SRH recombination

W.K. Chen Electrophysics, NCTU 1

Chapter 6Nonequilibrium excess carrier in semiconductor


Ambipolar transport

Excess electrons and excess holes do not move independently of each other. They diffuse, drift, and recombine with same effective diffusion coefficient, drift mobility and lifetime. This phenomenon is called ambipolar transport.

Two basic transport mechanismsDrift: movement of charged due to electric fields

Diffusion: the flow of charges due to density gradient

We implicitly assume the thermal equilibrium during the carrier transport is not substantially disturbed


Outline

Carrier generation and recombination

Characteristics of excess carrier

Ambipolar transport

Quasi-Fermi energy levels

Excess carrier lifetime

Surface effect

Summary


6.1 Carrier generation and recombination

Generation

Generation is the process whereby electrons and holes are created

Recombination

Recombination is the process whereby electrons and holes are annihilated

Any deviation from thermal equilibrium will tend to change the electron and hole concentration in a semiconductor.

(thermal exitation, photon pumping, carrier injection)

When the external excitation is removed, the concentrations of electron and hole in semiconductor will return eventually to their thermal-equilibrium values


pono GG =

pono RR =

ponopono RRGG ===

6.1.1 The semiconductor in equilibriumThermal-equilibrium concentrations of electron and hole in conduction and valence bands are independent of time.

Since the net carrier concentrations are independent of time, the rate at which the electrons and holes are generated and the rate at which they recombine must be equal.

For direction band-to-band transition

Direct bandgap semiconductor


6.1.2 Excess carrier generation and recombination

bandgap)(direct '' pn gg =

ppp

nnn

o

o

δ

δ

+=

+=

Excess electrons and excess holes

When external excitation is applied, an electron-hole pair is generated. The additional electrons and holes are called excess electrons and excess holes.

Generation rate of excess carriers

For direct band-to-band generation, the excess electrons and holes are created in pairs

2ioo npnnp =≠



Excess carriers recombination rate

In the direct band-to-band recombination, the excess electrons and holes recombine in pairs, so the recombination rate must be equal

bandgap)(direct ''pn RR =

Using the concept of collision model, we assume the rate of pair recombination obeys

)('

carriers excess of rateion Recombinat

mequilibriuat holes and electrons of rateion Recombinat

riumnonequilibunder holes and electrons of rateion Recombinat

2

2

iro

irooro

r

nnpRRR

npnR

npR

−=−=

==

=

α

αα

α


)]()([)( 2 tptnn

dt

tdnir −=α

)()( )()( tpptptnntn oo δδ +=+=

))]())[((

))]())((([)( 2

tnpntn

tpptnnndt

tdn

oor

ooir

δδα

δδα

++−=

++−=


noor ttp enentn τα δδδ /)0()0()( −− ==

orno pα

τ 1=

Low-level injectionLow-level injection

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

)( material type-p assume

2oir

oo

ptδntptnndt

tdn

np

<<−=

>>

Qα

injection) level-(low)()(

tnpdt

tndor δαδ

−=

The solution to this equation is an exponential decay from initial excess carrier concentration

Excess minority lifetime


The recombination rate of excess carriers

injection) level-(low )(

)()(

'no

or

tntnp

dt

tndR

τδδαδ

=+=−=

nopn

tnRRR

τδ )(

''' ===

popn

tpRRR

τδ )(

''' ===

( p-type, low level injection)

( p-type, low level injection)

n-type material, no>>po


6.2.1 Continuity equation

dxx

xFxFdxxF px

pxpx ⋅∂

∂+=+

+++ )(

)()(

dxdydzp

dxdydzgdydzdxxFxFdxdydzt

p

pnppxpx τ

−++−=∂∂ ++ )]()([

The net increase in the number of holes in the differential volume per unit time

From the calculus, the Taylor expansion gives

g R

Hole flux generation recombination

Flux in Flux out

dxdydzp

dxdydzgdxdydzx

xFdxdydz

t

p

pnp

px

τ−+

∂

∂−=

∂∂ + )(

dA


s)-(holes/cm )( 2

ptp

px pg

x

xF

t

p

τ−+

∂

∂−=

∂∂ +

s)-/cm(electrons )( 2

ntn

n pg

x

xF

t

n

τ−+

∂∂

−=∂∂ +

Continuity equation for holes

Continuity equation for electrons

:pt

p

τThe recombination rate holes including thermal-equilibrium recombination and excess recombination

