P-n junction - albany.edusoktyabr/NNSE618/NNSE618-L... · Let’ apply bias and calculate current...

NNSE 618 Lecture #20

1

Lecture contents

• P-n junction

– Current : Shockley model

– Generation-recombination current

– High injection


2

First we will use Shockley approximation:

1. Low injection = Fermi level do not change in the depletion layer =

minority carrier densities << majority carrier densities = majority

carrier density equals to doping concentration: pn<<nn0 =Nd (in n-

region and similar in p-region)

2. Non-degenerate = Boltzmann statistics

3. Drift-diffusion current (diffusion in quasi-neutral regions)

4. Long-base diode: length of the quasi-neutral regions is much larger

than the diffusion length of the minority carriers Ln, Lp.

5. No generation/recombination in the depletion layer

6. Abrupt depletion layer approximation

Let’ apply bias and calculate current through the p-n junction

Currents in p-n junction

From Sze, 1981


-xp

-xp xn

xn

3 Shockley model: Carrier concentrations and currents (Nd>Na)

Minority carrier current

Majority carrier

concentrations

Total current

is constant

throughout

the device

Recombination region

Recombination region

Majority carrier

current

Minority carrier

concentrations

Depletion region

s

pp0

nn0

From Colinge, Colinge


4

Definitions of quasi-Fermi levels:

Potential across the junction:

Currents in p-n junction: Shockley theory

Minority carrier concentrations at

the edges of the depletion region

(will serve as boundary conditions):

From Sze, 1981

2

( ) a tV Vip p

a

nn x e

N

ai Vq

ta VV

d

inn e

N

nxp

2

)(

kT

EEN

kT

EEnn cFn

ciFn

i expexp

kT

EEN

kT

EEnp

Fpv

v

Fpi

i expexp

FnFpa EEqV


5

Let’s find how minority carriers currents

recombine:

Continuity equation for electrons in p-region:

And diffusion current (no drift in the quasi-

neutral region):

Combining:

And similar for n-region:


Currents

From Sze, 1981

nqDnqJ nnn

nnn Jq

RGt

n

1

x

J

q

nnn

n

pp

10

0

x

nqDJ nn

n

ppp

n

nn

x

nD

0

2

2

p

nnnp

pp

x

pD

0

2

2

p n


6

Integrating minority carrier

concentration

with boundary condition

And the minority electron current

using diffusion length:

And similar for minority hole current:


From Sze, 1981

nn

ppp

D

nn

x

n

0

2

2

ta VV

a

ipp e

N

nxn

2

)(

nnptaDxxVV

ppp eennxn

1)( 00

nptaLxxVV

n

pn

nn eeL

nDq

x

nqDJ

1

0

nnn DL

pntaLxxVV

p

np

p eeL

pDqJ

1

0


7

If we can ignore the change of current in the

depletion region (no generation/recombination),

we can obtain total current as a sum of minority

carrier currents at respective edges of depletion

region:

Combining:

With saturation current:

(Mostly determined by low-doped side!)

I-V characteristics of p-n junction: Shockley model

From Sze, 1981

1a tV V

tot sJ J e

n

pn

p

np

sL

nDq

L

pDqJ

00

( ) ( )tot n p p nJ J x J x

a

i

n

n

N

nDq

2

p n


8

Bias and currents in p-n junction

Bias Potential

barrier at the

junction

Energy band

Equilibrium

V=0

Forward bias

Vf

Reverse bias

Vr

Hole diffusion

Hole drift

Electron diffusion

Electron drift

Hole diffusion

Hole drift

Electron diffusion

Electron drift

Hole diffusion

Hole drift

Electron diffusion

Electron drift


9 Shockley model and its limitations

From Sze, 1981

I-V characteristics of an ideal diode

Shockley model works for narrow-bandgap

semiconductors at low current densities (e.g.

Ge at room temperature) when depletion

region width is much smaller than diffusion

length of minority carriers, and the device is

much longer than the diffusion lengths.

Also the drift-difusion approximation should be

valid low doping level at least from one

side

Departures from the ideal equation is due to

• Short base

• Generation-recombination in the depletion

region

• Surface effects: leakage currents due to

band bending

• Tunneling of carriers between states in

bandgap

• Series resistance

I-V characteristics of an ideal diode: semilog plot

60mV/decade slope at RT:

1 1 decadeslope

ln10 60mVB

q

k T


10 Realistic p-n junction characteristics

I-V characteristics of a Si diode

From Sze, 1981

G-R

Series

resistance

High

injection


11 Generation-recombination in the depletion region

Generation/recombination in the depletion region

violates one of the Shockley approximations (#5)

• Radiative recombination in the depletion region:

• is almost constant throughout the depletion

region

• Integration of the recombination rate in the

depletion region increases the diode current

but still has 60mV/dec. slope:

• Trap-assisted generation-recombination : Shockley-

Hall-Reed (SHR) processes (with a trap close to

intrinsic Fermi level Et = Ei ) :

• If lifetimes are equal, the trap-assisted

recombination is the highest for

• The SHR recombination current is obtained

by integration of the rate over the depletion

region (120 mV/dec. slope):

B- radiative recombination constant

G-R

2

0 0

iSHR

i n i p

np nU

p n n n

Effective width x’<W

dx

dJ

qRG n

nn

10

ta VVienpn

2

o

VVi

SHR

taenU

2

12

max

From Van Zeghbroeck, 1998


12

Generation-recombination in the depletion region

• Total diode current with contribution of

radiation/recombination:

• At reverse bias the carrier concentrations in the

depletion region are small:

• The generation of carriers is the dominant

process

• The total reverse current will be higher than in

Shockley model:

0

22

2

'

xqnBWn

N

nDqJ i

iD

i

p

p

reverse

2innp

Dominant term when G-R in

depletion region takes place


13 High injection (direct bias)

High injection violates one of the Shockley approximations that the

majority carriers are determined by thermal equilibrium:

but they are not. If neutrality should lead to

Solving quadratic equations:

The current will be (also ignoring recombination in the

depletion region):

For large direct bias

ai Vq


Reduced slope,

120mV/dec.

0( ) ( )p p p p ap x p x N

0( ) ( ) ( )p p p p p pp x p x n x 0( ) ( )p p p pn x p x


14

Definitions of quasi-Fermi levels:

Or

Electrostatic potential across the junction:

Quasi-Fermi level can be related to the drift-

diffusion current density:

Constant quasi-Fermi level implies that the

electrostatic force equals the diffusion force:

Quasi-Fermi levels


kT

EEN

kT

EEnn cFn

ciFn

i expexp

kT

EEN

kT

EEnp

Fpv

v

Fpi

i expexp

iiFn

n

nkTEE ln

FnFpa EEqV

nnnni

nFn

n Jdx

dn

q

kTnq

dx

dn

n

nkT

dx

dEn

dx

dEn

dx

dEnJ Fn

nn


15

Junction breakdown occurs at high reverse bias.

• Avalanche multiplication (impact ionization of electron-hole pairs)

• Tunneling through the junction – Zener breakdown (at high doping concentrations), identical to that of tunneling in a metal-semiconductor junction.

• Thermal: junction temperature increases leading to increase of saturation current

Junction breakdown

From Sze, 1981 From Van Zeghbroeck, 1998

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