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```									             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

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
ND  N A                                         ND  N A
ax

p-side                      n-side             p-side                 n-side
EEE 531: Semiconductor Device Theory I – Dragica Vasileska
(A) Equilibrium analysis of step junctions
EC
qVbi
(a) Built-in voltage Vbi:
qVbi   Ei  E F  p   E F  Ei n
Ei
EF
nn0  ni exp E F  Ei  k BT 
EV
p-side            n-side
W                             p p 0  ni expEi  E F k BT 
(x)
qND                        k BT  p p 0 nn0          N AND 
+                        Vbi      ln            VT ln       
q    n2                n2 
-qNA -                      x                         i                i  
V (x)                             (b) Majority- minority carrier
Vbi
relationship:
E (x)                         x         pn0  p p 0 exp Vbi / VT 
 xp           xn                    n p 0  nn0 exp Vbi / VT 
Emax                          x

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: V ( x p )  0, V ( xn )  Vbi , E ( xn )  E ( x p )  0

p-side: V p ( x) 
qN A
2k s  0
x  x p 2
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 
V p (0)  Vn (0)   W  2k s 0 ( N A  N D )Vbi

               qN A N D
N A x p  N D xn 
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:
dV                     qN A N DW
Emax                      
dx      x 0      k s 0 ( N A  N D )
(e) Carrier concentration variation:
15

N A  N D  1015 cm 3
10
Concentration [cm-3]

13
10
Wcalc  1.23 mm
Emax( DC )  9.36 kV / cm
11
10                                                 -3
n [cm ]
-3

Emax( sim )  8.93 kV / cm
9                                      p [cm ]
10

7
10

5
10
0   0.5    1   1.5     2     2.5      3        3.5
Distance [mm]
EEE 531: Semiconductor Device Theory I – Dragica Vasileska
N A  N D  1015 cm 3
Wcalc  1.23 mm, Emax( DC )  9.36 kV / cm, Emax( sim )  8.93 kV / cm

15
10                                                                            0

Electric field [kV/cm]
14                                                                      -2
(x)/q [cm ]

5x10
-3

-4
0
-6
14
-5x10
-8

15
-10                                                                          -10
0   0.5   1   1.5   2    2.5   3   3.5                                  0   0.5   1   1.5   2   2.5   3   3.5

Distance [mm]                                                           Distance [mm]

EEE 531: Semiconductor Device Theory I – Dragica Vasileska
N A  1016 cm3 , N D  1018 cm3
Wcalc  0.328 mm, Emax( DC )  49.53 kV / cm, Emax( sim )  67 kV / cm

17
10                                                                          10

Electric field [kV/cm]
0
16
5x10                                                                          -10
(x)/q [cm ]
-3

-20
0                                                                     -30
-40
16
-5x10                                                                         -50
-60
17
-10                                                                         -70
0.6     0.8     1      1.2    1.4                                     0.6   0.8   1    1.2   1.4

Distance [mm]                                                       Distance [mm]

EEE 531: Semiconductor Device Theory I – Dragica Vasileska
(f) Depletion layer capacitance:
 Consider a p+n, or one-sided junction, for which:
2k s 0 Vbi  V 
W
qN D
 The depletion layer capacitance is calculated using:
dQc qN D dW   qN D k s 0   1 2(Vbi  V )
C                          2
dV    dV      2(Vbi  V )  C   qN D k s 0
1 C2
Measurement setup:
1
slope 
ND                              W
dW
Reverse
bias                   Forward bias           vac ~
V                 V
Vbi  V

EEE 531: Semiconductor Device Theory I – Dragica Vasileska
(B) Equilibrium analysis of linearly-graded junction:
12 k s 0 Vbi  V 
1/ 3
(a) Depletion layer width:                 W                     
         qa         
qaW 2
(c) Maximum electric field: Emax                   
8k s 0
1/ 3
            
qaks 0
2 2
(d) Depletion layer capacitance:                      C            
12Vbi  V 