:ptτ The recombination lifetime which includs thermal-equilibrium carrier lifetime and excess carrier lifetime

g R


6.2.2 Time-dependent diffusion equation

x

neDneJ

x

peDpeJ nnnppp ∂

∂+=

∂∂

−= E , E μμ

x

nDnF

e

J

x

pDpF

e

Jnnn

nppp

p

∂∂

−−==−∂

∂−==

+−+ E

)( , E

)(μμ

ptp

px pg

x

xF

t

p

τ−+

∂

∂−=

∂∂ + )(

ntn

n pg

x

xF

t

n

τ−+

∂∂

−=∂∂ + )(

dxx

xFxFdxxF px

pxpx ⋅∂

∂+=+

+++ )(

)()(

The current density in material is

By dividing current density the charge of each individual particle, we obtain particle flux


ntnnn

ptppp

ng

x

nD

x

n

t

n

pg

x

pD

x

p

t

p

τμ

τμ

−+∂∂

+∂

∂+=

∂∂

−+∂∂

+∂

∂−=

∂∂

2

2

2

2

E)(

E)(

Thus the continuity equations can be rewritten as

xn

x

n

x

n

xp

x

p

x

p

∂∂

+∂∂

=∂

∂∂∂

+∂∂

=∂

∂ EE

E)( and

EE

E)(Q

t

nng

xn

x

n

x

nD

t

ppg

xp

x

p

x

pD

ntnnn

ptppp

∂∂

=−+⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂

+∂∂

∂∂

=−+⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂

−∂∂

τμ

τμ

EE

EE

2

2

2

2


t

nng

xn

x

n

x

nD

t

ppg

xp

x

p

x

pD

ntnnn

ptppp

∂∂

=−+⎟⎠⎞

⎜⎝⎛

∂∂

+∂

∂+

∂∂

∂∂

=−+⎟⎠⎞

⎜⎝⎛

∂∂

+∂

∂−

∂∂

)(E)(E

)(

)(E)(E

)(

2

2

2

2

δτ

δμδ

δτ

δμδ

The thermal equilibrium concentrations, no and po, are not function of time. For homogeneous semiconductor, no and po are also independent of space coordinates

)(

)(

tppp

tnnn

o

o

δδ+=+=

Homogeneous semiconductor


6.3 Ambipolar transport

appint EE <<

Ambipolar transport When excess carriers are generated, under external applied field the excess holes and electrons will tend to drift in opposite directions

However, because the electrons and holes are charged particles, any separation will induce an internal field between two sets of particles, creating a force attracting the electrons and holes back toward each other

Only a relatively small internal electric field is sufficient to keep the excess electrons and holes drifting and diffusing together

The excess electrons and holes do not move independently of each other, but they diffuse and drift together, with the same effective diffusion coefficient and with the same effective mobility. This phenomenon is called ambipolar transport.

appintapp EEEE ≈−=

0E)(

E intint ≈

∂∂

=−

=⋅∇t

npe

sεδδ

W.K. Chen Electrophysics, NCTU 18t

nRg

xn

x

n

x

nD

t

nRg

xp

x

n

x

nD

nn

pp

∂∂

=−+⎟⎠⎞

⎜⎝⎛

∂∂

+∂

∂+

∂∂

∂∂

=−+⎟⎠⎞

⎜⎝⎛

∂∂

+∂

∂−

∂∂

)(E)(E

)(

)(E)(E

)(

2

2

2

2

δδμδ

δδμδ

⎪⎪

⎩

⎪⎪

⎨

⎧

=

====

≡=

pn

Rp

Rn

R

ggg

ptp

ntn

pn

δδ

ττ

For ambipolar transport, the excess electrons and excess holes generate and recombine together

ptnt ττ ≠

t

nng

xn

x

n

x

nD

t

ppg

xp

x

p

x

pD

ntnnn

ptppp

∂∂

=−+⎟⎠⎞

⎜⎝⎛

∂∂

+∂

∂+

∂∂

∂∂

=−+⎟⎠⎞

⎜⎝⎛

∂∂

+∂

∂−

∂∂

)(E)(E

)(

)(E)(E

)(

2

2

2

2

δτ

δμδ

δτ

δμδ

ppp

nnn

o

o

δδ+=+=


t

npn

Rgpnx

nnp

x

npDnD

pn

pnnpnppn

∂∂

+=

−++⎟⎠⎞

⎜⎝⎛

∂∂

−−∂

∂+

)()(

))(()(

E))(()(

)(2

2

δμμ

μμδμμδμμ

t

nRg

x

n

x

nD

∂∂

=−+⎟⎠⎞

⎜⎝⎛

∂∂

−∂

∂ )()(

)(E'

)('

2

2 δδμδ

)('

pn

pDnDD

pn

nppn

μμμμ

+

+=

)(

))(('

pn

np

pn

np

μμμμ

μ+

−=

⇒== kT

e

DD p

p

n

nμμ

QpDnD

pnDDD

pn

pn

+

+=

)(' (ambipolar diffusion coefficient)

