EEE 531: Semiconductor Device Theory I – Dragica Vasileska PN Junctions Theory Dragica Vasileska Department of Electrical Engineering Arizona State University 1. PN-Junctions: Introduction to some general concepts 2. Current-Voltage Characteristics of an Ideal PN-junction (Shockley model) 3. Non-Idealities in PN-Junctions 4. AC Analysis and Diode Switching
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EEE 531: Semiconductor Device Theory I – Dragica Vasileska
PN Junctions TheoryDragica Vasileska
Department of Electrical EngineeringArizona State University
1. PN-Junctions: Introduction to some general concepts
2. Current-Voltage Characteristics of an Ideal PN-junction (Shockley model)
3. Non-Idealities in PN-Junctions4. AC Analysis and Diode Switching
EEE 531: Semiconductor Device Theory I – Dragica Vasileska
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
p-side n-side
AD NN
p-side n-side
AD NN ax
Step junction Linearly-graded junction
EEE 531: Semiconductor Device Theory I – Dragica Vasileska
(A) Equilibrium analysis of step junctions
(a) Built-in voltage Vbi:
(b) Majority- minority carrier relationship:
CE
VE
iEFE
biqV
W)(x
x-qNA
qND
)(xV
x
biV
)(xE
xmaxE
px nx
+-
p-side n-side
2200
0
0
lnln
expexp
i
DAT
i
npBbi
BFiip
BiFin
niFpFibi
nNNV
n
npqTkV
TkEEnpTkEEnn
EEEEqV
Tbinp
Tbipn
VVnnVVpp
/exp/exp
00
00
EEE 531: Semiconductor Device Theory I – Dragica Vasileska
(c) Depletion region width: Solve 1D Poisson equation using depletion charge approximation, subject to the following boundary condi-tions:
p-side:
n-side:
Use the continuity of the two solutions at x=0, and charge neutrality, to obtain the expression for the depletion region width W:
0)()(,)(,0)( pnbinp xExEVxVxV
202
)( ps
Ap xx
kqNxV
bins
Dn Vxx
kqNxV
2
02)(
DA
biDAs
nDpA
np
pn
NqNVNNkW
xNxNVV
Wxx)(2)0()0( 0
EEE 531: Semiconductor Device Theory I – Dragica Vasileska
(d) Maximum electric field:The maximum electric field, which occurs at the metallurgical junction, is given by:
(e) Carrier concentration variation:
)(00max
DAs
DA
x NNkWNqN
dxdVE
105
107
109
1011
1013
1015
0 0.5 1 1.5 2 2.5 3 3.5
n [cm-3]
p [cm-3]
Con
cent
ratio
n [c
m-3
]
Distance [mm]
cmkVEcmkVE
mWcmNN
sim
DC
calc
DA
/93.8/36.9
23.110
)max(
)max(
315
m
EEE 531: Semiconductor Device Theory I – Dragica Vasileska
cmkVEcmkVEmWcmNN
simDCcalc
DA/93.8,/36.9,23.1
10
)max()max(
315
m
-1015
-5x1014
0
5x1014
1015
0 0.5 1 1.5 2 2.5 3 3.5
(x)
/q [
cm-3
]
Distance [mm]
-10
-8
-6
-4
-2
0
0 0.5 1 1.5 2 2.5 3 3.5E
lect
ric fi
eld
[kV
/cm
]Distance [mm]
EEE 531: Semiconductor Device Theory I – Dragica Vasileska
cmkVEcmkVEmWcmNcmN
simDCcalc
DA/67,/53.49,328.0
10,10
)max()max(
318316
m
-1017
-5x1016
0
5x1016
1017
0.6 0.8 1 1.2 1.4
(x)
/q [
cm-3
]
Distance [mm]
-70
-60
-50
-40
-30
-20
-10
0
10
0.6 0.8 1 1.2 1.4E
lect
ric fi
eld
[kV
/cm
]Distance [mm]
EEE 531: Semiconductor Device Theory I – Dragica Vasileska
(f) Depletion layer capacitance: Consider a p+n, or one-sided junction, for which:
The depletion layer capacitance is calculated using:
D
bisqN
VVkW 02
02
0 )(21)(2
sD
bi
bi
sDDckqN
VVCVV
kqNdV
dWqNdVdQC
VVbi V
21 C
DNslope 1
Forward biasReverse bias
Measurement setup:
WdW
~
V
vac
EEE 531: Semiconductor Device Theory I – Dragica Vasileska
(B) Equilibrium analysis of linearly-graded junction:
(a) Depletion layer width:
(c) Maximum electric field:
(d) Depletion layer capacitance:
Based on accurate numerical simulations, the depletion layer capacitance can be more accurately calculated if Vbi is replaced by the gradient voltage Vg:
3/1012
qaVVkW bis
3/120
2
12
VV
qakCbi
s
0
2
max 8
skqaWE
30
2
8ln
32
i
TsTg
qnVkaVV
EEE 531: Semiconductor Device Theory I – Dragica Vasileska
much smaller than the majority carrier density• No generation-recombination within the space-charge
region (SCR)
(a) Depletion layer:
CE
VE
FnEqV
FpE
W
Tnnn
Tppp
Ti
VVpxp
VVnxnVVnnp
/exp)(
/exp)(/exp
0
0
2
px nx
EEE 531: Semiconductor Device Theory I – Dragica Vasileska
(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:
npT
pnT
LxxVVpp
LxxVVnn
eenxn
eepxp/)(/
0
/)(/0
)1()(
)1()(
)(xnp )(xpn
0np0pn
px nx x
Forward bias
Reverse bias
Space-chargeregion W
EEE 531: Semiconductor Device Theory I – Dragica Vasileska
• Corresponding minority-carriers diffusion current densities are:
npT
pnT
LxxVV
n
pndiffn
LxxVV
p
npdiffp
eeL
nqDxJ
eeL
pqDxJ
/)(/0
/)(/0
)1()(
)1()(
px nx x
diffpJ minoritydiff
nJ minority
)()( pdiffnn
diffptot xJxJJ
No SCR generation/recombination
driftn
diffn JJ majority drift
pdiffp JJ majority
totJ
Shockley model
EEE 531: Semiconductor Device Theory I – Dragica Vasileska
(c) Total current density:• Total current equals the sum of the minority carrier diffu-
sion currents defined at the edges of the SCR:
• Reverse saturation current IS:
1)()(
/00
TVV
n
pn
p
np
pdiffnn
diffptot
eLnD
LpD
qA
xIxII
An
n
Dp
pi
n
pn
p
nps NL
DNL
DqAn
LnD
LpD
qAI 200
I
V
Ge Si GaAs
EEE 531: Semiconductor Device Theory I – Dragica Vasileska
(d) Origin of the current flow:
CE
VE
FnEqV
FpE
W
Forward bias:
CE
VEFnE
qV
FpE
W
Reverse bias:
VVq bi VVq bi
Reverse saturation current is due to minority carriers being collected over a distance on the order of the diffusion length.
Ln
Lp
EEE 531: Semiconductor Device Theory I – Dragica Vasileska
(e) Majority carriers current:• Consider a forward-biased diode under low-level injection
conditions:
• Total hole current in the quasi-neutral regions:
)(xpn0np
nx x
0nn)(xnn
Quasi-neutrality requires:
This leads to:
)()( xpxn nn
)()( xJDD
xJ diffp
p
ndiffn
)()()()( xJxJxJxJ diffp
driftp
diffp
totp
EEE 531: Semiconductor Device Theory I – Dragica Vasileska
• Electron drift current in the quasi-neutral region:
)()(
1)(),(1)( xJxqn
xExJDD
JxJ diffn
n
diffp
p
ntot
diffn m
x)(xJ diff
p
)(xJ diffn
)(xJ driftn
)()()( xJxJxJ driftn
diffn
totn
totJ
)()( xJxJ diffp
diffn
EEE 531: Semiconductor Device Theory I – Dragica Vasileska
(f) Limitations of the Shockley model:• The simplified Shockley model accurately describes IV-
characteristics 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.
EEE 531: Semiconductor Device Theory I – Dragica Vasileska
3. Non-Idealities in PN-junctions:(A) Generation and Recombination Currents
Continuity equation for holes:
Steady-state and no light genera-
tion process:
• Space-charge region recombination current:
scrJ
ppp RG
xJ
qtp
1
0,0 pGtp
n
p
n
p
n
px
xpscr
x
xpppnp
x
xp
dxRqJ
dxRqxJxJxdJ )()()(
EEE 531: Semiconductor Device Theory I – Dragica Vasileska
Reverse-bias conditions:• Concentrations n and p are negligible in the depletion
region:
• Space-charge region current is actually generation current:
• Total reverse-saturation current:
TkEE
TkEEn
pnn
RB
tin
B
itpg
g
i
np
i expexp,11
2
Generation lifetime
VVWqn
JWqn
JJ big
igen
g
igenscr
gensVVscrVV
s JJJeJJT
T
1/
EEE 531: Semiconductor Device Theory I – Dragica Vasileska
• Generation current dominates when ni is small, which is always the case for Si and GaAs diodes.