Based on accurate numerical simulations, the depletion
layer capacitance can be more accurately calculated if Vbi
is replaced by the gradient voltage Vg:
2       a 2 k s 0VT 
Vg  VT ln           3 
3       8qni 
EEE 531: Semiconductor Device Theory I – Dragica Vasileska
(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                                 np  ni2 exp V / VT 
E Fp
E Fn          n p (  x p )  n p 0 exp V / VT 
EV                                                pn ( xn )  pn 0 exp V / VT 

 xp       xn
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:
V / VT              ( x  xn ) / L p
pn ( x )  pn 0 ( e             1)e
V / VT           ( x  x p ) / Ln
n p ( x )  n p 0 ( e           1)e

n p (x )                          pn (x )
Space-charge                     Forward bias
region W

pn 0
n p0
x
 xp                  xn
Reverse bias
EEE 531: Semiconductor Device Theory I – Dragica Vasileska
• Corresponding minority-carriers diffusion current densities
are:
diff             qD p pn 0              V / VT              ( x  xn ) / L p
Jp      ( x)                      (e                1)e
Lp
diff             qDn n p 0              V / VT             ( x  x p ) / Ln
Jn      ( x)                     (e                 1)e
Ln
Shockley model
diff               diff
J tot  J p ( xn )  J n (  x p )
diff  drift
diff
majority J p      Jp
drift                              majority J n  J n
J tot

diff
diff
minority J n                                             minority J p

x
 xp                         xn
No SCR generation/recombination
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:
I
diff         diff
I tot    I p ( xn )  I n (  x p )                            Ge Si GaAs

 D p pn 0 Dn n p 0  V / V
 qA
 L

Ln 
 e T 1            
     p             
V
• Reverse saturation current IS:

 D p pn 0 Dn n p 0       2  Dp     Dn 
I s  qA                    qAni              
 L         Ln              L N    Ln N A 
     p                      p D          

EEE 531: Semiconductor Device Theory I – Dragica Vasileska
(d) Origin of the current flow:

Forward bias:                                    Reverse bias:

W                            EC
EC                                                          Ln         qV
qVbi  V                                      qVbi  V 
qV                           E Fp
E Fn       EV
E Fp                                                                                 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.
EEE 531: Semiconductor Device Theory I – Dragica Vasileska
(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 )

pn (x )                                        diff            Dn diff
pn 0                 Jn      ( x)      J p ( x)
xn
x                                  Dp

• Total hole current in the quasi-neutral regions:
tot           diff              drift         diff
J p ( x)      Jp      ( x)    J p ( x)      Jp      ( x)

EEE 531: Semiconductor Device Theory I – Dragica Vasileska
• Electron drift current in the quasi-neutral region:
 Dn    diff                     1
diff
Jn      ( x )  J tot      1 J p ( x ), E ( x )               diff
J n ( x)
D                           qn( x )m n
  p   

drift
J n (x )             J tot
tot        diff       drift
J n ( x)  J n ( x)  J n ( x)
diff      diff
J n ( x)  J p ( x)          diff
J p (x )
x
diff
J n (x )

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
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
 dJ p ( x ) J p ( xn )  J p (  x p )   q  R p dx
xp                                                xp
xn
J scr  q  R p dx
xp

EEE 531: Semiconductor Device Theory I – Dragica Vasileska
Reverse-bias conditions:
• Concentrations n and p are negligible in the depletion
region:
 ni2        ni               Et  Ei       Ei  Et 
R                                    k T   n exp k T 
  ,  g   p exp                       
 p n1  n p1    g               B             B 

• Space-charge region current is actually generation current:
qniW           qniW
J scr   J gen            J gen        Vbi  V
g             g
• Total reverse-saturation current:
J  Js e    V / VT

 1  J scr    J s  J gen
V VT
   
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)           EC

E Fp
AJ s
V (log-scale)           EV
E Fn

AJ gen                                                        W

IV-characteristics                               Generated carriers are
swept away from the
under reverse bias conditions                           depletion region.