(ambipolar mobilty)

ambipolar transport equation


t

nng

xn

x

n

x

nD

ntnnn ∂

∂=−+⎟

⎠⎞

⎜⎝⎛

∂∂

+∂

∂+

∂∂ )(E)(

E)(

2

2 δτ

δμδ

t

nRg

x

n

x

nD

∂∂

=−+⎟⎠⎞

⎜⎝⎛

∂∂

−∂

∂ )()(

)(E'

)('

2

2 δδμδ ambipolar transport equation

continuity equation for electrons

t

ppg

xp

x

p

x

pD

ptppp ∂

∂=−+⎟

⎠⎞

⎜⎝⎛

∂∂

+∂

∂−

∂∂ )(E)(

E)(

2

2 δτ

δμδ continuity equation for holes


6.3.2 Ambipolar transport under low injection

pon

opn

opon

oopn DnD

nDD

ppDnnD

ppnnDDD =≈

+++

+++=

)()(

)]()[('

δδδδ

injection) (low

typenFor

o

o

npn

pn o

<<=>>−δδ

t

pRg

x

p

x

pD

∂∂

=−+⎟⎠⎞

⎜⎝⎛

∂∂

−∂

∂ )()(

)(E'

)('

2

2 δδμδ

)(

))(('

pn

np

pn

np

μμμμ

μ+

−=

pDnD

pnDDD

pn

pn

+

+=

)('

pn

np

pn

np

n

n

pn

npμ

μμμ

μμμμ

μ −=−

≈+

−=

)(

))((

][

))[('

Ambipolar transport


)()( ''ppoppopp RRgGRgRg +−+=−=−

popo RG =Qp

ppp

pgRgRg

τδ

−=−=− '''

excess hole generation rate

Thermal-equilibrium hole generation rate

Thermal-equilibrium hole recombination rate

excess hole recombination rate

For n-type,Under low injection, the concentration of majority carriers electrons will be essentially constant. Then the probability per unit time of a minority carrier hole encountering a majority carrier electron will remain almost a constant

ppt ττ =Q (minority carrier hole lifetime)

So the net generation rate,


t

ppg

x

p

x

pD

popp ∂

∂=−+

∂∂

−∂

∂ )('

)(E

)(2

2 δτδδμδ n-type, homogeneous and low-

injection ambipolar transport

t

pRg

x

p

x

pD

∂∂

=−+⎟⎠⎞

⎜⎝⎛

∂∂

−∂

∂ )()(

)(E'

)('

2

2 δδμδ

pDD ='

pμμ −=' pppp

pgRgRg

τδ

−=−=− '''

For ambipolar transport , the transport and recombination parameters are governed by minority carriers

The excess majority carriers diffuse and drift with the excess minority carriers; thus the behavior of the excess majority carriers is determined by the minority carrier parameters


injection) (low

typepFor

o

o

pnp

np o

<<=>>−δδ

For p-type semiconductor

t

nRg

x

n

x

nD

∂∂

=−+⎟⎠⎞

⎜⎝⎛

∂∂

−∂

∂ )()(

)(E'

)('

2

2 δδμδ

)(

))(('

pn

np

pn

np

μμμμ

μ+

−=

pDnD

pnDDD

pn

pn

+

+=

)('

p-type, homogeneous ambipolar transport

nop

opn

opon

oopn DpD

pDD

ppDnnD

ppnnDDD =≈

+++

+++=

)()(

)]()[('

δδδδ

np

np

pn

np

p

p

pn

npμ

μμμ

μμμμ

μ =≈+

−=

)(

))((

][

))[('


)()( ''nnonnonn RRgGRgRg +−+=−=−

nono RG =Qn

nnn

ngRgRg

τδ

−=−=− '''

excess electron generation rate

Thermal-equilibrium electron generation rate

Thermal-equilibrium hole recombination rate

excess hole recombination rate

For p-type,Under low injection, the concentration of majority carriers holes will be essentially constant. Then the probability per unit time of a minority carrier electron encountering a majority carrier hole will remain almost a constant

nnt ττ =Q (minority carrier electron lifetime)

So the net generation rate,


n

nDD

μμ ==

'

'

t

nRg

x

n

x

nD

∂∂

=−+⎟⎠⎞

⎜⎝⎛

∂∂

−∂

∂ )()(

)(E'

)('

2

2 δδμδ

t

nng

x

n

x

nD

nonn ∂

∂=−+

∂∂

+∂

∂ )()(E

)( '2

2 δτδδμδ p-type, homogeneous and low-

injection ambipolar transport

nnnn

ngRgRg

τδ

−=−=− '''

For ambipolar transport , the transport and recombination parameters are governed by minority carriers

