Lecture 10

The pn Junction (II)

1

Contents

1. pn junction under bias

2. I-V characteristics

2

Key questions

Why does the pn junction diode exhibit current

rectification（整流）?

Why does the junction current in forward bias

increase

What are the leading dependences of the

saturation current (the factor in front of the

exponential)?

3

~ expqV

kT

1. PN junction under bias

Upon application of voltage:

Electrostatics upset:

depletion region widens or

shrinks

Current flows (with

rectifying behavior)

Carrier charge storage

4

5

Fig PN junction and its associated energy band diagram for (a) zero bias, (b)

reverse bias, and (c) forward bias

If we apply a potential between the p and n regions, we will

no longer be in an equilibrium condition-the Fermi energy

level will no longer be constant through the system.

6

Fig. Energy-band diagram of a pn

junction under reverse bias

• Under reverse bias of applied

voltage VR, the pn junction is

not in thermal equilibrium.

• The quasi-Fermi energy levels

for electrons and holes are

0ln nFn Fi

i

nE E kT

n

0ln

p

Fp Fi

i

pE E kT

n

•The difference between the two

energy level is equal to the applied

voltage in units of energy

Fp Fn RE E eV

• The total potential barrier is

total Fn Fp RV V

total bi RV V V

7

Assume: No Current Flows

- Due to small leakage current

Substitute

j B DV

0 0d p nox x x

2 s B D a d

a d

V N N

qN N

8

Characteristics

ID=0

ID>0

VD>0.7 V

ID<0

VD<VBD

Forward Bias

Reverse Bias

Breakdown

9

To model IV characteristics we need 2 concepts

• The Law of the Junction

• Steady State Diffusion

/

0( 1)D TV nV

DI I e

Carrier profiles in thermal equilibrium:

10

Inside SCR in thermal

equilibrium:

dynamic balance between

drift and diffusion for electrons

and holes.

drift diffJ J

Carrier concentrations in pn junction under bias:

p drift pJ qp E n drift nJ qn E

11

0, B SCR driftfor V V E J

0

2

( )

B a d

s a d

q N NE

N N

Forward bias

12

drift diffJ J

Current balance in SCR broken:

Net diffusion current in SCR

⇒minority carrier injection into QNR’S

⇒excess minority carrier concentrations in QNR’S

Lots of majority carriers in QNR’s ⇒current can be high

Current balance in SCR broken:

Net drift current in SCR

=> minority carrier extraction

from QNR’s

deficit of minority carrier

concentrations in QNR’s

Few minority carriers in

QNR’s

=>current small.

13

0, B SCR driftfor V V E J

drift diffJ J

Reverse bias

What happens if minority carrier concentrations in QNR

change from equilibrium?

=>Balance between generation and recombination broken

• In thermal equilibrium: rate of break up of Si-Si bonds balanced

by rate of formation of bonds

• If minority carrier injection:

=> carrier concentration above equilibrium

=> recombination prevails

• If minority carrier extraction:

=> carrier concentrations below equilibrium

=> generation prevails

14

Where does generation and recombination take place?

1. Semiconductor bulk

2. Semiconductor surfaces & contacts

In modern devices, recombination and generation mainly takes place

at surfaces:

• perfect crystalline periodicity broken at a surface

=>lots of broken bonds: generation and recombination

centers

• modern devices are very small

high area to volume ratio.

High generation and recombination activity at surfaces

=> carrier concentrations cannot deviate much from equilibrium

values:

15

0 0( ) , ( )n s n p s p

Complete physical picture for pn diode under bias:

• Forward bias: injected minority carriers diffuse through

QNR => recombine at semiconductor surface

At semiconductor surface:

carrier concentration

unchanged from

equilibrium.

16

• Reverse bias: minority carriers extracted by SCR

=> generated at surface and diffuse through QNR

At semiconductor surface:

carrier concentration

unchanged from

equilibrium.

17

18

The current view:

• Forward bias:

19

• Reverse bias:

20

What limits the magnitude of the diode current?

• not generation or recombination rate at surfaces

• not injection or extraction rates through SCR

• diffusion rate through QNR’s

2. I-V characteristics

The "Short" Diode

-- the width of P and N regions Wn and Wp are smaller than

the minority carrier diffusion length Ln and Lp.

