Band structure

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: k -vector ( hk is momentum)

Bloch: k -vector ( hk is the crystal momentum)
and 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 ∂ε n (k ) d hk
= v n (k ) = Ψ nk v Ψ nk =
ˆ ; = − e [ E(r, t ) + v n (k ) × B(r, t ) ]
dt h ∂k 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 εn (k ) (band structure) must be
used: e
H classical = εn [ hk + eA (r ) ] − eϕ(r ) + B× k
L
2m
dp d ( hk ) ∂H dr ∂H ∂ε
= =− , =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.
n = const (no interband transitions)
•Limits of validity:
ε gap (k )
2
eB ε gap (k )
2

eEa = , hωc = = , ε gap (k ) = ε n (k ) − ε n′ (k )
εF m εF
hω =?ε gap , λ a

What is the range of quantum numbers?

Sommerfeld: k runs through all of
k-space consistent with the Born-von
Karman periodic boundary
conditions:
Ψ ( x + L, y , z ) = Ψ ( x, y , z ) 
 2π
Ψ ( x + L, y , z ) = Ψ ( x, y + L , z )  ⇒ k =
L
( nx , ny , nz ) ↔ nx , ny , nz = 0, ±1, ±2,K
Ψ ( x, y , z ) = Ψ ( x , y , z + L )  

Ψ ( x + L, y , z ) = Ψ ( x , y , z ) = 0 
 π
Ψ ( x + L, y, z ) = Ψ ( x, y + L, z ) = 0  ⇒ k = ( nx , n y , nz ) ↔ nx , n y , nz = 1, 2,K
L
Ψ ( x, y , z ) = Ψ ( 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 Born-
von Karman periodic boundary conditions; n runs through an
infinite set of discrete values.

What are the energy levels?

Sommerfeld: 2 2
hk
ε (k ) =
2m
Bloch: For a given band index n, ε n (k ) has
no simple explicit form. The only general
property is periodicity in the reciprocal space:

ε n (k + G ) = ε n (k )

What is the velocity of electron?

Sommerfeld: The mean velocity of an electron in a
level with wave vector k
is:
hk 1 ∂ε
v= =
m h ∂k
Bloch: The mean velocity of an electron in a level with
band index n
and wave vector k
is:
1 ∂ε n (k )
Conductivity of a perfect crystal: v n (k ) =
σ →∞ h ∂k
NOTE: Quantum mechanical definition of a mean velocity
h∇
v ≡ Ψ v Ψ = ∫dr Ψ (r )
ˆ *
Ψ r)
(
mi

What is the Wave function

Sommerfeld: The wave function of an electron with
wave vector kis: 1
kΨ (r ) = e ikr

V
Bloch: The wave function of an electron with band index n
and wave vector k
is:

Ψk (r ) = e ϑnk (r )
ikr

where the function ϑnk ( r ) has no simple explicit form. The
only general property is its periodicity in the direct lattice
(i.e., real space): ϑnk (r + R ) = ϑnk (r )


Sommerfeld vs. Bloch: Density of States

Sommerfeld → Bloch
2 2
D(ε ) =
( 2π )
d∫ dk δ ( ε − ε (k ) ) → D(ε ) = ( 2π ) ∑ ∫ dk δ ( ε − ε (k ) )
d
n B.Z .
n


Bloch: van Hove singularities in the
DOS
2 2 dS
D(ε ) = ∑∫ dk δ ( ε − ε n (k ) ) = ∑ ∫ dk ∇ ε (k )
( 2π ) ( 2π )
d d
n B.Z . n Sn ( E ) k n

1
D(ε )d ε =
( 2π ) ∫ dS ∆k
d

d ε = ( ∇ k ε n (kΔk
)) = ∇ k εk( ) ∆ k
n

D(ε )


Bloch: van Hove singularities in the
DOS of Tight-Binding Hamiltonian
ε n (k ) = − 2t ( cos(k x a) + cos(k y a) + cos(k z a) ) ⇒ ∇ k ε n (k ) = 2ta ( sin(k x a) + sin( k y a) + sin( k z a) )

k = (0, 0, 0) max

  π π π
∇k ε n (k ) = 0 for k =  ± , ± , ± ÷min
1D 

 a
 π
a
 
a
π   π
k =  ± , 0, 0 ÷,  0, ± , 0 ÷,  0, 0, ± ÷saddle
  a   a   a

3D

Local DOS: ρ (r, ε ) = ∑ Ψα ( r ) δ (ε − ε α )
2

α

DOS: D(ε ) = ∫ dr ρ (r, ε )


Sommerfeld vs. Bloch: Fermi surface
•Fermi energy ε F = µ (T = 0) represents the sharp occupancy
cut-off at T=0 for particles described by the Fermi-Dirac statitics.
•Fermi surface is the locus of points in reciprocal space where ε (k ) = ε F

εF No Fermi surface for insulators!

