Document 7420715
Download
Report
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
Project CommentsForward derivative
• 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
Project CommentsCentral derivative
• 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
Project CommentsBackward derivative
• 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
lifetimes,
taken from
Shur**
p. 101.
L 14 Oct 9
7
Minority
electron
lifetimes,
taken from
Shur**
p. 101.
L 14 Oct 9
8
Lifetimes
from data
vs. that
used in
simulators
L 14 Oct 9
Minority
electron
lifetimes,
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
2Vbi 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 expEFn 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
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
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
Steady State conditions
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 expEFn 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,
Addison-Wesley, 1996
** Physics of Semiconductor Devices, M. Shur,
Wiley.
L 14 Oct 9
26