pn junction diodesocw.snu.ac.kr/sites/default/files/note/7011.pdf · 2018-01-30 · pn junction...

Chapter 5. PN Junction Diodes

PN Junction Diodes

Sung June [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

S 0Kre

Ñ × =ES 0

ddx K

re

=E1-Dimension

Ks 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 Nr = - + -

• Qualitative Solutionü Let us assume an equilibrium conditions

dxdEtoalproportionisr


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

Ec

Ef

Ev

N

Ec

Ef

Ev


Equilibrium P-N Junction

P N

Ec

Ef

Ev

Ec

Ef

Ev

V=0


Forward Biased P-N Junction

P N

Ec

Ef

Ev

Ec

Ef

Ev

V>0


Reverse Biased P-N Junction

P N

Ec

Ef

Ev

Ec

Ef

Ev

V<0


1 ( )c refV E Eq

= - -

Eref

ü V versus x relationship must have the same functional form as the “ upside-down” of Ec


dVdx

= -E

S 0

ddx K

re

=E

ü The voltage drop across the junction under equilibrium conditions and the appearance of charge near the metallurgical boundaryü Where does this charge come from?


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

hole electronCharge redistribution

Charge density

Space charge region ordepletion region


ü The build-up of charge and the associated electric field continues until the diffusion is precisely balanced by the carrier driftü The individual carrier diffusion and drift components must of course cancel to make JN and JP separately zero

• The Built-in Potential (Vbi)üConsider a nondegenerately-doped junction


dVdx

= -E

n n

p p

( )

n p bi( )( ) ( )

x V x

x V xdx dV V x V x V

- -- = = - - =ò òE

ü Integrating

N n N 0dnJ q n qDdx

m= + =E

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

N

n

/ /D dn dx kT dn dxn q nm

= - = -E


n n

p p

( ) nbi ( )

p

( )ln( )

x n x

x n x

n xkT dn kTV dxq n q n x- -

é ù= - = = ê ú

-ê úë ûò òE

2i

n D pA

( ) , ( ) nn x N n xN

= - =Q

A Dbi 2

i

ln N NkTVq n

æ ö= ç ÷

è ø


• The Depletion Approximationü It is very hard to solve

(1) The carrier concentrations are negligible in (2) The charge density outside the depletion region=0

)(0

ADs

NNnpkq

dxdE

-+-=e

np xxx ££-


S 0

D AS 0

( )

ddx K

q p n N NK

re

e

=

= - + -

E

ü Exact

ü Depletion Approximation

D A p nS 0

p n

( ) . . .

0 . . . and

q N N x x xd Kdx x x x x

eì - - £ £ï@ íï £ - ³î

E


q Quantitative Electrostatic Relationships

• Assumptions/definitions


• Step Junction with VA=0ü Solution for r

ü Solution for E

A p

D n

p n

. . . 0

. . . 00 . . . and

qN x xqN x x

x x x xr

- - £ £ìï

£ £íï £ - ³î

A S p

D S n

p n

/ . . . 0

/ . . . 00 . . . and

qN K x xd qN K x xdx

x x x x

e

e0

0

- - £ £ìï

£ £íï £ - ³î

E


p

( ) A0

S 0

x x

x

qNd dxK e-

¢ = - ¢ò òE

E

ü For the p-side of the depletion region

ü Similarly on the n-side

n0 D( )

S 0

x

x x

qNd dxK e

¢ = - ¢ò òEE

Dn n

S 0

( ) ( ) . . . 0qNx x x x xK e

= - - £ £E

A p D nN x N x=

Ap p

S 0

( ) ( ) . . . 0qNx x x x xK e

= - + - £ £E


ü Solution for V ( )/dV dx= -E

ü With the arbitrary reference potential set equal tozero at x=-xp and Vbi across the depletion region equilibrium conditions

p

bi n

0 at

at

V x xV V x x

= = -

= =

Ap p

S 0

Dn n

S 0

( ) . . . 0

( ) . . .

qN x x x xKdVqNdx x x x xK

e

e

ì + - £ £ïï= íï - 0 £ £ïî


ü For the p-side of the depletion region

p

( ) Ap0

S 0

( )V x x

x

qNdV x x dxK e-

¢ = + ¢ ¢ò ò2A

p pS 0

( ) ( ) . . . 02qNV x x x x xK e

= + - £ £

ü Similarly on the n-side of the junction

2Dbi n n

S 0

( ) ( ) . . . 02qNV x V x x x xK e

= - - £ £

2 2A Dp bi n

S 0 S 02 2qN qNx V xK Ke e

= - @ x=0


ü Solution for xn and xp

A p D nN x N x=Q

1/ 2

S 0 An bi

D A D

2( )

K Nx Vq N N N

eé ù= ê ú+ë û

1/ 2

S 0D n Dp bi

A A A D

2( )

KN x Nx VN q N N N

eé ù= ê ú+ë û

1/ 2

S 0 A Dn p bi

A D

2K N NW x x Vq N N

eé ùæ ö+º + = ê úç ÷

è øë ûDepletion width


• Step Junction with VA¹0ü When VA > 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=xn, becomes Vbi-VA

1/ 2

S 0 Dp bi A

A A D

2 ( )( )

K Nx V Vq N N N

eé ù= -ê ú+ë û

1/ 2

S 0 An bi A

D A D

2 ( )( )

K Nx V Vq N N N

eé ù= -ê ú+ë û

1/ 2

S 0 A Dbi A

A D

2 ( )K N NW V Vq N N

eé ùæ ö+= -ê úç ÷

è øë û


• Examination/Extrapolation of Resultsü Depletion widths decrease under forward biasing and increase under reverse biasing üA decreased depletion width when VA > 0 means less charge around the junction and a correspondingly smaller -field. Similarly, the potential decreases at all points when VA >0ü The Fermi level is omitted from the depletion region

E

AFnFp qVEE -=-


pn junction energy band diagrams.


Summary

31


Summary

32

pn junction diodesocw.snu.ac.kr/sites/default/files/note/7011.pdf · 2018-01-30 · pn junction...

Documents