-- the minority carrier concentration becomes a linear

function of distance

The “long" Diode

-- the width of P and N regions Wn and Wp are longer than

the minority carrier diffusion length Ln and Lp.

-- the minority carrier concentration is an exponential

function of distance

21

"Short" Diode

22

Development of analytical current model:

1. Calculate concentration of minority carriers at edges

of SCR, p(xn) and n(−xp)

2. Calculate minority carrier diffusion current in each

QNR, In and Ip

3. Sum electron and hole diffusion currents, I = In + Ip

23

Step 1: computation of minority carrier boundary

conditions at edges of SCR

In thermal equilibrium in SCR,

Define

Recall

Rewrite

th

kTV

q

drift diffJ J

2 2

0 0/ /n i d p i ap n N and n n N

2ln a d

B

i

N NkT

q n

ln lnp

d aB th B th

po no

N NV and V

n

24

• Solving for the equilibrium minority carrier concentrations in

terms of the built in potential,

0 0

0 0

(60 ) log (60 ) logp n

B

n p

p nmV mV

p n

Special case of Boltzman statics.

B B

th thV V

no a po dp N e and n N e

(60 ) log o

i

nmV

n

This result relates the minority carrier concentration on one side

of the junction to the majority carrier concentration on the other

side of the junction.

25

Under bias in SCR,

But if difference small with respect to absolute values of current:

This is called quasi-equilibrium.

• The new potential barrier

φj = (φB- VD)

is substituted for the

thermal equilibrium barrier

to find the new minority

carrier concentrations at the

SCR edges.

In quasi-equilibrium

drift diffJ J

2

1 1 2 2( ) ( ) ( ) ( ) in x p x n x p x n

26

At edges of SCR, then:

This is the low-level injection approximation

Charge neutral at each side:

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

Since 0( )p p pn x p

[ ( ) ( )]( ) ( )exp exp

( )

n pn B

p

q x xn x q V

n x kT kT

[ ( ) ( )]( ) ( )exp exp

( )

n pn B

p

q x xp x q V

p x kT kT

( ) ( )p p a n n dp x N and n x N

Law of the Junction

27

( )

B D D

th th th

V V

V V V

p p d pon x N e e n e

( )

B D D

th th th

V V

V V V

n n a nop x N e e p e

2 2

where and i ipo no

a d

n nn p

N N

and

28

•The minority carrier concentration at the SCR is an

exponential function of applied bias. It changes one decade for

every 60 mV change in VD.

•Law of the Junction is valid if minority carrier concentration is

less than equilibrium majority concentration. This condition is

called Low Level Injection.

/

0

log / /60mv

0 0

( )

10 10

D th

D th D

V V

n n n

e V V V

n n

p x p e

p p

n no p pop n and n p

29

Voltage dependence:

• Equilibrium (V = 0):

• Forward (V >0):

Lots of carriers available for injection:

=> V increasing, concentration of injected carriers increases

=> forward current can be high and increases with V.

2 2

( ) ( )i ip n

a d

n nn x p x

N N

2 2

( ) ( )i ip n

a d

n nn x p x

N N

( )

D

th

V

V

p p pon x n e

( )

D

th

V

V

n n nop x p e

30

• Reverse (V <0):

Few carriers available for extraction:

=> reverse current is small.

Minority carrier concentration becomes vanishingly small:

=> reverse current saturates.

Rectification property of pn diode arises from minority carrier

boundary conditions at edges of SCR.

2 2

( ) 0 ( ) 0i ip n

a d

n nn x p x

N N

31

Step 2: Diffusion current in QNR:

Diffusion equation (for electrons in p-QNR):

Inside p-QNR, electrons diffuse to reach and recombine

at contact =>Jn constant in p-QNR => n(x) linear.

