PN Junction: Physical and Mathematical Description of Operation Dragica Vasileska Professor Arizona State University • • • • • PN Junctions – General Considerations Ideal Current-Voltage Characteristics Generation and Recombination Currents Breakdown Mechanisms AC-Analysis.
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Transcript PN Junction: Physical and Mathematical Description of Operation Dragica Vasileska Professor Arizona State University • • • • • PN Junctions – General Considerations Ideal Current-Voltage Characteristics Generation and Recombination Currents Breakdown Mechanisms AC-Analysis.
PN Junction: Physical and
Mathematical Description of Operation
Dragica Vasileska
Professor
Arizona State University
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PN Junctions – General Considerations
Ideal Current-Voltage Characteristics
Generation and Recombination Currents
Breakdown Mechanisms
AC-Analysis and Diode Switching
1. PN-junctions - General Consideration:
• PN-junction is a two terminal device.
• Based on the doping profile, PN-junctions can be
separated into two major categories:
- step junctions
- linearly-graded junctions
ND N A
ND N A
ax
n-side
p-side
Step junction
p-side
n-side
Linearly-graded junction
(A) Equilibrium analysis of step junctions
EC
qVbi
Ei
EF
EV
p-side
(x)
n-side
p p 0 ni expEi EF k BT
W
+
qND
-qNA -
x
V (x)
Emax
N AND
k BT p p 0 nn0
Vbi
ln
V
ln
T
n2
n2
q
i
i
(b) Majority- minority carrier
relationship:
Vbi
E (x)
xp
(a) Built-in voltage Vbi:
qVbi Ei EF p EF Ei n
nn0 ni exp EF Ei k BT
x
xn
x
pn0 p p 0 exp Vbi / VT
n p 0 nn0 exp Vbi / VT
(c) Depletion region width:
Solve 1D Poisson equation using depletion charge
approximation, subject to the following boundary conditions: V ( x p ) 0, V ( xn ) Vbi , E ( xn ) E ( x p ) 0
qN A
p-side: V p ( x)
x x p 2
2k s 0
qN D
n-side: Vn ( x)
xn x 2 Vbi
2k s 0
Use the continuity of the two solutions at x=0, and
charge neutrality, to obtain the expression for the depletion
region width W:
xn x p W
2k s 0 ( N A N D )Vbi
V p (0) Vn (0)
W
qN A N D
N A x p N D xn
(d) Maximum electric field:
The maximum electric field, which occurs at the
metallurgical junction, is given by:
Emax
dV
dx
x 0
qN A N DW
k s 0 ( N A N D )
(e) Carrier concentration variation:
15
N A N D 1015 cm 3
Wcalc 1.23 mm
Emax( DC ) 9.36 kV / cm
Concentration [cm-3]
10
13
10
11
10
-3
n [cm ]
-3
Emax( sim) 8.93 kV / cm
p [cm ]
9
10
7
10
5
10
0
0.5
1
1.5
2
2.5
Distance [mm]
3
3.5
(f) Analytical vs. numerical data
N A N D 1015 cm3
1.23 mm, Emax( DC ) 9.36 kV / cm, Emax( sim) 8.93 kV / cm
Wcalc
5x10
15
0
Electric field [kV/cm]
-3
(x)/q [cm ]
10
14
0
-5x10
14
-10
15
0
0.5
1
1.5
2
2.5
Distance [mm]
3
3.5
-2
-4
-6
-8
-10
0
0.5
1
1.5
2
2.5
Distance [mm]
3
3.5
N A 1016 cm3 , N D 1018 cm3
0.328 mm, Emax( DC ) 49.53 kV / cm, Emax( sim) 67 kV / cm
Wcalc
10
Electric field [kV/cm]
5x10
17
16
-3
(x)/q [cm ]
10
0
-5x10
-10
16
17
0.6
0.8
1
1.2
Distance [mm]
1.4
0
-10
-20
-30
-40
-50
-60
-70
0.6
0.8
1
1.2
Distance [mm]
1.4
(g) Depletion layer capacitance:
Consider a p+n, or one-sided junction, for which:
