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Example 5-3
Find an expression for the electron current in the n-type material of a forward-biased p-n
junction.
The total current is
 Dp
Dn  qV / kT

I  qA
p 
n (e
 1)
 L n L p
n
 p

The hole current on the n side is
Dp
x / L
I p ( xn )  qA
pn e n p (e qV / kT  1)
Lp
Thus the electron current in the n material is
 Dp
Dn  qV / kT
 xn / L p
I n ( xn )  I  I p ( xn )  qA
(1  e
) pn 
n p  (e
 1)
Ln 
 L p
This expression includes the supplying of electrons for recombinat ion with the
injected holes, and the injection of electrons across the junction into the p side.
Solid State Electronic Devices
3. 3. Reverse Bias
If V  Vr
 p negatively biased with respect to n
When Vr 
kT
q
pn  pn (e q ( Vr ) / kT  1)   pn
pn  qA
Dp
Lp
pn e
 xn / L p
 qA
Dp
Lp
(  pn )e
 xn / L p
Physically, extraction occurs because
minority carriers at the edges of the
depletion region are swept down the
barrier at the junction by the E field,
Fig. 18. Reverse-biased p-n junction: minority carrier
and holes in the n region diffuse toward
distributions near the reverse-biased junction
the junction.
Solid State Electronic Devices
Example 5-4
Consider a volume of n-type material of area A, with a length of one hole
diffusion length Lp. The rate of thermal generation of holes within the
volume is
ALp
pn
p
since gth   n   r nn pn 
2
r i
pn
p
Assume that each thermally generated hole diffuses out of the volume
before it can recombine. The resulting hole current is I=qALppn/τp, which is
the same as the saturation current for a p+-n junction. We conclude that
saturation current is due to the collection of minority carriers thermally
generated within a diffusion length of the junction.
Solid State Electronic Devices
4. Reverse-bias Breakdown
< Preface >
•
If the current is not limited externally, the junction can be damaged by excessive
reverse current, which overheats the device as the maximum power rating is
exceeded.
•
It is important to remember, however, that such destruction of the device is not
necessarily due to mechanisms unique to reverse breakdown.
•
The first mechanism, called the Zener effect, is operative at low voltages(up to a few
volts reverse bias).
•
The breakdown occurs at higher voltages(from a few volts to thousands of volts), the
mechanism is avalanche breakdown.
Solid State Electronic Devices
4. 1. Zener Breakdown
Heavily doped junction → High electric fields → Tunneling effect occurs
High electric field makes steep energy band, and reverse bias makes narrower width of barrier.
Fig. 20. The Zener effect: (a) heavily doped junction at equilibrium; (b) reverse bias with
electron tunneling from p to n; (c) I-V characteristic
Solid State Electronic Devices
4. 2. Avalanche Breakdown
nout  nin (1  P  P 2  P 3  ...)
M n  (Electron multiplica tion)

nout
 1  P  P 2  P3  
nin
1
1 P
1
M
1  (V / Vbr ) n

•
Lightly doping
•
Breakdown mechanism is the impact ionization of host atoms by energetic carriers.
Fig. 21. Electron-hole pairs created by impact ionization : (a) a single ionizing collision by an
incoming electron in the depletion region of the junction; (b) primary, and tertiary collisions
Solid State Electronic Devices
4. 2. Avalanche Breakdown
•
In general, the critical reverse
voltage for breakdown increases
with the band gap of the material,
since more energy is required for an
ionizing collision.
•
Vbr
decreases
as
the
doping
increases, as Fig. indicates.
Fig. 22. Variation of avalanche breakdown voltage in abrupt p+-n junctions, as a function of
donor concentration on the n side, for several semiconductors.
Solid State Electronic Devices
4. 3. Rectifiers
•
Most forward-biased diodes exhibit an offset voltage E0, which can be approximated in a
circuit model by a battery in series with the ideal diode and resister R.
Fig. 23. Piecewise-linear approximations of junction diode characteristics : (a) the ideal diode;
(b) ideal diode with an offset voltage; (c) ideal diode with an offset voltage and a resistance to
account for slope in the forward characteristic.
Solid State Electronic Devices
4. 3. Rectifiers
A short, lightly doped region → The reason of punch-through
It is possible for W to increase until it fills the entire length of this region.
→ The result of punch-through is a breakdown below the value of Vbr
Fig. 24. Beveled edge and guard ring to prevent edge breakdown under reverse bias : (a)
diode with beveled edge; (b) closeup view of edge, showing reduction of depletion region
near the bevel; (c) guard ring
Solid State Electronic Devices
4. 4. Breakdown Diode
•
It is designed for a specific breakdown voltage(higher doping). Such diodes are also called
Zener diodes(several hundred voltages).
•
It can be used as voltage regulators in circuits with varying inputs.
Fig. 26. A breakdown diode : (a) I-V characteristic; (b) application as a voltage regulator
Solid State Electronic Devices
5. Transient and A-C Conditions
< Preface >
• Since most solid state devices are used for switching or for processing a-c
signals, we cannot claim to understand p-n junctions without knowing at
least the basics of time dependent processes.
• In this section we investigate the important influence of excess carriers in
transient and a-c problems.
• The switching of a diode from its forward state to its reverse state is
analyzed to illustrate a typical transient problem.
Solid State Electronic Devices
5. 1. Time Variation of Stored Charge
Δx
Jp(x+Δx)
Diffusion and recombinat ion : The continuity equation
p
t
x  x  x
1 J p ( x)  J p ( x  x) p


