CH05 PN Junction Diodes [호환 모드]ocw.snu.ac.kr/sites/default/files/NOTE/3627.pdf · 2018-01-30 · Chapter 5. PN Junction Diodes 9Th b ildThe build-up ofh dth itd l tifild ti

Chapter 5. PN Junction Diodes

Chapter 5.

PN Junction Electrostatics

Sung June [email protected]

http://helios.snu.ac.krp


Subjectj

Analysis of Field and potential distribution in the junction.

- Quantitative Electrostatic Relationships


PreliminariesPreliminaries

• Junction Terminology/Idealized Profiles

N t d i filNet doping profile


Step (abrupt) junction Linearly graded junction

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


• Poisson’s Equation

S 0Kρε

∇ ⋅ =ES 0

ddx K

ρε

=E1-Dimension

Ks is the semiconductor dielectric constant and ε0 is the permittivity of free space. ρ is the charge density (charge/cm3)of free space. ρ is the charge density (charge/cm )

D A( )q p n N Nρ = − + −

dxdEtoalproportionisρ

• Qualitative SolutionLet us assume an equilibrium conditions


It i bl t t i f d f th t ll i lIt is reasonable to expect regions far removed from the metallurgical junction to be identical to an isolated semiconductor.


U d ilib i diti th F i l l i t tUnder equilibrium conditions, the Fermi level is a constant


P-N Diode Junction Energy Bandgy

P N

Ec Ec

Ef

Ef

Ev Evv v


Equilibrium P-N Junctionq

V 0V=0

P N

Ec

E

Ec

EEf

Ev

Ef

Ev


Forward Biased P-N Junction

V 0V>0

P N

Ec

E

Ec

EEf

Ev

Ef

Ev


Reverse Biased P-N Junction

V 0V<0

P N

Ec

E

EcEf

Ev

Ef

Ev


V l ti hi t h th f ti l f thV versus x relationship must have the same functional form as the “ upside-down” of Ec

1 ( )c refV E Eq

= − −

Eref

q


dVdVdx

= −E

d ρ=

E

S 0dx K ε

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

Where does this charge come from?


Ch t lit i d t il i th i l t d if lCharge neutrality is assumed to prevail in the isolated, uniformly doped semiconductors

hole electronCharge redistribution

Space chargeSpace charge region ordepletion region

Charge density


Th b ild f h d th i t d l t i fi ld ti tilThe build-up of charge and the associated electric field continues until the diffusion is precisely balanced by the carrier drift

The individual carrier diffusion and drift components must of course pcancel to make JN and JP separately zero

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


dVdx

= −Edx

Integrating

n n

p p

( )

n p bi( )( ) ( )

x V x

x V xdx dV V x V x V

− −− = = − − =∫ ∫E

p p

N N 0dnJ q n qDμ= + =EN n N 0J q n qDdx

μ +E

Solving for and making use of the Einstein relationship, we obtainEg g p,E

N / /D dn dx kT dn dxn q nμ

= − = −En n q nμ


n n( ) n( )lx n x n xkT dn kTV d

⎡ ⎤⎢ ⎥∫ ∫E

p p

nbi ( )

p

( )ln( )x n x

V dxq n q n x− −

= − = = ⎢ ⎥−⎢ ⎥⎣ ⎦

∫ ∫E

22i

n D pA

( ) , ( ) nn x N n xN

= − =QA

A Dl N NkTV⎛ ⎞⎜ ⎟A D

bi 2i

lnVq n

= ⎜ ⎟⎝ ⎠


• The Depletion ApproximationIt is very hard to solve

)( AD NNnpqd−+−==

ρE )(00

ADss

NNnpKKdx

+εε

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

np xxx ≤≤


Exact

S 0

ddx K

ρε

=E

S 0

D AS 0

( )q p n N NK ε

= − + −S 0

D l ti A i tiDepletion Approximation

D A( ) . . .q N N x x xd⎧ − − ≤ ≤⎪E D A p n

S 0

p n

( ) . . .

