Band Theory

1
Band Theory

• The calculation of the allowed electron states in
  a solid is referred to as band theory or band
  structure theory.
• To obtain the full band structure, we need to
  solve Schrödingers equation for the full lattice
  potential. This cannot be done exactly and
  various approximation schemes are used. We will
  introduce two very different models, the nearly
  free electron and tight binding models.
• Considerably more mathematical detail is given in
  the set of notes on the web than in the slides.
  You do not need to learn most of the mathematical
  proofs but you do need to understand in principle
  how calculations are done. See past exam
  questions for what is required.
• We will continue to treat the electrons as
  independent, i.e. neglect the electron-electron
  interaction.

2
Energy Levels and Bands

• Isolated atoms have precise allowed energy
  levels.
• In the presence of the periodic lattice
  potential bands of allowed states are separated
  by energy gaps for which there are no allowed
  energy states.
• The allowed states in conductors can be
  constructed from combinations of free electron
  states (the nearly free electron model) or from
  linear combinations of the states of the isolated
  atoms (the tight binding model).

3
Influence of the lattice periodicity
In the free electron model, the allowed energy
states are   where for periodic boundary
conditions   nx , ny and ny positive or negative
integers.
Periodic potential Exact form of potential is
complicated Has property V(r R) V(r) where R
m1a m2b m3c where m1, m2, m3 are integers
and a ,b ,c are the primitive lattice vectors.
E
4
Waves in a periodic lattice
Recall X-ray scattering in Solid State
I Consider a wave, wavelength l moving through a
1D lattice of period a. Strong backscattering for
l 2a Backscattered waves constructively
interfere. Wave has wavevector k 2p/l.
Scattering potential period a
1D Reciprocal lattice vectors are G n.2p/a
n integer Bragg condition is k G/2 3D
lattice Scattering for k to k' occurs if k' k
G where G ha1 ka2 la3 h,k,l integer and
a1 ,a2 ,a3 are the primitive reciprocal lattice
vectors
k'
G
k
5
2.1 Bragg scattering energy gaps

• 1D potential period a. Reciprocal lattice
  vectors G 2n p/a
• A free electron of in a state exp( ipx/a), (
  rightward moving wave) will be Bragg reflected
  since k G/2 and a left moving wave exp( -ipx/a)
  will also exist.
• In the nearly free electron model (see notes for
  details) allowed un-normalised states for k p/a
  are
• ?() exp(ipx/a) exp( - ipx/a) 2 cos(px/a)
• ?(-) exp(ipx/a) - exp( - ipx/a) 2i sin(px/a)

N.B. Have two allowed states for same k which
have different energies
6
Cosine solution lower energy than sine solution
Cosine solution ?() has maximum electron
probability density at minima in potential.
Sine solution ?(-) has maximum electron
probability density at maxima in potential.
Cos(px/a)
Sin(px/a)
Cos2(px/a)
In a periodic lattice the allowed wavefunctions
have the property where R is any real lattice
vector.
Sin2(px/a)
7
Magnitude of the energy gap
Let the lattice potential be approximated
by   Let the length of the crystal in the
x-direction to be L. Note that L/a is the number
of unit cells and is therefore an integer.
Normalising the wavefunction ?()
Acos(px/a) gives   so   The expectation value
of the energy of an electron in the state ?()
is    
8
Gaps at the Brillouin zone boundaries
At points A ?() 2 cos(px/a) and
E(hk)2/2me - V0/2 . At points B ?(-)
2isin(px/a) and E(hk)2/2me V0/2 .
9
2.2 Bloch States

• In a periodic lattice the allowed wavefunctions
  have the property
• where R is any real lattice vector.
• Therefore
•  
• where the function ?(R) is real, independent of
  r, and dimensionless.
•  
• Now consider ?(r R1 R2). This can be written
•  
• Or
• Therefore
• a (R1 R2) ?(R1) ?(R2)
•  
• ?(R) is linear in R and can be written ?(R)
  kxRx kyRy kzRz k.R. where
• kx, ky and kz are the components of some
  wavevector k so
• (Blochs Theorem) 

10
Alternative form of Blochs Theorem

• (Blochs Theorem)
• For any k one can write the general form of any
  wavefunction as
•  
• Therefore we have
•  
• and
•  
• for all r and R. Therefore in a lattice the
  wavefunctions can be written as
•  
• where u(r) has the periodicity ( translational
  symmetry) of the lattice. This is an alternative
  statement of Blochs theorem.

Real part of a Bloch function. ? eikx for a
large fraction of the crystal volume.
11
Bloch Wavefunctions allowed k-states

• ?(r) expik.ru(r)

Periodic boundary conditions. For a cube of side
L we require ?(x L) ?(x) etc.. So but
u(xL) u(x) because it has the periodicity of
the lattice therefore Therefore i.e. kx 2p
nx/L nx integer. Same allowed k-vectors for Bloch
states as free electron states. Bloch states are
not momentum eigenstates i.e. The allowed states
can be labelled by a wavevectors k. Band
structure calculations give E(k) which determines
the dynamical behaviour.
12
2.3 Nearly Free Electrons
Construct Bloch wavefunctions of electrons out of
plane wave states.

• Need to solve the Schrödinger equation. Consider
  1D
• write the potential as a Fourier sum
• where G 2?n/a and n are positive and negative
  integers. Write a general Bloch function
• where g 2?m/a and m are positive and negative
  integers. Note the periodic function is also
  written as a Fourier sum
• Must restrict g to a small number of values to
  obtain a solution.
• For n 1 and 1 and m0 and 1, and k p/a
• obtain E(hk)2/2me or - V0/2 (see notes)

13
2.5 Tight Binding Approximation

• NFE Model construct wavefunction as a sum over
  plane waves.
• Tight Binding Model construct wavefunction as a
  linear combination of atomic orbitals of the
  atoms comprising the crystal.
• Where f(r) is a wavefunction of the isolated atom
• rj are the positions of the atom in the crystal.