− kF + kF
Points of Fermi
“Surface” in 1D

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 ε n (k ) = ε F in kvariable.


Is there a Fermi energy of intrinsic Semiconductors?

•If ε 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, ε n (k ) = ε F does not have a solution in a
solid with a gap, which therefore has no Fermi
surface!

DOS of real materials: Silicon, Aluminum, Silver


Colloquial Semiconductor “Terminology” in Pictures

←PURE

DOPPED→


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 (ε kin ) = µ D (−ε bin ) f (−ε bin ) µ D (ε kin + φ − hω ) f (ε kin + φ − hω )
τ (ε kin )

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.

Fourier analysis of systems living on
periodic lattice
f (r ) = f (r + R ) ⇒ f (r ) = ∑ f G e iGr
1 G

∫ ∫
−iGr
fG = dr e f (r ); e iGr dr = 0
V pcell pcell pcell
φ(k ) =φ(k +G ) ⇒φ(k ) = ∑ R eiRk φ
R
dk
φR =V pcell
BZ
∫ ( 2π ) 3
e −iRkφ(k )
Born-von Karman: f (r ) = f (r + N i ai ) ⇒
3
mi 1
f (r ) = ∑ f k e ikr
, k =∑ bi ; fk = ∫ dr e −ikr f (r ); ∫ eikr dr = 0
k i =1 Ni Vcrystal crystal crystal

∑e ikR
= N δk ,0 
 3

 R =∑ i ai , 0 ≤ni < N i , N = N1 N 2 N 3 ; k ∈BZ
R
n
∑e
k
ikR
= N δR ,0 

i=1


Fouirer analysis of Schrödinger equation

U (r ) = U (r + R ) ⇒U (r ) = ∑U G eiGr , G × = 2π m
R
G
 h2 2 
ˆ Ψ(r ) ≡ −
H  2m ∇ + U (r )  Ψ (r ) = εΨ (r ), Ψ (r ) = ∑ Ck eikr
  k

h2 k 2
∑ 2m
k
Ck e + ∑Ck ′U G e
ikr

k ′G
i ( k ′+G)r
= ε ∑Ck eikr
k
  h2k 2  
k′ → k − G : ∑ e 
ikr
− ε ÷Ck + ∑ U G Ck-G  = 0, for all r
k   2m  G 
 h2k 2 
 − ε ÷Ck + ∑ U G Ck-G = 0, for all k ⇒ ε k = ε (k )
 2m  G
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.

Fourier analysis, Bloch theorem, and
its corollaries
 − iGr  ikr
Ψ k (r ) = ∑ Ck −G e i ( k −G ) r
=  ∑ Ck −G e ÷e
G  G 744 
644 ↓ 8
ϑ (r ) = ϑ (r + R )
ˆ
 H Ψk +G (r ) = ε (k + G )Ψ k +G (r )
1. Ψ k +G (r ) = Ψk (r ) ⇒ 
ˆ
 H Ψk (r ) = ε (k + G )Ψ k (r ) → ε (k ) = ε (k + G )

 h2 
( ∇ + ik ) + U (r)  ϑnk (r) = ε n (k )ϑnk (r)
2
2.  − •Each zone n is indexed by a k
 2m  vector and, therefore, has as
many energy levels as there
 h2  are distinct k vector values
( ∇ − ik ) + U (r)  ϑn,−k (r) = ε n (k )ϑn,−k (r)
2
3.  −
 2m  within the Brillouin zone, i.e.:

4. ϑn ,k (r ) = ϑn ,− k (r ) , ε n (k ) = ε n ( −k ) (Kramers theorem)
* N = N1 N 2 N3

“Free” Bloch electrons?
h2k 2
•Really free electrons → Sommerfeld ε n (k ) = continuous spectrum with infinitely
degenerate eigenvalues. 2m
• ε n (k )=ε n (k + G ) does not mean that two electrons with wave vectors k and k + G
ε (
have the same energy, but that any reciprocal lattice point can serve as the originnofk )
.

•In the case of an infinitesimally
small periodic potential there is
periodicity, but not a real
potential. The ε 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!