q
x
p
Jp(x)
Rate of
increase of hole concentra-
recombination
Fig. 4-16
Hole buildup
tion in ΔxA per unit time
rate
We can obtain each component of the current
at position x and time t from above eq.
J p ( x, t )
p( x, t )
p( x, t )
We can integrate both sides at time t to obtain

q
q
x
p
t
x  p ( x, t )
p ( x, t ) 
J p (0)  J p ( x)  q  

 dx
0
t 
  p
Fig. 4-16. Current entering and leaving a volume ΔxA.
Solid State Electronic Devices
5. 1. Time Variation of Stored Charge
All hole current at p  /n, a long n region, xn  0.
Also, xn  , hole current is zero(0).
qA 
 
i (t )  i p ( xn  0, t ) 
p( xn , t )dxn  qA  p( xn , t )dxn

0
p
t 0
 i (t ) 
Q p (t )
p

dQ p (t )
dt
The hole current injected across the p  / n junction is
(1) Recombinat ion of excess carriers : Q p /  p
(2) Increasing or decreasing excess carriers :
dQ p
dt
(1)  (2) : providing minority carriers
Solid State Electronic Devices
5. 1. Time Variation of Stored Charge
Stored charges are
recombination with electrons
Carrier distributi on of stored charges after recombinat ion
Laplace Tranformat ion : i (t ) 
Q p (t )
p

dQ p (t )
dt
i (t  0)  0, Q p (0)  I p
0
1
p
Q p ( s )  sQ p ( s )  I p
Q p (s) 
I p
s 1/ p
 Q p (t )  I p e
t /  p
Fig. 27. Effects of a step turn-off transient in a p+-n diode: (a) current through the diode; (b)
decay of stored charge in the n-region; (c) excess hole distribution in the n-region as a
function of time during the transient.
Solid State Electronic Devices
5. 1. Time Variation of Stored Charge
The excess hole concentrat ion at xn  0 during the transien t,
pn (t )  pn (e qv(t ) / kT  1)
Finding pn will easily give us the transien t voltage .
Obtaining pn is not simple because hole distributi on does not remain in
the convenient exponentia l form it has in steady state.
 An approximat e solution for v(t ) can be obtained by assuming Quasi - steady state
p( xn , t )  pn (t )e
 xn / L p
We have for the stored charge at any instant,

Q p (t )  qA pn (t )e
0
 xn / L p
dxn  qALp pn (t ), pn (t )  pn (e
qv ( t ) / kT
 1) 
Q p (t )
qALp

kT  I p
t /  p
 v(t ) 
ln
e
 1

q  qALp pn

Solid State Electronic Devices
5. 2. Reverse Recovery Transient
= p(xn)-pn
t=0, p-n diode has forward-bias.
Ir=-E/R, when stored charges are
totally recombination.
It’s desirable that tsd is small
compared with the switching time.
Fig. 28. Stored delay time in a p+-n diode: (a) circuit and input square wave; (b) hole
distribution in the n-region as a function of time during the transient; (c) variation of current
and voltage with time; (d) sketch of transient current and voltage on the device I-V
characteristic
Solid State Electronic Devices
5. 2. Reverse Recovery Transient
tsd is storage delay tim e.
The critical parameter determinin g tsd is the carrier lifetime.
 1  I f
t sd  τ p erf 
I I

