CHAPTER 4

CONDUCTION IN SEMICONDUCTORS

Prof. Dr. Beşire GÖNÜL

CHAPTER 4CONDUCTION IN SEMICONDUCTORS

• Carrier drift• Carrier mobility• Saturated drift velocity• Mobility variation with temperature• A derivation of Ohm’s law• A derivation of Ohm’s law• Drift current equations• Semiconductor band diagrams with an electric field present• Carrier diffusion• The flux equation• The Einstein relation• Total current density• Carrier recombination and diffusion length

• We now have some idea of the number densityof charge carriers (electrons and holes) presentin a semiconductor material from the work wecovered in the last chapter. Since current is therate of flow of charge , we shall be able calculate

Drift and DiffusionDrift and Diffusion

covered in the last chapter. Since current is therate of flow of charge , we shall be able calculatecurrents flowing in real devices since we knowthe number of charge carriers. There are twocurrent mechanisms which cause charges tomove in semiconductors. The two mechanismswe shall study in this chapter are driftdrift andanddiffusiondiffusion.

• Electron and holes will move under the influence of anapplied electric field since the field exert a force oncharge carriers (electrons and holes).

• These movements result a current of ;

F qE=

Carrier DriftCarrier Drift

dI• These movements result a current of ;dI

d dI nqV A=:dI

:dV

drift current number of charge carriers per unit volume

charge of the electrondrift velocity of charge carrier

area of the semiconductor:A

:q

:n

Carrier Mobility ,Carrier Mobility , µ

dV Eµ=

:E:µ

applied field

mobility of charge carrier

[ ]

−=

SecV

cm2

µ µ is a proportionality factor

=E

Vdµ

So is a measure how easily charge carriers move under the influence ofan applied field or determines how mobile the charge carriers are.

µµ

n n -- type Sitype Si

+ -V

n – type Si

e-

V

L

VE =

e

dV

Electric field

Electron movement

Current flow

Current carriers are mostly electrons.

+ -V

p – type Si

holeV

p p -- type Sitype Si

L

VE =

hole

dV

Electric field

Hole movement

Current flow

Current carriers are mostly holes.

Carrier MobilityCarrier Mobility

E

Vd=µ

Macroscopic understanding

In a perfect Crystal

Microscopic understanding? (what the carriers themselves are doing?)

*

q

m

τµ =

0ρσ

=

→ ∞

It is a superconductor

*mgeneralinmm he

** ⟨

*

*

;

;

e

h

m n type

m p type

−

−

• A perfect crystal has a perfect periodicity and thereforethe potential seen by a carrier in a perfect crystal iscompletely periodic.

• So the crystal has no resistance to current flow andbehaves as a superconductor. The perfect periodicpotential does not impede the movement of the charge

µ

potential does not impede the movement of the chargecarriers. However, in a real device or specimen, thepresence of impurities, interstitials, subtitionals,temperature , etc. creates a resistance to current flow.

• The presence of all these upsets the periodicity of thepotential seen by a charge carrier.

The mobility has two component

The mobility has two componentsThe mobility has two components

Impurity interaction component

Lattice interaction component

• Assume that s/c crystal is at thermodynamic equilibrium (i.e. there isno applied field). What will be the energy of the electron at a finitetemperature?

• The electron will have a thermal energy of kkTT//22 per degree offreedom. So , in 3D, electron will have a thermal energy of

Thermal velocityThermal velocity

* 23 1 3 3kT kT kT= ⇒ = ⇒ =* 23 1 3 3

2 2 2th th

kT kT kTE m V V

m ∗= ⇒ = ⇒ =

( ) 21

*

2

1

:

−

mV

TV

electronofvelocitythermalV

th

th

th

α

α

• Since there is no applied field, the movement ofthe charge carriers will be completely random.This randomness result no net current flow. As

Random motionRandom motion result result no currentno current..

This randomness result no net current flow. Asa result of thermal energy there are almost anequal number of carriers moving right as left, inas out or up as down.

