The p-n diode

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Semiconductor Devices
A brief review
Dr. K. Fobelets
Purpose of the course
• Study bipolar devices in more detail
– Diodes and BJTs
– Closer to reality: recombination
– What causes the delays in these devices when
switching?
The most frequently used sentence in
this course will be:
Excess minority carrier
concentration
Structure
• 1. Lectures : 10 hrs
– Basic principles based on Q&A session
– Recombination and how does it impact the
characteristics
– LONG pn diode – correct and approximated
solutions
– LONG BJT
– Switching of pn diodes and BJTs
• 2. Classes: solving past exam papers
Review
• Electrons and holes
• Minority and majority carriers
• Energy band diagram
Free charged carriers in Si
Covalent bond
Si
Intrinsic Si
Thermal energy: kT
Movement: kT
Si Si Si Si
Si Si Si Si
Si Si Si Si
Si Si Si Si
Si Si Si Si
Si Si Si Si
Si Si Si Si
Si Si Si Si
Si Si Si Si
Extrinsic Si
Extrinsic Si
Si Si Si Si
Si Si Si Si
Si Si
Si As Si Si
Si Si Si Si
Si Si Si Si
NA
p-type
ND
n-type
Si B
Extrinsic Si
Obtained by doping
B
As
In semiconductors two types of free charged carriers exist: electrons and holes.
Q1: What are holes?
Si Si Si
+
Si Si Si Si
a) Spherical voids in a semiconductor
b) A positively charged Si atom that has lost its electron
c) A positively charged particle that is the result of quantum mechanics Si Si
Si Si Si
Si Si Si Si
Si Si
C
The two charged particles describe together the conduction in semiconductors.
Electron e- with charge q=-e
and mass mn = m0 m*n
Hole h+
with charge q=+e
and mass mp = m0 m*p
Intrinsic silicon (Si) has a small number of both free electrons and holes such that ni=pi.
In order to increase the free carrier concentration, the semiconductor can be doped. With
donors ND more electrons are created, with acceptors NA more holes are generated.
Q2: When intrinsic Si is doped with donor atoms, which of the following statements is
correct?
a)
b)
c)
d)
n = p = n i = pi
n > ni & p < ni
n > p > ni
p > n > ni
n: electron concentration
p: hole concentration
ni: intrinsic electron concentration
pi: intrinsic hole concentration
B
n > ni & p < ni in an n-type semiconductor.
n-type semiconductor
n = ND
p = ni2/ND
p-type semiconductor
n = ni2/NA
p = NA
By heart
The concept of majority carrier and minority carrier is important in semiconductor devices.
Majority carrier is the carrier type in a doped semiconductor with the highest
concentration. Minority carrier is the carrier type with the lowest concentration.
Q3: True or False?
The holes are the majority carriers in a p-type semiconductor (doped with acceptor atoms
NA).
TRUE
p-type semiconductor
p
p
>
n
p
p-type
p-type
semiconductor electron semiconductor
hole
concentration
concentration
n-type semiconductor
n
n
>
p
n
n-type
n-type
semiconductor hole
semiconductor
electron
concentration
concentration
MAJORITY CARRIERS
MINORITY CARRIERS
Drift and diffusion
• Two types of carrier movement
– As a result of an electric field → DRIFT
– As a result of a carrier gradient → DIFFUSION
Drift of carriers under influence of
an electric field: E
+
E
-
+
E
-
J  q  number of carriers  v
J  q  number of carriers   E
Diffusion of carriers due to a carrier
gradient
x
J  q  diffusion
J  qD
d
dx
constant  concentrat ion gradient
number of carriers
The purpose of semiconducting devices is to generate a current/voltage in response to an
applied voltage/current. Two different types of current can exist in a semiconductor: drift
and diffusion current. The expression of the total current that can flow in a semiconductor
is given by the drift-diffusion equation:
dn ( x )
J n ( x )  e  n n ( x ) E ( x )  eD n
dx
J p ( x )  e  p p ( x ) E ( x )  eD p
(1)
dp ( x )
dx
(2)
Q4: Which statement is true?
a)
b)
c)
d)
Term (1) is drift current and (2) diffusion current
Term (2) is drift current and (1) diffusion current
Only term (1) can exist in a semiconductor
Only term (2) can exist in a semiconductor
A
Drift current is proportional to the carrier concentration and the electric field
Diffusion current is proportional to the carrier gradient.
E(x)
n(x)
p(x)
Jndrift
Jpdrift
Jndiff
Jpdiff
Motion of free charged carriers in a semiconductor.
Q5: If a p-type semiconductor at room temperature is conducting carriers due to drift,
which of the following motion paths would be followed by the holes?
E
E
+
+
a)
c)
(b)
(d)
+
E
-
+
E
-
B
When carriers move in a semiconductor they are scattered along the
way. This means that they will be accelerated by the electric field (in
this case) and then interact with atoms, impurities, other carriers
that makes them lose some of their kinetic energy = scattering.
Therefore the carriers will travel with an average velocity in
amplitude and direction.
v   E
 