EEE 531: Semiconductor Device Theory I – Dragica Vasileska
Forward-bias conditions:
• Concentrations n and p are large in the depletion region:

np
2 V / VT
 ni e        R

2 V /V
ni e T  1             
 p n  n1    n  p  p1 
• Condition for maximum recombination rate:
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     qniW V / 2VT
J scr           e
 rec

EEE 531: Semiconductor Device Theory I – Dragica Vasileska
• Exact expression for the recombination current:
qni  V / 2VT           1          qN D 2Vbin  V 
J scr         e        ,    VT     , Enp 
rec               2    Enp               k s 0

• Corrections to the model:
qni  V / mrVT
J scr          e
 rec

• Total forward current:


J  Js e
V / VT

1 
qni  V / mrVT
 rec
e         J s,eff e      
V / VT
1
  ideality factor. Deviations of  from unity represent
an important measure for the recombination current.

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)

AJ scr
AJ

V
AJ d

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.
Zero-bias band diagram:                       Forward-bias band diagram:

EFn
EF              EFp
EC
EC

EV
EV
W                                               W
EEE 531: Semiconductor Device Theory I – Dragica Vasileska
Reverse-bias band diagram:               • Tunneling current (obtained by
using WKB approximation):
* 3      4 2 m* E 3 / 2 
exp 
2m q FcrVA               g 
It    2 2 1/ 2
4  E g         3qFcr 
                
EF                                         Fcr  average electric field in
p
the junction
EFn
EC      • The critical voltage for
tunneling breakdown, VBR, is
estimated from:
EV
I t (VBR )  10 I S

• With T, Eg and It .

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 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 
 qlF 
ai  exp       
     cr 

- Ei  critical energy for impact ionization to occur
- Fcr  critical electric field
- l  mean-free path for carriers

EEE 531: Semiconductor Device Theory I – Dragica Vasileska
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
EEE 531: Semiconductor Device Theory I – Dragica Vasileska
• Description of the avalanche process:
dJ n      dJ p
 0,      0
Jn                  J n  a n J n dx          dx        dx
dx                              dJ n    dJ p
J p  an J n dx                  Jp                      -
dx       dx

Impact ionization initiated by electrons.
J  J n  J p  const.

Jn                  J n  a p J p dx        Multiplication factors for
dx                             electrons and holes:
J p  a p J p dx                 Jp                      J n (W )        J p (0)
Mn           , Mp 
J n (0)         J p (W )
Impact ionization initiated by holes.

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 an and ap:

                  an  a p dx'
x
W
 dJ n                     1  1
  an e 0
 dx    an J n  a p J p   M
dx

 dJ                            n   0
  an  a p dx'
x
 p  an J n  a p J p        1    W
 dx                       1        a pe 0                 dx
 Mp 0


The breakdown condition does not depend on which
type of carrier initiated the process.

EEE 531: Semiconductor Device Theory I – Dragica Vasileska
• Limiting cases:
(a) an=ap (semiconductor with equal ionization rates):
      1    W                    1
 1  M   a n dx  M n  W
       n   0               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        W
Mn  e 0              1   an dx
0

EEE 531: Semiconductor Device Theory I – Dragica Vasileska
Breakdown voltages:
(a) Step p+n-junction                          • For one sided junction we can make
the following approximation:
W  Wn  W p  Wn
                              n         • Voltage drop across the depletion
p
region on the n-side:
1              1
Vn  FmaxWn  VBD  FmaxW
 Wp                Wn                     2              2
• Maximum electric field:
 F (x )                               qN DW              k s 0 2
 Fmax                                        Fmax           VBD           Fmax
k s 0           2qN D
• Empirical expression for the
breakdown voltage VBD:
3/ 2
x                      Eg             ND     kV 
VBD        1.1 
 60                16 
 10     cm 
 
     
EEE 531: Semiconductor Device Theory I – Dragica Vasileska
(b) Step p+-n-n+ junction
• Extension of the n-layer large:
1
             n                                     VBD     FmaxWm
p                                 n                              2
• Extension of the n-layer small:
 Wp
VP  FmaxWm  F1 Wm  W1 
W1 Wm                       1        1
 F (x )
2        2
 Fmax                                        • Final expression for the punch-
through voltage VP:
 F1                                          W1     W1 
VP  VBD    2 
 W 
Wm        
x                            m