*

0

2

? 300 1.18

? 1000 / 0.15 /( )

th e

d

V RT K m m

V E V m m V sµ

= = =

= = = −

• Calculate the velocity of an electron in a piece of n-typesilicon due to its thermal energy at RT and due to theapplication of an electric field of 1000 V/m across thepiece of silicon.

CalculationCalculation

5

? 1000 / 0.15 /( )

31.08 10 / sec

150 / sec

d

th th

d d

V E V m m V s

kTV V x m

m

V E V m

µ

µ

∗

= = = −

= ⇒ =

= ⇒ =

How long does a carrier move in time before collision ?

The average time taken between collisions is called as relaxation time, (or mean free time)

Microscopic understanding of mobilityMicroscopic understanding of mobility??

τHow far does a carrier move in space (distance) before a collision?

The average distance taken between collisions is called as mean free path, .l

*d

FV

mτ= ×

CalculationCalculation

Drift velocity=Acceleration x Mean free time

Force is due to the applied field, F=qE

* *d

d

F q EV

m m

qV E

m

τ τ

τµ µ

∗

= × =

= ⇒ =

Force is due to the applied field, F=qE

*

2 2

12 13

? ? 1 .18 0.59

0.15 /( ) 0 .0458 /( )

10 sec 1.54 10 sec

e o h o

e h

e e h h

l m m m m

m V s m V s

m mx

τ

µ µ

µ µτ τ

∗

∗ ∗− −

= = = =

= − = −

= = = =

• Calculate the mean free time and mean free path forelectrons in a piece of n-type silicon and for holes in apiece of p-type silicon.

CalculationCalculation

12 13

5 5

5 12 7

10 sec 1.54 10 sec

1.08 10 / 1 .052 10 /

(1 .08 10 / )(10 ) 10

(1.052 10

elec hole

elec

hole

e e h he h

th th

e th e

h th h

m mx

q q

v x m s v x m s

l v x m s s m

l v x

µ µτ τ

τ

τ

− −

− −

= = = =

= =

= = =

= = 5 13 8/ )(1 .54 10 sec) 2 .34 10m s x x m− −=

Saturated Drift VelocitiesSaturated Drift Velocities

dV Eµ=

dV for efor e--

So one can make a carrier go as fast as we like just by increasing the electric field!!!

dV

E E

for efor e--

for holesfor holes

(a) Implication of above eqn. (b) Saturation drift velocity

• The equation of does not imply that VVddincreases linearly with applied field EE..

•• VVdd increases linearly for low values of EE and then itsaturates at some value of VVdd which is close VVthth at

EVd .µ=

Saturated Drift VelocitiesSaturated Drift Velocities

saturates at some value of VVdd which is close VVthth athigher values of EE.

• Any further increase in EE after saturation point does notincrease VVdd instead warms up the crystal.

Mobility variation with temperatureMobility variation with temperature

µ µ

TTTT

High temperature Low temperature

)ln(µ

ln( T )ln( T )

Peak depends on the density of

impurities

High temperature Low temperature

IµLµ

1 1 1

T L Iµ µ µ= +

This equation is called as Mattheisen’s rule.

Variation of mobility with temperatureVariation of mobility with temperature

At high temperature (as the lattice warms up)

Lµ component becomes significant.

Lµ decreases when temperature increases.

2

3

2

3

1

−−⇒×= TTCLµ

It is called as a power law.

CC11 is a constant.

1.5T −

Carriers are more likely scattered by the lattice atoms.

At low temperatures

Variation of mobility with temperatureVariation of mobility with temperature

Iµ component is significant.

Iµ decreases when temperature decreases.

2

3

2 TCI ×=µ CC22 is a constant.

Carriers are more likely scattered by ionized impurities.

The peak of the mobility curve depends on the number density of ionized impurities.

Variation of mobility with temperatureVariation of mobility with temperature

Highly doped samples will therefore cause more scattering, and have a lower mobility, than low doped

samples.

This fact is used in high speed devices called High Electron Mobility Transistors (HEMTs) where electrons are made to

move in undoped material, with the resulting high carrier mobilities!

HEMTs are high speed devices.

samples.

