Semiconductor DevicesRevision energy band diagram

Dr. K. FobeletsRoom 714

k.fobelets@imperial.ac.ukbb.imperial.ac.uk

De Broglie“all particles can behave like waves”

Thus electrons in a semiconductor lattice have an energy associated to their electron wave.

Particle-wave duality↓

Quantum mechanics

Wave-particle duality - lightNewton Huygens

1650

T. Young1800

particles waves

Wave-particle duality - lightPlanck Einstein

1900

fhE

Waves can behave as particlesWorkfunction = minimum energy to release electron from material

Wave-particle duality - matterDe Broglie Einstein

1924

Matter (particles) can behave as waves.

hp

420

22 cmcpE

Classical description (1D): EVmpPEKE 2

2

Re-written by Schrödinger (1D): )()()()(2 2

22

xExxVdxxd

m

Schrodinger 1925

Schrödinger equationWave equation for particles

2)(xBack to particles: probability of finding the particle at x

mExVx)()( wave function of particle

electrostatic potential seen by particletotal energy of particlemass of particlereduced Planck constant

Solving the Schrödinger equationWhat is V(x) in a periodic lattice?

Electrostatic potential in lattice: -V(x)

a

V(x): electrostatic potential = work done/q = PE/q

xkqQPE potential energy related to the work done by bringing an e- (q<0) towards

an atom (Q>0)

+e-

Simplifying problemStep 1 – electron in a box (quantum well – QW):

PE

∞∞

-a/2

)()(2 2

22

xEdxxd

m

0)2/,2/( aax

mkkE

evennank

kxa

x

2)(

)sin(2)(

22

a/20

mkkE

oddnank

kxa

x

2)(

)cos(2)(

22

∞∞

x

Simplifying problemStep 1 – relevance

,2,1 nank

Discrete energy levels En within QW!

22

222

22n

mah

mkEn

∞∞

-a/2 a/2

∞∞

x

Ener

gy

E1

E2

E3

Simplifying problemStep 2 – electrons in different boxes

V(x)∞∞ ∞

electrons in different boxes: Wells not ∞ highThus e- will influence each otherPauli’s exclusion principleThus small changes to energy values for each e-

↓Splitting of levels in to bands of closely spacedenergy levels.

Ener

gyE1

E2

E3

~1022 cm-3 of Si atoms

each

with

14 e

-

Distribution of e-

Ener

gy

E1

E2

E3

Pauli’s exclusion principle~1022 cm-3 of Si atoms each with 14 e-

Schrödinger in periodic lattice: e- on discrete energy levels in bands

→ Bands of discrete energy levels

T=0K

How will all these e- be distributed?

• There are a discreet number of energy levels: STATEs• Two electrons per state (Pauli exclusion principle). • Quantum mechanics shows that available states occur in bands with gaps where no

states exist → bandgap• Electrons bound closest to nucleus will sit on the lowest state.• Valence electrons will fill the upper levels.

Completely filled with e- at T=0K

Ener

gy

Eg

Ec and Ev – band edgesEn

ergy

T=0K

→ Valence band

→ Conduction band

→ EV

→ EC→ EG → Forbidden band → bandgap

• Shine light with hn>Eg• or with sufficient thermal energy

Ener

gy

Eg

free electronsfree holes

CONDUCTION BAND

VALENCE BAND

• With doping: n-type

Ener

gy

Eg

More free electronsfree holes

• With doping: p-type

Ener

gy

Eg

free electronsMore free holes

What can energy bands teach us?

empty

filled

empty

filled

partially filled

filled

or

overlap

Ec

EvEg

Ec

Ev

Ec

Ev

Ec

Ev

Eg

metal

Small bandgapConduction band partially filled

No bandgap

What can energy bands teach us?

empty

filled

empty

filled

partially filled

filled

or

overlap

Ec

EvEg

Ec

Ev

Ec

Ev

Ec

Ev

Eg

metal insulator

Very large bandgapConduction band empty

What can energy bands teach us?

empty

filled

empty

filled

partially filled

filled

or

overlap

Ec

EvEg

Ec

Ev

Ec

Ev

Ec

Ev

Eg

metal semiconductor insulator

Bandgap ~ kT/qSome free electrons and holes at room temperature

Result of simplification

A silicon lattice is 3D V(x,y,z) much more complex.Thus energy band diagram much more complex.

We lost the information on the mass of the carriers (electronsand holes). More accurate calculations show:1. The existence of 2 types of carriers

electrons in conduction bandholes in valence band

2. Mass of electrons and holes is differentmcarrier for Si: mn < mp

Carrier statistics

• How many carriers are available and how are they distributed in energy?– Density of states– Occupation probability of states

Density of states

Solution of the 1-D Schrödinger equation: discrete energy levels

n

mkkE

nank

2)(

,2,122

EmEg23

22

22

1)(

Density of states = number of statesper unit volume, per unit energy

Each integer value of n = called state

Occupation probability of states

Electrons are fermions, they must obey Pauli’s exclusion principle: no two of them may occupy the same state.Statistical mechanics → Fermi-Dirac distribution function f(E).Gives the probability of finding an electron at a certain energy E.