2ks 0 Vbi V
W
qN D
The depletion layer capacitance is calculated using:
dQc qN D dW
qN D ks 0
1 2(Vbi V )
C
2
dV
dV
2(Vbi V ) C
qN D ks 0
1 C2
Measurement setup:
1
slope
ND
Reverse
bias
vac ~
Forward bias
Vbi V
W
V
V
dW
(B) Equilibrium analysis of linearly-graded junction:
12k s 0 Vbi V
W
qa
1/ 3
(a) Depletion layer width:
(c) Maximum electric field: Emax
(d) Depletion layer capacitance:
qaW 2
8k s 0
1/ 3
C
12
V
V
bi
qaks202
Based on accurate numerical simulations, the depletion
layer capacitance can be more accurately calculated if Vbi
is replaced by the gradient voltage Vg:
a 2k s 0VT
2
Vg VT ln
3
3
8qni
(2) Ideal Current-Voltage Characteristics:
Assumptions:
• Abrupt depletion layer approximation
• Low-level injection injected minority carrier density
much smaller than the majority carrier density
• No generation-recombination within the space-charge
region (SCR)
(a) Depletion layer:
W
EC
qV
E Fn
E Fp
np ni2 exp V / VT
n p ( x p ) n p 0 exp V / VT
pn ( xn ) pn 0 exp V / VT
EV
xp
xn
(b) Quasi-neutral regions:
• Using minority carrier continuity equations, one arrives at
the following expressions for the excess hole and electron
densities in the quasi-neutral regions:
V / VT
pn ( x ) pn 0 ( e
V / VT
n p ( x ) n p 0 ( e
1)e
( x xn ) / L p
1)e
n p (x)
( x x p ) / Ln
pn (x )
Forward bias
Space-charge
region W
pn 0
n p0
xp
xn
x
Reverse bias
• Corresponding minority-carriers diffusion current densities
are:
diff
J p ( x)
diff
J n ( x)
qD p pn 0
qDn n p 0
Lp
Ln
V / VT
(e
V / VT
(e
1)e
1)e
( x xn ) / L p
( x x p ) / Ln
Shockley model
diff
diff
J tot J p ( xn ) J n ( x p )
diff
majority J p
majority J ndiff J ndrift
drift
Jp
J tot
diff
minority J p
minority J ndiff
xp
xn
No SCR generation/recombination
x
(c) Total current density:
• Total current equals the sum of the minority carrier diffusion currents defined at the edges of the SCR:
I tot
I
diff
diff
I p ( xn ) I n ( x p )
Ge Si GaAs
D p pn 0 Dn n p 0 V / V
e T 1
qA
L
L
p
n
• Reverse saturation current IS:
D p pn 0 Dn n p 0
Dp
D
2
n
qAni
I s qA
L
L N
L
L
N
p
n
n A
p D
V
(d) Origin of the current flow:
Reverse bias:
Forward bias:
W
EC
Ln
qVbi V
qV
E Fp
EC
E Fn
qV
qVbi V
E Fp
EV
E Fn
EV
Lp
W
Reverse saturation current is
due to minority carriers being
collected over a distance on the
order of the diffusion length.
(e) Majority carriers current:
• Consider a forward-biased diode under low-level injection
conditions:
Quasi-neutrality requires:
nn (x )
nn 0
nn ( x ) pn ( x )
This leads to:
pn (x )
pn 0
x
xn
diff
J n ( x)
Dn diff
J p ( x)
Dp
• Total hole current in the quasi-neutral regions:
tot
J p ( x)
diff
drift
J p ( x) J p ( x)
diff
J p ( x)
• Electron drift current in the quasi-neutral region:
diff
J n ( x)
J tot
Dn
diff
1
diff
1 J p ( x ), E ( x )
J n ( x)
D
qn( x )m n
p
J ndrift (x )
J tot
J ntot ( x ) J ndiff ( x ) J ndrift ( x )
diff
diff
J n ( x) J p ( x)
diff
J p (x )
x
J ndiff (x )
(f) Limitations of the Shockley model:
• The simplified Shockley model accurately describes IVcharacteristics of Ge diodes at low current densities.