e
m
Q6: Solve diffusion processes
p+
1.
2.
n
p
Draw arrows indicating the direction of diffusion of carriers.
Identify the type of carriers that is diffusing.
Solution
p+
Holes
Electrons
n
p
p+
n
p
Q7: Why is there no net current while diffusion is happening?
1. Because hole diffusion and electron diffusion cancel each other.
2. Because an internal electric field is built up across each junction
causing drift of holes/electrons that cancel the diffusion of
.holes/electrons.
3. Because holes and electrons diffuse automatically back to where they
came from.
- E +
p+
Holes
+E n
p
Holes
diffusion
Electrons
drift
Electrons
2. Because an internal electric field is built up across each junction
causing drift of holes/electrons that cancel the diffusion of
.holes/electrons.
Depletion
+
Si Si Si Si Si
Si Si B Si Si
- Si Si Si Si
B
E
-
ND
p-Si
-
Capacitive effect
Capacitive effect
NA
-
B : boron atom ionised
n-Si
Si Si Si Si Si
Si
B
Si As+ Si Si Si
+ Si Si Si Si
As
E
+
+
As : arsenic atom ionised
As
Q8: True - False
Ec
EF
Ev
The position of the Fermi level EF determines the type of the
semiconductor.
Q9: Multiple choice
Ec
EF
Ev
1. This is the energy band diagram of an n-type semiconductor.
2. This is the energy band diagram of a p-type semiconductor.
3. This is the energy band diagram of an intrinsic semiconductor.
Ec
EF
Bottom of conduction band
EG Bandgap. No energy levels in this energy region.
Ei
Intrinsic “level”. Is the position of the Fermi level
EF when the semiconductor is intrinsic.
Ev
Top of valence band
Position of Fermi level is determined by the doping type and density
For n-type Si:
  E c  E F  
n  N C exp 