The excess majority carriers diffuse and drift with the excess minority carriers; thus the behavior of the excess majority carriers is determined by the minority carrier parameters


6.3.3 Applications of the ambipolar transport equation


Example 6.1

timeoffunction a asion concentratcarrier excess theCalculate

0for 0g'

carriers excess ofion concentrat uniform 0,At

conditioninjection -lowunder

field electrical applied zero

tor semiconduc type-n Homogenous

⇒>=

=t

t

Solution:

t

pp

x

pD

pop ∂

∂=−

∂∂ )()(

2

2 δτδδ

Uniform excess carriers

t

ppg

x

p

x

pD

popp ∂

∂=−+

∂∂

−∂

∂ )('

)(E

)(2

2 δτδδμδ

E=0 g’=0


po

p

t

p

τδδ

−=∂

∂ )(

poteptp τδδ /)0()( −=

potentn τδδ /)0()( −=

From the charge neutrality condition, the excess majority electron concentration is given by


Example 6.2

timeoffunction a asion concentratcarrier excess theCalculate

carriers excess of generation uniform 0,At

conditioninjection -lowunder



⇒>t

t

ppg

x

p

x

pD

popp ∂

∂=−+

∂∂

−∂

∂ )('

)(E

)(2

2 δτδδμδ

Solution:

0')(

)(

' =−+∂

∂⇒

∂∂

=− gp

t

p

t

ppg

popo τδδδ

τδ

Uniform generation

)1(')( / potpo egtp ττδ −−=

' )( ,0)(

state,steady At

gtpt

tppoτδδ

=∞==∂

∂


Example 6.3

offunction a asion concentratcarrier excess statesteady theCalculate

directions and bith thein diffuse then

condition

injection -lowunder only, 0at generated being are carriers excess the


tor semiconduc type-p Homogenous

x

xx

x

⇒−+

=

Solution:

t

nng

x

n

x

nD

nonn ∂

∂=−+

∂∂

+∂

∂ )()(E

)( '2

2 δτδδμδ

E=0 Steady state


⎪⎩

⎪⎨⎧

≤=

≥=⇒

+=

+

−

−−

0 )0()(

0 )0()(

)(

/

/

//

xenxn

xenxn

BeAexn

Lnx

Lnx

LnxLnx

δδ

δδ

δ

0)(

0At

0)(

0,At

2

2

'2

2

=−∂

∂≠

=−+∂

∂=

non

non

n

x

nDx

ng

x

nDx

τδδ

τδδ

0)(

2

2

=−∂

∂

nonD

n

x

n

τδδ

nonn DL τ=2

Minority carrier diffusion length

The general solution

p-typelog scale


Example 6.4

tx

tg'

x

t

xi

and offunction a asion concentratcarrier excess theCalculate

0for 0

0 and

0at ously instantane generated is pairs hole-electron of numbers finite

direction- n the E field electrical appliedconstant


o

⇒>=

==

Solution:

t

ppg

x

p

x

pD

popp ∂

∂=−+

∂∂

−∂

∂ )('

)(E

)(2

2 δτδδμδ


t

pp

x

p

x

pD

popp ∂

∂=−

∂∂

−∂

∂ )()(E

)(o2

2 δτδδμδ

potetxptxp τδ /),('),( −=

t

txp

x

txp

x

txpD pp ∂

∂=

∂∂

−∂

∂ ),('),('E

),('o2

2

μ

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −−=

tD

tx

tDtxp

p

p

p 4

)E(exp

4

1),('

2oμ

π

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −−==

−−

tD

tx

tD

eetxptxp

p

p

p

tt

po

po

4

)E(exp

4),('),(

2o

// μ

πδ

ττ


Zero applied field constant applied field


6.3.4 Dielectric relaxation time constantHow is the charge neutrality achieved and how fast ?

In previous analysis, we have assume a quasi-neutrality condition exists-that is, the concentration of excess holes is balanced by an equal concentration of excess electrons

Suppose that we have a situation in which a uniform concentration of δp is suddenly injected into a portion of the surface of a semiconductor. We will have instantly have a concentration of excess holes and a net positive charge density δp that is not balance by a concentration of excess electrons. How is the charge neutrality achieved and how fast ?


ερ

=⋅∇ E

tJ

∂∂

−=⋅∇ρ

Eσ=J

Poisson’s equation

Continuity equation(neglecting the effects of generation and recombination)

ttJ

∂∂

−==⋅∇⇒∂∂

−=⋅∇ρ

ερσσρ

E)(

0=+∂∂ ρ

εσρ

tdtet τρρ /)0()( −=

σετ =d

Dielectric relaxation time constant

pe δρ )(+=

( the time constant is related to dielectric constant)


Example 6.5 Dielectric relaxation time constant

constant timerelaxation dielectric theCalculate

cm 10

tor semiconduc type-n Homogenous316

⇒= −

dN

Solution:

ps 0.539 )10)(1200)(106.1(

)1085.8)(7.11(1619

14

=×

×=== −

−

dn

ord Neμ

εεσετ

In approximately four time constants (2 ps), the net charge density is essentially zero

The relaxation process occur very quickly (τd ≈0.5 ps) compared to the normal excess carrier lifetime (τ =0.1 μs).