I (log-scale)
V (log-scale)sAJ
genAJ
CE
VEFnE
FpE
W
IV-characteristicsunder reverse bias conditions
Generated carriers are swept away from the
depletion region.
EEE 531: Semiconductor Device Theory I – Dragica Vasileska
Forward-bias conditions:• Concentrations n and p are large in the depletion region:
• Condition for maximum recombination rate:
• Estimate of the recombination current:
11
/2/2 1
ppnnen
Rennpnp
VViVV
i
TT
nprecVV
rec
i
np
VVi
VVi
TT
T
en
pnen
R
enpn
,2//2
max
2/
TVV
rec
iscr e
WqnJ 2/max
Recombination lifetime
EEE 531: Semiconductor Device Theory I – Dragica Vasileska
• Exact expression for the recombination current:
• Corrections to the model:
• Total forward current:
ideality factor. Deviations of from unity represent an important measure for the recombination current.
0
2/ 2,12
,
s
binDnp
npT
VV
rec
iscr k
VVqNEE
VeqnJ T
TrVmV
rec
iscr e
qnJ /
11 /,
//
TTrT VVeffs
VmV
rec
iVVs eJe
qneJJ
EEE 531: Semiconductor Device Theory I – Dragica Vasileska
• Importance of recombination effects:
Low voltages, small ni recombination current dominates
Large voltages diffusion current dominates
log(I)
VdAJ
scrAJAJ
EEE 531: Semiconductor Device Theory I – Dragica Vasileska
(B) 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
EEE 531: Semiconductor Device Theory I – Dragica Vasileska
Tunneling breakdown:• Tunneling breakdown occurs in heavily-doped pn-
junctions in which the depletion region width W is about 10 nm.
W
EF
EC
EV
Zero-bias band diagram: Forward-bias band diagram:
W
EFn
EC
EV
EFp
EEE 531: Semiconductor Device Theory I – Dragica Vasileska
EFnEC
EV
EFp
Reverse-bias band diagram: • Tunneling current (obtained by using WKB approximation):
Fcr average electric field in the junction
• The critical voltage for tunneling breakdown, VBR, is estimated from:
• With T, Eg and It .
cr
g
g
crt qF
Em
E
VAFqmI
324
exp4
22/3*
2/122
3*
SBRt IVI 10)(
EEE 531: Semiconductor Device Theory I – Dragica Vasileska
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 n and p, 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 critical energy for impact ionization to occur- Fcr critical electric field- mean-free path for carriers
cr
ii Fq
Eexp
EEE 531: Semiconductor Device Theory I – Dragica Vasileska
EFnEC
EV
EFp
Expanded view of thedepletion region
Avalanche mechanism:
Generation of the excess electron-hole pairs is due to impact ionization.
EEE 531: Semiconductor Device Theory I – Dragica Vasileska
• Description of the avalanche process:
dxJJ nnn dxnJ
dxJJ nnp pJ
Impact ionization initiated by electrons.
dxJJ ppn dxnJ
dxJJ ppp pJ
Impact ionization initiated by holes.
.
0,0
constJJJ
dxdJ
dxdJ
dxdJ
dxdJ
pn
pn
pn
-
Multiplication factors forelectrons and holes:
)()0(
,)0()(
WJJ
MJ
WJM
p
pp
n
nn
EEE 531: Semiconductor Device Theory I – Dragica Vasileska
• 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 n and p:
W dx
pp
W dx
nn
ppnnp
ppnnn
dxeM
dxeM
JJdx
dJ
JJdx
dJ
xpn
xpn
0
'
0
'
0
0
11
11
The breakdown condition does not depend on whichtype of carrier initiated the process.
