible wedge” of the Brillouin
zone. Any other point
of the zone which is not in this wedge can be rotated into a
k-vector inside the wedge by a symmetry operation that leaves the crystal invariant.
k
along  :   (k )  t 5  4 cos  2  
1/ 2
 0    1
middle section :   (k )  t 3  4 cos  2 / 3   2 cos  4 / 3  
1/ 2
along  :   (k )  t 3  2 cos  4 / 3   4 cos  2 / 3  
1/ 2
PHYS 624: Electronic Band Structure of Solids
43
Graphite band structure in pictures:
Pseudo-Potential Plane Wave Method
Electronic Charge Density:
In the plane of atoms
In the plane
perpendicular to atoms
PHYS 624: Electronic Band Structure of Solids
44
Diamond vs. Graphite: Insulator vs. Semimetal
PHYS 624: Electronic Band Structure of Solids
45
Carbon Nanotubes
•Mechanics: Tubes as ultimate fibers.
•Electronics: Tubes as quantum wires.
•Capillary: Tubes as nanocontainers.
PHYS 624: Electronic Band Structure of Solids
46
From graphite sheets to CNT
C-C distance: a  1.421 A
1 3
primitive vectors: a1  (1,0) a 2   , 
2 2 
chiral vector: ch  n1a1  n2a 2
circumference: L  a 3(n12  n1n2  n22 )
d is highest common divisor of (2n1  n2 , 2n2  n1 )
translation vector of 1D unit cell along the axis: R  t1a1  t2a 2
3L
d
4(n12  n1n2  n22 )
number of atoms per unit cell: N 
d
modulues of translation vector: R 
•Single-wall CNT consists of rolling the
honeycomb sheet of carbon atoms into
a cylinder whose chirality and the fiber
diameter are uniquely specified by the
vector:
h
1 1
2 2
c na n a
chiral angle:   arctan
PHYS 624: Electronic Band Structure of Solids
3n2
2n1  n2
47
Metallic vs. Semiconductor CNT
The 1D band on CNT is obtained by slicing
the 2D energy dispersion relation of the
graphite sheet with the periodic boundary
conditions:
chk  2 m  chK   2 m  2n1  n2  3m
Conclusion:
n1  n2 are metallic
•The chiral CNT with 2n  n  3m
1
2
•The armchair CNT
are moderate band-gap semiconductors.
Metallic 1D energy bands are generally unstable under a Peierls distortion →
CNT are exception since their tubular structure impedes this effects making their
metallic properties at the level of a single molecule rather unique!
PHYS 624: Electronic Band Structure of Solids
48
CNT Band structure
PHYS 624: Electronic Band Structure of Solids
49