Band Theory of Solids

(Garcia Chapter 24)

Electrons in Solids• Considering electrons in metals as free

particles, electron gas in a box explains many experimental results which?

• Still, why are some solids metals and others insulators?– For metals we assumed that electrons are

free– We know, however, that there are no

free electrons in insulators

• QM give us the answer! – Need a more realistic potential for electrons

reflecting periodic ionic structure of solids

“Realistic” Potential in Solids

– ni are integers

• Example: 2D Lattice

cnbnanT

TrUrU

321

)()(

3 ;2 21

21

nn

bnanT

“Realistic” Potential in Solids• For one dimensional case where atoms

(ions) are separated by distance d, we can write condition of periodicity as

)()( ndxUxU

“Realistic” Potential in Solids

• Multi-electron atomic potentials are complex

• Even for hydrogen atom with a “simple” Coulomb potential solutions are quite complex

• So we use a model one-dimensional periodic potential to get insight into the problem

Bloch’s Theorem• Bloch’s Theorem states that for a

particle moving in the periodic potential, the wavefunctions ψ(x) are of the form

• uk(x) is a periodic function with the periodicity of the potential– The exact form depends on the potential

associated with atoms (ions) that form the solid

)()(

function periodic a is )( ,)()(

dxuxu

xuwhereexux

kk

kikx

k

Bloch’s Theorem

• Bloch’s Theorem imposes very special conditions on any solution of the Schrödinger equation, independent of the form of the periodic potential

• The wave vector k has a two-fold role: 1. It is still a wave vector in the plane wave

part of the solution2. It is also an index to uk(x) because it

contains all the quantum numbers, which enumerate the wavefunction

Bloch’s Theorem

• What is probability density of finding particle at coordinate x?

2

**

*

*2

)()(

)()()()()(

])([])([)(

)()()()(

xuxP

xuxueexuxuxP

exuexuxP

xxxxP

k

kkikxikx

kk

ikxk

ikxk

• But |uk(x)|2 is periodic, so P(x) is as well

2)(xuk

Bloch’s Theorem

The probability of finding an electron at any atom in the solid

is the same!!!

• Each electron in a crystalline solid “belongs” to each and every atom forming the solid

)()( dxPxP

Covalent Bonding Revisited

• When atoms are covalently bonded electrons supplied by atoms are shared by these atoms since pull of each atom is the same or nearly so– H2, F2, CO,

• Example: the ground state of the hydrogen atoms forming a molecule– If the atoms are far apart there is very

little overlap between their wavefunctions

– 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 the atoms

0/

30

11 ar

s ea

Covalent Bonding Revisited

Schrödinger Equation Revisited

• If a wavefunctions ψ1(x) and ψ2(x) are solutions for the Schrödinger equation for energy E, then functions– -ψ1(x), -ψ2(x), and ψ1(x)±ψ2(x) are also

solutions of this equations

– the probability density of -ψ1(x) is the same as for ψ1(x)

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

2

)()]()[()(

2

2121221

22

2,12,122,1

22

xxExxxUdx

xxdm

xExxUdx

xd

m

• Consider an atom with only one electron in s-state outside of a closed shell

• Both of the wavefunctions below are valid and the choice of each is equivalent

• If the atoms are far apart, as before, the wavefunctions are the same as for the isolated atoms

Band Theory of Solids

0

0

/

/

)()(

)()(naZr

s

naZrs

erAfr

erAfr

Band Theory of Solids

• The sum of them is shown in the figure

• These two possible combinations represent two possible states of two atoms system with different energies

• Once the atoms are brought together the wavefunctions begin to overlap– There are two possibilities

1. Overlapping wavefunctions are the same (e.g., ψs+

(r))

2. Overlapping wavefunctions are different

Tight-Binding Band Theoryof Solids

Garcia Chapter 24.4 and 24.5

Electron in Two Separated Potential Wells

Potential Wells Moved Closer

Tight-Binding Approximationfirst two states in infinite and finite potential wells

Symmetric and Anti-symmetric Combinations of Ground State

Eigenfunctions

Six States for Six Atom Solid

Splitting of 1s State of Six Atoms

Atoms and Band Structure• Consider multi-electron

atoms:1. The outer electrons (large n

and l) are “closer” to each other than the inner electrons• Thus, the overlap of the wave-

functions of the outer electrons is stronger than overlap of those of inner electrons

• Therefore, the bands formed from outer electrons are wider than the bands formed from inner electrons

• Bands with higher energies are therefore wider!

Splitting of Atomic Levels in Sodium

Occupation of Bands Sodium

Splitting of Atomic Levels in Carbon

Occupation in Carbon at Large Atomic Separation

Actual Occupation of Energy bands in Diamond

Insulators, Semiconductors, Metals

• The last completely filled (at least at T = 0 K) band is called the Valence Band

• The next band with higher energy is the Conduction Band – The Conduction Band can be empty or

partially filed

• The energy difference between the bottom of the CB and the top of the VB is called the Band Gap (or Forbidden Gap)

• Can be found using computer • In 1D computer simulation of

light in a periodic structure, we found the frequencies and wave functions

• Allowed modes fall into quasi-continuous bands separated by forbidden bands just as would be expected from the tight binding model

Computer simulation can give exact solution in simple cases

Insulators, Semiconductors, Metals

• Consider a solid with the empty Conduction Band

• If apply electric field to this solid, the electrons in the valence band (VB) cannot participate in transport (no current)

Insulators, Semiconductors, Metals

• The electrons in the VB do not participate in the current, since– Classically, electrons in the

electric field accelerate, so they acquire [kinetic] energy

– In QM this means they must acquire slightly higher energy and jump to another quantum state

– Such states must be available, i.e. empty allowed states

– But no such state are available in the VB!

This solid would behave as an insulator

Insulators, Semiconductors, Metals

• Consider a solid with the half filled Conduction Band (T = 0K)

• If an electric field is applied to this solid, electrons in the CB do participate in transport, since there are plenty of empty allowed states with energies just above the Fermi energy

• This solid would behave as a conductor (metal)

Band Overlap• Many materials are

conductors (metals) due to the “band overlap” phenomenon

• Often the higher energy bands become so wide that they overlap with the lower bands– additional electron energy

levels are then available

Band Overlap• Example: Magnesium (Mg; Z =12):

1s22s22p63s2

– Might expect to be insulator; however, it is a metal

– 3s-band overlaps the 3p-band, so now the conduction band contains 8N energy levels, while only have 2N electrons

– Other examples: Zn, Be, Ca, Bi

Band Hybridization

• In some cases the opposite occurs– Due to the overlap, electrons from

different shells form hybrid bands, which can be separated in energy

– Depending on the magnitude of the gap, solids can be insulators (Diamond); semiconductors (Si, Ge, Sn; metals (Pb)

Insulators, Semiconductors, Metals• There is a qualitative difference

between metals and insulators (semiconductors)– the highest energy band “containing”

electrons is only partially filled for Metals (sometimes due to the overlap)• Thus they are good conductors even at very low

temperatures• The resisitvity arises from the electron

scattering from lattice vibrations and lattice defects

• Vibrations increases with temperature higher resistivity

• The concentration of carriers does not change appreciably with temperature

Insulators, Semiconductors, Metals• The difference between Insulators and

Semiconductors is “quantitative”– The difference in the magnitude of the

band gap

• Semiconductors are “Insulators” with a relatively small band gap– At high enough temperatures a fraction

of electrons can be found in the conduction band and therefore participate in transport

Insulators vs Semiconductors

• There is no difference between Insulators and Semiconductors at very low temperatures

• In neither material are there any electrons in the conduction band – and so conductivity vanishes in the low temperature limit

Insulators vs Semiconductors

• Differences arises at high temperatures– A small fraction of the electrons is thermally

excited into the conduction band. These electrons carry current just as in metals

– The smaller the gap the more electrons in the conduction band at a given temperature

– Resistivity decreases with temperature due to higher concentration of electrons in the conduction band *

21mnq

Holes

• Consider an insulator (or semiconductor) with a few electrons excited from the valence band into the conduction band

• Apply an electric field– Now electrons in the valence band have

some energy sates into which they can move

– The movement is complicated since it involves ~ 1023 electrons

Concept of Holes• Consider a semiconductor with a small

number of electrons excited from the valence band into the conduction band

• If an electric field is applied, – the conduction band electrons will participate in

the electrical current– the valence band electrons can “move into” the

empty states, and thus can also contribute to the current

Holes from the Band Structure Point of View

• If we describe such changes via “movement” of the “empty” states – the picture can be significantly simplified

• This “empty space” is a Hole – “Deficiency” of negative charge – holes are

positively charged– Holes often have a larger effective mass

(heavier) than electrons since they represent collective behavior of many electrons

Holes• We can “replace” electrons at the

top of eth band which have “negative” mass (and travel in opposite to the “normal” direction) by positively charged particles with a positive mass, and consider all phenomena using such particles

• Such particles are called Holes• Holes are positively charged and

move in the same direction as electrons “they replace”

Hole Conduction• To understand hole motion, one

requires another view of the holes, which represent them as electrons with negative effective mass

• To imagine the movement of the hole think of a row of chairs occupied by people with one chair empty

• To move all people rise all together and move in one direction, so the empty spot moves in the same direction

Concept of Holes• If we describe such changes via

“movement” of the “empty” states – the picture will be significantly simplified

• This “empty space” is called a Hole – “Deficiency” of negative charge can be

treated as a positive charge– Holes act as charge carriers in the sense

that electrons from nearby sites can “move” into the hole

– Holes are usually heavier than electrons since they depict collective behavior of many electrons

Conduction

Electrical current for holes and electrons in the same direction