kT
N
 E  E F   N C
exp  c
 C

kT
n
ND


 N
E c  E F  k T ln  C
 ND




Devices
• A combination of n and p type
semiconductors plus ohmic contacts to
apply the external voltages/currents makes
devices
• When combining a-similar materials
diffusion will occur and as a result an
internal electric field will be built up to an
amount that opposes diffusion current.
Energy band diagram
e.g.
p-Si – n-Si
p-Si – n-Si – p-Si
It is possible to start from the knowledge on
workfunctions, f and the energy reference:
the vacuum level, Evac. The workfunction is
dependent on the doping concentration!
Evac
Evac
p-Si
e×fn-Si
n-Si
e×fp-Si
EF
EF
Evac
Evac
p-Si
e×fn-Si
n-Si
e×fp-Si
EF
EF
Depleted region on both sides
Evac
e  V 0  e  f p  Si  f n  Si 
Evac
p-Si
e×fp-Si
e×fn-Si
n-Si
Ec
EF
Ec
EF
Ev
Ev
Diffusion and drift can occur at the
same
time.
E
A charge packet
Both also always occur across junctions
A look at the short pn-diode
PN diode
I
p
p
n
n
E
V
p
n
Short PN diode
DIFFUSION
I
p
p
n
n
E
V
p
n
Short PN diode
DIFFUSION
I
p
p
n
n
E
V
p
n
Short PN diode
DIFFUSION
p
p
n
V
n
Minority carrier concentration
I
How do we find the current?
E
p
n
distance
Linear variation of minority carrier concentration
Apply diffusion current formula to the minority carrier variation
Short PN diode
I
p
p
n
n
V
E
p
n
Only few carriers can contribute to the current
Contents of course this year
• Long pn diode
– Introducing the concept of recombination of carriers.
– Switching of the pn diode, where does the delay come
from?
• Bipolar junction transistor
– Internal functioning
– Switching delays
But what happens in a long pn diode?
p
n
p
n
Short
Long
Ln
Lp
Minority carrier diffusion length
In long semiconductors recombination of
the minority carriers will occur whilst
diffusing
Excess holes,  p in an n-type semiconductor will recombine
with the large amount of available electrons.
Loss of both carrier type, but felt most in excess
minority carriers. Remember: the amount of majority carriers is
much larger than the excess.
In long semiconductors recombination of
the minority carriers will occur whilst
diffusing
Excess holes,  p in an n-type semiconductor will recombine
with the large amount of available electrons.
Loss of both carrier type, but felt most in excess
minority carriers. Remember: the amount of majority carriers is
Injection of carriers
much larger than the excess.
x
Lp
• Diffusing minority
carriers (e.g. holes)
recombine with majority
carriers (electrons) within
a diffusion length Lp
Generation-recombination
• Generation of carriers and recombination is
continuously happening at the same time
such that the equilibrium carrier
concentrations are maintained.
R=G
Charge neutral
Recombination - generation
• In case there is an excess carrier
concentration then the recombination rate R
of the excess, will be larger than its
generation rate, G: R>G
When there is a shortage, then G > R
Recombination - generation
• Simple model: Recombination/generation rate is proportional
to excess carrier concentration.
• Thus no net recombination/generation takes place if the
carrier density equals the thermal equilibrium value.
Recombination of e- in p-type
semiconductor
h+
Recombination of in n-type
semiconductor
U n  Rn  G n 
U
p
 Rp  Gp 
n p  n p0
n
p n  p n0

p

n p

n
p n

p
Diffusion, drift and recombination of
carriers
What is the consequence of this recombination on the
characteristics of the pn diode with neutral regions
larger than the diffusion lengths of the minority
carriers?
In the pn diode the carrier gradient
determines the current thus we have
to find the function p(x) of the
minority carrier concentration.
• Note, reasoning done for p(x). For n(x) analogous approach.
Mathematical description of diffusion
and recombination
Jp (x+Dx)
Jp(x)
A
x
x
p
t
Rate of hole
variation

x x  Dx
=
x+Dx
1 J p ( x)  J p ( x  Dx)
q
Dx

p

p
Variation of hole
Recombination
concentration in + rate
Dx x A/s
Mathematical description of diffusion
and recombination
p
t
Dx  0 :

1 J p ( x)  J p ( x  Dx)
Dx
x x  Dx
q
p ( x, t )
 1 J p
t

q
x


p

p
p

p
p  p 0   p = bulk defined + excess concentration
p
t

 1 J p
q
x
with

p

p
Jp : total current = drift + diffusion
Neglect drift current (no electric field applied)
2
p 0  p n0 
ni
ND
Mathematical description of diffusion
and recombination
J p ( x )   eD p
p
t

 1 J p
q
x

p
p
p  p0  p
p
t
2
 Dp
x
2

dx
p
p
= bulk defined+ excess concentration
 p
2
 Dp
 p
dp ( x )
x
2

p
p
with
2
p 0  p n0 
ni
ND
Solve equation in steady state
p
t
 p
0
2
x
2

p
p
D p

p
2
Lp
p
Boundary conditions:
General solution of
2nd
Diffusion length
contact
x  X n  p  0
0
x  0  p  D p
order differential equation:
p ( x ) 
Dp
Dp
 Xn
sinh 
 L
p

x X
sinh 
 L

p




n




Xn x
 x


 p ( x )  C 1 sinh
 C2 
L

 p

Too complicated
• Short approximation
• Long approximation
Xn << Lp
Xn >> Lp
p ( x ) 
LINEAR
Dp
 Xn
sinh 
 L
p

