A return to density of states & how to calculate energy

bands

Making HW and test corrections due March 19

Learning Objectives for Today

After today’s class you should be able to: Relate DOS to energy bands Compare two main methods for

calculating energy bands Be able to use tight binding model to

calculate energy bands Understand basics of Dirac notation

The nearly-free-electron model

1 electron per atom:When EF is well away

from a gap, dispersion is similar to free-electron

case, but with slight change in curvature

(electrons in metals act free)

As we increase the number of electrons per atom (or per unit cell), EF

moves up the dispersion relation.

E

kEF

dispersion relation

4

The nearly-free-electron model

E

k

EF

When EF is close to or within a gap, major

changes in the material

properties occur...

Valence Band (HOMO)

Conduction Band(LUMO)

HOMO: Highest Occupied Molecular OrbitalLUMO: Lowest Unoccupied Molecular Orbital

How DOS g(E) relates to Dispersion

There are more states in a given energy interval at the top and bottom of this band.

In general, DOS(E) is proportional to the inverse of the slope of E(k) vs. k

The flatter the band, the greater the density of states at that energy.

The density-of-states curve

counts levels.

DOS curves plot thedistribution of

electrons in energy

DOS g(E) is the number of electron states per unit volume per unit energy at energy E

To find the energy density of states g(E), we need:the density of states in k-space g(k) and the energy bands.

Not quite so simple

SiliconVan Hove points are discontinuities of

the first derivative of g(E)

critical points of the Brillouin zone

How would you experimentally determine

the density of states?

X-ray Photoemission Spectroscopy

Like a fancy photoelectric effect

How would you theoretically determine

the energy bands?

Methods to calculate bandstructures Solve the Schrodinger Equation and apply the Bloch theorem

Simplify the complicated crystal potential to something solvable. E.g. Kronig-Penney model.

Treat the complicated crystal potential as a sum of a simpler potential (solvable Schrodinger Eqn) and potential perturbation. E.g. near free-electron model (plane waves + perturbation), k.p perturbation theory and tight-binding model (atomic orbitals + perturbation).

Numerical methods, e.g. density function theory and quantum Monte Carlo

For more information, see the following websites for instance, http://en.wikipedia.org/wiki/Density_functional_theory http://en.wikipedia.org/wiki/Electron_configuration

The main property of solids that determines their electrical properties is the distribution of their electrons.

Two main models for electron distribution:1) Nearly Free Electron Approximation

Valence electrons are assumed trapped in a box (the sample) with a periodic potential

2) The Tight Binding ApproximationValence electrons are assumed to occupy molecular

orbitals delocalized throughout the solid

Summary of Nearly Free Electron Model (What We’ve Already Done!)

Nearly free electron Model Electrons nearly free due to very

large overlap (opposite from TB) Wave functions approximated by

plane waves (free electrons) Assume energy is unchanged and

solve for 1st order correction Works for upper states even if tightly

bound electrons in lower states

)( tkxiAe

m

kE

2

22

My Summary of the Two Main Approaches

Nearly free e-’s +

pseudopotential

Electrons nearly free due

(opposite from TB)

Wavefunctions ~ plane waves

Assume energy is unchanged

and correct to first order

Works for upper states even if

TB electrons in lower states

Tight-binding or LCMO

Assume some electrons

independent of each other

(often true, tight core)

Linear combination of atomic

wave functions (Wannier)

Each Wannier function is

equal to the unperturbed

atomic orbital (LCAO approx)

Comparing Chemists and PhysicistsCalculation theories fall into 2 general categories, which have their roots in 2 qualitatively very different physical pictures for e- in solids (earlier):

“Physicist’s View” - Start from an “almost free” e- & add the periodic potential

Nearly free electrons, Pseudopotential methods

“Chemist’s View” - Start with atomic energy levels & build up the periodic solid by decreasing distance between atoms

Now, we’ll focus on the 2nd method.

Method #2 (Qualitative Physical Picture #2)

“The Chemists’ Viewpoint” Start with the atomic/molecular picture of a solid. The atomic energy levels merge to form molecular levels, & merge to

form bands as periodic interatomic interaction V turns on.

TIGHTBINDING or

Linear Combination of Atomic Orbitals (LCAO) method.

This method gives good bands, especially valence bands! The valence bands are almost the same as those from the

pseudopotential method! Conduction bands are not so good because electrons act free!

The Tightbinding Method

Some believe the Tightbinding / LCAO method gives a clearer physical picture (than pseudopotential method does) of the causes of the bands & the gaps.

In this method, the periodic potential V is discussed as in terms of an Overlap Interaction of the electrons on neighboring atoms.

