#### Transcript Document 7420715

```EE 5340
Semiconductor Device Theory
Lecture 14 - Fall 2003
Professor Ronald L. Carter
[email protected]
http://www.uta.edu/ronc
L 14 Oct 9
1
• A plot of
r  dV/d[ln(C)] vs. V has
slope = -1/M, and
intercept = VJ/M
• Forward der. of data gives ri’ =
dV/d(ln(C))=[Vi+1-Vi]/[ln(Ci+1)-ln(Ci)], at
Vi’ = [Vi+1+Vi]/2
L 14 Oct 9
2
• A plot of
r  dV/d[ln(C)] vs. V has
slope = -1/M, and
intercept = VJ/M
• Central der. of data gives ri’ =
dV/d(ln(C))=[Vi+1-Vi-1]/[ln(Ci+1)-ln(Ci-1)],
at Vi’ = [Vi+1-Vi-1]/2 (= Vi only if all DV
are equal.
L 14 Oct 9
3
• A plot of
r  dV/d[ln(C)] vs. V has
slope = -1/M, and
intercept = VJ/M
• Backward der. of data gives ri’ =
dV/d(ln(C))=[Vi-Vi-1]/[ln(Ci)-ln(Ci-1)], at
Vi’ = [Vi+Vi-1]/2
L 14 Oct 9
4
Choosing the data
range for r vs. V
1
M
(Cj0/Cj)^1/M
 Cj 


C 
 j0 

30
25
M = 1/2
20
M = 1/3
15
10
5
0
-10
L 14 Oct 9
-8
-6
-4
Va (Volts)
-2
0
2
5
Choosing the data
range for r vs. V
3
1
M
(Cj0/Cj)^1/M
 Cj 


C 
 j0 

y = -1.7632x + 1.0161
2
R = 0.9995
2
M = 1/2
M = 1/3
1
y = -1.0671x + 0.9592
2
R = 0.9907
0
-1.0
L 14 Oct 9
-0.5
0.0
Va (Volts)
0.5
6
Minority
hole
taken from
Shur**
p. 101.
L 14 Oct 9
7
Minority
electron
taken from
Shur**
p. 101.
L 14 Oct 9
8
from data
vs. that
used in
simulators
L 14 Oct 9
Minority
electron
taken
from
Shur**
p. 101.
9
The Continuity
Equation (cont.)
The Continuity Equations are thus

n dn 1
dn

   J n , where
 U , and
t dt q
dt

p dp 1
dp

   J p , where
 U , and
t dt q
dt
the U - values can be modeled by SRH.
L 14 Oct 9
10
Review of depletion
approximation
qVbi
EFp
Ec
EFn
EFi
Ev
-xpc -xp 0 xn
L 14 Oct 9
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
11
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
L 14 Oct 9

12
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
L 14 Oct 9
-xp
0
xn

Ev
Vt
xnc
1
x

13
Law of the junction: “Remember
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 
L 14 Oct 9
14
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
L 14 Oct 9
15
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

L 14 Oct 9


also ppnp
ni2
nno
e
xp
Vbi

Vt
Va
Vt
Va
 ni2e Vt
16
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
L 14 Oct 9
17
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 injections apply so that
np < ppo for -xpc < x < -xp, and
pn < nno for xn < x < xnc
L 14 Oct 9
18
Ideal Junction
Theory (cont.)
Apply the Continuity Eqn in CNR

p dp 1
0

   J p , x n  x   x nc
t dt q
and

n dn 1
0

   J n , - x pc  x   x p
t dt q
L 14 Oct 9
19
Ideal Junction
Theory (cont.)
dn
Since Ex  0 in the CNR, Jnx  qDn
dx
dp
and Jpx  qDp
giving
dx
d2 pn 
dx2
2
 
d np
dx
L 14 Oct 9
pn

 0, for xn  x  xnc , and
Dp p
2

np
Dn n
 0, for - xpc  x  xp
20
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)
L 14 Oct 9
21
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 
0.1
45 sec
2
1  7.7E  18Nim  4.5E  36Nim
1.E+13 1.E+14 1.E+15 1.E+16 1.E+17 1.E+18 1.E+19 1.E+20
L 14 Oct 9
Doping Concentration (cm^-3)
22
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


L 14 Oct 9
23

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
L 14 Oct 9
-xp
0
xn

Ev
Vt
xnc
1
x

24
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
L 14 Oct 9
-xp 0
xn
x
xnc
25
References
* Semiconductor Device Fundamentals, by Pierret,