EEF

f(E)

1T=0K

T1

T2>T1

kTEE

EfFexp1

1)(

1/2

Probability function and band diagramRoom temperature

Energy

EF

f(E) 1 1/2 0

EG

empty

full

insula

torsem

icond

uctor

metal w

ith

overl

appin

g ban

dsGrey area is full at T=0

Density of carriers • Carriers can only occupy the available states (energy

levels)• Thus no carriers in the bandgap• Density of states:

EEmh

Eg

EEmh

Eg

vpv

cnc

323

323

28)(

28)(

• The Fermi-Dirac distribution function f(E) will give the probability a state is filled with an electron

• Integrate gc(E)f(E) over the band gives the number of free electrons

• Integrate gv(E)[1-f(E)] over the band gives the number of free holes

Density of carriers

v

c

E

v

Ec

dEEfEgp

dEEfEgn

)(1)(

)()(

kTEENp

kTEENn

vFV

FcC

exp

exp

Density of free electrons and holes

23

2

*

23

2

*

22

22

hkTm

N

hkTm

N

pV

nC

Effective density of states

For kTEEkTEE vFFc &

Density of carriers

Fc EE EcEF

Ev

n

kTEENn Fc

C exp

vF EE

Ec

EFEv

p

kTEENp vF

V exp

Density of carriers

kTEE

np

kTEE

nn

Fii

iFi

exp

expEi is the position of the Fermi level when the material is intrinsic. Lies ± midgap.

Density of free electrons and holes

EcEF

Ev

Ei

How conductive is the semiconductor?

• Fermi level represents character of semiconductor Intrinsic n-type p-type

EcEF

Ev

Ec EF

Ev

Ec

EFEv

Position of Fermi level: EF

n=p=ni n>p n<p

Material dependent

kTEENn Fc

C exp

kTEENp vF

V exp

Carriers: graphically (see B. Streetman)

EF

E

1/2 F(E)1

E E

Ec

Ev

g(E) carrierconcentration

a) intrinsic

electrons

holes

EF

E

1/2 F(E)1

E E

Ec

Ev

g(E) carrierconcentration

b) n-type

electrons

holes

EF

E

1/2 F(E)1

E E

Ec

Ev

g(E) carrierconcentration

c) p-type

electrons

holes

Extrinsicsemiconductor

Intrinsicsemiconductor

Density of free carriers and position of the Fermi-level are related.

g(E) Density of states

f(E) Fermi-Dirac distribution function

T>0K

f(E)

f(E)

f(E)

Linking potential energy in the energy band diagram E(x) with electrostatic potential V(x) and electric field .

)(xEPE

exExV

)()(

dxxdVx )()( E

)(xE

Energy band diagram.

What does an energy band diagram describe?It gives the potential energy of free carriers

above Ec for free electrons below Ev for free holes

In between is bandgap in which no carriers can reside

EcEF

Ev

EiEG

Influence of an electric field on the energy band diagram

Ec

Ev

EG

Represents the potential energy of the carriers

x

When an electric field, E is applied, carriers gain potential energy (represented by E) in the direction of the field.

Ei

If Ethen Higher potential

energy E here.

Since electrons drift in opposite direction as electric field.

E

Influence of an electric field on the energy band diagram

Represents the energy of the carriers

When an electric field is applied, carriers gain potential energy (represented by E) in the direction of the field.

E(x)Thus:Ec(x)

Ev(x)EG

Ei(x)

Influence of an electric field on carrier movement in the energy band diagram

• Energy band diagram of a semiconductor under an electric field

EEc

EvEG

e- opposite direction to electric field.

h+ in direction of electric field.

Influence of an electric field on the energy band diagram: maths for e-

E Ec

EvEG

Electric field E points “uphill” for electrons!

exExV

)()(

dxxdVx )()( E

dxxdE

ex )(1)( E

dxxdE

ex c )(1)( E

)(xEPE

Excess carriers Internal electric field

• & Vext = 0V ; I = 0A

• Current densities:Jn = emnnE + eDndn/dxJp = emppE - eDpdp/dx

• Dn dn/dx = -mnn E Dp dp/dx = mpp E

• E = -(Dn/mnn) dn/dx E = (Dp/mpp) dp/dx

n(x)

x

If

ThenE(x)

Ec

EvEg

EF

xTVq

kTD

m

e- diffusion

e- drift

=0=0

All transport equations

qxExV

dxxdVx

xNxNxpxnexdxxd

dxxnd

dxxpd

dxxdpeDxxpexJ

dxxdneDxxnexJ

AD

p

n

ppp

nnn

)()(

)()(

)()()()()()(

0)(

0)(

)()()()(

)()()()(

2

2

2

2

E

E

E

E

m

m

)()(

)(

)()(

)()(

)(

)()(

2

2

xpxN

nxp

xNxp

xnxN

nxn

xNxn

nD

in

Ap

pA

ip

An

Steady state (DC) in short layers Low carrier injection at 300K

Conclusions

• Two types of carriers in a semiconductor– Electrons negative charge and electron mass mn

– Holes positive charge and hole mass mp

• Two types of current in semiconductors– Drift current, caused by external electric field– Diffusion current, caused by excess carrier

concentration.

+

• Free electrons reside in the conduction band, have a negative charge and an effective mass mn

• Free holes reside in the valence band, have a positive charge and a different effective mass mp

• The simplified energy band gives the potential energy of the carriers

• No carriers can reside in the bandgap, EG

• The position of the Fermi level, EF defines the type and density of free carriers.

• An electric field (internal or external) causes a tilt in the band diagram.

For the exams you do not have to be able to:

• Solve the Schrodinger equation• Derive the density of states• Derive the expression for the electron and hole concentration.

The solutions for these are derived in the study group questions, but the purpose is solely to satisfy your curiosity.The equations will be given in a formulae sheets and you have to be able to apply them, but not derive them.