1

Chapter 7. The pn Junction

1. Basic Structure

In thermal equilibrium, the diffusion force & the E-field force exactly balance each other

2

2. Built-in Potential Barrier

• In thermal equilibrium,

the Fermi energy is constant

throughout the entire system

• The built-in potential barrier, Vbi

FpFnbiV

kT

EEn

kT

EENn FiF

iFc

co exp)(

exp

kT

enn

EEe

Fnio

FFiFn

exp

i

dFn

n

N

e

kTln

i

aFpFFiFp

FFiiao

n

N

e

kTEEe

kT

EEnNp

ln

exp

In the n region

If we define Fn as

Taking the natural log of both sides & setting no = Nd

Similarly,

∴ The built-in potential barrier for the step junction

22i

dat

i

daFnFpbi

n

NNV

n

NN

e

kTV lnln

where Vt = kT/e, thermal voltage

3

dx

xdEx

dx

xd

s

)()()(2

2

1

)(Cx

eNdx

eNdx

xE

s

a

s

a

s

3. Electric Field

Assuming uniform doping & abrupt junction approximation

d

a

eNx

eNx

)(

)(

)(

0, 1

p

s

a

p

s

ap

xxeN

E

xeN

CExx

From Poisson’s equation,

dx

xdEx

dx

xd

s

)()()(2

2

Since the charge densities are

-xp < x < 0

0 < x < xn

The electric field in the p region, E, is

Since the currents are zero in thermal equilibrium, the electric field is assumed to be zero

in the neutral p region.

-xp ≤ x ≤ 0

4

In the n region,

Since E=0 at x=xn, C2=-eNdxn/s

The electric field is continuous at x=0

2CxeN

dxeN

Es

d

s

d

)( n

s

d xxeN

E

ndpa

n

s

dp

s

a

xNxN

xeN

xeN

)(

0 ≤ x ≤ xn

5

The maximum electric field occurs at the metallurgical junction

(p region) ⇒

'

1

2

)2

(

)()()(

CxxxeN

dxxxeN

dxxEx

p

s

a

p

s

a

If we arbitrarily set the potential equal to zero at x=-xp

2

2'

1

'

1

2

2

)(2

)(

2

20

p

s

a

p

s

a

p

p

s

a

xxeN

x

xeN

C

CxxeN

-xp ≤ x ≤ 0

6

Since the potential is a continuous function, at x=0,

At x=xn ,

22

'

2

2

2)

2()(

2)0(

p

s

an

s

d

p

s

a

xeNx

xxeN

x

CxeN

)(2

2)

2()(

22

22

2

pand

s

p

s

ann

s

dn

xNxNe

xeNx

xeN

x

0 ≤ x ≤ xn

(n region) ⇒ '

2

2

)2

()()( Cx

xxeN

dxxxeN

x n

s

dn

s

d

)(2

22

pand

s

xNxNe

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4. Space Charge Width

2/1

222

2

2

222

12

)(22

2)(

2

dad

abisn

nda

a

d

s

n

a

dd

s

n

a

dand

s

pand

s

bi

n

a

dpndpa

NNN

N

e

Vx

xNNN

Nex

N

NN

e

xN

NNxN

exNxN

eV

xN

NxxNxN

Similarly,

2/1

12

daa

dbisp

NNN

N

e

Vx

8

Total depletion width or space charge width, W

2/1

2/1

2/1

112

2

12

)1(

da

bis

da

dabis

a

da

dad

abis

a

dnpn

NNe

V

NN

NN

e

V

N

NN

NNN

N

e

V

N

NxxxW

※ [Example 7.1, 7.2] doping, temperature, ni condition Vbi, W, xn, xp, Emax

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5. Reverse Applied Bias

W

VV

NN

NNVVe

xeNxeNE

Rbi

da

da

s

Rbi

s

pa

s

nd

)(2

)(22/1

max

“ The space charge width W increases with an increasing reverse-bias voltage VR ”

• The magnitude of the electric field in the depletion region increases with an applied reverse-bias voltage.

2/1

11)(2

da

Rbis

RbiRFpFntotal

NNe

VVW

VVVV

Q. Uniform E field의 경우 E=V/d=V/W

인데, pn junction에서는 왜 E=2V/W

의모습을가지나?

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6. Junction Capacitance

An increase in the reverse bias voltage dVR will uncover

additional positive charges in the n region and additional

negative charges in the p region.

2/1

2/1

2/1

2/1

))((2

)(2

112'

1)(2

''

daRbi

das

Rbi

dad

asd

dad

aRbisn

R

nd

R

NNVV

NNe

VVNNN

N

eeNC

NNN

N

e

VVx

dV

dxeN

dV

dQC

depletion layer approximation

또는 C’ = s/W

pand dxeNdxeNdQ ' where

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7. One-sided Junctions

If Na≫ Nd (p+-n junction)

2/1

2/1

1)(2

11)(2

d

Rbis

da

Rbis

Ne

VV

NNe

VVW

ds

Rbi

Rbi

ds

Ne

VV

C

VV

NeC

)(2

'

1

)(2

2

2/1

xn≫ xp⇒W ≈ xn

Almost the entire space charge layer extends into the low-doped region of the junction

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Breakdown voltage (항복전압) : 역방향전류가갑자기증가

8. Junction Breakdown

Zener breakdown: 도핑이높을경우

tunneling mechanism을통해발생

low breakdown voltage

Avalanche breakdown: 도핑이높을경우

high electric field에의해발생

high breakdown voltage

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Avalanche Breakdown : 공핍영역(depletion region)에서 carrier들의 collision process에의해발생

의미수를의생성되는의해서정공에전자와단위길이당

이온화율이며과

은여기서

과정에서

pair hole-electron

,

factor, tionmultiplica

tionmultiplica Avalanche

정공의전자와는여기서 pn

ppnnn

nonn

nonn

dxxIdxxIxdI

IIM

IMWI

)()()(

)0(

)(

어떤지점 x에서의전류증분

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0 0

0

0

0

( . 9.28)

11

n

n n p p

n p

n

p n n p

p n

n

W W

n n n

Wn no no

n no

W

n

dI xI x I x

dx

I I x I x

dI xI x I

dx

dI xI

dx

dI x I W I I dx

M I Idx

I

M I I refer to Fig

dxM

로가정하면

이므로

그러므로 avalanche breakdown 전압은

Mn이 ∞에 접근할 때의 전압으로 정의

될 수 있으므로 avalanche breakdown

조건은다음과같이표현될수있음

01

W

dx

15

2

2 22 2

max max 2

2

max

2

maxmax

2 1

,

2

2

s bi Rd n d dn

s s s d

bi bi R R

d R

R critical R B

d

one sided p n junction

V VeN x e N eNE E x

e N

V V V V

eE N V

EV E E V V

eN

을고려하면

를무시하면

Breakdown voltage는임계전계(critical electric field)에서발생하며, 도핑농도와연관