EE 5340
Semiconductor Device Theory
Lecture 15 - Fall 2009
Professor Ronald L. Carter
ronc@uta.edu
http://www.uta.edu/ronc
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
2
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
3
References for Part A: 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.
1. Device Electronics for Integrated Circuits, 3rd ed., by Richard S. Muller, Theodore I. Kamins,
and Mansun Chan, John Wiley and Sons, New York, 2003.
2. Mark E. Law, E. Solley, M. Liang, and Dorothea E. Burk, “Self-Consistent Model of MinorityCarrier Lifetime, Diffusion Length, and Mobility, IEEE ELECTRON DEVICE LETTERS, VOL.
12, NO. 8, AUGUST 1991.
3. Note: This article is removed from the list and items 6 and 7 are added. 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.
4. 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.
5. 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.
6. D.B.M. Klaassen, “A Unified Mobility Model for Device Simulation – I. Model Equations and
Concentration Dependence”, Solid-State Electr. Vol. 35, pp. 953-959, 1992. See below.
7. D.B.M. Klaassen, “A Unified Mobility Model for Device Simulation – II. Temperature
Dependence of Carrier Mobility and Lifetime”, Solid-State Electr. Vol. 35, pp. 961-967, 1992.
Download at DbmK.pdf.
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4
Taken from Synopsys [1] manual
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5
Taken from Synopsys [1] Table 3-6.
Default … parameters –
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6
Part of a SPICE model for the Motorola
1N5233 Zener diode is shown in Table 1.
For purposes of this assignment, this means
that
1. IS may be interpreted as the multiplier of the
(exp(vD/NVt) – 1) term in the diffusion current.
2. The multiplier of the exp(vD/(NRVt)) term in the
recombination current may be interpreted as ISR.
3. The M value implies that this is essential a step
diode.
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7
Table 1. A SPICE model for the
Motorola 1N5233 diode
.model D1N5233
Is=629E-18
Rs=1.176
N=1
Xti=3
Eg=1.11
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Cjo=140p
M=.5369
Vj=.75
Isr=1.707n
Nr=2
BV = 6
8
Use the information given to make the best estimate of
the following:
1. Diode area.
2. Concentration of donors or acceptors on the lightly
doped side. Support your conclusion as to the type of
Si on the lightly doped side.
3. Concentration and type of the heavily doped side.
4. Estimate the value IKF might have. The multiplier of
the exp(vD/(2NVt)) term in the high level injection
current may be interpreted as √(IS×IKF).
5. Length of the charge neutral region on the lightly
doped side.
6. Show that the estimates are self-consistent for all
regions of diode operation – especially capacitance, BV,
recombination, and diffusion ranges.
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9
Injection
Conditions
 Va - Vbi 
 giving
 pno  pn  ppo exp
 Vt 
 Va -Vbi 
 -Vbi 




 pn  ppoe  Vt   pno , pno  ppoe  Vt  ,

 Va  
 so pn  pno exp   1, at x  xn
 Vt  


 Va  
 sim. np  npo exp   1, at x  xp
V



t
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10
Ideal Junction
Theory
•
•
•
•
•
Assumptions
Ex = 0 in the chg neutral reg. (CNR)
MB statistics are applicable
Neglect gen/rec in depl reg (DR)
Low level injection applies so that
np < ppo for -xpc < x < -xp, and
pn < nno for xn < x < xnc
Steady State conditions
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11
Ideal Junction Theory (cont.)
p n
In the steady state (static) case, 
 0 , and
t t
applying the Continuity Equation to the CNR

p dp 1
0

   Jp , x n  x  x nc , and
t dt q

n dn 1
0

   Jn , - x pc  x  x p
t dt q
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12
Ideal Junction
Theory (cont.)
dn
Since Ex  0 in the CNR, Jnx  qDn
dx
dp
and Jpx  qDp
giving
dx
d2 pn 
dx2
2
pn

 0, for xn  x  xnc , and
Dp p
 
d np
dx
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2

np
Dn n
 0, for - xpc  x  xp
13
Ideal Junction
Theory (cont.)
2
2
Define Ln  Dn n and Lp  Dp p . So
pn  x   Ae
x
Lp
 Be
x
np  x   Ce Ln  De
x
x
Lp
, xn  x  xnc
Ln , - x  x   x .
pc
p




pn xn  np  xp
Va Vt
with B.C.

