Recombination processes
Electronic Materials and Devices
BECE201L
Module 3: Carrier Transport Mechanism
n0
Equillibrum state:
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Charge Carriers & their Transport
gT
 Charge Carriers Generation in Semiconductors
 Charge Carriers Recombination in Semiconductors
gT – rate of electron-hole pairs
thermal generation
rT
WV
p0
rT – rate of electron-hole pairs
thermal anihilation
gT = r T
Steady state
Dr. K. Govardhan,
School of Electronics Engineering,
VIT University
60
constant carrier concentrations
61
Recombination processes
n0 + Δn
gr
p0 + Δn
gT
n0 + Δn
Non-equillibrum state:
gT – rate of electron-hole pairs
thermal generation
gr – rate of electron-hole pairs
radiative generation
r – rate of electron-hole pairs
anihilation
Dr
h
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K
Recombination processes
r
WV
Steady state
WC
gT
p0 + Δn
gr + gT = r
constant carrier concentrations
r
WV
Non-equillibrum state:
gT – rate of electron-hole pairs
thermal generation
r – rate of electron-hole pairs
anihilation
Transient state
gT < r
vary carrier concentrations
R = r - gT
R – recombination rate
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Recombination processes
Recombination processes
 Phonon
n = n0 + n
dn
Δn

dt
τ
d(  n)
Δn
 dt
τ
n0
 - lifetime
 Auger (impact) Recombination – the whole energy
Wg is taken be third carrier, electron or hole, called
the hot carrier (RA , A)
n
 Radiative Recombination – the whole
taken be a new created photon that
crystal e.g. as the visible radiation (Rr , r)
n = n0 exp (-t/)

n(3) = 0.05n0
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R = Rph + RA + Rr
t
1


1
 ph

energy Wg is
can leave the
1
A

1
r
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Wc
B1
Wt
A2
Rf 
B2
Wv
τf 
Model constants:
 p0 ,  n0
- efficiency of B2 and A1
n1 , p1
- position of Wt inside Wg
τ f   p0
Wt – recombination centre
level
Dr
A1
Phonon recombination – SRH model
K
Phonon Recombination – SRH model
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energy
Wg
of
atoms
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R  -
Recombination
–
the
whole
is
taken
by
the
thermal
vibration
in the lattice, called phonons (Rph , ph)
(np - n 2i )
 p0 (n  n1 )   n0 (p  p1 )
● low injection level - Δn << max (n0, p0)
n-type
Δn
n  n1   n
p  p1  n
  p0 0
  n0 0
Rf
n 0  p 0  n
n 0  p 0  n
 W - Wt 
n 1  N c exp  c

 kT 
n 0  n1  n
p  p1  n
  n0 0
n 0  p 0  n
n 0  p 0  n
n0 >> max( p0,ni)
 W - Wv 
p1  N v exp t

 kT 
τ f   p0
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τ f   p0
n 0  n1
p  p1
  n0 0
n 0  p0
n 0  p0
usually n1 = p1 = ni
p-type
p0 >> max( n0,ni)
τ f   n0
Auger (impact) recombination
Phonon recombination – SRH model
n 0  n1  n
p  p1  n
  n0 0
n 0  p 0  n
n 0  p 0  n
● high injection level - Δn >> max (n0, p0, ni)
reeh = CAnn2p
CAn – Auger recombination constant
τ f   p0   n0

 p0   n0
 p0
WV
electron-hole-hole process
rehh =

1
10
n/n 0
CApnp2
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rehh Wg
WV
CAp – Auger recombination constant
for e-h-h process
hot hole
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Auger (impact) recombination
Steady-state impact electron-hole
pair generation and annihilation
in e-e-h processes
RAn = reeh0 – geeh0= 0
geeh0 = reeh0 = CAnn02p0
electron-hole-hole process
in steady state:
WC
reeh0
gehh0 = rehh0 = CApn0p02
RAn = reeh – geeh0 =
= CAnn2p – CAnn02p0
geeh0
WV
Recombination rate in the electronhole-hole process:
Steady-state impact electron-hole
pair generation and annihilation
in e-h-h processes
RAp = rehh0 – gehh0 = 0
RAp = rehh – gehh0 =
= CApnp2 – CApn0p02
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rehh0
hot electron
Recombination rate in the electronelectron-hole process:
Dr
electron-electron-hole process
in steady state:
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Auger (impact) recombination
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WC
reeh W
g
for e-e-h process
n type
0.1
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hot electron
electron-electron-hole process
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τ f   p0
gehh0
WV
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reeh W
g
geeh0
WV
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rehh Wg
gehh0
WV
hot hole
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Auger (impact) recombination

● General expresission:


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● General expresission:
R A  C An n 2 p - n 02 p 0  C Ap np 2 - n 0 p 02
● low injection level
n type
Auger (impact) recombination


 A   An 
n0=ND>>max(p0,Δn)
p0=NA>>max(n0,Δn)
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
 A   Ap 
 As
Δn>>max(n0,p0)
1
1

C An n 20 C An N 2D
1
1

C Ap p 20 C Ap N 2A

1

(C An  C Ap )(n) 2
1
1

C A (n) 2
CAn 2
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Radiative recombination
At the surface, the number of recombination centres responsible for phonon recombination
is larger than in the whole volume due to the larger number of defects and outside agents
interference.
n0 + Δn
WC
As result, in the border layer of semiconductor structure, the recombination rate, R(x)
increases (R2=var.) in comparison to its value inside the structure where usually it is
constant (R1=const.),
Dr
Relative emission efficiency
ν – frequency of emitted
electromagnetic waves
determining the colour of
emitted light
Surface recombination
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hν – photon – quant of radiative
energy
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
● high injection level
 A   As 
p type

R A  C An n 2 p - n 02 p 0  C Ap np 2 - n 0 p 02
InSb
hν = Wg
p0 + Δn
R1=const.
WV
R(x)
Radiative recombination gives almost
monochromatic radiation with the
colour depended on Wg
n0
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R2=var
Surface recombination
Surface recombination
The surface recombination rate is defined by the expression:
When the excess carriers, Δn and Δp, occur in the semiconductor structure, their
recombination is faster in the border layer than in the whole volume disrupting their
homogeneous distribution.
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Rs = s ns
As result, the diffusion current of electron-hole pairs from inside towards the border
layer appears.
ns – excess carrier concentration
at the surface [cm-3]
s – coefficient of surface
recombination [cm/s]
Since the electon-hole pair is electrically neutral, its move gives no electrical current,
lowering the inside excess carrier concentration only.
R1=const.
from 102 cm/s – for etched surface
to 104 cm/s – for a surface after
sanding
R(x)
n(x)
n = p
n0
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Coefficient s can change in wide range
depending on the surface state, e.g. in Ge:
R2=var
R(x) = const.
n(x)
Rs
n = p
Δns
n0
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Carrier recombination and
diffusion length
Recombination rate
K
• Minority carriers find themselves surrounded by very high concentration of
majority carriers and will readily recombine with them.
• The recombination rate is proportional to excess carrier density,
.
• By means of introducing excess carriers into an intrinsic s/c,
the number of majority carriers hardly changes, but the
number of minority carriers increases from a low- to highvalue.
Dr
d p
1
 
 p
dt
 p
• When we illuminate our sample (n-type silicon with 1015 cm-3 )
with light that produces 1014 cm-3 electron-hole pairs.
 t
 p(t )   p(0)exp   
 
Excess hole concentration when t=0
Lifetime of holes
• 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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p
Excess hole concentration decay exponentially with time.
Similarly, for electrons;
d n
1
 
 n
dt
n
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 t
 n(t )   n(0)exp   
 
Diffusion length, L
Excess Carrier Concentrations
When excess carriers are generated in a specimen, the minority carriers
diffuse a distance, a characteristic length, over which minority carriers can
diffuse before recombining majority carriers. This is called as a diffusion
length, L.
equilibrium values
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n  n  n0
p  p  p0
Excess minority carriers decay exponentially with diffusion distance.
 x 
 L 
p 


x 

 Ln 
 p( x)   p(0)exp  
 n( x)   n(0)exp  
Excess electron concentration when x=0
Charge neutrality condition:
n  p
Diffusion length for holes
Diffusion length for electrons
Ln  Dnn
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Lp  Dpp
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“Low-Level Injection”
Indirect Recombination Rate
K
Suppose excess carriers are introduced into an n-type Si sample
(e.g. by temporarily shining light onto it) at time t = 0.
How does p vary with time t > 0?
• Often the disturbance from equilibrium is small, such that the
majority-carrier concentration is not affected significantly:
Dr
– For an n-type material:
1.Consider the rate of hole recombination via traps:
| n || p | n0 so n  n0
– For a p-type material:
p
t R
| n || p | p0 so p  p0
2.Under low-level injection conditions, the hole generation rate is
not significantly affected:
However, the minority carrier concentration can be
significantly affected.
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 c p N T p
p
t G
83

p
t G  equilibrium
  pt
R  equilibrium
 c p N T p0
Minority Carrier (Recombination)
Lifetime
 p  c 1N
The net rate of change in p is therefore
p
t R G
p
t R G