x X
sinh 
 L

p




n




EXPONENTIAL
Short semiconductor
• Xn ≤ Lp carriers do not have time to recombine (=∞) !
• Taking linear approximation.
pn(x)
pn(x)
p’n
pn(x)
Dp
pn(x)= pn0+ Dp (1–x/Xn)
NO recombination : variation
of the excess carrier
concentration linear
pn0
0
Xn
x
Contact imposes pn(Xn)=0
Diffusion and recombination
• Xn >> Lp carriers do have time to recombine (t<∞) !
• Taking exponential approximations
pn(x)
pn(x)
p’n
Dp
pn0
pn(x)
pn(x)=pn0+
0
Lp
Dp
 X
1  exp   n
 L
p

Xn x

 exp





 X
x 

  exp   n
 L 
 L
p 
p





 
When recombination occurs
and Xn >> Lp variation of the
excess carrier concentration is
exponential
Contact imposes pn(Xn)=0
pn still too complex for quick
calculations
• Take really extreme case
• Xn >>> Lp or Xn → ∞
pn(x)=
Dp
 X
1  exp   n
 L
p


 exp





 X
x 

  exp   n
 L 
 L
p 
p





 

x 


D p exp 
 L 
p 

Note: I and Q of both expressions of pn(x) for Xn → ∞ the same
I for pn(x)=

x 

D p exp  
 L 
p 

same as for linear approximation when Xn=Lp
Diffusion and recombination
• Xn >>> Lp carriers do have time to recombine (t<∞) !
• Taking exponential approximations
pn(x)
p’n
pn(x)=pn0+Dp e-x/Lp
pn0
pn(x)
Dp
When recombination occurs
and Xn → ∞ variation of the
excess carrier concentration is
exponential
0
Lp
∞ x
Imposes pn(Xn)=0
SHORT ↔ LONG
approximation
pn(x)
Lp=200 nm, Xn=20nm
Correct solution
Exponential solution
Linear solution
pn(x)
Lp=Xn=200nm
Boundary of short
Short
x
x
pn(x)
Lp=200 nm, Xn=400nm
pn(x)
Intermediate
x
Lp=200 nm, Xn=1000nm
Long
x
• Calculation of currents in pn diode with
neutral regions larger than the diffusion
length, using the long semiconductor
approximation
→
• Exponential variation of the excess minority
carrier concentration.
Carrier injections: forward bias
e-diff
p
• Carrier injection across
junction
n
h+diff
-wp 0 wn
 V 

n ' p  n p 0 exp 

 VT 
np(-x) pn(x)
n’p
p’n
np0
 V 

p ' n  p n 0 exp 

 VT 
• Creates minority carrier
concentration gradients
pn0
-x
x
np0=ni2/NA & pp=NA
pn0= ni2/ND & nn=ND
Carrier injections: reverse bias
e-drift
n
+
h drift
p
-wp
0
np(-x)
wn
pn(x)
np0
V
n ' ' p  n p 0 exp 
 V
 T




V
p ' ' n  p n 0 exp 
 V
 T




pn0
n’’p
-x
• Minority carriers are swept
across junction V<0
p’’n
x
• Small amount of minority
carriers → small current
Thus
e-diff
p
n
h+diff
-wp 0 wn
np(-x)
n’p
np0
-x
Dnp
pn(x)
Dpn
p’n
pn0
x
( x)
n p ( x )  D n p e
Ln
Dnp = np0 (eeV/kT -1)
(x)
p n ( x )  D p n e
Lp
Dpn = pn0 (eeV/kT -1)
Two methods to calculate current
-wp
np
0
wn
Slope
I
pn
Qn
Dnp
-x
0
Qp
Dpn
0
1. Gradient excess carrier concentration
2. Re-supply of recombined excess charge
x
x
1. Excess carrier concentration gradient
Maximum diffusion currents at the edges of the transition region
np
e-x
Slope
Dnp
0
-wp
In = e A Dn dnp/dx
= max @ x=0
pn
h+
Dpn
0
wn
x
Ip = -e A Dp dpn/dx
= max @ x=0
1. Excess carrier concentration gradient
Fill in expression for excess carrier concentration
e