Schrödinger equation for “free” Bloch
electrons
h 2
 0 h2
 2m ( ∇ + ik ) + ε (k )  θ k (r ) = 0 ε (k ) = 2m ( k + G )
2 0 0 2

  1 iGr
Counting of Quantum States:
ϑk (r ) =
0
e
V
Extended Zone Scheme: FixG (i.e., the BZ) and then count k vectors within the
region corresponding to that zone.
Reduced Zone Scheme: Fix k in any zone and then, by changing G , count all
equivalent states in all BZ.


“Free” Bloch electrons at BZ boundary

πx πx
Ψ + : ( eiGx / 2 + e −iGx / 2 ) : cos Ψ− : ( eiGx / 2 − e −iGx / 2 ) : sin
a a


•Second order perturbation theory, in crystalline potential, for the reduced
zone scheme: 2
H+k U G+k
ε k (G ) = ε (G ) + G + k U G + k + ∑
0
k + ... + O(U 3 )
H ε k0 (G ) − ε k0 (H)
1 i (G +k)r
r G + k = ϑ (r )e 0
nk
ikr
= e
V
1
G + k U H + k = ∑ U G′ ∫ e i ( − H+G+G ′ )
dr = ∑ U G′δ G ′,G-H = U G-H
V G′ G′

1
G + k U H + k = ∫ U (r )dr
V U G′
2

second order correction = ∑ 0
G′ ε k (G ) − ε k (G ′ + G )
0


•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
ε (G = 0) − ε (G ′) =
0 h  2
0
k − ( G′ + k )
2
 = h ( − G ′ 2 − 2G ′k )
2m   2m
k k

0 at BZ boundary ↔ Laue diffraction!
Ψ = α eikr + β ei (G+k)r
1 0 
(ε
(k ) − ε (k ) ) + 4 U G′ 
2 2
ε ±
G=0 (k ) = ε G = 0 (k ) − ε G′ (k ) ±
0 0
G=0
0
G′
2 
G' G' G'
k = ⇒ ε G = 0 (k ) = ε G = 0 (k = ) + U G′ ; ε G = 0 (k ) = ε G = 0 (k = ) − U G′
+ 0 − 0

2 2 2

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.


Fermi surface in 2D for free Sommerfeld electrons

2
 2π  4π 2
S BZ =b =
2
÷ = , Scell = a 2
 a  Scell
S BZ 4π 2 2π 2
S state = = = , N = N1 N 2
2 N 2 NScell Scrystall
2π 2 S BZ Sb π
S Fermi = N e S state = 2 = ⇒ kF = = 0.798
a 2 2π a

π
S Fermi = 2 N e S state = S BZ ⇒ k F = 1.128
a
S BZ
S Fermi = zN e S state =z , for crystal with z-valence atoms
2

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.


Fermi surface is orthogonal to the BZ boundary

( ε 0 (k ) − ε 0 (k + G ) )
2
ε (k ) + ε (k + G )
0 0
2
ε (k ) = ± + UG
2 4
1 (ε (k ) − ε 0 (k + G ) ) ( ∇ k ε 0 (k ) − ∇ k ε 0 (k + G ) )
0
1
∇ k ε (k ) = ( ∇ k ε 0 (k ) + ∇ k ε 0 ( k + G ) ) ± 2
2 2
( ε 0 (k ) − ε 0 (k + G) )
2
2
+ UG
4

at the BZ boundary : ε 0 (k ) = ε 0 (k + G )
h2 h2  k 
∇ k ε (k ) = ( k + k + G ) ¬  + G ÷×G = 0
2m m2 


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.


Constructing Bloch functions from
atomic orbitals


From localized orbitals to wave functions overlap


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 = ϕ( x − R i )

One-dimensional case

→Assuming that only nearest neighbor orbitals overlap:


One-dimensional examples:
s-orbital band vs. p-orbital band
tp >0
ts < 0

ts < t p

+∞
Ψ kl ( x) = ∑
n = −∞
eikxφ l ( x − na )

¬ Re [ Ψ kl ( x)] →

bandwidth: W = 4dtl = 2 ztl

Wannier Functions

→It would be advantageous to have at our disposal localized wave functions with vanishing
overlap i j = δij : Construct Wannier functions as a Fourier transform of Bloch wave
functions!


Wannier functions as orthormal basis set

Ri − a Ri + a

1D example: decay as power law,
so it is not completely localized!


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,
r
designated as the A and B sublattices. The vector τ connects the two r r
sublattices and is not a translation vector. Primitive translation vectors are a, b
.