• For Si and Ge diodes, one needs to take into account
several important non-ideal effects, such as:
Generation and recombination of carriers within the
depletion region.
Series resistance effects due to voltage drop in the
quasi-neutral regions.
Junction breakdown at large reverse biases due to tun-
neling and impact ionization effects.
(3) Generation and Recombination Currents
J scr
Continuity equation for holes:
p
1 J p
Gp Rp
t
q x
Steady-state and no light genera-
tion process: p t 0 , G p 0
• Space-charge region recombination current:
xn
xn
xp
xp
dJ p ( x ) J p ( xn ) J p ( x p ) q R p dx
xn
J scr q R p dx
xp
Reverse-bias conditions:
• Concentrations n and p are negligible in the depletion
region:
ni2
ni
E t Ei
Ei E t
n exp
R
, g p exp
p n1 n p1
g
k BT
k BT
Generation lifetime
• Space-charge region current is actually generation current:
J scr J gen
qniW
qniW
J gen
Vbi V
g
g
• Total reverse-saturation current:
V / VT
J Js e
1 J scr J s J gen
V VT
• Generation current dominates when ni is small, which is
always the case for Si and GaAs diodes.
I (log-scale)
AJ s
V (log-scale)
AJ gen
IV-characteristics
under reverse bias conditions
EC
E Fp
EV
E Fn
W
Generated carriers are
swept away from the
depletion region.
Forward-bias conditions:
• Concentrations n and p are large in the depletion region:
np
2 V / VT
ni e
2 V /V
ni e T 1
R
p n n1 n p p1
• Condition for maximum recombination rate:
V / 2VT
Recombination lifetime
n p ni e
V /V
ni2 e T
ni V / 2VT
Rmax
e
, rec p n
n p p n rec
• Estimate of the recombination current:
max
J scr
qniW V / 2VT
e
rec
• Exact expression for the recombination current:
J scr
qni V / 2VT
1
qN D 2Vbin V
e
,
VT
, Enp
rec
2
Enp
k s 0
• Corrections to the model:
J scr
qni V / mrVT
e
rec
• Total forward current:
V / VT
J Js e
qni V / mrVT
V / VT
1
e
J s,eff e
1
rec
ideality factor. Deviations of from unity represent
an important measure for the recombination current.
• Importance of recombination effects:
Low voltages, small ni recombination current dominates
Large voltages diffusion current dominates
log(I)
AJ scr
AJ
V
AJ d
(4) Breakdown Mechanisms
• Junction breakdown can be due to:
tunneling breakdown
avalanche breakdown
• One can determine which mechanism is responsible for the
breakdown based on the value of the breakdown voltage
VBD :
VBD < 4Eg/q tunneling breakdown
VBD > 6Eg/q avalanche breakdown
4Eg/q < VBD < 6Eg/q both tunneling and
avalanche mechanisms are responsible
Tunneling breakdown:
• Tunneling breakdown occurs in heavily-doped pnjunctions in which the depletion region width W is about
10 nm.
Zero-bias band diagram:
EF
EC
W
EV
Forward-bias band diagram:
EFn
EFp
EC
EV
W
• Tunneling current (obtained by
using WKB approximation):
Reverse-bias band diagram:
It
EF
p
EFn
EC
EV
* 3
2m q FcrVA
2 2 1/ 2
4 E g
4 2m* E 3 / 2
g
exp
3qFcr
Fcr average electric field in
the junction
• The critical voltage for
tunneling breakdown, VBR, is
estimated from:
I t (VBR ) 10 I S
• With T, Eg and It .