 eV 
 n p (  x )  n p 0  exp 
  1   e
 kT 


In
 
max
diff
h+
( x)


 p n ( x )  p n 0  exp 
  1   e
k
T




Ln


 eV 
dn p 0  exp 
  1   e
 kT 


 eAD n
dx
 eV 
(x)
( x)


 eV 
  e
dp n 0  exp 

1

 kT 


Ln
Ip
    eAD
max
diff
Lp
p
 
max
diff
Dn 

 eV 
 exp 


1

Ln
 kT 


eAn
p0
In
Lp
dx
x0
In
( x)
x0
Ip
 
max
diff
eAp n 0 D p 

 eV 
 exp 

  1 
Lp
 kT 


Ip
Changing gradient!
→
Changing diffusion current density
p
Itot
n
Ip
In
Itot=In + Ip
Ip
In
In
 
Ip
  I
x
diff
x
drift
eAn
tot

 eV 
p0 D n 
 exp 
  e

1

Ln
 kT 


 In
 
x
diff
( x)
Ln
eAp n 0 D p 

 eV 
 exp 

  1   e
Lp
 kT 


Ip
 
In
  I
x
diff
x
drift
tot
 Ip
 
x
diff
(x)
Lp
2. Re-supply of recombined excess carriers
-wp
0
wn
np
-x
x
pn
Ip
np = Dnp e-(-x)/Ln
np0
I
Dnp
Qn
In
pn = Dpn e-(x)/Lp
Dpn
Qp
0
0
pn0
x
Excess carrier charge Q recombines every  seconds (carrier life time).
For steady state Q has to be re-supplied every  seconds → current
2. Re-supply of recombined excess carriers
Charge – minority carrier life time ratio
np
np = Dnp
pn
Ip
e-(-x)/Ln
Qn
np0
Dnp
In
pn = Dpn e-(x)/Lp
Dpn
Qp
-x
0
-wp
0
pn0
x
0
wn
Charge = area under excess carrier concentration: integrate
-∞ and + ∞ are the contacts: excess charge = 0!
0
Qn = -e A ∫-∞ np dx
In = Qn/n = e A Ln Dnp /n
∞
Qp = e A ∫0 pn dx
Ip = Qp/p = e A Lp Dpn /p
Total current
Same equation as short diode with
length exactly equal to the
minority carrier diffusion lengths
• I = Ip(0) + In(0) = e A (Dp pn0 /Lp + Dn np0/Ln )(eeV/kT -1)
• I = I0 (eeV/kT -1)
• With I0 = e A (Dp pn0/Lp + Dn np0/Ln)
Reverse bias current
SHORT ↔ LONG
approximation error on current calculation:
ratio of currents
10
9
Error on linear and
exponential approximation
same when Xn=Lp
8
Ireal/Iapprox
7
6
Ireal/Iexp
5
Ireal/Ilin
4
3
2
1
0
0
1
2
3
Xn/Lp
4
5
• Non-idealities in the pn diodes
Log(I)
ideal
real
c)
b)
a)
V
(a) Low voltage: low injection of carriers
Log(I)
ideal
real
a)
V
V
(c) High voltage: high injection of carriers
Log(I)
ideal
real
n’p ≈ pp
p’n ≈ nn
c)
V
I tot
  eV
nkT
 I s  e





 a) n=2
 1  b) n=1

 c) n=2
(d) Higher currents
Log(I)
ideal
real
d)
Current determined by resistance
V
Switching of p-n diodes
• When a p-n diode is forward biased, excess carrier
concentrations exists at both sides of the depletion region
edge.
• To switch the diode from forward to off or reverse bias,
this excess carrier concentration needs to be removed.
• The transients resulting from the time it takes to remove
the excess carriers will lead to the equivalent capacitance.
np
pn
p
-wp
0 wn
n
np
p
eh+
pn n
Switching off
i
on
Steady state snap shots
off
-wp 0 wn
0
t
p
To
this?
How
do we go from this:
-wp
p+pno
Excess carrier concentration
Dpn
pno
Off: NO current flows!!!
x
n
0
wn
Variation of the excess carrier concentration
as a function of time.
p(x,t)
p
t
 1 J p