 e
1,
pno
npo


and pn xnc   np  xpc  0, (contacts)
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14
Diffusion Length model
Diffusion Length, L (microns)
1000.0
electrons
holes
100.0
10.0
1.0
L = (D)1/2
Diffusion Coeff. is
Pierret* model
min 
45 sec
2
1  7.7E  18Nim  4.5E  36Nim
0.1
1.E+13 1.E+14 1.E+15 1.E+16 1.E+17 1.E+18 1.E+19 1.E+20
Doping Concentration (cm^-3)
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15
Excess minority
carrier distr fctn
For xn  x  xnc , Wn  xnc  xn ,



sinh xnc  x  Lp  Va V
 e t  1
pn  x   pno


sinh Wn Lp


and for - xpc  x  xp , Wp  xpc  xp ,



 


sinh x  xpc Ln  Va V
 e t  1
np  x   npo


sinh Wp Ln


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16

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
xnc
1
x

17
Carrier
Injection
ln(carrier conc)
ln Na
 Va V

t

np  xp  npo e
 1






~Va/Vt
ln Nd
 Va V

t
pn xn   pno  e
 1




ln ni
~Va/Vt
ln ni2/Nd
ln ni2/Na
-xpc
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-xp 0
xn
x
xnc
18
Minority carrier
currents
Jp  x  
dpn 
qDp dx ,
2
qni Dp

for xn  x  xnc


cosh xnc  x  Lp  Va V
 e t  1



NdLp
sinh Wn Lp


Jn  x   qDn

 
d np
dx


, for - xpc  x  xp
 


cosh x  xpc Ln  Va V
 e t  1



NaLn
sinh Wp Ln


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qni2Dn

19
Evaluating the
diode current
Assu min g no gen/rec in DR, then

 Va V
J  Jp xn   Jn  xp  Js  e t  1 




where Js  Jsn  Jsp with definitions

Jsn / sp 
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2
qni

Dn / p
Na / dLn / p

coth Wp / n Ln / p

20
Special cases for
the diode current
Long diode : Wn  Lp , or Wp  Ln
Jsn 
2
qni
Dn
2 Dp
, and Jsp  qni
NaLn
NdLp
Short diode : Wn  Lp , or Wp  Ln
Jsn 
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qni2
Dn
2 Dp
, and Jsp  qni
NaWp
NdWn
21
Ideal diode
equation
• Assumptions:
–
–
–
–
–
low-level injection
Maxwell Boltzman statistics
Depletion approximation
Neglect gen/rec effects in DR
Steady-state solution only
• Current dens, Jx = Js expd(Va/Vt)
– where expd(x) = [exp(x) -1]
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22
Ideal diode
equation (cont.)
• Js = Js,p + Js,n = hole curr + ele curr
Js,p = qni2Dp coth(Wn/Lp)/(NdLp)
= qni2Dp/(NdWn), Wn << Lp, “short”
= qni2Dp/(NdLp), Wn >> Lp, “long”
Js,n = qni2Dn coth(Wp/Ln)/(NaLn)
= qni2Dn/(NaWp), Wp << Ln, “short”
= qni2Dn/(NaLn), Wp >> Ln, “long”
Js,n << Js,p when Na >> Nd
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Diffnt’l, one-sided
diode conductance
Static (steadystate) diode I-V
characteristic
 Va 
ID  Is exp d 
 Vt 
ID
 dI D 
gd  

 dVa VQ
IQ
Va
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VQ
24
Diffnt’l, one-sided
diode cond. (cont.)
ID  JA  JsA exp dVa Vt   Is exp dVa Vt 


Is exp VQ Vt
 dID 
gd VQ  

. If Va  Vt ,

Vt
 dVa  VQ
 
 
then gd VQ 
IDQ
 
, where IDQ  ID VQ .
Vt
Vt
1
The diode resistance, rd VQ 

gd IDQ
 
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25
Charge distr in a (1sided) short diode
pn
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Wn = xnc- xn • Assume Nd << Na
• The sinh (see L10)
pn(xn)
excess minority
carrier distribution
Q’p
becomes linear for
Wn << Lp
pn(xn)=pn0expd(Va/Vt)
x
• Total chg = Q’p =
x
xnc
Q’p = qpn(xn)Wn/2
n
26
Charge distr in a 1sided short diode
pn p (x ,V +V) • Assume Quasin n a
pn(xn,Va)
• Q’p =
+qpn(xn,Va)Wn/2
Q’p
• Q’p =q(W/2) x
{pn(xn,Va+V)

p
(x
,V
)}
n
n
a
x
• Wn = xnc - xn (Va)
xnc
27
Q’p
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xn
static charge
distributions
Cap. of a (1-sided)
short diode (cont.)
Qp  Q'p A, A  diode area. Define Cd 
dQp
dVa

d  qApn0 Wn
 qApn (xn )Wn 




exp
d
V
V



a t 
2
2


 dVa 
IDQ Wn2 IDQ
When Va  Vt , Cd VQ 

transit .
Vt 2Dp
Vt
d
dVa
 
xnc
pn
Wn2
So, rd VQ Cd VQ  transit   q
dx 
2Dp
xn J p
   
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References
[1] Taurus Medici Medici User Guide Version A-2008.09, September
2008, ©SYNOPSYS Inc pg 3-306 – 3-315. This reference also
quotes [2] below.
[2] D.J Roulston, N.D. Arora and S. G Chamberlain, “Modeling and
Measurement of Minority-Carrier Versus Doping in Diffused Layers
of n+-p Silicon Diodes,” IEEE Trans, Electron Devices, Vol. ED-29,
pp. 284-291, Feb. 1982.
[3] Semiconductor Device Fundamentals , 2nd edition, by Robert F.
Pierret, Addison Wesley, New York, 1996.
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