EEE 531: Semiconductor Device Theory I – Dragica Vasileska
• Limiting cases:
(a) n=p (semiconductor with equal ionization rates):
(b) n>>p (impact ionization dominated by one carrier):
W
W
p
ppp
W
W
n
nnn
dxMdx
M
dxMdx
M
0
0
0
0
1
111
1
111
W
n
dx
n dxeM
Wn
010
EEE 531: Semiconductor Device Theory I – Dragica Vasileska
Breakdown voltages:
(a) Step p+n-junction
pW nW
p n
)(xFmaxF
• For one sided junction we can make the following approximation:
• Voltage drop across the depletion region on the n-side:
• Maximum electric field:
• Empirical expression for the breakdown voltage VBD:
npn WWWW
WFVWFV BDnn maxmax 21
21
x
2max
0
0max 2
FqNk
Vk
WqNF
D
sBD
s
D
cmkVNE
V DgBD 16
2/3
101.160
EEE 531: Semiconductor Device Theory I – Dragica Vasileska
(b) Step p+-n-n+ junction
pW 1W
p n
)(xFmaxF
• Extension of the n-layer large:
• Extension of the n-layer small:
• Final expression for the punch-through voltage VP:
mBD WFV max21
11max 21
21 WWFWFV mmP
x
mW
n
1F
mmBDP W
WWW
VV 11 2
EEE 531: Semiconductor Device Theory I – Dragica Vasileska
(4) 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 carriers:
• Applied voltages, currents and solution for pn:p
nnp
n p
dx
pdD
dtpd
2
2
0110
0110,)(,)(
JJeJJtJVVeVVtV
ti
ti
ti
nnsn expxptxp )()(),( 1
EEE 531: Semiconductor Device Theory I – Dragica Vasileska
• Equation for pn1(x):
• Boundary conditions:
• Final expression for pn1(x):
0)(
0)(1
2'
12
12
121
2
p
nnn
pp
pn
L
xp
dx
pdxp
Di
dx
pd
TT
nn
T
ti
nn
nnn
VV
VVp
pV
eVVptp
pptp
0101
100
10
exp)0(exp),0(
0)(),(
'
0101 expexp),(
pTT
nn L
xVV
VVp
txp
EEE 531: Semiconductor Device Theory I – Dragica Vasileska
• Small-signal hole current:
• Low-frequency limit for the admittance Y:
• RC-constant:
1010
0
11 exp1 YV
VV
iVL
VpAqDdx
dpAqDI
Tp
Tp
np
x
np
pTT
pTp
npdif
TT
VVs
TTp
npd
difdpTTp
np
VI
VV
VLpAqD
C
IdVdI
VI
VeI
VV
VLpAqD
G
CiGiVV
VLpAqD
Y
T
21exp
21
current Forward,exp
211exp
00
/00
00
0
2p
difdCR
The characteristic time constant is on the order of the minority carriers lifetime.
EEE 531: Semiconductor Device Theory I – Dragica Vasileska
• Equivalent circuit model for forward bias:
• Bias dependence:
deplC
difC
dd G
R 1
sR sL
C
aV
difC
deplC
EEE 531: Semiconductor Device Theory I – Dragica Vasileska
(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:
p
ppp
p
pp
p
p
npDpp
n
Qdt
dQtItI
QtI
dtdQ
px
Jq
RJqdt
pd
)()()(
11 1
Valid for p+n diodeQp(t) = excess hole charge
EEE 531: Semiconductor Device Theory I – Dragica Vasileska
Diode turn-on:
• For t<0, the switch is open, and the excess hole charge is:
• At t=0, the switch closes, and we have the following boundary condition:
0)0()0( pp QtQ
0)0()0( pp QQ
p+ n
t=0
IF
pp tFp
tp eIBeAtQ
//1)(
• Final expression for the excess hole charge:
EEE 531: Semiconductor Device Theory I – Dragica Vasileska
S
FTa
VVpnp
LxVVnn
II
VV
eLAqpQeepxp TapTa
1ln
11)( /0
//0
• Graphical representation:
• Steady state value for the bias across the diode:
)(tQp
t
Fp I
),( txpn
x0np
t increasing
Slope almost constant
EEE 531: Semiconductor Device Theory I – Dragica Vasileska
Diode turn-off:
• For t<0, the switch is in position 1, and a steady-state situation is established:
• At t=0, the switch is moved to position 2, and up until time t=t1 we have:
• The current through the diode until time t1 is:
RV
I FF
0),0( 0 ann Vptp
p+ n
t=0
VF
R
VR
R
1 2
RV
I RR
EEE 531: Semiconductor Device Theory I – Dragica Vasileska
• 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:
• This gives the following final solution:
• Diode switching time:
Fpppp
ppR IQQ
Qdt
dQI
)0()0(,
ptRFpRpp eIIItQ
/)(
R
Fprrrrp I
IttQ 1ln0)(
EEE 531: Semiconductor Device Theory I – Dragica Vasileska