q
x

p

p
Relationship for charge Qp
contact

eA
p ( x, t )
t
0
dQ p ( t )
dt
dQ p ( t )
dt

eA
e
 Ip 
dx  
eA
contact

e
Jp
contact
Q p (t )

p

J p
x
0
eA
e
Jp
0

contact
dx 

0
Q p (t )

p
eA
p

p
dx
Transient during switching off
Recombination term
i(t)= I + dQ/dt = Q/ + dQ/dt Charge depletion term (or buildup)
Excess charge due to charge injection at any instance of time
Average lifetime of minority carriers
For switch from on to off:
At t<0 → Ion=Ion (Von)
At t≥0 → Ioff = 0 (Voff = 0)
And at t=-0 Q(0)=Ion 
At t→∞ Q(∞)=0
t>0
0 = Q/ + dQ/dt
Q(t)=Ion e-t/
Since no current in “off”, charge has to disappear by
recombination!
Transient during switching off
variation of the excess carrier concentration as a function of time
Qp(t)=eA∫p(x,t)dx=Ippe-t/p
p
Dpn
Variation in time
gradient→ i≠0
i=0→gradient=0
t=0
x
A voltage, vd will exists across the diode as long as charge remains
p(x,t)=Dp(vd(t)) e-x/Lp
Revision
• When a pn diode switches, the excess minority
carrier concentration needs to change. The removal
of the excess minority carrier concentration causes
the delay in the pn diode.
• The variation of the excess carrier concentration as
a function of time given by:
i p (t ) 
Q p (t )

p

dQ p ( t )
dt
ON-OFF (open circuit)
take: p+n → Itot ≈ Ip
p+
vd
i p (t ) 
n
Q p (t )

Ip

dt
p

@ t  0 ; i p ( 0 )  I ON 
R
dQ p ( t )
V
R
@ t  0; i p ( 0 )  0
t=0
V
0
Q p (t )

p

dQ p ( t )
dt
t
Q p ( t )  I ON  p exp 

 p






Q p (0 )

p
OFF (open circuit) → ON
take: p+n → Itot ≈ Ip


@ t  0; i p (0 )  0; Q p (0 )  0


@ t  0 ; i p ( 0 )  I ON 
p+
vd
I ON 
n
Ip
Q p (t )
dQ p ( t )


p
  p I ON  Q p ( t )

dt
Q p ( t )   p I ON


ln Q p ( t )   p I ON
t=0
V

dt
dQ p ( t )
R
R
dQ p ( t )
p

i p (t ) 
V
Q p (t )



t
0
p
dt

integrate
p

t

p


ln Q p ( t )   p I ON  ln   p I ON  
 Q p ( t )   p I ON
ln 

  p I ON

t

p

t


p


t 
  I
Q p ( t )   p I ON   p I ON exp  
p ON
  
p 




 1  exp   t  
  

p 



dQ p ( t )
dt
Reverse recovery transient
Switch the diode from forward to reverse bias
ei
h+
np
on
Steady state snap shots
pn n
p
-wp 0 wn
0
e-
t
h+
off
To this?
How
do we go from this:
p Excess carrier concentration
-wp
0
Dpn
Dpn
0
x
Reverse bias current flows!!!
wn
Transients when switching to reverse bias
e(t)
i(t) R
E
e(t)
p
n
If≈E/R
If
I
t
Ir≈-E/R
-E
V
p
If → gradient≠0
Ir → gradient≠0
-Ir
i(t)
t
t
x
v(t)
t
-E
Storage delay time: tsd
If
i(t)
v(t)
t
-Ir
tsd
Time required for the stored charge to disappear
tsd = minority carrier ln(1 + If/Ir)
Calculate storage delay time: tsd
IF
i p (t ) 
i(t)
Q p (t )


dQ p ( t )
dt
p





@ t  0 ; i(0 )  I F  Q (0 )   p I F
v(t)
t

@ t  0 ; i(0 )   I R X
 Q ( 0 )   p I R !
@ t  t sd ; Q ( t sd )  0
-IR
tsd
0  t  t sd  i p ( t )   I R
 IR 
Q p (t )

