PN Junction Diodes

PN Junction Diodes

Sung June Kim

[email protected]

http://helios.snu.ac.kr

Chapter 5.

Contents

q Drift

q Diffusion

q Generation-Recombination q Equations of State

2

q Preliminaries

• Junction Terminology/Idealized Profiles

Net doping profile

ü The step junction is an acceptable approximation to an ion- implantation or shallow diffusion into a lightly doped starting wafer

Step (abrupt) junction Linearly graded junction

• Poisson’s Equation

K

r Ñ × E = e

S 0

d

dx K r

= e E

1-Dimension

K_s is the semiconductor dielectric constant and e₀ is the permittivity of free space. r is the charge density (charge/cm³)

D A

( )

q p n N N

r = - + -

• Qualitative Solution

ü Let us assume an equilibrium conditions

dx to dE al

proportion

r is

ü It is reasonable to expect regions far removed from the metallurgical junction to be identical to an isolated semiconductor.

ü Under equilibrium conditions, the Fermi level is a constant

P-N Diode Junction Energy Band

P

E_c

E_f E_v

N

E_c E_f

E_v

Equilibrium P-N Junction

P N

E_c

E_f E_v

E_c E_f

E_v

V=0

Forward Biased P-N Junction

P N

E_c

E_f E_v

E_c E_f

E_v

V>0

Reverse Biased P-N Junction

P N

E_c

E_f E_v

E_c E_f

E_v

V<0

1 (

)

V E E

= - q -

E_ref

ü V versus x relationship must have the same functional form as the

“ upside-down” of Ec

dV

= - dx E

S 0

d

dx K r

= e E

ü The voltage drop across the junction under equilibrium conditions and the appearance of charge near the metallurgical boundary

ü Where does this charge come from?

ü Charge neutrality is assumed to prevail in the isolated, uniformly doped semiconductors

hole electron

Charge redistribution

Charge density Space charge

region or

depletion region

ü The build-up of charge and the associated electric field continues until the diffusion is precisely balanced by the carrier drift

ü The individual carrier diffusion and drift components must of course cancel to make J_N and J_P separately zero

• The Built-in Potential (V

)

üConsider a nondegenerately-doped junction

dV

= - dx E

n n

p p

( )

n p bi

( )

( ) ( )

x V x

x

dx

dV V x V x V

- -

- ò ^E = ò = - - =

ü Integrating

N n N

dn 0

J q n qD m dx

= E + =

ü Solving for and making use of the Einstein relationship, we obtain

E

N n

/ /

D dn dx kT dn dx

n q n

= - m = -

E

n n

p p

( )

n

bi ( )

p

ln ( )

( )

x n x

x n x

n x

kT dn kT

V dx

q n q n x

- -

é ù

= - = = ê ú

ê - ú

ë û

ò ^E ò

2 i

n D p

A

( ) , ( ) n

n x N n x

Q = - = N

A D

bi 2

i

ln N N V kT

q n

æ ö

= ç ÷

è ø

• The Depletion Approximation ü It is very hard to solve

(1) The carrier concentrations are negligible in

(2) The charge density outside the depletion region=0

) (

0

A D

s

N N

n k p

q dx

dE = - + -

e

n

p

x x

x £ £

-

S 0

D A

S 0

( )

d

dx K

q p n N N K

r e e

=

= - + -

E

ü Exact

ü Depletion Approximation

D A p n

S 0

p n

( ) . . .

0 . . . and

q N N x x x

d K

dx x x x x

e

ì - - £ £

@ í ï

ï £ - ³ î

E

q Quantitative Electrostatic Relationships

• Assumptions/definitions

• Step Junction with V

=0 ü Solution for r

ü Solution for

E

A p

D n

p n

. . . 0

. . . 0

0 . . . and

qN x x

qN x x

x x x x

r

- - £ £ ì

ï £ £ í

ï £ - ³ î

A S p

D S n

p n

/ . . . 0

/ . . . 0

0 . . . and

qN K x x

d qN K x x

dx x x x x

e e

0 0

- - £ £

ì

ï £ £ í

ï £ - ³ î

E

p

( ) A

0 S 0

x x

x

d qN dx

K e

¢ = -

¢

ò

^E ò

ü For the p-side of the depletion region

ü Similarly on the n-side

0 n

D

( ) S 0

x

x x

d qN dx

K e

¢ = - ¢

ò

E ò

D

n n

S 0

( ) qN ( ) . . . 0

x x x x x

K e

= - - £ £ E

A p D n

N x = N x

A

p p

S 0

( ) qN ( ) . . . 0

x x x x x

K e

= - + - £ £

E

ü Solution for V ( )

E = - dV dx /

ü With the arbitrary reference potential set equal to zero at x=-x_p and V_bi across the depletion region equilibrium conditions

p

bi n

0 at at

V x x

V V x x

= = -

= =

A

p p

S 0 D

n n

S 0

( ) . . . 0

( ) . . .

qN x x x x

dV K

qN

dx x x x x

K e e

ì + - £ £ ï ï

= í

ï - 0 £ £

ï î

ü For the p-side of the depletion region

p

( ) A

0 p

S 0

( )

V x x

x

dV qN x x dx

K e

¢ =

+ ¢ ¢

ò ò

A 2

p p

S 0

( ) ( ) . . . 0

2

V x qN x x x x

K e

= + - £ £

ü Similarly on the n-side of the junction

D 2

bi n n

S 0

( ) ( ) . . . 0

2

V x V qN x x x x

K e

= - - £ £

2 2

A D

p bi n

S 0 S 0

2 2

qN qN

x V x

K e ^{= -} K e

ü Solution for x_n and x_p

A p D n

N x = N x Q

1/ 2

S 0 A

n bi

D A D

2

( )

K N

x V

q N N N

e

é ù

= ê ú

ë + û

1/ 2

D n S 0 D

p bi

A A A D

2

( )

K

N x N

x V

N q N N N

é e ù

= ê ú

ë + û

1/ 2

S 0 A D

n p bi

A D

2K N N

W x x V

q N N

é e æ + ö ù

º + = ê ç ÷ ú

è ø

ë û

Depletion width

• Step Junction with V

¹0

ü When V

> 0, the externally imposed voltage drop lowers

the potential on the n-side relative to the p-side

ü The voltage drop across the depletion region, and hence the boundary condition at x=x_n, becomes V_bi-V_A

1/ 2

S 0 D

p bi A

A A D

2 ( )

( )

K N

x V V

q N N N

e

é ù

= ê - ú

ë + û

1/ 2

S 0 A

n bi A

D A D

2 ( )

( )

K N

x V V

q N N N

é e ù

= ê - ú

ë + û

1/ 2

S 0 A D

bi A

A D

2 K N N ( )

W V V

q N N

é e æ + ö ù

= ê ç ÷ - ú

è ø

ë û

• Examination/Extrapolation of Results

ü Depletion widths decrease under forward biasing and increase under reverse biasing

üA decreased depletion width when V

> 0 means less charge around the junction and a correspondingly smaller -field.

Similarly, the potential decreases at all points when V

>0 ü The Fermi level is omitted from the depletion region

E

A Fn

Fp

E qV

E - = -

pn junction energy band diagrams.

Summary

31

Summary

32