As we’ll see, we can define these interactions in terms of a small number of parameters.

Tightbinding/LCAO

Assume the atomic orbitals ~ unchanged

bare atoms

solid

Atomic energy levels merge to form molecular levels & merge to form

bands as periodic interatomic interaction V turns on.

Covalent Bonding Revisited

When atoms are covalently bonded electrons are shared by atoms

Example: the ground state of the hydrogen atoms forming a molecule If atoms far apart, little overlap If atoms are brought together the

wavefunctions overlap and form the compound wavefunction, ψ1(r)+ψ2(r), increasing the probability for electrons to exist between atoms

)]()([)]()()[()]()([

2

)()]()[()(

2

2121221

22

2,12,122,1

22

xxExxxUdx

xxdm

xExxUdx

xd

m

These two possible combinations

represent 2 possible states of two atoms

system with different energies

LCAO: Electron in Hydrogen Atom(in Ground State)

1)(1xKAer

Second hydrogen atom

-10 -5 0 5 10

0.0

0.5

1.0

1.5

2.0

1(r

)

r (aB)

2)(2xKAer

Approximation: Only Nearest-Neighbor interactions

Chain of 5 H atoms

HH22 MoleculeMolecule

EE

Bonding

Antibonding

Nonbonding

00

11

22

33

44

# of Nodes

Group: For 0, 2 and 4 nodes, determine wavefunctions

If there are N atoms in the chain there will be N energy levels and N electronic states (molecular orbits). The wavefunction for each electronic state is:

k = eiknan

Where:a is the lattice constant, n identifies the individual atoms within the chain, n represents the atomic orbitals

k is a quantum # that identifies the wavefunction and tells us the phase of the orbitals.

k=0

k=/a

k=/2a

a

The larger the absolute value of k, the more nodes one has

Infinite 1D Chain of H atoms

k = /a

/a = 0+(exp{i})1 +(exp{i2})2 +(exp{i3})3+(exp{i4})4+…

/a = 0 - 1 + 2 - 3 + 4 +…

k = /2a

/2a = = 00+(exp{i+(exp{i/2})/2})1 1 +(exp{i+(exp{i})})2 2

+(exp{i3+(exp{i3/2})/2})33+(exp{i2+(exp{i2})})44+…+…

/2a = = 0 0 + 0 - + 0 - 2 2 + 0 + + 0 + 4 4 +…+…

k = 0 k = 0

00 = = 00++1 1 ++2 2 ++3 3 ++4 4 +…+… k=0

k=/a

k=/2a

a

k=0 k=0 orbital phase does not change when we translate byorbital phase does not change when we translate by a ak=k=/a /a orbital phase reverses when we translate byorbital phase reverses when we translate by a a

Infinite 1D Chain of H atoms

What would happen if consider k>/a? If not obvious, try in groups k=2/a. What

is the wavefunction?

k = eiknan

LCAO +Bloch notation

& potential U(r)=U(r+Na) obeys the Born-von Karman condition

)()(

,)()(

axuxu

exux

nknk

ikxnknk

ikaNnik eec /2Where:

Combining the Bloch theorem and the above gives that

Let’s simplify by considering just two states. (Dirac notation)

2211 cc

Na

a

a

a 2

1

~The collection of all functions of x constitutes a vector space. But to present a possible physical state, the wave functions must be normalized:

1)(2

dxx

Brief Summary of Dirac Notation

Dirac Notation Wave Mechanics (Position Space)

Dirac ket: |(t) System State (x,t) wavefunction

Linear operator A Measurement A differential or multiplicative operator

Eigenvalue Equation

Expectation Value (average of many identical measurements)

nnn aA )x(a)x(A nnn eigenvalue

gives possible results of a measurement

eigenstate or eigenket eigenfunction gives probability of measurement result an

AA dx)x(A)x(Axall

*

Dirac Notation with LCAO Approximation

Dirac Notation for 2 atoms:

If we assume little overlap, can expand to

and

Or expanding

2211 cc

111 EH

EH

222 EH

EcHcHc 1212111

EcHcHc 2222121

Eigenvalue problem with two solutions. E1 and E2

are the unperturbed atom energies. Cross term is the

overlap.

Solution:

EcHcHc 1212111

EcHcHc 2222121

EcVcEc 112211

EcEcVc 122121 *

Lower energy result is the bonding state

V12 is overlap integral

Example similar to homework

Find the energies at the H point of BCC.

H

'

''2

2

0))(2

(K

KkKKKk cUcKkm

h

0)( '0 Kkkkk cUc

How do we plot the Empty Lattice Bands?

The limit of a vanishing potential is called the “empty lattice”, and the empty-lattice

bands are often plotted for comparison with the energy

bands of real solids.

