PN Junction: Physical and Mathematical Description of

Operation

Dragica VasileskaProfessor

Arizona State University• 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

p-side n-side

AD NN

p-side n-side

AD NN ax

Step junction Linearly-graded junction

(A) Equilibrium analysis of step junctions

(a) Built-in voltage Vbi:

(b) Majority- minority carrier relationship:

CE

VE

iE

FE

biqV

W)(x

x-qNA

qND

)(xV

x

biV

)(xE

xmaxE

px nx

+

-

p-side n-side

2200

0

0

lnln

exp

exp

i

DAT

i

npBbi

BFiip

BiFin

niFpFibi

n

NNV

n

np

qTk

V

TkEEnp

TkEEnn

EEEEqV

Tbinp

Tbipn

VVnn

VVpp

/exp

/exp

00

00

(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

kqN

xV

bins

Dn Vxx

kqN

xV

2

02)(

DA

biDAs

nDpA

np

pn

NqNVNNk

W

xNxN

VVWxx

)(2)0()0( 0

(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

dxdV

E

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

cen

trat

ion

[cm

-3]

Distance [mm]

cmkVE

cmkVE

mW

cmNN

sim

DC

calc

DA

/93.8

/36.9

23.1

10

)max(

)max(

315

m

cmkVEcmkVEmW

cmNN

simDCcalc

DA

/93.8,/36.9,23.1

10

)max()max(

315

m

cmkVEcmkVEmW

cmNN

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 3.5

Ele

ctric

fiel

d [

kV/c

m]

Distance [mm]

(f) Analytical vs. numerical data

cmkVEcmkVEmW

cmNcmN

simDCcalc

DA

/67,/53.49,328.0

10,10

)max()max(

318316

m

cmkVEcmkVEmW

cmNcmN

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 1.4

Ele

ctric

fiel

d [

kV/c

m]

Distance [mm]

(g) Depletion layer capacitance:

Consider a p+n, or one-sided junction, for which:

The depletion layer capacitance is calculated using:

D

bis

qNVVk

W02

02

0 )(21)(2

sD

bi

bi

sDDc

kqNVV

CVVkqN

dVdWqN

dVdQ

C

VVbi V

21 C

DNslope

1

Forward biasReverse bias

Measurement setup:

WdW

~

V

vac

(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

qaVVk

W bis

3/120

2

12

VV

qakC

bi

s

0

2

max 8

skqaW

E

30

2

8ln

32

i

TsTg

qn

VkaVV

(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:

CE

VE

FnEqV

FpE

W

Tnnn

Tppp

Ti

VVpxp

VVnxn

VVnnp

/exp)(

/exp)(

/exp

0

0

2

Tnnn

Tppp

Ti

VVpxp

VVnxn

VVnnp

/exp)(

/exp)(

/exp

0

0

2

px nx

(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 nxx

Forward bias

Reverse bias

Space-chargeregion W

• Corresponding minority-carriers diffusion current densities are:

npT

pnT

LxxVV

n

pndiffn

LxxVV

p

npdiffp

eeL

nqDxJ

eeL

pqDxJ

/)(/0

/)(/0

)1()(

)1()(

px nxx

diffpJ minoritydiff

nJ minority

)()( pdiffnn

diffptot xJxJJ

No SCR generation/recombination

driftn

diffn JJ majority drift

pdiffp JJ majority

totJ

Shockley model

(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

eL

nD

L

pDqA

xIxII

An

n

Dp

pi

n

pn

p

nps NL

D

NL

DqAn

L

nD

L

pDqAI 200

I

V

Ge Si GaAs

(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

(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

)()( xJD

DxJ diff

pp

ndiffn

)()()()( xJxJxJxJ diffp

driftp

diffp

totp

• Electron drift current in the quasi-neutral region:

)()(

1)(),(1)( xJ

xqnxExJ

D

DJxJ diff

nn

diffp

p

ntot

diffn m

x)(xJ diff

p

)(xJ diffn

)(xJ driftn

)()()( xJxJxJ driftn

diffn

totn

totJ

)()( xJxJ diffp

diffn

(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.

(3) Generation and Recombination Currents

Continuity equation for holes:

Steady-state and no light genera-

tion process:

• Space-charge region recombination current:

scrJ

ppp RG

x

J

qtp

1

0,0 pGtp

n

p

n

p

n

px

xpscr

x

xpppnp

x

xp

dxRqJ

dxRqxJxJxdJ )()()(

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:

Tk

EE

Tk

EEn

pn

nR

B

tin

B

itpg

g

i

np

i expexp,11

2

Generation lifetime

VVWqn

JWqn

JJ big

igen

g

igenscr

gensVVscrVV

s JJJeJJT

T

1/

• 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.

