EE 5340
Semiconductor Device Theory
Lecture 14 – Spring 2011
Professor Ronald L. Carter
ronc@uta.edu
http://www.uta.edu/ronc
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 “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 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/tp, (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/tn, (prop to exc min carr)
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Minority hole lifetimes
Mark E. Law, E. Solley,
M. Liang, and Dorothea
E. Burk, “SelfConsistent Model of
Minority-Carrier
Lifetime, Diffusion
Length, and Mobility,
IEEE ELECTRON
DEVICE LETTERS,
VOL. 12, NO. 8,
AUGUST 1991
The parameters used in
the fit are
τo = 10 μs,
Nref = 1×1017/cm2, and
CA = 1.8×10-31cm6/s.
τp 
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τo
1  ND Nref  τ oC AND2
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Minority electron lifetimes
Mark E. Law, E. Solley,
M. Liang, and Dorothea
E. Burk, “SelfConsistent Model of
Minority-Carrier
Lifetime, Diffusion
Length, and Mobility,
IEEE ELECTRON
DEVICE LETTERS,
VOL. 12, NO. 8,
AUGUST 1991
The parameters used in
the fit are
τo = 30 μs,
Nref = 1×1017/cm2, and
CA = 8.3×10-32 cm6/s.
τn 
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τo
1  ND Nref  τ oC AND2
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Minority Carrier Lifetime, Diffusion Length and
Mobility Models in Silicon
A. [40%] Write a review of the model equations for minority
carrier (both electrons in p-type and holes in n-type
material) lifetime, mobility and diffusion length in silicon.
Any references may be used. At a minimum the material
given in the following references should be used.
Based on the information in these resources, decide which
model formulae and parameters are the most accurate for
Dn and Ln for electrons in p-type material, and Dp and Lp
holes in n-type material.
B. [60%] This part of the assignment will be given by 10/12/09.
Current-voltage data will be given for a diode, and the
project will be to determine the material parameters (Nd,
Na, charge-neutral region width, etc.) of the diode.
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References for Part A
Device Electronics for Integrated Circuits, 3rd ed., by Richard S.
Muller, Theodore I. Kamins, and Mansun Chan, John Wiley and Sons,
New York, 2003.
Mark E. Law, E. Solley, M. Liang, and Dorothea E. Burk, “SelfConsistent Model of Minority-Carrier Lifetime, Diffusion Length,
and Mobility, IEEE ELECTRON DEVICE LETTERS, VOL. 12, NO. 8,
AUGUST 1991.
D.B.M. Klaassen; “A UNIFIED MOBILITY MODEL FOR DEVICE
SIMULATION”, Electron Devices Meeting, 1990. Technical Digest.,
International 9-12 Dec. 1990 Page(s):357 – 360.
David Roulston, Narain D. Arora, and Savvas G. Chamberlain “Modeling
and Measurement of Minority-Carrier Lifetime versus Doping in
Diffused Layers of n+-p Silicon Diodes”, IEEE TRANSACTIONS ON
ELECTRON DEVICES, VOL. ED-29, NO. 2, FEBRUARY 1982, pages
284-291.
M. S. Tyagi and R. Van Overstraeten, “Minority Carrier Recombination
in Heavily Doped Silicon”, Solid-State Electr. Vol. 26, pp. 577-597,
1983. Download a copy at Tyagi.pdf.
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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.)
dn dp
 The LHS,
or
 -U, 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 equations as :

p dp 1
dp
δp

   J p , where
 U   and
t dt q
dt
τp

n dn 1
dn
δn

   J n , where
 U  
t dt q
dt
τn
Which can be interprete d as :
Local rate of change 
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net generation rate  rate of inflow
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Review of depletion
approximation
EFp
qVbi
Ec
EFn
EFi
Ev
-xpc -xp 0 xn
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•
•
•
•
xnc x •
Depletion Approx.
pp << ppo, -xp < x < 0
nn << nno, 0 < x < xn
0 > Ex > -2Vbi/W,
in DR (-xp < x < xn)
pp=ppo=Na & np=npo=
ni2/Na, -xpc< x < -xp
nn=nno=Nd & pn=pno=
ni2/Nd, xn < x < xnc
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Review of
D. A. (cont.)
-xpc-xp
Ex
xn
xnc
2Vbi  Va 
W
, W  xp  xn ,
qNeff
x
Neff
NaNd

, Na xp  Ndxn ,
Na  Nd
Ex  0, x  xp
q
Ex  - Na x  xp , xp  x  0,

q
Ex  Na x  xn , 0  x  xn ,

Ex  0, x  xn

-Emax
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
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Forward Bias Energy Bands

nnon equil  ni expEFn  EFi  / kT   n p  n p 0 eVa Vt  1
q(Vbi-Va)
Imref, EFn
Ec
EFN
EFi
EFP qVa
Imref, EFp




pnon equil  ni exp EFi  EFp / kT  pn  pn 0 eVa
-xpc
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-xp
0
xn

Ev
Vt
1
x

xnc
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References
1 and M&KDevice
Electronics for Integrated
Circuits, 2 ed., by Muller and Kamins, Wiley,
New York, 1986. See Semiconductor Device
Fundamentals, by Pierret, Addison-Wesley,
1996, for another treatment of the m model.
2Physics of Semiconductor Devices, by S. M. Sze,
Wiley, New York, 1981.
3 and **Semiconductor Physics & Devices, 2nd ed.,
by Neamen, Irwin, Chicago, 1997.
Fundamentals of Semiconductor Theory and
Device Physics, by Shyh Wang, Prentice Hall,
1989.
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