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:      dp
2
d p      p
n    Dp            n        n
dt              dx   2           p
• Applied voltages, currents and solution for pn:
V (t )  V0  V1eit , V1  V0                                                            it
it                               pn ( x, t )  pns ( x )  pn1 ( x )e
J (t )  J 0  J1e , J1  J 0
EEE 531: Semiconductor Device Theory I – Dragica Vasileska
• Equation for pn1(x):
d 2 pn1 1  i p                 d 2 pn1 pn1 ( x )
          pn1 ( x )  0          2 0
dx  2    Dp p                    dx  2
L p'

• Boundary conditions:
pn (, t )  pn 0  pn1 ()  0
 V0  V1eit 
pn (0, t )  pn 0 exp                p (0)  pn 0V1 exp V0 
 
                 n1               V 
     VT                  VT        T

• Final expression for pn1(x):
pn 0V1     V0   x 
pn1 ( x, t )         exp  exp 
V 

VT        T    L 
   p' 

EEE 531: Semiconductor Device Theory I – Dragica Vasileska
• Small-signal hole current:
dpn1       AqD p pn 0V1               V0 
I1   AqD p                        1  i p exp   YV1
V 
dx x 0     L pVT                    T
• Low-frequency limit for the admittance Y:
 V0  1
exp 1  i p   Gd  iCdif
AqD p pn 0
Y                                  
L pVT          VT  2     
AqD p pn 0     V0  I s eV0 / VT    I   dI
Gd             exp                           , I  Forward current
L pVT         VT       VT        VT dV
1 AqD p pn 0          V0  1 I
Cdif                p exp            p
2 L pVT               VT  2 VT
• RC-constant:
p            The characteristic time constant is on
Rd Cdif                    the order of the minority carriers lifetime.
2
EEE 531: Semiconductor Device Theory I – Dragica Vasileska
• Equivalent circuit model for forward bias:
Cdepl

Rs           Ls
Cdif

1
Rd 
Gd

• Bias dependence:                 C                              Cdif

Cdepl

Va
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:
d p n       1                 1D        1 J p pn
    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
EEE 531: Semiconductor Device Theory I – Dragica Vasileska
Diode turn-on:
• For t<0, the switch is open, and
p+   n
the excess hole charge is:
Q p (t  0)  Q p (0 )  0
t=0
• At t=0, the switch closes, and
we have the following boundary                              IF
condition:
Q p (0  )  Q p (0  )  0

• Final expression for the excess hole charge:
t /  p          1  e t /  p 
Q p (t )  A  Be                pIF

               


EEE 531: Semiconductor Device Theory I – Dragica Vasileska
• Graphical representation:
Q p (t )                                  pn ( x , t )
Slope almost constant

pIF                                                          t increasing

pn 0
t                                            x

• Steady state value for the bias across the diode:
pn ( x )  pn 0 e   Va / VT

1 e
 x / Lp

 Q p  Aqpn 0 L p e
Va / VT

1

 IF         
Va  VT ln 1 
            

     IS     
EEE 531: Semiconductor Device Theory I – Dragica Vasileska
Diode turn-off:
• For t<0, the switch is in position
p+   n
1, and a steady-state situation is
established:                                          t=0
VF
IF                                         1       2
R                                VF               VR
• At t=0, the switch is moved to
R          R
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
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:
dQ p       Qp
 IR                    , Q p (0  )  Q p (0  )   p I F
dt       p
• This gives the following final solution:
t /  p
Q p (t )    p I R   p  I F  I R e

• Diode switching time:
 IF 
Q p (trr )  0  trr        p ln1  
 IR 
EEE 531: Semiconductor Device Theory I – Dragica Vasileska
• Graphical representation:
Va (t )

pn ( x , t )                                                                     t
Slope almost
constant
t=0
 VR

pn 0    t=ts
IF
ttrr
x
ts   trr
t
 0.1I R
ts  switching time
trr  reverse recovery time
 IR

EEE 531: Semiconductor Device Theory I – Dragica Vasileska

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