EE 5340
Semiconductor Device Theory
Lecture 15 – Spring 2011
Professor Ronald L. Carter
ronc@uta.edu
http://www.uta.edu/ronc
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
2
Law of the junction: “Remember
to follow the minority carriers”
N N 


p


n
po
a
d   V ln
no .


Vbi  Vt ln

V
ln
t 
t 


 n2 
pno 
n

po


 i 
pno npo
 - Vbi 
,
Invert to get

 exp
ppo nno
 Vt 
pn np
 Va - Vbi 

and when Va  0,

 exp
pp nn
 Vt 
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Law of the
junction (cont.)
 Switched to non - eq. not'n for Va  0 .
 So pn  pno  pn , nn  nno  nn ,
and np  npo  np , pp  ppo  pp .
 Assume nn  pn and np  pp .
 Assume low - level injection 
pp  ppo  Na and nn  nno  Nd
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Law of the
junction (cont.)
 So for pn  ppe
 We have pn 
Va -Vbi
Vt
npo
nno
ppo e
and npo  nno e
Va
Vt

 the Law of the Junction

Va
pnnn x  ni2e Vt ,
n


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
also ppnp
ni2
nno
e
xp
Vbi

Vt
Va
Vt
Va
 ni2e Vt
5
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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6
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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7
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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8
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
2
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
np
Dn n
 0, for - xpc  x  xp
9
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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10
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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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
12
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
13
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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14

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

15
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
16
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

17
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

18
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 
qni2
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Dn
2 Dp
, and Jsp  qni
NaWp
NdWn
19
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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20
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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21
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
22
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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23
Charge distr in a (1sided) short diode
pn
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
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n
24
Charge distr in a 1sided short diode
pn p (x ,V +V)• Assume Quasi-static
n n a
pn(xn,Va)
charge distributions
• Q’p = +qpn(xn,Va)Wn/2
Q’p • Q’ =q(W/2) x
p
Q’p
{pn(xn,Va+V)
pn(xn,Va)}
x• Wn = xnc - xn (Va)
xn
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xnc
25
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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26
Evaluating the diode current density
Assumin g no gen/rec in DR, then
 Va Vt

iD Va   Jp x n   Jn  x p A  Js A e  1 


where Js  Jsn  Jsp with the definition s
Dn
Jsn  qn
cothWp L n ,
NaL n
2 Dp
Jsp  qni
cothWn L p 
NdL p
2
i
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27
General timeconstant
For all diodes, long or short, the conductance
 
gd VQ


 d Jn  Jp 
 dID 

 A
 gn  gp


 dVa  VQ
 dVa  VQ
There is always a characteristic time so that
dQp
dQn
pgp  Cp 
, and n gn  Cn 
, and the
dVa
dVa
total diode capacitance C  Cp  Cn
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General timeconstant (cont.)
For the short diode side, p  p,trans

Wn2
,
2Dp
and n  n,trans 
2
Wp
2Dn
, the
physical charge transit times. For the
long diode side, p  p0 and n  n0 ,
the respective min. carr. life - times.
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General timeconstant (cont.)
Practical diodes are usually one - sided
The effective transition time is the
1
1
1
average given by


and
F min  transit
Cd  gd F
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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  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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