EE 4345 - Semiconductor
Electronics Design Project
Spring 2002 - Lecture 04
Professor Ronald L. Carter
ronc@uta.edu
http://www.uta.edu/ronc/
L 04 24Jan02
1
Practical
Junctions
• Junctions are formed by diffusion or
implantation into a uniform
concentration wafer. The profile can
be approximated by a step or linear
function in the region of the junction.
• If a step, then previous models OK.
• If not, 1/2 --> M, 1/3 < M < 1/2.
L 04 24Jan02
2
Law of the junction
(injection of minority carr.)
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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3
Carrier Injection and diff.
ln(carrier conc)
 Va V

t
pn xn   pno  e
 1




ln Na
ln Nd
 Va V

t

np  xp  npo e
 1






~Va/Vt
ln ni
~Va/Vt
ln ni2/Nd
ln ni2/Na
-xpc
L 04 24Jan02
-xp 0
xn
x
xnc
4
Ideal diode
equation
• I = Is [exp(Va/nVt)-1], Is = Isn + Isp
Long diode : Wn  Lp , or Wp  Ln
Dp
Dn
2
2
I sn  qni A
, and I sp  qni A
Na Ln
Nd Lp
Short diode : Wn  Lp , or Wp  Ln
Dp
Dn
2
2
I sn  qni A
, and I sp  qni A
NaWp
Nd Wn
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5
Diffnt’l, one-sided
diode conductance
Static (steadystate) diode I-V
characteristic
ID

 Va
 I s exp
 nVt

ID
 
  1
 
 dID 
gD  

dV
 a  VQ
IQ
Va
L 04 24Jan02
VQ
6
Diffnt’l, one-sided
diode cond. (cont.)
ID  JA  J s AexpVa nVt   1  I s expVa nVt   1
I s expVQ nVt
dID 
gd VQ  

. If Va  Vt ,

Vt
dVa V

 

Q
 
then gd VQ 
IDQ
 
, where IDQ  ID VQ .
nVt
nVt
1
The diode resistance, rd VQ 

gd IDQ
 
L 04 24Jan02
7
Charge distr in a (1sided) short diode
pn
Wn = xnc- xn • Assume Nd << Na
• The sinh (see L12)
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
8
Charge distr in a 1sided short diode
pn p (x ,V +V) • Assume Quasin n a
static charge
pn(xn,Va)
distributions
• Q’p = Q’p =
Q’p
qp
(x
)W
/2
n
n
n
Q’p
• dpn(xn) = (W/2)*
{pn(xn,Va+V)
L 04 24Jan02
x
x
xnc
- pn(xn,Va)}
9
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
   
L 04 24Jan02
10
Diode equivalent
circuit (small sig)
ID
IQ
dID
1
 gd  
dVD V
rd Vt
Q
 is the practical
“ideality factor”
IQ
VD
VQ
rdCd   , (  tr for short, min for long)
Cdiffusion 
L 04 24Jan02
IQ
Vt
, rdiff 
Vt
IQ
11
Small-signal eq
circuit
Va
Cdiff
rdiff
 Va
Cdepl  C j  C jo  1 
 Vbi
L 04 24Jan02



Cdepl
1 / 2
Cdiff and Cdepl
are both
charged by
Va = VQ
,Va  VQ
12
Reverse bias
junction breakdown
• Avalanche breakdown
– Electric field accelerates electrons to
sufficient energy to initiate
multiplication of impact ionization of
valence bonding electrons
– field dependence shown on next slide
• Heavily doped narrow junction will
allow tunneling - see Neamen*, p. 274
– Zener breakdown
L 04 24Jan02
13
Ecrit for reverse
breakdown (M&K**)
Taken from p. 198, M&K**
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14
Reverse bias
junction breakdown
• Assume -Va = VR >> Vbi, so Vbi-Va-->VR
• Since
Emax= 2(Vbi-Va)/W ,
when Emax = Ecrit
BV = e (Ecrit )2/(2qN-)
L 04 24Jan02
15
BV for reverse
breakdown (M&K**)
Taken from
Figure 4.13,
p. 198, M&K**
Breakdown
voltage of a
one-sided, plan,
silicon step
junction showing
the effect of
junction
curvature.4,5
L 04 24Jan02
16
References
* Semiconductor Physics and Devices,
2nd ed., by Neamen, Irwin, Boston,
1997.
**Device Electronics for Integrated
Circuits, 2nd ed., by Muller and
Kamins, John Wiley, New York, 1986.
L 04 24Jan02
17
Diode Switching
• Consider the charging and discharging
of a Pn diode
– (Na > Nd)
– Wd << Lp
– For t < 0, apply the Thevenin pair VF and
RF, so that in steady state
• IF = (VF - Va)/RF, VF >> Va , so current source
– For t > 0, apply VR and RR
• IR = (VR + Va)/RR, VR >> Va, so current source
L 04 24Jan02
18
Diode switching
(cont.)
 VF 
It  0   IF   
 RF 
F: t < 0
RF
Sw
VF,VR >>
Va
R: t > 0
+ VF
RR
+
VR
D
I t     Is 
L 04 24Jan02
for t  0,
VF  Va VF
IQ 

RF
RF
19
Diode charge
for t < 0
pn
pxn , VF   pno e VF / Vt , pno 
2
ni
ND
dp  IF
dp

, since Jp  qDp
dx qADp
dx
Q'p,diff  qpn xn , VF Wn  IF TR ,
pno
L 04 24Jan02
xn
xnc x
TR
WN2

2Dp
20
Diode charge for
t >>> 0 (long times)
pn

VF / Vt 


p xn , t  0  pno e
IS
dp
dp

, since Jp  qDp
 Js
dx qADp
dx
pno p x, t   
xn
L 04 24Jan02
xnc x
21
Equation
summary
dp
1

Jp,
dx R,t qDp
Is

, Is  JsA
AqDp
IF
VF
dp


, IF 
dx F,t  0 qADp
RF
For t small, but  0, a current, IR  VR / RR
flows to discharge Q
L 04 24Jan02
22
Snapshot for t
barely > 0
pn
pxn , t  0   pno e VF / Vt 
px, t  0  Total charge removed,
px, ts 
Qdis=IRt
dp  IF

dx qADp
IR
dp

dx qADp
pno
L 04 24Jan02
xn
xnc x
23
I(t) for diode
switching
ID
IF
- 0.1 IR
-IR
ts
ts+trr
t
Dp
2
Is  qni
Lp tanh Wn / Lp
IR
dp
For 0  t  ts ,

is a constant,
dx qADp
L 04 24Jan02
Qdischarge  IR ts


24
References
* Semiconductor Physics and Devices,
2nd ed., by Neamen, Irwin, Boston,
1997.
**Device Electronics for Integrated
Circuits, 2nd ed., by Muller and
Kamins, John Wiley, New York, 1986.
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25