Avalanche breakdown:
• Most important mechanism in junction breakdown, i.e. it
imposes an upper limit on the reverse bias for most diodes.
• Impact ionization is characterized by ionization rates an and
ap, defined as probabilities for impact ionization per unit
length, i.e. how many electron-hole pairs have been
generated per particle per unit length:
Ei
ai exp
qlFcr
- Ei critical energy for impact ionization to occur
- Fcr critical electric field
- l mean-free path for carriers
Avalanche mechanism:
EF
p
EFn
EC
EV
Generation of the excess electron-hole
pairs is due to impact ionization.
Expanded view of the
depletion region
• Description of the avalanche process:
Jn
J p an J n dx
dx
J n an J n dx
Jp
Impact ionization initiated by electrons.
Jn
J p a p J p dx
dx
J n a p J p dx
Jp
Impact ionization initiated by holes.
dJ p
dJ n
0,
0
dx
dx
dJ p
dJ n
dx
dx
J J n J p const.
Multiplication factors for
electrons and holes:
J p (0)
J n (W )
Mn
, Mp
J n (0)
J p (W )
• Breakdown voltage voltage for which the multiplication
rates Mn and Mp become infinite. For this purpose, one
needs to express Mn and Mp in terms of an and ap:
x
an a p dx '
W
1
1
0
dJ n
a
e
dx
a
J
a
J
n
M
n n
p p
dx
0
n
dJ
x
an a p dx '
W
p an J n a p J p
1
dx
a pe 0
dx
1
M p 0
The breakdown condition does not depend on which
type of carrier initiated the process.
• Limiting cases:
(a) an=ap (semiconductor with equal ionization rates):
W
1
1
1 M a n dx M n W
0
n
1 a n dx
0
W
1
1
1
a p dx M p W
Mp 0
1 a p dx
0
(b) an>>ap (impact ionization dominated by one carrier):
W
an dx
Mn e 0
W
1 a n dx
0
Breakdown voltages:
(a) Step p+n-junction
p
• For one sided junction we can make
the following approximation:
W Wn Wp Wn
• Voltage drop across the depletion
region on the n-side:
n
Wp
Fmax
1
1
Vn FmaxWn VBD FmaxW
2
2
Wn
• Maximum electric field:
k s 0 2
qN DW
Fmax
VBD
Fmax
k s 0
2qN D
• Empirical expression for the
breakdown voltage VBD:
F (x)
x
VBD
Eg
60
1.1
3/ 2
ND
16
10
-3 / 4
kV
cm
(b) Step p+-n-n+ junction
• Extension of the n-layer large:
p
n
n
VBD
1
FmaxWm
2
• Extension of the n-layer small:
Wp
Fmax
1
1
VP FmaxWm F1 Wm W1
2
2
W1 Wm
F (x)
• Final expression for the punchthrough voltage VP:
F1
x
W1
W1
2
VP VBD
Wm Wm
• Doping-dependence of the breakdown voltage VBD:
Width of the n-layer W1 increases
One-sided abrupt
junction
Tunneling breakdown
Log-scale
VBD
p+-n-n+
Log-scale
ND
• Temperature dependence:
As temperature increases, lattice scattering increases which
makes impact ionization less probable. As a result of this,
the breakdown voltage increases.
(c) Plane vs. planar or cylindrical junction
• Plane junction:
Maximum electric field:
p+
n
• Planar junction:
Fmax
Except for surface effects, this is an
ideal junction.
Maximum electric field:
rj
p+
W
qN DW
Q
k s 0
k s 0
Fmax
n
qN DW
W
1
k s 0 2r j
The smaller the radius rj, the larger
the electric field crowding.
(5) AC-Analysis and Diode Switching
(a) Diffusion capacitance and small-signal equivalent
circuit
• This is capacitance related to the change of the minority
carriers. It is important (even becomes dominant) under
forward bias conditions.