14
2.5.1 Molecular orbitals and bonding

• Consider a electron in the ground, 1s, state of a
  hydrogen atom
• The Hamiltonian is
• The expectation value of the electron energy is
• This give ltEgt E1s -13.6eV

15
Hydrogen Molecular Ion

• Consider the H2 molecular ion in which
• one electron experiences the potential
• of two protons. The Hamiltonian is
• We approximate the electron wavefunctions as
• and

16
Bonding andanti-bonding states

• Expectation value of the energy are (see notes)
• E E1s g(R) for
• E E1s g(R) for
• g(R) a positive function
• Two atoms original 1s state
• leads to two allowed electron
• states in molecule.
• Find for N atoms in a solid have N allowed energy
  states

17
2.5.2 Tight binding approximation

• Write wavefunction as a linear combination of
  atomic orbitals
• Where f(r) is a wavefunction of the isolated
  atom. rj are the positions of the atom in the
  crystal. We will consider s-states which have
  spherical symmetry. To be consistent with Blochs
  theorem.
• N is the number of atoms in the crystal. Term
  for normalisation
• Check
• Let rm rj - R

18

• The expectation value of the energy is
• This can be written in terms of the relative
  atomic position ?m rj rm
• The sum over j gives N since there are N atoms in
  the crystal.
• As the integral is over all space integration
  over (r-rm) give same answer as integration over
  r. This gives
• Each term in the sum corresponds to a lattice
  vector from a lattice site to a neighbouring
  lattice site.

19
The tight binding approximation for s states
First term give binding energy of the isolated
atoms

• Further terms involve overlap integrals between
  orbitals on more and more distant neighbouring
  sites.
• Approximation Consider only rm values for
  nearest neighbours.

1D rm a or a
20
E(k) for a 3D lattice

• Simple cubic nearest neighbour atoms at
•  
• So E(k) - a -2g(coskxa coskya coskza)
• Minimum E(k) - a -6g
• for kxkykz0
• Maximum E(k) - a 6g
• for kxkykz/-p/2
• Bandwidth Emav- Emin 12g
• For k ltlt p/a
• cos(kxx) 1- (kxx)2/2 etc.
• E(k) constant (ak)2g/2
• c.f. E (hk)2/me

Behave like free electrons with effective mass
h/a2g
21
Each atomic orbital leads to a band of allowed
states in the solid
22
Independent Bloch states
Solution of the tight binding model is periodic
in k. Apparently have an infinite number of
k-states for each allowed energy state. In fact
the different k-states all equivalent.

• Bloch states
• Let k k? G where k? is in the first
  Brillouin zone
• and G is a reciprocal lattice vector.
• But G.R 2?n, n-integer. Definition of the
  reciprocal lattice. So
• k? is exactly equivalent to k.

The only independent values of k are those in the
first Brillouin zone.
23
Reduced Brillouin zone scheme
The only independent values of k are those in the
first Brillouin zone.
Results of tight binding calculation
2p/a
-2p/a
Results of nearly free electron calculation
Reduced Brillouin zone scheme
24
Extended, reduced and periodic Brillouin zone
schemes
Periodic Zone Reduced
Zone Extended Zone All
allowed states correspond to k-vectors in the
first Brillouin Zone. Can draw E(k) in 3
different ways
25
The number of states in a band

• Independent k-states in the first Brillouin zone,
  i.e. ?kx? lt ?/a etc.
• Finite crystal only discrete k-states allowed
• Monatomic simple cubic crystal, lattice constant
  a, and volume V.
• One allowed k state per volume (2?)3/V in
  k-space.
• Volume of first BZ is (2?/a)3
• Total number of allowed k-states in a band is
  therefore

Precisely N allowed k-states i.e. 2N electron
states (Pauli) per band This result is true for
any lattice each primitive unit cell
contributes exactly one k-state to each band.
26
Metals and insulators

• In full band containing 2N electrons all states
  within the first B. Z. are occupied. The sum of
  all the k-vectors in the band 0.
• A partially filled band can carry current, a
  filled band cannot
• Insulators have an even integer number
• of electrons per primitive unit cell.
• With an even number of electrons per
• unit cell can still have metallic behaviour
• due to ban overlap.
• Overlap in energy need not occur
• in the same k direction

27
EF
EF
INSULATOR METAL METAL or
SEMICONDUCTOR or SEMI-METAL
28
Bands in 3D
Germanium

• In 3D the band structure is much more complicated
  than in 1D because crystals do not have spherical
  symmetry.
• The form of E(k) is dependent upon the direction
  as well as the magnitude of k.

Figure removed to reduce file size
29
Bound States in atoms
Electrons in isolated atoms occupy discrete
allowed energy levels E0, E1, E2 etc. . The
potential energy of an electron a distance r from
a positively charge nucleus of charge q is
30
Bound and free states in solids
The 1D potential energy of an electron due to an
array of nuclei of charge q separated by a
distance a is Where n 0, /-1, /-2 etc. This
is shown as the black line in the figure.
0
V(r)
E2 E1 E0
V(r) Solid
V(r) lower in solid (work function). Naive
picture lowest binding energy states can become
free to move throughout crystal
r
0
31
Electron probability density has the same
symmetry as the lattice
In a periodic lattice the allowed wavefunctions
have the property where R is any real
lattice vector.
32
2.2 Bloch States

KEY POINT