Conduction in Semiconductor

7/27/2019 Conduction in Semiconductor

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Carrier drift Carrier mobility Saturated drift velocity

Mobility variation with temperatureA derivation of Ohms 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


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We now have some idea of the number density ofcharge carriers (electrons and holes) present in asemiconductor material from the work we covered

in the last chapter. Since current is the rate of flowof charge , we shall be able calculate currentsflowing in real devices since we know the numberof charge carriers. There are two currentmechanisms which cause charges to move insemiconductors. The two mechanisms we shallstudy in this chapter are drift and diffusion.

Drift and Diffusion


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Electron and holes will move under the influence of an

applied electric field since the field exert a force on chargecarriers (electrons and holes).

These movements result a current of ;

F qEdI

Carrier Drift

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


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Carrier Mobility ,

dV E:E:

applied field

mobility of charge carrier

SecV

cm2

is a proportionality factor

E

Vd

So is a measure how easily charge carriers move under the influence of

an applied field or determines how mobile the charge carriers are.


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n - type Si

L

VE

+ -V

n type Si

e-

dV

Electric field

Electron movement

Current flow

Currentcarriers are mostly electrons.


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+ -V

p type Si

hole

dV

Electric field

Hole movement

Current flow

Current carriers are mostly holes.

p - type Si

L

VE


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Carrier Mobility

E

Vd

Macroscopic understanding

In a perfect Crystal

0

It is a superconductor

Microscopic understanding? (what the carriers

themselves are doing?)

*

q

m

generalinmm he**

*

*

;

;

e

h

m n type

m p type


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Aperfect crystal has a perfect periodicity and therefore thepotential seen by a carrier in a perfect crystal is completelyperiodic.

So the crystal has no resistance to current flow and behavesas a superconductor. The perfect periodic potential doesnot impede the movement of the charge carriers. However,in a real device or specimen, the presence of impurities,interstitials, subtitionals, temperature , etc. creates aresistance to current f low.

The presence of all these upsets the periodicity of thepotentialseen by a charge carrier.


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The mobility has two component

The mobility has two components

Impurity interaction

component

Lattice interaction

component


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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 finite

temperature?

The electron will have a thermal energy ofkT/2 per degree of freedom.So , in 3D, electron will have a thermal energy of

Thermal velocity

* 23 1 3 3

2 2 2th th

kT kT kT E m V V

m

21

*

2

1

:

mV

TV

electronofvelocitythermalV

th

th

th


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Since there is no applied field, the movement ofthe charge carriers will be completely random.

This randomness result no net current flow. As aresult of thermal energy there are almost an equalnumber of carriers moving right as left, in as out orup as down.

Random motion result no current.


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*

0

2

5

? 300 1.18

? 1000 / 0.15 /( )

31.08 10 / sec

150 / sec

th e

d

th th

d d

V RT K m m

V E V m m V s

kTV V x m

m

V E V m

Calculate the velocity of an electron in a piece of n-type

silicon due to its thermal energy at RT and due to theapplication of an electric field of 1000 V/m across the pieceof silicon.

Calculation


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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)

How far does a carrier move in space (distance) before acollision?

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

Microscopic understanding of mobility?


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*d

FV

m

Calculation

* *d

d

F qEV

m m

qV E

m

Drift velocity=Acceleration x Mean free time

Force is due to the applied field, F=qE


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*

2 2

12 13

5 5

5 12 7

? ? 1.18 0.59

0.15 /( ) 0.0458 /( )

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 o h o

e h

e e h he h

th th

e th e

h th h

l m m m m

m V s m V s

m mx

q q

v x m s v x m s

l v x m s s m

l v x

5 13 8/ )(1.54 10 sec) 2.34 10m s x x m

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

of p-type silicon.

Calculation


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Saturated Drift Velocities

d

V E

dV

EE

for e-

for holes

So one can make a carrier go as fast as we like just by

increasing the electric field!!!

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


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The equation of does not imply that Vdincreases linearly with applied field E.

Vd increases linearly for low values of E and then it

saturates at some value ofVd which is close Vthat highervalues ofE.

Any further increase in E after saturation point does not

increaseVdinstead warms up the crystal.

EVd .

Saturated Drift Velocities


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Mobility variation with temperature

)ln(

TT

ln( T )

Peak depends

on the density of

impurities

High temperature Low temperature

I

L1 1 1

T L I

This equation is called as

Mattheisens rule.


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2

3

2

3

1

TTCL

Variation of mobility with temperature

At high temperature(as the lattice warms up)

L

1.5T

component becomes significant.

L decreases when temperature increases.

It is called as a power law.

C1is a constant.

Carriers are more likely scattered by the lattice atoms.


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At low temperatures


I component is significant.

I

decreases when temperature decreases.

2

3

2 TCI C2is a constant.

Carriers are more likely scattered by ionized impurities.


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This fact is usedin high speed devicescalled High Electron

Mobility Transistors (HEMTs) where electrons are made to

move in undoped material, with the resulting highcarrier mobilities!

The peak of the mobility curve depends on the number

density of ionized impurities.


HEMTs are high speed devices.

Highly doped samples will therefore cause more

scattering, and have a lower mobility, than low doped

samples.


