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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 Schrödinger’s 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.

Band Theory+

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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).

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

Scattered waves moving to left

a

Waves in a periodic latticeRecall 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

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/aare

ψ(+) = exp(ipx/a) + exp( - ipx/a) = 2 cos(px/a)

ψ(-) = exp(ipx/a) - exp( - ipx/a) = 2i sin(px/a)

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2.1 Bragg scattering & energy gapsN.B. Have two allowed states for same k which have different energies

Cosine solution lower energy than sine solution

Cosine solution ψ(+) has maximum electron probability density at minima in potential.

Sine solution ψ(-) has maximum electron probability density at maxima in potential.

Cos(px/a)

Sin(px/a)

In a periodic lattice the allowed wavefunctions have the property

where R is any real lattice vector.

Cos2(px/a)

Sin2(px/a)

Check this!

Magnitude of the energy gapLet 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 ψ(+) = Acos(px/a) gives

so

The expectation value of the energy of an electron in the state ψ(+) is

Gaps at the Brillouin zone boundaries

At points A ψ(+)= 2 cos(px/a) andE=(hk)2/2me - V0/2 .

At points B ψ(-) = 2isin(px/a)and E=(hk)2/2me + V0/2 .

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 ψ(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

(Bloch’s Theorem)

2.2 Bloch States(Bloch’s 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 Bloch’s theorem.

Alternative form of Bloch’s Theorem

Real part of a Bloch function. ψ ≈ eikx for a large fraction of the crystal volume.

ψ(r) = exp[ik.r]u(r)Bloch Wavefunctions: allowed k-states

Periodic boundary conditions. For a cube of side L we require

ψ(x + L) = ψ(x) etc.. So

but u(x+L) = u(x) because it has the periodicity of the lattice therefore

Therefore i.e. kx =2pnx/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.

Need to solve the Schrödinger equation. Consider 1D

write the potential as a Fourier sum

where G = 2n/a and n are positive and negative integers. Write a general Bloch function

where g = 2m/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 m=0 and 1, and k ~p/a

obtainE=(hk)2/2me+ or - V0/2 (see notes)

2.3 Nearly Free ElectronsConstruct Bloch wavefunctions of electrons out of plane wave states.

NFE Model: construct wavefunction as a sum over plane waves.

Tight Binding Model: construct wavefunction as a linear combination of atomic orbitalsof 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.

2.5 Tight Binding ApproximationConsider a electron in the ground, 1s, state of a hydrogen atom

The Hamiltonian is

The expectation value of the electron energy is

This give <E> = E1s = -13.6eV

E1s

F(r)

+

V(r)

2.5.1 Molecular orbitals and bondingConsider the H2+ molecular ion in which

one electron experiences the potential

of two protons. The Hamiltonian is

We approximate the electron wavefunctions as

and

e-

r

R

p+

p+

Hydrogen Molecular IonExpectation 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

V(r)

Bonding andanti-bonding statesWrite 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 Bloch’s theorem.

N is the number of atoms in the crystal. Term for normalisation

Check

Let rm = rj - R

Bloch’s theorem Correct

2.5.2 Tight binding approximationThe expectation value of the energy is

This can be written in terms of the relative atomic position ρm = 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.

Further terms involve “overlap integrals” between orbitals on more and more distant neighbouring sites.

Approximation: Consider only rm values for nearest neighbours.

Constant as f depends only on magnitude or (r-rm)

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Nuclear positions

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The tight binding approximation for s statesFirst term give binding energy of the isolated atoms

1D: rm = +a or –a

Simple cubic: nearest neighbour atoms at

So E(k) =- a -2g(coskxa + coskya + coskza)

Minimum E(k) =- a -6g

for kx=ky=kz=0

Maximum E(k) =- a +6g

for kx=ky=kz=+/-p/2

Bandwidth = Emav- Emin = 12g

For k << p/a

cos(kxx) ~ 1- (kxx)2/2 etc.

E(k) ~ constant + (ak)2g/2

c.f. E = (hk)2/me

- = 10

g = 1

E(k)

-p/a

p/a

k [111] direction

E(k) for a 3D latticeBehave like free electrons with “effective mass” h/a2g

Gap: no allowed states

Band of allowed states

Gap: no allowed states

Band of allowed states

Each atomic orbital leads to a band of allowed states in the solidBloch states

Let k = ḱ + G where k is in the first Brillouin zone

and G is a reciprocal lattice vector.

But G.R = 2n, n-integer. Definition of the reciprocal lattice. So

k is exactly equivalent to k.

- = 10

g = 1

E(k)

-p/a

p/a

k [111] direction

Independent Bloch statesSolution 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.

The only independent values of k are those in the first Brillouin zone.

|k| > p/a

Displace into

1st B. Z.

Reduced Brillouin zone schemeThe 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

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

Independent k-states in the first Brillouin zone, i.e. kx</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/Vin k-space.

Volume of first BZ is (2/a)3

Total number of allowed k-states in a band is therefore

The number of states in a bandPrecisely 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.

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

EF

Metal due to overlapping bands

Metals and insulatorsEmpty Band

Partially Filled Band

Part Filled Band

Energy Gap

Full Band

Energy Gap

Full Band

EF

EF

INSULATOR METAL METAL

or SEMICONDUCTOR or SEMI-METAL

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.

Germanium

Bands in 3DGermanium

Figure removed to reduce file size

V(r)

E2

E1

E0

Increasing Binding Energy

r

Bound States in atomsElectrons 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

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Nuclear positions

a

Bound and “free” states in solidsThe 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

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.

2.2 Bloch States

KEY POINT

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