Institute of Solid State PhysicsTechnische Universität Graz

21.Electron Bands

June 12, 2018

Tight binding

Tight binding does not include electron-electron interactions

222

02 4A

MOAe A

Z eHm r r

1 2 3 unit_cell 1 2 3, ,

1 expkl m n

i lk a mk a nk a r la ma naN

unit_cell i i ii

r c r r

Atomic wave functionsThis is the tight-binding wave function.

1 2 3exppqs k kT i pk a qk a sk a

Tight binding, one atomic orbital

For only one atomic orbital in the sum over valence orbitals

1 2 3nearest neighbors

exp( ( ))k a a a a a MO a a a MO mm

E c c H c H i hk a jk a lk a

mikk

mE t e

a MO ar H r

a MO a mt r H r

1 2 3nearest neighbors

exp( ( )) small terms

small terms

a a MO a m a MO mm

k a a a

c H c H i hk a jk a lk a

E c

On-site energy Overlap integral

Tight binding, fcc

lmnik

lmn

E t e

Density of states (fcc)

Calculate the energy for every allowed k in the Brillouin zone

http://lamp.tu-graz.ac.at/~hadley/ss1/bands/tbtable/tbtable.html

Tight binding, fcc

Christian Gruber, 2008

Tight binding, fcc

http://www.phys.ufl.edu/fermisurface/

http://lamp.tu-graz.ac.at/~hadley/ss1/bands/tbtable/tbtable.html

http://www.springermaterials.com/navigation/bookshelf.html#l_2_106048_

Tight binding

Harrison, Electronic Structure, Freeman 1980

Tight binding 1-d

Finite potential well

bTight binding fails when the electrons aren't mostly localized.

Tight binding 1-d

k = 0

k = /a

k = 0

first band

second band

2N electron states per band

( ) ikna

n

x e x na

k = /a

2 electron states/band

The N atomic orbitals in the tight binding wave function can hold 2Nelectrons

N atoms in the crystal

N translational symmetries

N allowed k states in the first Brillouin zone

these k's can be put in one-to-one correspondence with the eigenvalues of the translation operator

N solutions to the Schrödinger equation of Bloch form labeled by k

2N electron states in the first Brillouin zone

2N electrons per band

2N states per Brillouin zone

A crystal L×L×L has unit cells.3

3

LNa

The first Brillouin zone contains k points.

3

3

3 3

2

2LaNa

L

There are N translational symmetries.

Each k state can hold 2 electrons (spin).

There are 2N per Brillouin zone.

Thermodynamic properties of metals

From the band structure measurements, we obtain the electron density of states.

Thermodynamic properties can be calculated from the tabulated data for the density of states

Metals, semiconductors, and insulators

From Ibach & Lueth

Insulators: band gap > 3 eV

Student Projects

Plot the dispersion relation for a 1-D square wave potential Using the Kronig-Penney model, the plane wave method, and tight binding.

Draw the missing Fermi surfaces

Use the tight-binding model to calculate a dispersion relation.

Make a similar table to the table of tight binding solutionsfor the plane wave method

Institute of Solid State PhysicsTechnische Universität Graz

Crystal Physics

Crystal physics explains what effects the symmetries of the crystal have on observable quantities.

An Introduction to Crystal Physics Ervin Hartmannhttp://ww1.iucr.org/comm/cteach/pamphlets/18/index.html

International Tables for Crystallographyhttp://it.iucr.org/

Kittel chapter 3: elastic strain

Strain

A distortion of a material is described by the strain matrix

ˆ ˆ ˆ(1 )ˆ ˆ ˆ(1 )ˆ ˆ ˆ(1 )

xx xy xz

yx yy yz

zx zy zz

x x y z

y x y z

z x y z

x

y

z

x

y

z

Stress

9 forces describe the stress

Xx, Xy, Xz, Yx, Yy, Yz, Zx, Zy, Zz

Xx is a force applied in the x-direction to the plane normal to x

Xy is a sheer force applied in the x-direction to the plane normal to y

yx z

x y z

yx z

x y z

yx z

x y z

XX XA A A

YY YA A A

ZZ ZA A A

stress tensor:

Stress is force/m2

Xx Xy

Xz

x

y

z

Stress and Strain

The stress - strain relationship is described by a rank 4 stiffness tensor. The inverse of the stiffness tensor is the compliance tensor.

ij ijkl kls

ij ijkl klc

Einstein convention: sum over repeated indices.

xx xxxx xx xxxy xy xxxz xz xxyx yx xxyy yy

xxyz yz xxzx zx xxzy zy xxzz zz

s s s s s

s s s s

Statistical Physics

Microcannonical Ensemble: Internal energy is expressed in terms of extrinsic quantities U(S, M, P, , N, V).

ij ij k K l ldU TdS d E dP H dM

ij K lij k l

U U U UdU dS d dP dMS P M

The normal modes must be solved for in the presence of electric and magnetic fields (Advanced Solid State Physics course).

Internal energy in an electric field

In an electric field, if the dipole moment is changed, the change of the energy is,

U E P

Using Einstein notation

This is part of the total derivative of U

ij ij k K l ldU TdS d E dP H dM

k kdU E dP

Statistical Physics

Microcannonical Ensemble: Internal energy is expressed in terms of extrinsic quantities U(S, M, P, , N, V).

Cannonical ensemble: At constant temperature, make a Legendre transformation to the Helmholtz free energy.

F = U - TS

F(V, T, N, M, P, )

Make a Legendre transformation to the Gibbs potential G(T, H, E, )

ij ij k K l lG U TS E P H M

ij K lij k l

ij ij k K l l

U U U UdU dS d dP dMS P M

dU TdS d E dP H dM

ij ijV

Gibbs free energy

ijij

G

kk

G PE

ll

G MH

G ST

ij ij k k l ldG SdT d P dE M dH

ij k lij k l

G G G GdG dT d dE dHT E H

ij ij k K l lG U TS E P H M

ij ij k K l ldU TdS d E dP H dM

ij ij ij ij k k k k l l l ldG dU TdS SdT d d E dP P dE H dM M dH

total derivative: