Calculating Band Structure

Nearly free electron• Assume plane wave solution for electrons

• Weak potential V(x)

• Brillouin zone edge

Tight binding method

• Electrons in local atomic states (bound states)

• Interatomic interactions >> lower potential

• Unbound states for electrons

• Energy Gap = difference between bound / unbound states

Crystal Field Splitting

• Group theory to determine crystalline symmetry

• Crystalline symmetry establishes relevant energy levels

• Field splitting of energy levels

However all approaches assume a crystal structures. Bands and energy gaps still exist without the need for crystalline structure. For these systems, Molecular Orbital theory is used.

Free Electron Model

• Energy bands consist of a large number of closely spaced energy levels.

• Free electron model assumes electrons are free to move within the

metal but are confined to the metal by potential barriers.

• This model is OK for metals, but does not work for semiconductors since

the effects of periodic potential have been ignored.

Kronig-Penny Model

• This model takes into account the effect of periodic arrangement of

electron energy levels as a function of lattice constant a

• As the lattice constant is reduced, there is an overlap of electron

wavefunctions that leads to splitting of energy levels consistent

with Pauli exclusion principle.

Energy bands for diamond versus lattice constant.

A further lowering of the

lattice constant causes the

energy bands to split again

Formation of Bands

Inter-atom interactions

Many more states

Periodic potential

Band gap

Conduction / valence bands

Bound states

Conduction band states

Free electron model

Valence band states

Conduction / valence bands

Conduction band states

Lowest Unoccupied

Molecular Level

(LUMO)

Valence band states

Highest Occupied

Molecular Orbital

(HOMO)

Electrons fill from bottom upSemiconductor = filled valence band

Example band structures

Find:

Valence bands?

Conduction bands?

Energy Gap?

Highest Occupied Molecular

Level (HOMO)?

Lowest Unoccupied Molecular

Level (LUMO)?

Ge Si GaAs

Simple Energy Diagram

A simplified energy band diagram used to describe semiconductors. Shown

are the valence and conduction band as indicated by the valence band edge,

Ev, and the conduction band edge, Ec. The vacuum level, Evacuum, and the

electron affinity, , are also indicated on the figure.

Metals, Insulators and

Semiconductors

Possible energy band diagrams of a crystal. Shown are: a) a half filled band,

b) two overlapping bands, c) an almost full band separated by a small

bandgap from an almost empty band and d) a full band and an empty band

separated by a large bandgap.

Semiconductors

• Filled valence band (valence = 4, 3+5, 2+6)

• Insulator at zero temperature

Semiconductors Si, Ge

Filled p shells

4 valence electrons

Metals

Free electrons

Valence not 4

Binary system III-V: GaAs, InP, GaN, GaP

Binary II-VI: CdTe, ZnS,

Eg Temperature Dependence

Eg Doping Dependence

Doping, N, introduces impurity bands that lower the bandgap.

Energy bands in Electric Field

Energy band diagram in the presence of a uniform electric field. Shown are

the upper almost-empty band and the lower almost-filled band. The tilt of

the bands is caused by an externally applied electric field.

Electrons travel down.

Holes travel up.

The effective massThe presence of the periodic potential, due to the atoms in the crystal without

the valence electrons, changes the properties of the electrons. Therefore, the

mass of the electron differs from the free electron mass, m0. Because of the

anisotropy of the effective mass and the presence of multiple equivalent band

minima, we define two types of effective mass: 1) the effective mass for density

of states calculations and 2) the effective mass for conductivity calculations.

Electron excited out of

valence bandTemperature

Light

Defect

…

Electron in conduction

band state

Empty state in valence

band (Hole = empty

state)

Motion of Electrons and Holes in Bands

Electrons - holes

Electron in conduction bandNOT localized

Hole in valence bandUsually less Mobile (higher

effective mass), but not always

Electron – hole pairs in different bands

large separation

Region Near Gap

In the region near the gap,

Local maximum / minimum

dE/dk = 0

effective mass m* = h2/(d2E/dk2)

Electrons

Minimum energy

Bottom of conduction band

Holes

Opposite E(k) derivative

“Opposite effective charge”

Top of valence band

kx

e(k)

Conduction

band

Valence

band

General Carrier Concentration

Gap

Conduction

band

Valence

band

Probability of hopping into state

n0 = (number of states / energy) * energy distribution

gc (E) = density of states

f (E) = energy distribution

Density of states

The density of states in a semiconductor equals the density per unit volume

and energy of the number of solutions to Schrödinger's equation.