0 . . . and

N N x x xd Kdx x x x x

ε ≤ ≤⎪≅ ⎨

⎪ ≤ − ≥⎩

E


Quantitative Electrostatic RelationshipsQuantitative Electrostatic Relationships

• Assumptions/definitions


• Step Junction with VA=0Solution for ρ

A p

D n

. . . 0

. . . 0

qN x x

qN x xρ

− − ≤ ≤⎧⎪

≤ ≤⎨ D n

p n0 . . . and q

x x x xρ ⎨

⎪ ≤ − ≥⎩

Solution for E

/ 0qN K x xε− − ≤ ≤⎧ A S p

D S n

/ . . . 0

/ . . . 0

qN K x xd qN K x xdx

ε

ε0

0

− − ≤ ≤⎧⎪

≤ ≤⎨⎪

E

p n0 . . . and dx

x x x x⎪ ≤ − ≥⎩


F th id f th d l ti i

( ) Ax x qNd dx′ = − ′∫ ∫E

E

For the p-side of the depletion region

p0S 0

xd dx

K ε−∫ ∫E

A( ) ( )qNE A

p pS 0

( ) ( ) . . . 0qNx x x x xK ε

= − + − ≤ ≤E

Similarly on the n-side

n0 Dx qNd d′ ′∫ ∫E D( )

S 0x x

qd dxK ε

′ = − ′∫ ∫EE

qNDn n

S 0

( ) ( ) . . . 0qNx x x x xK ε

= − − ≤ ≤E

N NA p D nN x N x=


S l ti f V ( )/dV dESolution for V ( )/dV dx= −E

A ( ) 0qN x x x x⎧ + ≤ ≤⎪A

p pS 0

D

( ) . . . 0x x x xKdVqNdx

ε+ − ≤ ≤⎪⎪= ⎨

⎪ Dn n

S 0

( ) . . . qNdx x x x xK ε

⎪ − 0 ≤ ≤⎪⎩

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

ilib i ditiequilibrium conditions

p0 at V x x= = −

bi nat V V x x= =


F th id f th d l ti iFor the p-side of the depletion region

( ) Ap( )

V x x qNdV x x dxK

′ = + ′ ′∫ ∫p

p0S 0

( )x K ε−∫ ∫

2A( ) ( ) 0qNV x x x x x= + − ≤ ≤p pS 0

( ) ( ) . . . 02

V x x x x xK ε

= + ≤ ≤

Similarly on the n-side of the junctionSimilarly on the n side of the junction

2Dbi( ) ( ) . . . 0qNV x V x x x x= − − ≤ ≤bi n n

S 0

( ) ( ) . . . 02

V x V x x x xK ε

≤ ≤

2 2A DqN qNx V x @ 0A Dp bi n

S 0 S 02 2q qx V xK Kε ε

= − @ x=0


S l ti f dSolution for xn and xp

A p D nN x N x=Q

1/ 2

S 0 An bi

2( )

K Nx VN N N

ε⎡ ⎤= ⎢ ⎥+⎣ ⎦D A D( )q N N N⎢ ⎥+⎣ ⎦

/1/ 2

S 0D n Dp bi

A A A D

2( )

KN x Nx VN q N N N

ε⎡ ⎤= ⎢ ⎥+⎣ ⎦A A A D( )q⎣ ⎦

1/ 22K N N⎡ ⎤⎛ ⎞S 0 A D

n p biA D

2K N NW x x Vq N N

ε⎡ ⎤⎛ ⎞+≡ + = ⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦Depletion width


• Step Junction with VA≠0When VA > 0, the externally imposed voltage drop lowers

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


Th lt d th d l ti i d h thThe voltage drop across the depletion region, and hence the boundary condition at x=xn, becomes Vbi-VA

1/ 2

S 0 D2 ( )K N V Vε⎡ ⎤⎢ ⎥

S 0 Dp bi A

A A D

2 ( )( )

K Nx V Vq N N N

ε⎡ ⎤= −⎢ ⎥+⎣ ⎦

1/ 2

S 0 An bi A

2 ( )( )

K Nx V Vq N N N

ε⎡ ⎤= −⎢ ⎥+⎣ ⎦D A D( )q N N N+⎣ ⎦

1/ 2

S 0 A Dbi A

A D

2 ( )K N NW V Vq N N

ε⎡ ⎤⎛ ⎞+= −⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦A Dq N N⎝ ⎠⎣ ⎦


• Examination/Extrapolation of ResultsDepletion widths decrease under forward biasing and increase

under reverse biasingunder reverse biasing A decreased depletion width when VA > 0 means less charge

around the junction and a correspondingly smaller -field. ESimilarly, the potential decreases at all points when VA >0

The Fermi level is omitted from the depletion region

AFnFp qVEE −=−



j ti b d dipn junction energy band diagrams.

Documents

CH05 PN Junction Diodes [호환 모드]ocw.snu.ac.kr/sites/default/files/NOTE/3627.pdf · 2018-01-30 · Chapter 5. PN Junction Diodes 9Th b ildThe build-up ofh dth itd l tifild ti