Here plotted in the reduced zone scheme (translations back

into the 1st BZ).

Example: 1D Empty Lattice

( ) ( ), ( )i k nG x ikx inGxnk nk nkk nG e e u x u x e

• V 0:

• We assume a periodicity of a. Define the reciprocal

lattice constant G = 2 / a. We can therefore

restrict k within the range of [-G/2, G/2].

2 2

( ) , ( )2

ikxkE k k e

m

2 2( )( )

2nk

k nGE x

m

Bloch’s theorem

implies

Sorry G=K

Free Electrons in 1DV 0:

22

( ) , ( )2

i k nG xnk nk

k nGE k k e

m

The symmetry of the reciprocal lattice requires:

)()( 1 GkEkE

Where k1 is a

wavevector lying in the 1st BZ.

2

1

2

1 2)()( Gk

mGkEkE

The sign is redundant.

Empty Lattice Bands for bcc Lattice

clbkahGhkl

General reciprocal lattice translation

vector:

Let’s use a simple cubic lattice, for which the reciprocal lattice is also simple cubic:

za

cya

bxa

a ˆ2

ˆ2

ˆ2

And thus the general reciprocal lattice translation vector is: zl

ayk

axh

aGhkl ˆ

2ˆ

2ˆ

2

For the bcc lattice, let’s plot the empty lattice bands along the [100] direction in reciprocal space.

We write the reciprocal lattice vectors that lie in the 1st BZ as:

zza

yya

xxa

k ˆ2

ˆ2

ˆ2

1

[100] 0 < x < 1

[110] 0 < x < ½, 0 < y < ½

The maximum value(s) of x, y, and z depend on the reciprocal lattice type and the direction within the 1st BZ. For example:

Remember that the reciprocal lattice for a bcc direct lattice is fcc!

Here is a top view, from the + kz direction:

ky

kx

H

H

N

a

2

a

2

Energy Bands in BCC

Group: Plot the Empty Lattice Bands for bcc Lattice

Thus the empty lattice energy bands are given by:

{G} = {000}

22222

2

1

2 2

22)( lzkyhx

amGk

mkE hkl

20

222 2

2xEx

amE

{G} = {110} 1111 20

220 xExEE)110()101()011()110(

)110()110()101()011( 211 20

2220 xExEE

)200( 20 2 xEE

1111 20

220 xExEE)011()101()101()011(

{G} = {200}

)002( 20 2 xEE

42 20

220 xExEE)200()002()020()020(

Along [100], we can enumerate the lowest few bands for the y = z = 0 case, using only G vectors that have nonzero structure factors (h + k + l = even, otherwise S=0):

Empty Lattice Bands for bcc Lattice: Results

Thus the lowest energy empty lattice energy bands along the [100] direction for the bcc lattice are:

2220)( lkhxEkE

0

1

2

3

4

5

6

0 0.2 0.4 0.6 0.8 1

x

E/E0

Series1

Series2

Series3

Series4

Series6

Series7

Series8

22222

2

1

2 2

22)( lzkyhx

amGk

mkE hkl

Summary Band Structures What is being plotted? Energy vs. k, where k is the

wavevector that gives the phase as well as the wavelength of the electron wavefunction (crystal momentum).

How many lines are there in a band structure diagram? As many as there are orbitals in the unit cell.

How do we determine whether a band runs uphill or downhill? By comparing the orbital overlap at k=0 and k=/a.

How do we distinguish metals from semiconductors and insulators? The Fermi level cuts a band in a metal, whereas there is a gap between the filled and empty states in a semiconductor.

Why are some bands flat and others steep? This depends on the degree of orbital overlap between building units.

Wide bands Large intermolecular overlap delocalized e-

Narrow bands Weak intermolecular overlap localized e-

How the energies split as we increase N

•Energy levels get closer as N increases.Energy levels get closer as N increases.•Degenerate pairs, except for ground state.Degenerate pairs, except for ground state.

Energy Bands and Fermi Surfaces in 2-D Square Lattice

Reminder: the Brillouin zones of the reciprocal lattice can be identified with a simple construction:

The 1st BZ is defined as the set of points reached from the origin without crossing any Bragg planes.

12

2

2

233

3 3

3

33

3 2 /a

2 /a

A truly free electron system would have a Fermi circle to define the locus of states at the Fermi energy.

Group Problem Show for a square lattice (2D) that the

kinetic energy at the corner of the 1st BZ is larger than that of an electron at the midpoint of a side face. By how much?

What is the corresponding factor in 3D? Draw the energy bands from the zone

center to these to points on the boundary. What bearing might this have on the

conductivity of divalent metals?