r
 f




2
Fig. 28. Effects of storage delay time on switching signal: (a) switching voltage; (b) diode current
Solid State Electronic Devices
Example 5-5
At the time t=0 the current is switched to –Ir at a forward biased p+-n diode.
Apply appropriate boundary condition and quasi-steady state approximation to find the tsd.
Q p (t )  I f  p e
From Eq. (5 - 47),
i (t ) 
Q p (t )
p

dQ p (t )
dt
for t  0, Q p  I f  p
 t / p
 I r p (e
 t / p
  p [  I r  ( I f  I r )e
t /  p
 1)
]
Assuming that Q p (t )  qAL p pn (t ) as in Eq. (5 - 52),
Using Laplace transfor ms,
Q p (s)
I
 r 
 sQ p ( s )  I f  p
s
p
Q p (s) 
I f p
s 1/ p

Ir
s(s  1 /  p )
pn (t ) 
p
qAL p
 I
r
 ( I f  I r )e
t /  p

This is set to equal zero when t  t sd , and we obtain :
 I 
 If 
t sd   p ln  r    p ln 1  
Ir 

 I f  I r 
Solid State Electronic Devices
5. 3. Switching Diodes
•
A diode with fast switching properties → either store very little charge in the
neutral regions for steady forward currents, or have a very short carrier lifetime, or
both.
The methods to improve the switching speed of a diode.
1. To add efficient recombination centers to the bulk material. For Si diodes, Au doping
is useful for this purpose. The carrier lifetime varies with the reciprocal of the
recombination center concentration.
2. To make the lightly doped neutral region shorter than a minority carrier diffusion
length. This is the narrow base diode. In this case the stored charge for forward
conduction is very small, since most of the injected carriers diffuse through the
lightly doped region to the end contact. → Very little time required to eliminate the
stored charge in the narrow neutral region.
Solid State Electronic Devices
5. 4. Capacitance of p-n Junctions
1) Junction capacitance : dominant under reverse bias
2) Charge storage capacitance : dominant under forward bias
Junction Capacitance
C
In equilibriu m,
W
2V0  N a  N d

q  Na Nd
dQ
dV



With bias,
W
2 V0  V   N a  N d

q
 Na Nd



Uncompen sated charge,
Q  qAxn 0 N d  qAx p 0 N a
 Nd Na 
Na
Nd
Nd Na

xn 0 
W , x p0 
W ,  Q  qA
W  A 2q V0  V 
Na  Nd
Na  Nd
Nd  Na
 Nd  Na 
Solid State Electronic Devices
5. 4. Capacitance of p-n Junctions
Voltage variatio n  barrier height change
Cj 
Nd Na
dQ
A
2 q

d (V0  V ) 2 (V0  V ) N d  N a
Voltage variable capacitanc e  V0  V 
1 / 2
Applicatio n : varactor
Using the form of the parallel plate capacitor formular
C j  A
Nd Na
q
A
dQ

same as C j 
2 (V0  V ) N d  N a W
dV
For p  /n junction, N a  N d , xn 0  W
Cj 
Applicatio n :
A 2 q
Nd
2 V0  V
p+
n
xp0
xn0
doping concentrat ion measuremen t via capacitanc e measuremen t
Solid State Electronic Devices
5. 4. Capacitance of p-n Junctions
Charge Storage Capacitance
Forward biased with a steady current
stored charge in the injected hole distributi on
Q p  I p  qApn L p  qALp pn e qV / kT
Capacitanc e due to small changes in this stored charge
q2
q
Cs 

ALp pn e qV / kT 
I p   p
dV
kT
kT
a - c conductanc e
dI qALp pn d qV / kT
q
Gs 

e

I
dV
 p dV
kT
dQ p


The charge storage capacitanc e limit switching effect for forward - biased p - n
junction in high - frequency circuit Cs  dQ / dV   good switching
Solid State Electronic Devices
5. 4. Capacitance of p-n Junctions
Fig. 30. Depletion capacitance of a junction: (a) p+-n junction showing variation of depletion
edge on n side with reverse bias. Electrically, the structure looks like a parallel plate capacitor
whose dielectric is the depletion region, and the plates are the space charge neutral regions;
(b) variation of depletion capacitance with reverse bias.
Solid State Electronic Devices
5. 4. Capacitance of p-n Junctions
Fig. 31. Diffusion capacitance in p-n junctions. (a) Steady-state minority carrier distribution for a
forward bias, V, and reduced forward bias, V-ΔV in a long diode; (b) minority carrier distributions in
a short diode; (c) diffusion capacitance as a function of forward bias in long and short diodes.
Solid State Electronic Devices