A Derivation of Ohm’s LawA Derivation of Ohm’s Law

d d d

dd

I nqV A V E

I qJ

A m

µ

τµ

τ

∗

= =

= =

2

2 1

x d x x

x x

nqJ nqV nq E J E

m

nqJ E

m

τµ

τσ σ ρ

σ

∗

∗

= = =

= = =[ ] [ ][ ] [ ]1 ( )

m

m

ρ

σ

= Ω −

= Ω −

A Derivation of Ohm’s LawA Derivation of Ohm’s Law

V

L

R

VI =

VA VI

L Rρ= =

area

current

This is in fact ohm’s law which is written slightly in a different form.

1x

x xx

I IV VJ E

A L A Lσ σ

ρ= = ⇒ =

Drift Current EquationsDrift Current Equations

For undoped or intrinsic semiconductor ; n=p=ni

For electron

n nJ nqEµ=

For hole

p pJ pqEµ=n nJ nqEµ=

drift current for electrons

number of free electrons per unit volume

mobility of electron

p pJ pqEµ=

drift current for holes

number of free holes per unit volume

mobility of holes

Drift Current EquationsDrift Current Equations

Total current density

i e h

i n P

J J J

J nqE pqE

n p n

µ µ

= +

= +

= =

( )

i

i i n p

n p n

J n q Eµ µ

= =

= +

since

For a pure intrinsic

semiconductor

Drift Current EquationsDrift Current Equations

?=totalJ for doped or extrinsic semiconductor

n-type semiconductor;

T n D nn p J nq E N q Eµ µ>> ⇒ ≅ =T n D nn p J nq E N q Eµ µ>> ⇒ ≅ =where ND is the shallow donor concentration

p-type semiconductor;

T p A pp n J pq E N q Eµ µ>> ⇒ ≅ =where NA is the shallow acceptor concentration

Variation of resistivity with temperatureVariation of resistivity with temperature

Why does the resistivity of a metal increase withincreasing temperature whereas the resistivity of asemiconductor decrease with increasingtemperature?

1 1ρ = =

nqρ

σ µ= =

This fact is used in a real semiconductor device called a

thermistor, which is used as a temperature sensing element.

The thermistor is a temperature – sensitive resistor; that is its terminalresistance is related to its body temperature. It has a negative temperaturecoefficient , indicating that its resistance will decrease with an increase in itsbody temperature.

Semiconductor Band Diagrams with Semiconductor Band Diagrams with Electric Field PresentElectric Field Present

At equilibrium ( with no external field )

EC

EĐ

E

All these energies are horizontal

Pure/undoped semiconductor

EVhorizontal

How these energies will change with an applied field ?

+ -

n – type

Electric field

Electron movementHole flow

EC

EĐ

EV

Efe-

hole

qV

• With an applied bias the band energies slope down for the givensemiconductor. Electrons flow from left to right and holes flow from rightto left to have their minimum energies for a p-type semiconductor biasedas below.

EC

e-+_

p – type

Electric field

Electron movementHole flow

EC

EĐ

EV

Ef

hole

qV

Under drift conditionsUnder drift conditions;;

••UnderUnder driftdrift conditionsconditions;; holes float and electrons sink.Since there is an applied voltage, currents are flowingand this current is called as drift current.

•There is a certain slope in energy diagrams and thedepth of the slope is given by qV, where V is the batteryvoltage.

work qE L

Vq L work qV gain in energyL

qVSlope of the band qE Force on the electron

= − ×

= − × ⇒ = − =

= − = − =

The work on the charge carriersThe work on the charge carriers

Work = Force x distance = electrostatic force x distance

qVSlope of the band qE Force on the electron

L

VE

L

= − = − =

=

Since there is a certain slope in the energies, i.e. theenergies are not horizontal, the currents are able to flow.

where L is the length of the s/c.