That is the reason why the continuity equation in calculating relaxation time does not contain any generation or recombination terms.


6.3.5 Haynes-Shockley experiment

Zero applied field constant applied field

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −−==

−−

tD

tx

tD

eetxptxp

p

p

p

tt

po

po

4

)E(exp

4),('),(

2o

// μ

πδ

ττ

The Haynes-Shockley experiment was one of the first experiment to actually measure excess-carrier behavior, which can determine

The minority carrier lifetime

The minority carrier diffusion coefficient

The minority carrier lifetime


Haynes-Shockley experiment

Excess-carrier pulse are effectively injected at contact A

Contact B is rectifying contactand is under reverse bias (do not perturb the electric field)

A fraction of excess carriers will be collected by contact B

The collect carriers will generate an output voltage Vo when flow through output resistance R2


The idealized excess minority carrier (hole) pulse is injected at contact A at time t=0

The excess carriers (holes) will drift along the semiconductor producing an output voltage as a function of time

The peak of pulse will arrive at contact B at time to

During the time period, the occurs diffusion and recombination

0 0 =−⇒=− tExtx opp μυ

oop t

d

E=μ


At t=to, the peak of pulse reaches contact B. where times t1 and t2, the magnitude of the excess concentration is e-1

If the time difference between t1

and t2, is not too large, the prefactor do not change appreciably during this time

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −−==

−−

tD

tx

tD

eetxptxp

p

p

p

tt

po

po

4

)E(exp

4),('),(

2o

// μ

πδ

ττ

212

o or ,4)E( ttttDtx pp ==− μ

12

22o ,

16

)()E(ttt

t

tD

o

pp −=Δ

Δ=

μ

From the broadened pulse width, we can obtain diffusion coefficient

diffusion coefficient


⎟⎟⎠

⎞⎜⎜⎝

⎛ −=⎟

⎟⎠

⎞⎜⎜⎝

⎛ −=

−=

pooppo

op

po

o dK

dK

tKS

τμτμ

τ Eexp

)E/(exp)exp(

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −−==

−−

tD

tx

tD

eetxptxp

p

p

p

tt

po

po

4

)E(exp

4),('),(

2o

// μ

πδ

ττ

The area S undrer the curve is proportional to the number of excess holes that have not recombined with majority carrier electrons

By varying the electric field, the area under the curve will change

A plot of ln(S) as a function of (d/μpEo) will yield a straight line whose slope is (1/τpo)


⎟⎟⎠

⎞⎜⎜⎝

⎛ −⋅+=

oppo

dKS

E

1)ln()ln(

μτ

)ln(S

opEμ

dy =

poτ1

We can determine the minority carrier lifetime


6.4 Quasi-Fermi energy levelsAt thermal equilibrium,

the electron and hole concentrations are functions of the Fermi-level.

The Fermi level remains constant throughout the entire material

The carrier concentrations is exponentially determined by the Fermi-level

2

exp exp

ioo

ffio

fifio

npn

kT

EEnp

kT

EEnn i

=

⎟⎟⎠

⎞⎜⎜⎝

⎛ −=⎟⎟

⎠

⎞⎜⎜⎝

⎛ −=


At non-thermal equilibriumIf excess carriers are created, thermal equilibrium no longer exists and Fermi energy is strictly no longer defined

We may define quasi-Fermi levels for electrons and holes to relate the concentrations for non-equilibrium semiconductor in the same form of equation as that in thermal equilibrium

In such a way, the quasi-Fermi levels for electrons and holes specified for non-thermal equilibrium conditions do not hold constants over the entire material

