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Recombination: Depletion Region, Bulk, Radiative, Auger, and Tunnelling Ch 140 Lecture Notes #13...
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Transcript of 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