the electronic structure of solids -...

The electronic structure of solids

� We need a picture of the electronic structure of solid that we can use to explain experimental observations and make predictions

� Why is diamond an insulator ?

� Why is sodium a metal ?

� Why does the conductivity of silicon increase when you heat it ?

Charge transport in solids

� The conductivity of a material is determined by three factors:

– the charge on the charge carriers

– the number of charge carriers

– the mobility of the charge carriers

� σ = n e µ

Conductivity of common materials

�Very large variation in conductivity

A simple model for metals

� Consider a metal such as Na, Mg or Al to be essentially a box in which the valence electrons of the metal are confined.

� The potential within the box is taken to be uniform and much lower than that in the surrounding medium.

� Treat quantum mechanically as a particle in a box

The Free Electron Theory

� You can get solutions of the form– ψn(r) = A sin(πnxx/L) sin(πnyy/L) sin(πnzz/L)– these solutions are standing waves

� Adjust the boundary conditions to get traveling waves– ψk(r) = exp (i k.r), k is the wavevector– εk = (h2/2m) (kx

2 + ky2 + kz

2)– k can not take all values but in many electron systems it

is almost continuous

k, the wavevector

� k is related to the momentum of the electrons in the orbital

– p = hk

� k is related to the wavelength of the electron wave

– |k| = 2 π / λ

Solutions to a three dimensional particle in a box problem

� E = (n2x + n2

y + n2z)h2 / (8ma2)

� For a large box the energy levels are going to be close together

� Consider energy levels as forming a continuous band

� How many energy levels do we have with energy less than some critical value ?

Number of electrons below Emax

� We can have two electrons per unique combination of nx, ny and nz

� Set R2 = n2x + n2

y + n2z

N = 2 (1/8) (4/3)ππππ R3max

= (8ππππ/3)(2mEmax/h2)3/2 a3

N(E) - Density of States (DOS)

� N(E) = 4π(2m/h2)3/2 E1/2

At temperatures above 0 K some higher energy states areoccupied, f(E) = {1+ exp[E-EF)/kT]}-1

� Spectra strongly resemble simple parabolic density of states predicted by free electron model

X-ray emission spectra for sodium and aluminum

Na

Al

Temperature dependence of electron distribution� At temperatures above 0K some electron promoted to

states with higher k.

( ) ( )[ ]KTEEEf

F /exp11−+

=

Electron distribution described by Fermi-Dirac distribution function

The wavevector k

� Classically the kinetic energy of an electron is given by E = p2 / 2m

� The free electron model gives the energy of an electron as, E = (n2

x + n2y + n2

z)h2 / (8ma2)

� The momentum p is usually expressed as kh so E = (k2

x + k2y + k2

z) h2 /2m

Electrical conductivity

� In the absence of an electric field states corresponding to k and -k are equally likely to be populated so there is no overall movement of charge

� In the presence of an electric field states with the same |k| but differing k do not necessarily have the same energy. This can lead to charge transport.

Electrical conductivity

Limitations of the free electron model

� Predicts all materials will be metals !

The tight binding approximation

� Consider a solid to be a large molecule and apply molecular orbital theory

MOs for evenly spaced H atoms

� Solids have MOs that are so close in energy they form continuous bands

H H2 H4 H9 Hn

Chains of CH units

� Consider polyene chains

(CH)H2 (CH)2H2 (CH)4H2 (CH)8H2 (CH)nH2

Evenly spaced

Bond alternation

Electronic structure of NaCl

The band gap

�The occurrence of groups or “bands” of orbitals with energy gaps in between them is common

The origin of band gaps

� The chemists view– atoms in solids have orbitals that overlap to produce

“large molecular orbitals”– These “molecular orbitals” do not occur at all energies

� The physicists view– we need to modify the theory to take into account the

periodicity of the structure– electron waves can be diffracted by a regular array on

ions in a solid

Variation of band width and overlap with interatomic distance�Pushing atoms closer together increases orbital

overlap and increases band widths

Calculated for Nausing TBA

Variation of conductivity with pressure� As pressure effects interatomic distances and band

widths it can have a profound influence on electronic conductivity

What are the coefficients for the orbitals in the bands?

� Consider a chain of atoms� Use LCAO, Ψ(x) = Σ cnψn(x)� The periodicity of the chain limits the

possible solutions for cn

� cn = exp(ikna)

Bloch functions for 1D chain

Conductivity of solids

� This approach to the electronic structure of solids naturally introduces electronic states (orbitals) with characteristic momentum p=hk

� Electrical conductivity can again be related to differing numbers of electrons in states with +k and -k

� Conductivity is limited by lattice vibrations (phonons) in metals

Bands in metals, semiconductors and insulators

E

Metal Intrinsicsemiconductor

Insulator

E E

Insulators

� All bands are fully occupied or empty making it impossible for more electrons to be in states with +k rather than -k

The band structure of group IV elements

Intrinsic and extrinsic semiconductors� In an intrinsic semiconductor the conduction band is

populated by thermal excitation of electrons from the valence band

� In an extrinsic semiconductor doping is used to produce partially occupied bands

Band gaps for Group IV Elements

Band gaps for inorganic compounds

Doping semiconductors

� The addition of very small amounts ofdopant can dramatically influence properties– P, As added to silicon gives n - type material– B, Al, Ga gives p - type material

� The conductivity of doped semiconductors varies less with temperature

Extrinsic semiconductors

� Doping can be used to increase the conductivity of a semiconductor

E

p doping

E

n doping

conduction band

valence band

Temperature dependence of electron distribution

Temperature dependence of conductivity� The conductivity of a metal decreases with increasing

temperature– mobile electrons are scattered by lattice vibrations

� The conductivity of a semiconductor increases with increasing temperature as more charge carriers become available

Doped graphites

� Graphite is a semimetal– doping with bromine or potassium improves its

conductivity

E

E

E

Br2

K

TiO2 , VO2 and TiS2

� TiO2 is an insulator as the d-bands are empty� VO2 at higher temps is metallic as the d-band is partly filled� TiS2 is a metal as S 3p and Ti 3d bands overlap

LixV2O5

� V2O5 has an empty d band and a layered structure

� Intercalation of Li into the material dopes the V2O5– puts electrons in to the empty d band

� This improves the solids conductivity

VO2

� VO2 has a rutile like structure– chains of edge sharing VO6 octahedra– V(IV) has d1 electron configuration– at low temperatures it displays localized metal-

metal bonds and is a semiconductor– at high temperatures the structural distortion

disappears and it is a metal

Phase transitions in VO2

High T Low T

E

metal d bandsplits

oxygen band

Polyacetylene

� Polyacetylene is a semiconductor because it displays bond alternation– without bond alternation it would be a metal

� It can be doped to make it conducting– use oxidizing agents, AsF5, I2 etc. to remove

electrons from the valence band

K2[Pt(CN)4]Br0.3.3H2O

� In KCP the Pt dz2 orbitals are in an evenly spaced chain (at room temp) forming a single band

K2[Pt(CN)4] Brominedoped

Insulator MetalE

Structure of K2[Pt(CN)4]Br0.3.3H2O

� 1D chain compound with overlapping dz2 orbitals

Peierls distortion

� On cooling below 150K the conductivity of KCP drops rapidly– this is associated with a lattice distortion

E

the electronic structure of solids -...

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