ECE 663

• So far, we looked at equilibrium charge distributions. Theend result was np = ni

2

• When the system is perturbed, the system tries to restoreitself towards equilibrium through recombination-generation

• We will calculate the steady-state rates

• This rate will be proportional to the deviation fromequilibrium, R = A(np-ni

2)

R-G processes

ECE 663

h

h

Real spaceEnergy space

Direct Band-to-band recombination

Lasers, LEDs, ..

ECE 663

h

h

Real spaceEnergy space

Direct excitonic recombination

Organic Solar cells, CNTs, wires (1-D systems)

ECE 663

R-G for a Direct Band-gap

Conserve momentum and Energy

1. Eg = h

2. k = kphoton = 2/c

= Eg/ħc

Eg ~ 1.2 eV

k ~ 6/m << BZ = 2/a [Å]

Optical transitions almost vertical !!

ECE 663

Indirect Band-gap

Since photons make vertical transitions,they won’t conserve momentum for indirect band-gaps (Si, Ge)

ECE 663

Indirect Band-gap: What about phonons?

Conserve momentum and Energy

1. Ephonon = hphonon

2. k = kphonon = 2phonon/vsound

= Ephonon/ħvsound ≈ 2/a [Å]

Because they involve atomic Displacements, wavelengths arecomparable

But energy very small (~ 10-100 meV).Not sufficient

ECE 663

Indirect Band-gap: Involving traps

Two step process !!

ktrap ≈ 2/a

ECE 663

Phonons Phonons

Real spaceEnergy space

Indirect (Trap-assisted) recombination

Nonradiative Recombination(Ge, Si FETs, solar cells )

ECE 663

Shockley-Read-Hall

ECE 663

Phonon Phonon

Real spaceEnergy space

Auger Recombination

III-Vs, highly doped samples(Opposite process is impact ionization)

XX

ECE 663

Real spaceEnergy space

Impact Ionization

Si, Ge, InP Lasers, FETs

X

X

ECE 663

The capture process

NT = nT + pT

5 2 35 3 2

ECE 663

The capture rate

Assume these c’s don’t change under bias

ECE 663

Net Recombination Rates

TpTpGR

P

TnTnGR

N

pepnctp

r

nenpctn

r

(1-f)

ECE 663

Net Recombination Rates

TpTpGR

P

TnTnGR

N

pepnctp

r

nenpctn

r

ECE 663

Problem Solving Strategy

•Look at equilibrium

en

•Steady-state: Set rN = rP

nT

•Plug back in:

R = rN, rP

Similar

prescription

in ECE 687

toget ballistic

current,

except

R = A(n-f),

not A(np-ni

2)

ECE 663

Look at Equilibrium first

Detailed Balance

ECE 663

Look at Equilibrium first

rN = rP = 0

en = cnpT0n0/nT0 = cnn1

ep = cpnT0p0/pT0 = cpp1

n1 = nie(E’T-Ei)/kT

p1 = nie(Ei-E’T)/kT

Charge densities if trapspin Fermi energy

ECE 663

)(

)(

1

1

pppnctp

r

nnnpctn

r

TTpGR

P

TTnGR

N

Substituting:

ECE 663

Now look at steady-state

ECE 663

Detailed Balance, Steady State

No net clockwise flow Steady clockwise flow

ECE 663

… and TransientsUnsteady flow

ECE 663

Steady State

TTT

TTpTTn

TTpPTTnN

nNp

pppncnnnpc

pppncrnnnpcr

)()(

)()(

11

11

)()( 11

1

ppcnnc

pNcnNcn

pn

TpTnT

ECE 663

Steady State Recombination Rate R

• Use:

• And

• And

)(

)(

1

1

pppncr

or

nnnpcr

TTpP

TTnN

211 i

TTT

npn

nNp

)()( 11

1

ppcnnc

pNcnNcn

pn

TpTnT

ECE 663

Steady State Recombination Rate R

)()( 11

2

ppnnnnp

rrRnp

iPN

Tnn Nc

1Tp

p Nc1

n1 = nie(E’T-Ei)/kT p1 = nie(Ei-E’T)/kT ci=ivth

Typical #s

NT ET TN TP SN SP

Material [cm-3] [eV] [m2] [m2] [m/s] [m/s]

Si, Ge 1e13 0.0 1e-15 1e-15 0.0 0.0

III-Vs 2e16 0.4 1e-14 1e-13 0.0 0.0

ECE 663

Let’s look at a few limits

n = n0 + n, n << n0

p = p0 + p, p << p0 << n0

Low-level injection, n-type material

n ~ pFew traps

Deep traps (midgap) n1 ≈ p1 ≈ ni

ECE 663

Low level injection

)()())((

)()(

1010

200

11

2

pppnnnnppnn

R

ppnnnnp

R

np

i

np

i

• Set n0p0 = ni2 cancels term in numerator

• Drop np term in numerator• n0p >> p0n in numerator for n-type material• Keep only n0 related term in denominator

ECE 663

• For n-type:

• For p-type

p

pR

n

nR

Low level injection

We’ll frequently adopt this approximation

ECE 663

Surface Recombination

Lattice periodicity broken at surface/interface – mid-gap E levelsCarriers generated-recombined per unit area

ECE 663

Surface States

Reconstruction

Expt (Akiyama et al, PRB 2000) Theory (Rakshit/Liang/Ghosh/Hersam/Datta, PRB 2005)

ECE 663

• Processes and descriptions analogous to bulk R-G using surface parameters

• For a single energy level surface state:

)(

)(

1

1

pppncr

nnnpcr

TTpP

TTnN

)(

)(

1

1

sTsssTpsPs

sTssTsnsNs

pppncr

nnnpcr

kTEEi

kTEEi

Ti

iT

epp

enn/)(

1

/)(1

'

'

kTEE

i

kTEEis

ITi

iIT

epp

enn/)(

1

/)(1

'

'

)()( 11

2

ppnnnnp

Rnp

i

)(1

)(1

11

2

ssn

ssp

iss

pps

nns

npnR

Surface Recombination

ECE 663

Surface Recombination

)(1

)(1

11

2

ssn

ssp

iss

pps

nns

npnR

CAVEAT !! sn and sp have units of length/time – surface recombination velocity for single level surface states

Tsnsn Ncs Tspsp Ncs

ECE 663

Multi-Level (more realistic)

DIT(E) – density of interface traps (per unit area-energy)DIT(E)dE – density of IT between E and E+dE (replaces NTs)

dEEDpp

cnn

c

npnR IT

E

Ess

nsss

ps

issC

V

)()(

1)(

111

2

In summary• R-G processes drive system towards equilibrium

R (np – ni2)

(In ECE687, restoration will be driven by the contacts)

• For indirect band-gap materials, SRH dominates

• Coeffs depend on minority carrier lifetime, a critical concept for this course

• Minority carrier lifetime depends on trap cross-section (size), trap density and electron thermal velocity

• When computing current, the drive forces (drift-diffusion) in the next chapter will be countered by these RG forces