p

dQ p ( t )
dt
Calculated storage delay time: tsd
 IR 

 dt
v(t)

t
p
t



dQ p ( t )
dt
p

dt
p

dQ p ( t )
dQ p ( t )
pIR
 Q p (t )





 ln  p I R  Q p ( t )
integrate
t
0
p
t
tsd


  p I R  Q p (t )
IF i(t)
-IR
Q p (t )

 ln  p I R  Q p ( t )  ln  p I R  Q p ( 0 )
p
t
exp 

 p





 I  Q p (t )
 p R

 pIR  pIF


Q p ( t )   p I R   p I R   p I F

t
exp 

 p





Calculated storage delay time: tsd
IF i(t)

Q p ( t )   p I R   p I R   p I F
v(t)
t  t sd
t

0   p I R   p I R   p I F

IR
t sd   p ln 
 I R  I F
-IR
tsd


t 


exp 
  
p 


 t
exp   sd
 
p






 I  I F
   p ln  R

 
IR





After: tsd
IF i(t)
Q p ( t sd )  0
vd  0
v(t)
t
Build-up of
depletion region
t bu  RC depl
-IR
vd   E
tsd
Small signal equivalent circuit
• Junction capacitance
• Diffusion capacitance
• Due to depletion region
• Due to charge storage effects
p
n
w
• Cj = e A/w
• w function of bias
→ C voltage variable capacitance
• Important in reverse bias
np
p
pn n
-wp 0 wn
• Cd = dQ/dV = d (I )/dV
= e/kT I 
• Important in forward bias
Equivalent conductances
• Diffusion conductance
• Series resistance rs
• gd = dI/dV = e/kT I0
≈ e/kT I
• Due to n and p region +
contact resistance
eeV/kT
• Vd = Vappl – rs I
• Slope of the current voltage
characteristic in forward
bias
rd
rs
Cj
Cd
Only linear circuit elements present
Large signal equivalent circuit
Rs
C Reverse bias: depletion capacitance
Forward bias: diffusion capacitance
Non-linear circuit elements present
Conclusions
• The characteristics in a pn diode are based
upon excess minority carrier diffusion.
– Excess carrier concentrations are being formed
by injection of carriers across the junction.
– The gradient of the excess minority carrier
concentration at the junction determines the
magnitude of the current.
– Delay times are due to the storage of excess
minority charge in the layers.
Revision
• When recombination is taken into account, the
excess minority carrier concentration reduces while
diffusing through the neutral regions of the diode.
• The variation of the excess carrier concentration is
then given by:
p
t
 p
2
 Dp
x
2

p

p
Lifetime of minority
carrier holes
Revision
• The steady state solution for the excess minority
carrier concentration is then:
p ( x ) 
Dp
 Xn
sinh 
 L
p

x X
sinh 
 L

p




n




• This is considered too complex for quick
calculations and approximations are used in the
case of a short or long neutral region.
Revision
• Short: Xn ≤ Lp
linear
pn(x)
pn(x)
p’n
pn(x)
Dp
pn(x)= pn0+ Dp (1–x/Xn)
pn0
0
Xn
x
Contact imposes pn(Xn)=0
Revision
• Long: Xn >>> Lp
exponential
pn(x)=pn0+
pn(x)
Dp
 X
1  exp   n
 L
p


 exp





 X
x 

  exp   n
 L 
 L
p 
p


Dp
pn0
pn(x)
p’n
0
pn(x)=pn0+Dp e-x/Lp
Lp
∞ x
Imposes pn(Xn)=0



 
Revision
• These approximation make some errors in the calculation
of the current and the charge stored in the neutral regions.
• However we will see that:
1. I and Q for simplified and non-simplified exponential variation
of pn(x) for Xn → ∞ is the same
2. I for pn(x)
when Xn=Lp

x 

= D p exp   
 Lp 
is same as for linear approximation
Errors on current
1400
Lp=20 nm
Current (a.u.)
1200
1000
Series1
Correct
Exponential
Series2
Linear
Series3
800
600
400
200
0
101
20
2
40
3
2004
Xn (nm)
Short = good approximation up to Xn = Lp
Long = good approximation up to Xn > 5 ×Lp
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