Boundary conditions:

Electron profile:

2

0( ) ip

a

nn x W n

N

2

( ) expip

a

n qVn x

N kT

( ) ( )( ) ( ) ( )

p p p p

p p p p

p p

n x n Wn x n x x x

x W

Forward bias

32

0( )( ) ( ) ( )n n n

n n n n

n n

p x pp x p x x x

W x

Similarly

Since QNR region remains charge neutral,

( ) ( )

( ) ( )

p a p

n d n

p x N n x

n x N p x

The majority carrier concentration

must increase by the same amount

as the injected minority carrier

concentration

excess minority carrier concentration

=excess majority carrier concentration

33

Electron current density:

Similarly for hole flow in n-QNR:

•Since the current is continuous,

the total current density J cannot

vary with position.

=> The total current density is

the sum of the minority carrier

diffusion current density at the

edge of the depletion region.

2 2

( ) ( )

exp

p p p p

n n n

p p

i i

a an

p p

n x n WdnJ qD qD

dx W x

n nqV

N kT NqD

W x

2

exp 1i nn

a p p

n D qVJ q

N W x kT

2

exp 1pi

p

d n n

Dn qVJ q

N W x kT

34

Step 3: sum both current components:

often written as:

with

2 1 1exp 1

pnn p i

a p p d n n

DD qVJ J J qn

N W x N W x kT

2 1 1exp 1

pni

a p p d n n

DD qVI qAn

N W x N W x kT

0 exp 1qV

I IkT

0 [ ]I saturation current A

Reverse bias

35

/

0 0( ) 0 for 0.1 VD thV V

n n n n Dp x p e p V

/

0 0( ) 0 for 0.1 VD thV V

p p p p Dn x n e n V

36

0 0

0 ( )( ) 0

=

p pn np n

n n p p

p n n p

n n p p

n Wp WJ qD qD

W x W x

D p D nq

W x W x

2

0

1 1 pni

a p p d n n

DDI qAn I

N W x N W x

The total current density:

The total current

Analyze current components in reverse bias:

The minority diffusion currents are extracted and

become the majority carrier current on the other side of

the junction.

37

The second methodThe ambipolar transport equation for excess minority carrier holes

in an n region

assume that the electric field is zero in both the quasi-neutral p and n

regions and steady state

An ohmic contact exists at x = (xn + Wn), implying an infinite surface

recombination velocity.

=> an excess minority carrier concentration is zero

Here, Wn is the width of n-QNR

0( )n n n np x x W p

0( ) exp an n n

eVp x p

kT

2

2 2

( )0 ( )n n

n

p

d p px x

dx L

2'

2

0

( ) ( ) ( )n n n np p

p

p p p pD E g

x x t

38

The solution is

If Wn << Lp. we can approximate the hyperbolic sine terms by

The minority carrier concentration becomes a linear function of

distanceCompare with the first method

n n nW W x

0

sinh ( ) /( ) exp 1

sinh( / )

n n pan n

n p

x W x LeVp x p

kT W L

( ) ( )sinh n n n n

p p

x W x x W x

L L

sinh n n

p p

W W

L L

0( ) exp 1a n nn n

n

eV x W xp x p

kT W

( ( ))np p

d p xJ eD

dx

0( ) exp 1

p n ap

n

eD p eVJ x

W kT

2

exp 1pi

p

d n n

Dn eVJ q

N W x kT

“Long" Diode

Step 1 Minority Carrier Distribution

Step 2 Diffusion current in QNR

Step 3 PN junction current

39

Long pn junction

Wn>>Lp and Wp>>Ln

40

Step 1 Minority Carrier Distribution

the ambipolar transport equation for excess minority carrier holes in an n

region

assume that the electric field is zero in both the quasi-neutral p and

n regions and steady state

where the diffusion length

Similarly, the excess minority carrier electron concentration in the p

region is determined from

2'

2

0

( ) ( ) ( )n n n np p

p

p p p pD E g

x x t

2

2 2

( )0 ( )n n

n

p

d p px x

dx L

2

2 2

( )0 ( )

p p

p

n

d n nx x

dx L

2

0p p pL D

41

The general solutions are

The boundary conditions for the total minority carrier

concentrations are

The minority carrier concentration

/ /

0

/ /

0

( ) ( ) ( )