p
t R

p
t G
  c p N T p  c p N T p0
p
1
c p NT
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Relaxation to Equilibrium State
Dr
K
Consider a semiconductor with no current flow in which
thermal equilibrium is disturbed by the sudden creation of
excess holes and electrons. The system will relax back to the
equilibrium state via the R-G mechanism:
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for electrons in p-type material
p
p

t
p
for holes in n-type material
n
T
 ranges from 1 ns to 1 ms in Si and depends on the density of
metallic impurities (contaminants) such as Au and Pt, and the
density of crystalline defects. These impurities/defects give rise
to localized energy states deep within the band gap. Such deep
traps capture electrons or holes to facilitate recombination and
are called recombination-generation centers.
85
n
n

t
n
T
The minority carrier lifetime  is the average time an excess
minority carrier “survives” in a sea of majority carriers
 c p N T ( p  p0 )   pp
where  p 
 n  c 1N
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np
t
 DN
 2 np
x 2

np
n
 GL
Numerical1 : A uniformly donor-doped silicon wafer maintained at room
temperature is suddenly illuminated with light at time t=0.
Assuming N D  1015 / cm3 ,  P  10 6 sec , and a light-induced creation of
electrons and holes per 1017 cm3  sec
throughout the semiconductor,
determine pn (t ) for t  0 .
pn
 pn pn
 DP

 GL
t
x 2
p
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Solution :
Step1 – Si, T=300K, N D  1015 / cm3, and
GL  1017 / cm3  sec
at all points inside the semiconductor.
Step2 -
ni  1010 / cm3 , N D  ni , n0  N D  1015 / cm3
p0  ni2 / N D  105 / cm3 with uniform doping, the
equilibrium
and
values are the same
everywhere throughout the semiconductor.
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pn is not function of position, the diffusion equation becomes an
ordinary differential equation and simplifies to
K
Step3 – Prior to t=0, equilibrium conditions prevail and pn  0
illumination -> pn increase.
excess carrier number ↑  thermal recombination rate ↑
∴ light generation and thermal recombination rates must
balance under steady state conditions, we can even state
Dr
The general solution is
GL  pn (t  ) /  P or pn (t  )  pn|max  GL P
if low-level injection prevails.
Step4 –
pn (t )  GL P  Ae t /  P
Applying the boundary condition yields
A  GL P
pn|max  GL P  1011 / cm3  n0  1015 / cm 3
and
pn
 2 p n p n
 DP

 GL
t
x 2
P
pn (t )  GL P (1  e t / P )  solution
the boundary condition
pn (t ) |t 0  0
90
dpn pn

 GL
dt
P
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Epilogue – Light output from the stroboscope can be modeled to first
order by the pulse train pictured below.
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Step5 – A plot of the solution equation is shown below.
Note that, pn (t ) starts from zero at t=0 and eventually
saturates at GL P after a few  P .
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pn (t )  GL P (1  e  t / P )
Dr
K
The upper equation approximately describes the carrier build-up
during a light pulse.
However, the stroboscope light pulses have a duration of t on  1 sec
compared to a minority carrier lifetime of  P  150  sec .
With t on /  P  1 , one sees only the very first portion of the Fig.
3.25 transient before the light is turned off and the semiconductor is
allowed to decay back to equilibrium.
Numerical 2: As shown below, the x=0 end of a uniformly doped semiinfinite bar of silicon with ND=1015 cm-3 is illuminated so as to create
pn0=1010 cm-3 excess holes at x=0. The wavelength of the illumination is
such that no light penetrates into the interior (x>0) of the bar. Determine
pn(x).
The light-off pn (t ) expression derivation closely parallels the lighton sample problem solution.
Exceptions -> pn at the beginning of the light-off = pn at the end
of the light-on.
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94
95
 So---Under steady state conditions with GL=0 for x > 0
 Solution: at x=0, pn(0)= pn0= 1010 cm-3 , and pn  0 as x  
 The light first creates excess carriers right at x=0
GL=0 for x>0
Diffusion and recombination
 As the diffusing holes move into the bar their numbers are reduced by
recombination
 Under steady state conditions it is reasonable to expect an excess
distribution of holes near x=0, with pn(x) monotonically decreasing from
pn0 at x=0 to pn0= 0 as x  
 Can we ignore the drift current? E  0 ? Yes
(pn max  ni )
because 1>Excess hole pile-up is very small , here
2> Also the majority carriers redistribute in such a way to partly cancel
the minority carrier charge. Thus use of diffusion eq. is justified.
pnx  0  pnx 0  pn0
for x  0
pnx  0
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d 2 pn pn

0
dx 2
p
 The general solution
pn ( x )  Ae  x / LP  Be x / LP
exp(x/Lp)   as x  
where
B0
 With x=0,
A  pn 0
pn ( x )  pn0 e  x / LP  solution
K
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Dr
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Dp
Lp  Dp p