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A Derivation of Ohms Law

2

2 1

d d d

dd

x d x x

x x

I nqV A V E

I qJ

A m

nqJ nqV nq E J E

m

nqJ E

m

1 ( )

m

m


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A Derivation of Ohms Law

R

VI

VA VI

L R

1x

x xx

I IV VJ E

A L A L

V

L

area

current

This is in fact ohms law which is written slightly in a different form.


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Drift Current Equations

For undoped or intrinsic semiconductor ; n=p=ni

For electron

n nJ nqE

drift

currentfor

electrons

number

of free

electrons

per unit

volume

mobility

ofelectron

For hole

p pJ pqEdrift

currentfor holes

number

of freeholes per

unit

volume

mobility

of holes


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Total current density

( )

i e h

i n P

i

i i n p

J J J

J nqE pqE

n p n

J n q E

since

For a pure

intrinsic

semiconductor


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?totalJ for doped or extrinsic semiconductor

n-type semiconductor;

T n D nn p J nq E N q E where ND is the shallow donor concentrationp-type semiconductor;

T p A pn J pq E N q E where NA is the shallow acceptor concentration


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Variation of resistivity with temperature

Why does the resistivity of a metal increase withincreasing temperature whereas the resistivity of a

semiconductor decrease with increasing

temperature?

1 1nq

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 temperature

coefficient , indicating that its resistance will decrease with an increase in its

body temperature.

Semicond ctor Band Diagrams ith


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Semiconductor Band Diagrams with

Electric Field Present

At equilibrium ( with no external field )

EC

E

EV

All these

energies

are

horizontal

Pure/undoped semiconductor

How these energies will change with an applied field ?

+ -

n type

Electric field

Electron movement

Hole flow

EC

E

EV

Efe-

hole

qV


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With an applied bias the band energies slope down for the given

semiconductor. Electrons flow from left to right and holes flow from rightto left to have their minimum energies for a p-type semiconductor biased

as below.

p type

Electric fieldElectron movement

Hole flow

EC

E

EV

Ef

e-

hole

qV

+_


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Under drift conditions;

Under drift conditions; holes float and electrons sink.Since there is an applied voltage, currents are flowing

and 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 battery

voltage.


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work qE L

Vq L work qV gain in energy

L

qVSlope of the band qE Force on the electron

L

V

E L

The work on the charge carriers

Work = Force x distance = electrostatic force x distance

Since there is a certain slope in the energies, i.e. the

energies are not horizontal, the currents are able to flow.

where L is the length of the s/c.


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The work on the charge carriers

1(1)

'

(2)

(1) (2) ,

i ix x

nx

in n

dVElectrostatic Force gradiant of potential energy

dx

dE dE qE E

dx q dx

one can define electron s electrostatic potential as

dVE

dx

comparison of equations and gives

EV is a relation between V an

q

id E


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Carrier Diffusion

Current mechanisms

Drift Diffusion

1

P nkT

dP dnkT

dx dx

dn dP

dx kT dx

photons

Contact with a metal


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Carrier Diffusion

Diffusion current is due to the movement of the carriers from high

concentration region towards to low concentration region. As thecarriers diffuse, a diffusion current flows. The force behind the

diffusion current is the random thermal mo t ion of carr iers.

1dn dP

dx kT dx A concentration gradient produces a

pressure gradient which produces the forceon the charge carriers causing to move

them.

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 semiconductorwith light.


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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 pairs

move towards to the lower concentration region until

they reach to their equilibrium values. So there is anumber of charge carriers crossing per unit area per

unit time, which is called as flux. Flux is proportional

to the concentration gradient, dn/dx.

n

dnFlux D

dx


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Flux

2 1

2

2

,

,th

n n n

p p p

n p

Flux m s

D l D m s

The current densities for electrons and holes

dn dnJ q D qD for electronsdx dx

dp dp

J q D qD for holesdx dx

J A m


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Einstein Relation

Einstein relation relates the two independent current

mechanicms of mobility with diffusion;

pn

n p

DD kT kTand for electrons and holes

q q

Constant value at a fixed temperature

2

2

/sec

sec

J K Kcm kT volt volt

cm V q C

25kT

mV at room temperatureq


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Total Current Density

When both electric field (gradient of electric potential) and

concentration gradient present, the total current density ;

n n n

p p p

total n p

dnJ q nE qD

dx

dpJ q pE qD

dx

J J J


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Carrier 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

cm-3 ) with light that produces 1014 cm-3 electron-hole pairs.

The electron concentration (majority carriers) hardly

changes, however hole concentration (minority carriers)goes from 1.96 x 105 to 1014 cm-3.


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Recombination rate

Minority carriers find themselves surrounded by very highconcentration of majority carriers and will readily recombine withthem.

The recombination rate is proportional to excess carrier density, .p

1

n

d nn

dt

( ) (0)exp

tn t n

( ) (0) expt

p t p

1

p

d pp

dt

Lifetime of holes

Excess hole concentration when t=0

Excess hole concentration decay exponentially with time.

Similarly, for electrons;

Diff i l th L


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Diffusion length,

When excess carriers are generated in a specimen, the minority

carriers 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.

( ) (0) expn

xn x n

L

( ) (0)exp

p

xp x p

L

n n nL D p p pL D

L

Excess electron concentration when x=0

Diffusion length for electrons

Diffusion length for holes

Conduction in Semiconductor

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