Chemistry of Graphite: sp 2 hybridization, covalent
bonds, and all of that

1 A, B 2 A, B
φ1A, B = s ± px
3 3
1 A , B 1 A , B 1 A, B
φ2A, B = s m px ± py
3 6 2
1 A, B 1 A, B 1 A, B
φ3A, B = s ± px ± py
3 6 2
φ4A, B = pzA, B

1 A 1 A
Ψ b ( onding )
i = ( φi +φi ) , Ψi
B a ( ntibonding )
= ( φi −φiB )
2 2

Truncating the basis to a single π orbital per atom

•The atomic s orbitals as well as the Eigenstates of translation operator:
p ,p
atomic carbon x y functions form strong
bonding orbitals which are doubly occupied
1
∑
and lie below the Fermi energy. They also
form strongly antibonding orbitals which are kA = eikmφA (r − m)
high up and empty. N m

•This leaves space on energy axis near the 1
Fermi level for π orbitals (they point
perpendicular to the direction of the bond
kB =
N m
∑ eikmφB (r − m )
between them)
φB (r ) = φA (rτ )
−
•The π orbitals form two bands, one
bonding band lower in energy which is
doubly occupied, and one antibonding band Bloch eigenstates:
higher in energy which is unoccupied.
•These two bands are not separated by a
kn = α kA + β kB
gap, but have tendency to overlap by a 2 2
small amount leading to a “semimetal”. α + β =1


Diagonalize 2 x 2 Hamiltonian

ˆ
 kA H kA ˆ
kA H kB 
H (k ) =  ÷
 kB H kA
ˆ ˆ kB ÷
kB H
 
1
kA H kA = ∑ eik ( m-m′) ∫ drφA (r − m′) HφA (r − m )
ˆ ˆ
N mm′
kB H kB = ∫ drφB (r ) HφB (r ) = kA H kA = ∫ drφA (r ) HφA (r )
ˆ ˆ ˆ ˆ
ˆ ˆ ˆ ˆ
kA H kB = ∫ drφA (r ) HφB (r ) + e −ika ∫ drφA (r ) HφB (r + a) + e −ikb ∫ drφA (r ) HφB (r + b)

−t
ε ± (k ) = ±t 1 + e −ika + e−ikb
ε ± (k ) = ±t [ 3 + 2 cos(ka) + 2 cos(kb) + 2 cos(k (a − b)) ]
1/ 2


Band structure plotting: Irreducible BZ

Ψ (r ) = { R | Ta } Ψ (r ) = Ψ ({ R | Ta }
−1
n
Rk
n
k
n
k r ) = Ψk ( R −1r − R −1a)
n

= { R | −R a}
−1
{ R | Ta } r = Rr + a, { R | Ta }
−1 −1 −1


Graphite band structure in pictures

•Plot ε (k ) 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 k
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.

along Σ : ε ± (k ) = ±t 5 + 4 cos ( 2πς )  ( 0 < ς < 1)
1/ 2
 
middle section : ε ± (k ) = ±t 3 − 4 cos ( 2πς / 3 ) + 2 cos ( 4πς / 3 ) 
1/ 2
 
along Λ : ε ± (k ) = ±t 3 + 2 cos ( 4πς / 3 ) + 4 cos ( 2πς / 3 ) 
1/ 2
 


Graphite band structure in pictures:
Pseudo-Potential Plane Wave Method

Electronic Charge Density:
In the plane of atoms In the plane
perpendicular to atoms


Diamond vs. Graphite: Insulator vs. Semimetal


Carbon Nanotubes

•Mechanics: Tubes as ultimate fibers.
•Electronics: Tubes as quantum wires.
•Capillary: Tubes as nanocontainers.


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: c h = n1a1 + n2a 2
circumference: L = a 3(n12 + n1n2 + n2 )
2

d is highest common divisor of (2n1 + n2 , 2 n2 + n1 )
translation vector of 1D unit cell along the axis: R = t1a1 + t2a 2
3L
modulues of translation vector: R =
d
4(n12 + n1n2 + n22 )
•Single-wall CNT consists of rolling the number of atoms per unit cell: N =
honeycomb sheet of carbon atoms into d
a cylinder whose chirality and the fiber 3n2
diameter are uniquely specified by the chiral angle: θ = arctan
vector: c = n a +n a 2n1 + n2
h 1 1 2 2


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 = 2π m ⇒ c h K ± = 2π m ⇔ 2n1 + n2 = 3m

Conclusion:
•The armchair CNT n1 = n2 are metallic
•The chiral CNT with 2n1 + n2 ≠ 3m
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!

CNT Band structure


Band structure

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