2

exp

exp

i

fpfioo

fifnio

nnp

kT

EEnppp

kT

EEnnnn

i

≠

⎟⎟⎠

⎞⎜⎜⎝

⎛ −=+=

⎟⎟⎠

⎞⎜⎜⎝

⎛ −=+=

δ

δ


Example 6.6 Quasi-Fermi level

energy Fermi-quasi theCalculate

cm 10pn rium,nonequilibIn

cm 10 and cm 10 ,cm 10

K 300Tat tor semiconduc type-n Homogenous

313

35310315

⇒==

===

=

−

−−−

δδoio pnn

Solution:

eV 2982.0)ln( exp ==−⇒⎟⎟⎠

⎞⎜⎜⎝

⎛ −=

i

ofif

fifio n

nkTEE

kT

EEnn

eV 179.0)ln( exp

eV 2984.0)ln( exp

=+

=−⇒⎟⎟⎠

⎞⎜⎜⎝

⎛ −=+=

=+

=−⇒⎟⎟⎠

⎞⎜⎜⎝

⎛ −=+=

i

ofpfi

fpfioo

i

ofifn

fifnio

n

ppkTEE

kT

EEnppp

n

nnkTEE

kT

EEnnnn

iδδ

δδ


At non-thermal equilibriumSince the majority carrier electron concentration does not change significantly for low-injection condition, the quasi-Fermi level for electrons here is not much different from the thermal-equilibrium Fermi level.

The quasi-Fermi level for minority carrier holes is significantly the Fermi level and illustrate the fact that we have deviate from the thermal equilibrium significantly

31335

315315

cm 10cm 10

cm 1001.1cm 100.1 :−−

−−

→=

×→×

op

n


6.5 Excess-carrier lifetimeIn a perfect semiconductor

The electronic energy states do not exist within the forbidden bandgap

In a real semiconductorDefects (traps) occur within the crystal, creating discrete electronic energy states or impurity energy bands within the forbidden-energy band

These defect energy states may be the dominant effect in determining the mean carrier lifetime in the real semiconductor


6.5.1 Shockley-Read-Hall theory of recombinationRecombination center

A trap within the forbidden bandgap may act as a recombination center, capturing both electrons and holes with almost equal probability.

Shockley-Read-Hall theoryAssume a single recombination center exists at an energy Et within the bandgap.

If the trap is an acceptor-like trap; that is , it is negatively charged when it contains an electron and is neutral when it does not contain an electron.

negatively charged electronNeutral empty state


Shockley-Read-Hall theoryThere are four basic processes in SRH theory

Process 1: The capture of an electron from the CB by a neutral empty trap

Process 2 The emission of an electron back into the CB

Process 3 The capture of a hole from the VB by a trap containing an electron

Process 4 The emission of a hole from a neutral trap into the VB


Process 1: electron capture

nEfNCR tFtncn ⋅−⋅⋅= ))](1([Electron capture rate

⎥⎦

⎤⎢⎣

⎡ −+

=

kT

EEEf

fttF

exp1

1)(

Process 2: electron emission

)]([ tFtnen EfNER ⋅⋅=

(at thermal equilibrium)encn RR =

)]([))](1([ tFotntFotn EfNEnEfNC ⋅⋅=⋅−⋅⋅

Is there relationship between capture constant and emission constant?


ntc

cn

nfoc

cfot

n

ntFo

nntFo

tFon

CkT

EENE

EkT

EEN

kT

EEC

EnEf

CEEf

nEfC

⎥⎦⎤

⎢⎣⎡ −−=

=⎥⎦

⎤⎢⎣

⎡ −−⋅−⎥

⎦

⎤⎢⎣

⎡ −+⋅

=⋅−⋅⇒=⋅−

⋅

exp

exp)1exp1(

)1)(

1(

)(

))(1(

nn CnE '= ⎥⎦⎤

⎢⎣⎡ −

=⎥⎦⎤

⎢⎣⎡ −−=

kT

EEn

kT

EENn it

itc

c expexp'

The relation between emission constant and capture constant is valid at all conditions including when Fermi level is right located at trap energy, in which the trap is the dominant process to provide the free electrons in the conduction band

The electron emission rate is increased exponentially as the trap energy level closes to the conduction band


pEfNCR tFtpcp ⋅⋅⋅= ))((

⎥⎦

⎤⎢⎣

⎡ −+

=

kT

EEEf

fttF

exp1

1)(

))](1([ tFtpep EfNER −⋅⋅=

Process 3: hole capture rate

Process 4: hole emission rate

epcp RR =

⎥⎦⎤

⎢⎣⎡ −

=

⎥⎦

⎤⎢⎣

⎡ −⋅=−⎥

⎦

⎤⎢⎣

⎡ −+⋅=⎥

⎦

⎤⎢⎣

⎡ −⋅

−⋅=⋅

−⋅⋅=⋅⋅⋅

kT

EENCE

kT

EEE

kT

EEE

kT

EENC

EfEpC

EfNEpEfNC

tpp

ftp

ftp

fp

tFpp

tFtptFtp

υυ

υυ

exp

exp)1exp1(exp

)1)(

1(

))](1([))((

(at thermal equilibrium)

'pCE pp = ⎥⎦⎤

⎢⎣⎡ −

=⎥⎦⎤

⎢⎣⎡ −

=kT

EEn

kT

EENp ti

it expexp' υ

υ


Capture constantsThe electron capture constant comes from a electron with velocity υth must come within a cross-sectional area σcn of a trap to be captured and thus sweep out an effective trap volume per second. The same is for hole capture constant.

