recombination: depletion region, bulk, radiative, auger, and tunnelling ch 140 lecture notes #13...

Recombination: Depletion Region, Bulk, Radiative, Auger, and Tunnelling

Ch 140

Lecture Notes #13

Prepared by David Gleason

Review of Depletion Region Recombination

• We assume:– Flat Quantum Fermi Levels

• Requires that the fastest recombination rate is slow with respect to diffusion

– kn ≈ kp = • There are an even distribution of traps where does not

depend on x

–

This leads to:

U(x)dx U(x)dx0

0

w

U total kTw

q(Vbi Vapp )Umax

Review of Depletion Region Recombination (cont.)

We also have

and JD/R = qUtotal

So

Umax 12 NTni exp( qVapp /2kT)

JDR kTw

4(Vbi Vapp )NTni exp( qVapp /2kT)

Quasi-Neutral Region

• The Quasi-neutral region is defined as a region of the semiconductor with an uneven distribution of carriers in a region of flat bands

• As pictured, the holes will diffuse away from w’ into the bulk where they will recombine

w w’

Ef,n

Ef,p

h+

h+

h+

h+

h+

h+

Quasi-Neutral Region

Bulk Recombination

At steady state

using Fick’s 1st Law

and Fick’s 2nd Law

We have

p(x)

tR(x) D(x) 0

Flux Do

Co(x,t)

(x)

Co

tDo

2Co(x,t) Do

2Co(x, t)

x 2

p(x)

t0

p(x) po(x)

p

Dp

2 p(x)

x 2

To solve this we must first establish some boundary conditions

p( w ) po exp( qVapp /kT)1.

2.

3.

Solving for p(x) yields

n( w ) p(w) ni2 exp( qVapp /kT)

p() po

p(x) po po exp( qVapp /kT) 1 exp x w

Lp

Where is the diffusion length

Lp Dp p

Bulk Recombination (cont.)

The bulk recombination current can be determined by

JBR = q flux

where the flux here is for all carriers at any point in the flat band region

This is solved easiest at w’ since there there is no electron movement to consider. At other values of x

JBR=-q fluxholes+q fluxelectrons

At x=w’ this simplifies to

Solving this and evaluating at x=w’ recognizing that the last term simplifies

We have

JBR qfluxholes qDp

p(x)

x

exp x w

Lp

1

JBR qDp po

Lp

exp qVapp

kT

1

From

We can substitute

To get

This is the bulk region recombination

JBR qDp po

Lp

exp qVapp

kT

1

no pp ni2 po

ni2

no

ni

2

ND

JBR qDpni

2

LpND

exp qVapp

kT

1

Radiative Recombination• Assume a perfect

semiconductor crystal– No surface state recombination– No depletion region

recombination ( is very small)

– No bulk recombination (Lp is very big)

• Generate carriers through light absorption or thermal excitation– Carriers diffuse until finally they

recombine in the inverse of the absorption reaction

– Light is emitted with h = Eg

e-e-

h+ h+

kr’ kr

h

Radiative Recombination (cont.)

• This process has been ignored until now because for indirect band gap semiconductors the carrier lifetime due to radiative recombination is really long.– 99.9% of bulk recombination in Si and Ge will

occur across trap states

• For direct gap semiconductors, including GaAs and porous Si, radiative recombination is more competitive– Leads to LEDs, lasers, ect.

Radiative Recombination Current

Rate of electron recombination given by

At equilibriumThis expression can be plugged into the rate

equation away from equilibrium to give

And finally

n

tnpkr kr

0 no pokr kr kr

ni2kr

n

tkr np ni

2

Jr qkr np ni2

kr’ kr

Determination of kr from the absorption spectra

• Indirect semiconductors can not be made pure enough to emit, so kr must be calculated from the absorption spectra– At equilibrium in a perfect sample, the rate of thermal

absorption must equal the rate of radiative recombination because they are inverse processes

– The thermal absorption is given by the overlap between the blackbody curve at temperature T and the absorption spectra at temperature T

Si absorption spectraGaAs absorption spectra

300K Blackbody

Determination of kr from the absorption spectra

Eg

GaAsEg

Si

Photons absorbed by Si

Photons absorbed by GaAs

Note that even though Si has a lower Eg than GaAs, less light is absorbed due to the shape of the absorption spectra caused by the indirect band gap of Si.At equilibrium, the amount absorbed is equal to that emitted through radiative recombination, so we can calculate kr, which is sometimes called B, and has units

of cm4s-1.

Auger Recombination

• Pronounced or

• Occurs at very high injection or doping conditions– This is a 3 body process whereby two majority

carriers collide • One looses energy Eg and combines with a minority

carrier

• The other gains energy Eg, which it subsequently looses through thermalization

Auger Recombination (cont.)• For n-type

– Auger lifetime

– Gn = recombination rate

• For p-type– Auger lifetime

– Gp = recombination rate• Gp 2x10-31 cm6/s for Si at

room temperature

• The dependence is on n or p, and therefore depends on the doping or excitation level

e- e-

+Eg

-Eg

h+

heat

A 1

Gnn2

A 1

Gp p2

e-

+Eg

-Eg

h+

heat

h+

Tunneling Current• Tunneling is only important at high

high dopant densities and low temperatures

• The tunneling probability is given by

• The tunneling probability is temperature independent, and since most other currents (thermionic emission) are highly temperature dependent it is only seen at low temps Ratio of tunneling current to thermionic current

for Si-Au barrier taken from Sze p. 264

Ttun exp 8w

3h2qme

* Vbi V 1

2

w 1

ND

where

Summary of Recombination

• Bulk

• Depletion Region

• Thermionic

• Radiative

• Auger

• Tunneling

JBR qDpni

2

LpND

exp qVapp

kT

1

JDR kTw

4(Vbi Vapp )NTni exp( qVapp /2kT)

Jth A*T 2 exp qb

kT

exp

qVapp

kT

1

Jr qkr np ni2

A 1

Gnn2

JT exp b / ND

Summary of Recombination (cont.)

• Bulk– A=1, depends ND

– Jo proportional to exp(-Eg/kT)

• Depletion Region– A=2

• Thermionic– A=1, does not depend on ND

– Jo proportional to exp(-qb/kT)

• Radiative– Insignificant for indirect gap semiconductors, – Strictly depends on excess carriers

• Auger– Only at really high carrier concentrations

• Tunneling– Only significant at low T and high ND or NA

– Constant with temperature