• The diffusion capacitance is obtained from the device
impedance, and using the continuity equation for minority
2
carriers:
dp
d p
p
n
dt
Dp
n
dx
2
n
p
• Applied voltages, currents and solution for pn:
it
V (t ) V0 V1e , V1 V0
it
J (t ) J 0 J 1e , J 1 J 0
pn ( x, t ) pns ( x ) pn1 ( x )eit
• Equation for pn1(x):
d 2 pn1 1 i p
d 2 pn1 pn1 ( x )
pn1 ( x ) 0
2 0
2
2
Dp p
dx
dx
L p'
• Boundary conditions:
pn (, t ) pn 0 pn1 () 0
V0 V1eit
pn 0V1
V0
pn (0, t ) pn 0 exp
pn1 (0)
exp
V
VT
VT
T
• Final expression for pn1(x):
x
pn 0V1
V0
pn1 ( x, t )
exp exp
L
VT
VT
p'
• Small-signal hole current:
AqD p pn 0V1
dpn1
V0
I1 AqD p
1 i p exp YV1
dx x 0
L pVT
VT
• Low-frequency limit for the admittance Y:
Y
AqD p pn 0
L pVT
V0 1
exp 1 i p Gd iCdif
VT 2
V0 I s eV0 / VT
I
dI
Gd
exp
, I Forward current
L pVT
VT
VT dV
VT
V0 1 I
1 AqD p pn 0
Cdif
p exp
p
2 L pVT
VT 2 VT
AqD p pn 0
• RC-constant:
Rd Cdif
p
2
The characteristic time constant is on
the order of the minority carriers lifetime.
• Equivalent circuit model for forward bias:
Cdepl
Rs
Cdif
Rd
• Bias dependence:
Ls
1
Gd
C
Cdif
Cdepl
Va
(b) Diode switching
• For switching applications, the transition from forward bias
to reverse bias must be nearly abrupt and the transit time
short.
• Diode turn-on and turn-off characteristics can be obtained
from the solution of the continuity equations:
dpn
1
1 J p pn
1D
J p R p
dt
q
q x
p
dQ p
Qp
dQ p Q p
I p (t )
I (t ) I p (t )
dt
p
dt
p
Qp(t) = excess hole charge
Valid for p+n diode
Diode turn-on:
• For t<0, the switch is open, and
the excess hole charge is:
Q p (t 0) Q p (0 ) 0
• At t=0, the switch closes, and
we have the following boundary
condition:
p+
t=0
IF
Q p (0 ) Q p (0 ) 0
• Final expression for the excess hole charge:
Q p (t ) A Be
t / p
t / p
pIF 1 e
n
• Graphical representation:
Q p (t )
pn ( x, t )
pIF
Slope almost constant
t increasing
pn 0
x
t
• Steady state value for the bias across the diode:
Va / VT
pn ( x ) pn 0 e
IF
Va VT ln 1
IS
1 e
x / Lp
Va / VT
Q p Aqpn 0 L p e
1
Diode turn-off:
• For t<0, the switch is in position
1, and a steady-state situation is
established:
VF
IF
R
p+
t=0
1
• At t=0, the switch is moved to
position 2, and up until time t=t1
we have:
pn (0, t ) pn 0 Va 0
• The current through the diode
until time t1 is:
VR
IR
R
VF
R
2
VR
R
n
• To solve exactly this problem and find diode switching
time, is a rather difficult task. To simplify the problem, we
make the crucial assumption that IR remains constant even
beyond t1.
• The differential equation to be solved and the initial
condition are, thus, of the form:
IR
dQ p
dt
Qp
p
, Q p (0 ) Q p (0 ) p I F
• This gives the following final solution:
Q p (t ) p I R p I F I R e
t / p
• Diode switching time:
Q p (trr ) 0 trr
IF
p ln 1
IR
• Graphical representation:
Va (t )
pn ( x, t )
t
Slope almost
constant
t=0
pn 0
VR
t=ts
ttrr
ts switching time
trr reverse recovery time
IF
x
0.1I R
IR
ts
t rr
t