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

ppnn

enRennp

np

VViVV

i

TT

nprecVV

rec

i

np

VVi

VVi

TT

T

en

pn

enR

enpn

,2//2

max

2/

TVV

rec

iscr e

WqnJ 2/max

Recombination lifetime

• 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

VVqNE

EVe

qnJ T

TrVmV

rec

iscr e

qnJ /

11 /,

//

TTrT VVeffs

VmV

rec

iVVs eJe

qneJJ

• Importance of recombination effects:

Low voltages, small ni recombination current dominates

Large voltages diffusion current dominates

log(I)

VdAJ

scrAJAJ

(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 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

EFn

EC

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

3

24exp

4

22/3*

2/122

3*

SBRt IVI 10)(

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

EFn

EC

EV

EFp

Expanded view of thedepletion region

Avalanche mechanism:

Generation of the excess electron-hole pairs is due to impact ionization.

• Description of the avalanche process:

dxJJ nnn dx

nJ

dxJJ nnp pJ

Impact ionization initiated by electrons.

dxJJ ppn dx

nJ

dxJJ ppp pJ

Impact ionization initiated by holes.

.

0,0

constJJJ

dx

dJ

dx

dJdx

dJ

dx

dJ

pn

pn

pn

-

Multiplication factors forelectrons and holes:

)(

)0(,

)0(

)(

WJ

JM

J

WJM

p

pp

n

nn

• 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

x

pn

x

pn

0

'

0

'

0

0

11

11

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

• 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

dx

MdxM

dx

MdxM

0

0

0

0

1

111

1

111

W

n

dx

n dxeM

W

n

010

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

FqN

kV

k

WqNF

D

sBD

s

D

cmkVNE

V/-

DgBD

43

16

2/3

101.160

(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

W

W

WVV 11 2

• Doping-dependence of the breakdown voltage VBD:

• Temperature dependence:

As temperature increases, lattice scattering increases which makes impact ionization less probable. As a result of this, the breakdown voltage increases.

BDV

DNLog-scale

Log

-sca

le

p+-n-n+

One-sided abruptjunction

Tunneling breakdown

Width of the n-layer W1 increases

(c) Plane vs. planar or cylindrical junction

• Plane junction:

• Planar junction:

n

Maximum electric field:

Except for surface effects, this is an ideal junction.

00max

s

D

s k

WqN

kQ

Fp+

n

p+

Maximum electric field:

The smaller the radius rj, the largerthe electric field crowding.

js

D

rW

k

WqNF

21

0max

rj

W

(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 carriers:

• Applied voltages, currents and solution for pn:p

nnp

n p

dx

pdD

dt

pd

2

2

0110

0110,)(,)(

JJeJJtJVVeVVtV

ti

ti

ti

nnsn expxptxp )()(),( 1

• 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

D

i

dx

pd

TT

nn

T

ti

nn

nnn

V

V

V

Vpp

V

eVVptp

pptp

0101

100

10

exp)0(exp),0(

0)(),(

'

0101 expexp),(

pTT

nn L

xV

V

V

Vptxp

• Small-signal hole current:

• Low-frequency limit for the admittance Y:

• RC-constant:

1010

0

11 exp1 YV

V

Vi

VL

VpAqD

dx

dpAqDI

Tp

Tp

np

x

np

pTT

pTp

npdif

TT

VVs

TTp

npd

difdpTTp

np

VI

VV

VL

pAqDC

IdVdI

VI

VeI

VV

VL

pAqDG

CiGiVV

VL

pAqDY

T

21

exp21

current Forward,exp

21

1exp

00

/00

00

0

2p

difdCR

The characteristic time constant is on the order of the minority carriers lifetime.

• Equivalent circuit model for forward bias:

• Bias dependence:

deplC

difC

dd G

R1

sR sL

C

aV

difC

deplC

(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

Q

dt

dQtItI

QtI

dt

dQ

p

x

J

qRJ

qdt

pd

)()()(

11 1

Valid for p+n diodeQp(t) = excess hole charge

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:

S

FTa

VVpnp

LxVVnn

I

IVV

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

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:

R

VI F

F

0),0( 0 ann Vptp

p+ n

t=0

VF

R

VR

R

1 2

R

VI R

R

• 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

Q

dt

dQI

)0()0(,

ptRFpRpp eIIItQ

/)(

R

Fprrrrp I

IttQ 1ln0)(

• Graphical representation:

),( txpn

x0np

t=0

Slope almostconstant

t=ts

ttrr

)(tVa

FI

RI

st

RI1.0rrt

RV

t

t

ts switching timetrr reverse recovery time

ts switching timetrr reverse recovery time