( ) ( ) ( )

p p

n n

x L x L

n n n n

x L x L

p p p p

p x p x p Ae Be x x

n x n x n Ce De x x

0 0

0 0

( ) ( ) exp 1 exp

( ) ( ) exp 1 exp

a nn n n n

p

pap p p p

n

eV x xp x p x p p

kT L

x xeVn x n x n n

kT L

0 0 0

0

0 0

2

0 0

( ) exp exp

( ) exp

( )

( )

a bin n n p n

aP p p

n n n d

ip p p

d

eV eVp x p n n

kT kT

eVn x n

kT

p x p n N

nn x n n

N

42

Step 2 Diffusion current in QNR

Electron and hole current densities through the

space charge region of a pn junction

43

Since the electron and hole currents are continuous functions

through the pn junction, the total pn junction current will be

the minority carrier hole diffusion current at x = xn, plus the

minority carrier electron diffusion current at x = -xp.

Since we are assuming uniformly doped regions, the thermal-

equilibrium carrier concentration is a constant

Similarly

( )( )

n

np n p

x x

dp xJ x eD

dx

( )( )

n

n

p n p

x x

d p xJ x eD

dx

0( ) exp( ) 1

p n ap n

p

eD p eVJ x

L kT

0( ) exp( ) 1

n p an p

n

eD n eVJ x

L kT

44

Step 3 PN junction current

The total current density in the pn junction is

If Va is more than a few kT/eV,

the forward-bias current is an

exponential function of the

forward-bias voltage.

0 0( ) ( ) exp( ) 1

p n n p ap n n p

p n

eD p eD n eVJ J x J x

L L kT

0 0p n n p

S

p n

eD p eD nJ

L L

exp( ) 1aS

eVJ J

kT

2

0

1 1 pni

a p p d n n

DDI qAn I

N W x N W x

Short

diode

45

Ideal electron and hole current components

through a pn junction under forward bias.

•The minority carrier diffusion current densities decay exponentially

in each region. However, the total current through the pn junction is

constant.

•The difference between total current and minority carrier diffusion

current is a majority carrier current.

Key conclusions

Application of voltage to pn junction results in disruption of balance between drift and diffusion in SCR:

– in forward bias, minority carriers are injected into quasi-neutral regions

– in reverse bias, minority carriers are extracted from quasi-neutral regions

In forward bias, injected minority carriers recombine at surface.

In reverse bias, extracted minority carriers are generated at surface.

46

Computation of boundary conditions across SCR

exploits quasi-equilibrium: balance between

diffusion and drift in SCR disturbed very little.

Rate limiting step to current flow: diffusion

through quasi-neutral regions.

I-V characteristics of p-n diode:

47

0 (exp 1)qV

I IkT

Example

Given a diode with the p-region doped with Na=1018cm-3 and

the n-region doped with Nd=1016cm-3,calculate the minority

carrier concentrations at the edge of the depletion region as

the function of the applied voltage VD=0.6V and VD=-0.6V

respectively.

48

49

Solution:

we begin by finding the minority carrier

concentrations in thermal equilibrium:

2 204 3

0 16

1010 cm

10

in

d

np

N

2 20

2 3

0 18

1010 cm

10

ip

a

nn

N

According to the law of the junction

14 3

60mV0 6 3

10 cm for 0.6V( ) 10

10 cm for 0.6V

DVD

n n n

D

Vp x p

V

12 3

60mV0 8 3

10 cm for 0.6V( ) 10

10 cm for 0.6V

DVD

p p p

D

Vn x n

V

Homework10

Consider a pn junction with doping Na=1016cm-3, and

Nd=1018cm-3. what the applied voltage is required for

the depletion width Xd=1.5 um.

Note:

50

10 310 /cmin 14 111.7 8.85 10 Fcm

Homework11

An IC diode is designed to have a room-temperature

saturation. current of I0=5×10-17A for a particular application.

The fabrication process results in the device dimensions and

physical parameters are listed below

51

Dimensions Doping Diffusion

coefficient

Wp=0.5 um Na=2.5×1017cm-3 Dn=14cm2s-1

Wn=1.0um Nd=4.0×1016cm-3 Dp=10cm2s-1

a) What diode area A is required for I0?

b) Find the current and minority carrier concentrations at the edges of the

depletion region for a forward bias VD=720 mV.

c) Plot the carrier concentration distribution along the diode.