Calculation of the number of states with wavenumber less than k

Fermi-surface (3-D)

• K-space

– Set of allowed k

vectors

• Fermi surface

– Electrons occupy

all kf2 states less

than Ef*2m/h

– kF ~ wavelength

of electron

wavefunction

2p/L

Allowed state

for k-vector

kx

ky

NLL

kFkF

3

63

3

2

3

)/2(

1

3

4pp

p

Area of sphere / k states in spheresVolume in lattice

Density of stateshttp://ece-www.colorado.edu/~bart/book/book/chapter2/ch2_4.htm

Number of states:

Density in energy:

Kinetic energy of electron:

Density of states / energy:

In conduction band, Nc:

3

6 2

3

2 LN Fk

p

Different m*

in conduction and

valence band

Density of States in 1, 2 and 3D

Probability density functionsThe distribution or probability density functions describe the probability that

particles occupy the available energy levels in a given system. Of particular

interest is the probability density function of electrons, called the Fermi function.

The Fermi-Dirac distribution function, also called Fermi function, provides the

probability of occupancy of energy levels by Fermions. Fermions are half-

integer spin particles, which obey the Pauli exclusion principle.

Fermi-Dirac vs other distributions

Intrinsic: Ec – Ef = ½ Eg

High temperature:

Fermi ~ Boltzmann

Maxwell-Boltzmann:

Noninteracting particles

Bose-Einstein: Bosons

Carrier DensitiesThe density of occupied states per unit volume and energy, n(E), ), is simply

the product of the density of states in the conduction band, gc(E) and the

Fermi-Dirac probability function, f(E).

Since holes correspond to empty states in the valence band, the probability

of having a hole equals the probability that a particular state is not filled, so

that the hole density per unit energy, p(E), equals:

Carrier Densities

Product of density of states and distribution

-- defines accessible bands

-- within kT of Ef

Carrier DensitiesElectrons

Holes

Limiting Cases

0 K:

Non-degenerate semiconductors: semiconductors for which the Fermi

energy is at least 3kT away from either band edge.

Intrinsic SemiconductorIntrinsic semiconductors are usually non-degenerate

Mass Action Law

The product of the electron and hole density equals the square of the

intrinsic carrier density for any non-degenerate semiconductor.

The mass action law is a powerful relation which enables to quickly

find the hole density if the electron density is known or vice versa

Doped SemiconductorAdd alternative element for electron/holes

Si valence = 4 P valence = 5 B valence=3

Si = Si = Si = Si =

== ==

Si = Si = Si = Si =

== ==

Si = Si = Si = Si =

== ==

Si = Si = Si = Si =

== === ===

Si = Si = Si = Si =

== ==

Si = Si = P = Si =

== ==

Si = Si = Si = Si =

== ==

Si = Si = Si = Si =

== === ===

Si = Si = Si = Si === ==

Si = Si -- B = Si === ==

Si = Si = Si = Si =

== ==

Si = Si = Si = Si =

== === ===

Pure Si

All electron paired

Insulator at T=0

Phosphorous

n-doped

Electron added to

conduction band

Boron

p-doped

Positive hole added to

conduction band

Hole

e-

Dopant Energy levels

Si

Eg=1.2eV

P 0.046eV

B 0.044eV

As 0.054eV

Cu 0.24eV

Cu 0.40eV

Cu 0.53eVAu 0.54eV

Au 0. 35eV

Au 0. 29eV

Energy required to donate hole

Energy required

to donate electronEasily ionized

= easily donate electrons

Large energy bad.Add scattering

Donate no carriers

Carrier concentration in thermal

equilibrium

• Carrier concentration vs. inverse temperature

1/T(K)

Thermally activated

Intrinsic carriers

N(carriers) = N(dopants)

Activation of dopants

Region of

Functional device

ne

Dopants and Fermi Level

• Free electron metal:

• Intrinsic semiconductor– n(electrons) = n(holes)

– Fermi energy = middle

• n-doped material – n(electrons) >> n(holes)

– Fermi level near conduction band

• p-doped materials– n(electrons) >> n(holes)

– Fermi level near conduction band

m

kkn F

FF

e2

,3

22

2

3

e

p

Ec

Ev

Ef

Ec

Ev

Ef

Ec

Ev

Ef

Fermi Energy is not material specific but depends on doping level and type

Mobility and Dopants

• Dopants destroy periodicity

– Scattering, lower mobility

e

1E14 1E15 1E16 1E17 1E18

Dopant Concentration (cm-3)

10000

1000

100

Mobility

(cm2/V-s)

GaAse

hSi

e

Doping / Implantation

• Simple bipolar transistor = 5 implants

• Complicated CMOS circuit >12

Implants:

(1)NBL (isolation)

(2) Deep n (Collector)

(3) Base well (p)

(4) Emitter (n)

(5) Base contact