The capture constants for electrons and holes proportion to respective cross-sectional area

Emission constantsThe electron emission rate from a trap is the product of electron capture rate and free carrier concentration when Ef=Et.

The electron emission rate is increased exponentially as the trap energy level closes to the conduction band

The hole emission rate is increased exponentially as the trap energy level closes to the valence band

'pCE pp =

nn CnE '=

cnthnC συ= cpthpC συ=

⎥⎦⎤

⎢⎣⎡ −−=

kT

EENn tc

c exp' ⎥⎦⎤

⎢⎣⎡ −

=kT

EENp tυ

υ exp'


nEfNCR tFtncn ⋅−⋅⋅= ))](1([

)]([ tFtnen EfNER ⋅⋅=

pEfNCR tFtpcp ⋅⋅⋅= ))((

))](1([ tFtpep EfNER −⋅⋅=


Under non-equilibrium condition

)]('))(1([

)]([))](1([

tFtFtnn

tFtntFtnn

encnn

EfnEfnNCR

EfNEnEfNCR

RRR

−−⋅=⋅⋅−⋅−⋅⋅=

−=

Net electron capture rate at acceptor trap

)]('))(1([ tFtFtnn EfnEfnNCR −−⋅=

))](1('))([ tFtFtpp EfpEpfNCR −−⋅=

⎥⎦⎤

⎢⎣⎡ −−=

kT

EENn tc

c exp'

⎥⎦⎤

⎢⎣⎡ −−=

kT

EENp t υ

υ exp'

)'()'(

')(

ppCnnC

pCnCEf

pn

pntF +++

+=

Net hole capture rate at acceptor trap (Process 3 & 4)

At steady state, pn RR =


We may note that

⎥⎦⎤

⎢⎣⎡ −−=⎥⎦

⎤⎢⎣⎡ −−⋅⎥⎦

⎤⎢⎣⎡ −−=

kT

EENN

kT

EEN

kT

EENpn c

cttc

cυ

υυ

υ expexpexp''

2'' inpn =

RppCnnC

nnpNCCRR

pn

itpnpn ≡

+++

−==

)'()'(

)( 2

For SRH-dominated recombination process, the recombination rate of excess carriers is

)'()'(

)( 2

ppCnnC

nnpNCCnR

pn

itpn

+++

−==

τδ

nRpR


SRH under low injection

trap)(deep ' ,'

injection) (low

type)-(n

pnnn

pn

pn

oo

o

oo

>>>>>>>>

δ

Case 1: n-type semiconductor with deep trap energy at low injection

pNCnC

pnNCC

ppCnnC

nnpNCCnR tp

n

tpn

pn

itpn δδ

τδ

=≈+++

−==

)'()'(

)( 2

pNCn

R tp δτδ

==

For n-type semiconductor with deep trap energy at low injection, the net recombination rate is limited by the hole capture process during SRH recombination

)3( kTEE tf >−


For n-type semiconductor with deep trap energy at low injection, the net recombination rate is limited by the hole capture process during SRH recombination

The recombination is related to the mean minority carrier lifetime

nRpR

fEtEpo

tp

ppNC

nR

τδδ

τδ

===

tppo NC

1=τ

If the trap concentration increases, the probability of excess carrier recombination increases; thus the excess minority carrier lifetime decreases

n-typelow injection


Case 2: p-type semiconductor with deep trap energy at low injection

trap)(deep ' ,'

injection) (low

type)-(p

nppp

np

np

oo

o

oo

>>>>>>>>

δ

tnno NC

1=τ

nRpR

fEtE

+

notn

nnNC

pR

τδδ

τδ

=== p-typelow injection


)'()'(

)( 2

ppCnnC

nnpNCCnR

pn

itpn

+++

−==

τδ

)'()'(

)( 2

ppnn

nnpnR

nopo

i

+++−

==τττ

δ

SRH recombination process

Case 3: n/p-type semiconductor with deep trap energy

General SRH recombination rate


Example 6.7 Intrinsic semiconductor

Solution: )'()'(

)( 2

ppnn

nnpnR

nopo

i

+++−

==τττ

δ

))(2(

)(2

)()(

))()( 22

nopoi

i

iinoiipo

iii

nn

nnn

nnnnnn

nnnnnR

ττδδδ

δτδτδδ

+++

=+++++

−+⋅+=

injection lowunder

''

levelenergy intrinsic toequalenergy aat located is trapdeep Assume

, ,

i

ooioo

npn

δpppδnnnnpn

==

+=+===

τδ

ττδ nn

Rnopo

=+

=)(

Under low injection inn <<δ

The excess-carrier lifetime increases as we change from an extrinsic to an intrinsic semiconductor

nopo τττ +=


6.6 Surface effectsSurface states

At the surface, the semiconductor is abruptly terminated. This disruption of periodic potential function results in allowed electronic energy states within the energy bandgap, which is called surface states

