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www.nanohub.org

NCN

Lecture20:Electrostaticsofpnjunctiondiodes

u amma s ra u amalam@purdue.edu

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Outline

1) Introductionto

p

n

junctions

2 Drawin banddia rams

3) Accuratesolutioninequilibrium

Ref.Semiconductor

Device

Fundamentals,

Chapter

5

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WhatisaDiodegoodfor.

solarcells GaAslasers

OrganicLED

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. .

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pnJunctionDevices

Symbols

AN

D

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TopicMap

signal

SignalDiode

Schottky

BJT/HBT

MOS

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DrawingBandDiagraminEquilibrium

( )+ = + D AD q p n N N equilibrium

N N N r g

t q= +

J

= =

1 = +JP P P

pr g

N N N

Smallsignaldn/dt~jtxnTransient fullsolution

P P Pqp E qD p= J

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PandNdopedMaterialSidebySide

NdopedE Pdoped

ConductionBand

EcEF

ConductionBand

EF

ValenceBandV

ValenceBand

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kko e ec ron rans er un er

ordinary circumstances!

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FormingaJunction

N N

= ND

Junction

x

2

0=

Ain n N

Actual Carrier

Concentrations

n-side p-side

n p

ln(n),

np

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Bulk Region Bulk RegionDepleted

Region x

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FormationofaJunction

DN AN

+

ln(ND)

ln(n)ln(p)

Q= n+N NqND DepletedRegion

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-qNA

xn xp

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SketchofElectrostatics

positionPotential

Vbi

E ieldmax = =os s

npo

A Dx N

q

E K xK N

q

position

p-n+ND-NA qND

Depleted

Region

10

Bulk Region

-qNABulk Region

xpxn

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SketchofElectrostatics

Potential xnxp

position

BandDiagram

position

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Outline

1) Introductiontopnjunctions

2) Drawingbanddiagrams

3) Analyticalsolutioninequilibrium

4) Banddiagramwithappliedbias

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ShortcuttoBanddiagram

Neutral Neutral

Space

Charge

DN AN

2C

EF

1

V

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BuiltinPotential:boundaryconditions@infinity

qVbi

11

Eg,2

2

1 1 2 2 2+ + = + gb ,i EqV

2 1 12 2 = +

g ,biEqV

( )2

1

1

2 2 = + + +

B B

Ag ,

V , C ,

Dk T ln k N

NT ln

N

NE

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( )2

2 1

12= + g , BB E

D

T

C

A

/ k

V , ,

k T ln

N N e

NN

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InterfaceBoundaryConditions

D

E=(D/ko)

pos on

xn xp

positionxn xp

01 0 21 2(0 ) (0 ) = = =EKD E DK

2

1

(0 ) (0 ) +=K

E E

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Displacementiscontinuousacrosstheinterface,fieldneednotbe..

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BuiltinvoltageforHomojunctions

Neutral Neutral

Space

Charge

DN AN

Vacuumlevel1bi

C

EV

EF

16

( )2

2 1

2 1

= + g , Bbi B E / k T V , C

A D

,

qV k T ln

N N e

N N2

= =g B

A AB BE / k T

C

D D

iV

k T ln k T ln

n N N e

NN NN

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AnalyticalSolutionofPoissonEquation

Depleted Region

Q=

n+N N

q D

qNAposition

xn xp

2d V + =

0 2S D Adx

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AnalyticalSolutionforHomojunctions

(Charge)p-n+ND-NA

qND -qNA ( )0

0

= Ds

nqNEk

x

xn xp

position

( )0

0

+ = As

p

qN

k

xE

Efield =D An pN x N x

positionPotential

( ) ( )0 0 += +

n p

bi

E x E xqVVbi

position 22

0 02 2

= + AD pns s

qN xxqN

k k

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DepletionRegionsinHomojunctions

Neutral Neutral

Space

Charge

xn xp

D A

=xN N x 2 k N

22A pnD

qNq xN x

pn

( )=

+D A

n

D

bixN Nq N

0 02 2 =bi s sk k ( )0

= +s

D

A A

p

D

bix N N Vq N

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HW: Solve the same problem for a hetero-junction

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CompleteAnalyticalSolution

N P

x

0=0=

-xp 0 xn

EV

xn

-xp -xp xn0x x x-xp xn

-qNA

..........

................

0

0

=

A

os

D

qNp

K

qNn

x x

dx x

E

............. ,0

os

p nx x x x x( )

00

= p

x xA

xS

qNd dx

K

E0

( )0

=

nx D

x xS

qd

Ndx

K

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..........

0

( ) ( ) 0

= + ppS

AqxN

x x x x

K

E ..........

0

( ) ( ) 0

= D nn

S

q x x x

Nx x

K

E

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Outline

1 Introductionto n unctiontransistors

2) Drawingbanddiagrams

4) Banddiagramwithappliedbias

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TopicMap

signal

SignalDiode

Schottky

BJT/HBT

MOS

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ApplyingBiastopnJunction

ln I

2

. us on m e

2. Ambipolar transport

6,7

3. High injection

-A

5. Breakdown

4

5

6. Trap-assisted R-G

7. Esaki Tunneling

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ForwardandReverseBias

DN AN

Forward Bias

DN

AN

Reverse Bias

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BandDiagramwithAppliedBias

( )D AD q p n N N +

= + Banddiagram(now)

N N N r g

t q= +

J

=

1P P P

pr g

= +J

N N N

Nextclass

P P Pqp E qD p= J

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ApplyingaBias:PoissonEquation

qVbi

EF-EV

EC-EF

bi

( ) in EF

q(Vbi-V)( )

( ) = p i

i

F

i

Ep x n e

C

-n

Fp-EV

-qV ( )2 = pn FFin p n e

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DepletionWidths

GND-V DN AN

xn xp

02

( )s

A bin

k NV Vx

=

22AD n p

qNqN xx =

D A pn D A D

0 02 2s skk ( )0

( )s D

bi

A A D

p Vq N N N V= +

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Whataboutheterojunctions?

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FieldsandDepletionatForward/ReverseBiasesVA0

NP

VA0

ElectricField

Position

atforwardbiases

Position

A0Significantincreaseof

peakfieldatreverse

bias

Potential VA0

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Conclusion

importanttopicyoulearninthiscourse.Banddiagram

sagrap ca wayo qu c yso v ng o ssonequat on.

2) If youconsistentlyfollowtherulesofdrawingband

diagrams,youwillalwaysgetcorrectresults.Tryto

followtherules,notguessthefinalresult.

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