Semiconductor Device Modeling
and Characterization – EE5342
Lecture 8 – Spring 2011
Professor Ronald L. Carter
ronc@uta.edu
http://www.uta.edu/ronc/
First Assignment
• e-mail to listserv@listserv.uta.edu
– In the body of the message include
subscribe EE5342
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Second Assignment
• Submit a signed copy of the document
that is posted at
www.uta.edu/ee/COE%20Ethics%20Statement%20Fall%2007.pdf
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Additional University Closure
Means More Schedule Changes
• Plan to meet until noon some days in
the next few weeks. This way we will
make up for the lost time. The first
extended class will be Monday, 2/14.
• The MT changed to Friday 2/18
• The P1 test changed to Friday 3/11.
• The P2 test is still Wednesday 4/13
• The Final is still Wednesday 5/11.
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Shockley-ReadHall Recomb
Indirect, like Si, so
intermediate state
ET
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E
Ec
Ef
Efi
Ec
Ev
Ev
k
5
S-R-H trap
1
characteristics
• The Shockley-Read-Hall Theory
requires an intermediate “trap” site in
order to conserve both E and p
• If trap neutral when orbited (filled)
by an excess electron - “donor-like”
• Gives up electron with energy Ec - ET
• “Donor-like” trap which has given up
the extra electron is +q and “empty”
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S-R-H trap
char. (cont.)
• If trap neutral when orbited (filled)
by an excess hole - “acceptor-like”
• Gives up hole with energy ET - Ev
• “Acceptor-like” trap which has given
up the extra hole is -q and “empty”
• Balance of 4 processes of electron
capture/emission and hole capture/
emission gives the recomb rates
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S-R-H
recombination
• Recombination rate determined by:
Nt (trap conc.),
vth (thermal vel of the carriers),
sn (capture cross sect for electrons),
sp (capture cross sect for holes), with
tno = (Ntvthsn)-1, and
tpo = (Ntvthsn)-1, where sn~p(rBohr)2
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S-R-H
recomb. (cont.)
• In the special case where tno = tpo = to
the net recombination rate, U is
d p
dn
URG  

dt
dt
U
pn  ni2 

 ET  Efi  
  to
p  n  2ni cosh kT



where n  no  n, and p  po  p, (n  p)
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S-R-H “U” function
characteristics
• The numerator, (np-ni2) simplifies in
the case of extrinsic material at low
level injection (for equil., nopo = ni2)
• For n-type (no > n = p > po = ni2/no):
(np-ni2) = (no+n)(po+p)-ni2
= nopo - ni2 + nop + npo + np
~ nop (largest term)
• Similarly, for p-type, (np-ni2) ~ pon
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S-R-H “U” function
characteristics (cont)
• For n-type, as above, the denominator
= to{no+n+po+p+2nicosh[(Et-Ei)kT]},
simplifies to the smallest value for
Et~Ei, where the denom is tono, giving
U = p/to as the largest (fastest)
• For p-type, the same argument gives
U = n/to
• Rec rate, U, fixed by minority carrier
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S-R-H net recombination rate, U
• In the special case where tno = tpo = to
= (Ntvthso)-1 the net rec. rate, U is
d p
dn
URG  

dt
dt
U
pn  ni2 

 ET  Efi  
  to
p  n  2ni cosh kT



where n  no  n, and p  po  p, (n  p)
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S-R-H rec for
excess min carr
• For n-type low-level injection and net
excess minority carriers, (i.e., no > n
= p > po = ni2/no),
U = p/to, (prop to exc min carr)
• For p-type low-level injection and net
excess minority carriers, (i.e., po > n
= p > no = ni2/po),
U = n/to, (prop to exc min carr)
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Minority
hole
lifetimes.
Taken
from
Shur3,
(p.101).
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Minority
electron
lifetimes.
Taken
from
Shur3,
(p.101).
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Parameter example
•
tmin =
(45 msec)
1+(7.7E-18cm3Ni+(4.5E-36cm6Ni2
• For Nd = 1E17cm3, tp = 25 msec
– Why Nd and tp ?
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M. E. Law, E. Solley, M. Liang, and D. E. Burk, “Self-Consistent Model of
Minority-Carrier Lifetime, Diffusion Length, and Mobility,” IEEE Electron
Device Lett., vol. 12, pp. 401-403, 1991.
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M. E. Law, E. Solley, M. Liang, and D. E. Burk, “Self-Consistent Model of
Minority-Carrier Lifetime, Diffusion Length, and Mobility,” IEEE Electron
Device Lett., vol. 12, pp. 401-403, 1991.
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S-R-H rec for
deficient min carr
• If n < ni and p < pi, then the S-R-H net
recomb rate becomes (p < po, n < no):
U = R - G = - ni/(2t0cosh[(ET-Efi)/kT])
• And with the substitution that the
gen lifetime, tg = 2t0cosh[(ET-Efi)/kT],
and net gen rate U = R - G = - ni/tg
• The intrinsic concentration drives the
return to equilibrium
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The Continuity
Equation
• The chain rule for the total time
derivative dn/dt (the net generation
rate of electrons) gives
dn n n dx n dy n dz




.
dt t x dt y dt z dt
The definition of the gradient is
      
n  
i
j
k n,
x
y
z 

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The Continuity
Equation (cont.)
The definition of the vector velocity is
dx  dy  dz 
v 
i
j
k.
dt
dt
dt

 
Since A B  AxBx  AyBy  AzBz ,

dn n
then

 n  v
dt t
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The Continuity
Equation (cont.)
The gradient operator can be distributed



as   n v  n  v  n  v .
Considering the second term on the RHS,
 dx  dy  dz
 v 


 0, since
x dt y dt z dt

 dx d x

 0, etc.
x dt dt x
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The Continuity
Equation (cont.)
Consequently, since

Jn

 qn v , we have


n 1
dn n
   J n . So
  n v 

t q
dt t


dp p 1
dn n 1
   Jp

   J n , and

dt t q
dt t q
are the " Continuity Equations".
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The Continuity
Equation (cont.)
dp
dn
 The LHS,
or
 -V, of the Continuity Eq.
dt
dt
represents the Net Generation Rate of n
or p at a particular point in space (x, y, z).
n p
 The first term on the RHS,
or , is
t t
the " explicit" Local Rate of Change of n or
p at (x, y, z).
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The Continuity
Equation (cont.)

1
The second term on the RHS,    J n
q

1
or    J p is the local rate of n or p
q
concentrations flowing " out of" the
point (x, y, z). Note the difference in
signs for electrons (-q) and holes (  q).
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The Continuity
Equation (cont.)
So, we can re - write the continuity
equation for the holes as :

p dp 1

   Jp
t dt q
Which can be interpreted as :
Local rate of change 
net generation rate  rate of inflow
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References
*Fundamentals of Semiconductor Theory and Device
Physics, by Shyh Wang, Prentice Hall, 1989.
**Semiconductor Physics & Devices, by Donald A.
Neamen, 2nd ed., Irwin, Chicago.
M&K = Device Electronics for Integrated Circuits,
3rd ed., by Richard S. Muller, Theodore I. Kamins, and
Mansun Chan, John Wiley and Sons, New York, 2003.
• 1Device Electronics for Integrated Circuits, 2 ed., by
Muller and Kamins, Wiley, New York, 1986.
• 2Physics of Semiconductor Devices, by S. M. Sze,
Wiley, New York, 1981.
• 3 Physics of Semiconductor Devices, Shur, PrenticeHall, 1990.
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