Since the density of traps at the surface is larger than the bulk, the excess minority carrier lifetime at the surface will be smaller than the corresponding lifetime in the bulk material

For n-type material

bulk) (in the po

B

poB

ppR

τδ

τδ

==

surface) (at the pos

ss

pR

τδ

=

popos ττ <<


Assume excess carriers are uniformly generated throughout the entire semiconductor material

At steady state, the generation rate is equal to recombination rate at a given position either in the bulk or at the surface

surface) (at the

bulk) (in the

pos

ss

po

BB

pRG

pRG

τδ

τδ

==

==

Bspopos pp δδττ <⇒<

pumping) (uniform sB RRG ==

For the case of uniform pumping


Example 6.8 Surface recombination

Solution:

ondistributiion concentratcarrier -excess stateSteady

pumping uniform and 0E Assume

/scm 10 s10 ,s10 ,cm10tor semiconduc type-n 270

6-314

⇒=

==== −−psppopB Dττδ

1-136

714 cm 10

)10(

)10()10()(

)0( surfaceat ion concentratcarrier -Excess

===⇒==

=

−

−

po

posBs

pos

s

po

B pppp

G

x

ττ

δδτδ

τδ

Q

pumping) (uniform 0')(

0)E(at equation ansport Carrier tr

2

2

=−+

=

pop

pg

dt

pdD

τδδ

1-3-206

14

s-cm 1010

10' where === −

po

Bpg

τδ


pp LxLxpo BeAegxp //')( −++= τδ

pumping) (uniform ')( / pLxpo Begxp −+= τδ

13314314

314

109 cm 10cm 10)0(

cm 10')(

:B.C.

×−=⇒=+==

===+∞−−

−

BBpp

gpp

s

poB

δδ

τδδ

)0.9-(110)( /14 pLxs epxp −== δδ

m31.6cm 31600.0)(10)(10 where -6 μτ ==== popp DL


6.6.2 Surface recombination velocityA gradient in the excess-carrier concentration existing near the surface leads to a diffusion of excess carriers from the bulk region toward the surface where they recombine.

))0(()0()(

Flux osurface

p ppspsdx

pdD −== δδ

Surface recombination rate


For the case of uniform pumping

pLxpo Begxp /')( −+= τδ

0')(

0)E(at equation ansport Carrier tr

2

2

=−+

=

pop

pg

dt

pdD

τδδ

)'

1(')( / pLx

pp

poppo e

sLD

Lsggxp −

+−=

ττδ

sLD

sgB

L

B

dx

pd

dx

pd

Bgp

pp

po

pxsurface

po

+

−=⇒

−==

+=

=

)/(

'

)()(

')0(

0

τ

δδ

τδQ

Uniform pumpingE=0Steady state

Uniform pumpingE=0Steady state


x

)(xpδ

0=s

Surface recombination velocity is sensitive to the surface conditionsFor sand-blasted surfaces, the typical values of s may be as high as 105 cm/s

For clean etched surfaces, this value may be as low as 10 to 100 cm/s states

pumping) (uniform cm/s 1)0(

'⎟⎟⎠

⎞⎜⎜⎝

⎛−=

p

g

L

Ds po

p

p

δτ

0if ')( == sgxp poτδ


Example 6.10 Surface recombination velocity

Solution:

tyion velocirecombinat surface theDetermine

m 10(0) and m 6.13 /s,cm 10,cm 10

s10 ,s10 ,cm10tor semiconduc type-n

oumping)(uniform 6.8 Examplein case For the

3-132314

70

63-14

⇒

====

===−

−−

cδp L Dg'τ pppo

sppopB

μ

ττδ

cm/s 1)0(

'⎟⎟⎠

⎞⎜⎜⎝

⎛−=

p

g

L

Ds po

p

p

δτ

cm/s 1085.2110

10

106.31

10 413

14

4×=⎟⎟

⎠

⎞⎜⎜⎝

⎛−

×= −s

⇒ pumping uniform


Short briefs for surface recombination

bulk) (in the po

B

poB

ppR

τδ

τδ

==

surface) (at the pos

ss

pR

τδ

=

))0(()0()(

Flux osurface

p ppspsdx

pdD −== δδ



Figure 6.19 Figure for problems 6.18 and 6.20


Figure 6.20 Figure for Problem 6.25









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SRH recombination