The work on the charge carriersThe work on the charge carriers

1(1)

'

i ix x

dVElectrostatic Force gradiant of potential energy

dx

dE dEqE E

dx q dx

one can define electron s electrostatic potential as

= − = −

− = − ⇒ = ⋅

'

(2)

(1) (2) ,

nx

in n

one can define electron s electrostatic potential as

dVE

dx

comparison of equations and gives

EV is a relation betweenV an

q

= −

= − id E

Carrier DiffusionCarrier Diffusion

Current mechanisms

Drift DiffusionP nkT

dP dnkT

=

=

1

dP dnkT

dx dx

dn dP

dx kT dx

=

=

photons

Contact with a metal

Carrier DiffusionCarrier Diffusion

Diffusion current is due to the movement of the carriers from highconcentration region towards to low concentration region. As thecarriers diffuse, a diffusion current flows. The force behind thediffusion current is the random thermal motion of carriers.

1dn dP

dx kT dx= ⋅

A concentration gradient produces apressure gradient which produces the force

dx kT dx= ⋅ pressure gradient which produces the force

on the charge carriers causing to movethem.

How can we produce a concentration gradient in a semiconductor?

1) By making a semiconductor or metal contact.2) By illuminating a portion of the semiconductor with light.

Illuminating a portion of the semiconductor with light

By means of illumination, electron-hole pairs canbe produced when the photon energy>Eg.

So the increased number of electron-hole pairsmove towards to the lower concentration region untilthey reach to their equilibrium values. So there is athey reach to their equilibrium values. So there is anumber of charge carriers crossing per unit area perunit time, which is called as flux. Flux is proportionalto the concentration gradient, dn/dx.

n

dnFlux D

dx= −

FluxFlux

[ ][ ]

2 1

2,th

Flux m s

D l D m s

The current densities for electrons and holes

dn dnJ q D qD for electrons

ν

− −= −

= =

= − − =

2

,

n n n

p p p

n p

J q D qD for electronsdx dx

dp dpJ q D qD for holes

dx dx

J A m

= − − =

= + − = −

=

Einstein RelationEinstein Relation

Einstein relation relates the two independent currentmechanicms of mobility with diffusion;

pn

n p

DD kT kTand for electrons and holes

q qµ µ= =

Constant value at a fixed temperature

( )( )2

2

/sec

sec

J K Kcm kTvolt volt

cm V q C= = =

−

2 5kT

m V a t ro om tem p era tu req

=

Total Current DensityTotal Current Density

When both electric field (gradient of electric potential) and concentration gradient present, the total current density ;

n n n

dnJ q nE qD

dxµ= +

p p p

total n p

dpJ q pE qD

dx

J J J

µ= −

= +

Carrier recombination and diffusion lengthCarrier recombination and diffusion length

• By means of introducing excess carriers into an intrinsics/c, the number of majority carriers hardly changes, butthe number of minority carriers increases from a low- tohigh-value.

• When we illuminate our sample (n-type silicon with 1015• When we illuminate our sample (n-type silicon with 1015

cm-3 ) with light that produces 1014 cm-3 electron-holepairs.

• The electron concentration (majority carriers) hardlychanges, however hole concentration (minority carriers)goes from 1.96 x 105 to 1014 cm-3.

Recombination rate• Minority carriers find themselves surrounded by very high

concentration of majority carriers and will readily recombine withthem.

• The recombination rate is proportional to excess carrier density, .pδ

( ) (0)expt

p t pδ δτ

= −

1

p

d pp

d t

δδ

τ= −

1

n

d nn

d t

δδ

τ= − ( ) (0)exp

tn t nδ δ

τ = −

pd t τ

Lifetime of holes

Excess hole concentration when t=0

Excess hole concentration decay exponentially with time.

Similarly, for electrons;

Diffusion length,Diffusion length,

When excess carriers are generated in a specimen, the minoritycarriers diffuse a distance, a characteristic length, over whichminority carriers can diffuse before recombining majority carriers.This is called as a diffusion length, L.

Excess minority carriers decay exponentially with diffusion distance.

L

( ) (0)expn

xn x n

Lδ δ

= −

( ) (0)exp

p

xp x p

Lδ δ

= −

n n nL Dτ= p p pL Dτ=

Excess electron concentration when x=0

Diffusion length for electronsDiffusion length for holes