PN JUNCTION THEORY - Polytech Nice Sophia ...users.polytech.unice.fr/~lorenz/PN_junction_eng.pdfPN Homojunction • Non linear device • rectifier devices (composants redresseur)

PN JUNCTION THEORY

1

Prof. Philippe LORENZINIPolytech-Nice Sophia

PN Homojunction• Non linear device• rectifier devices (composants redresseur)• 2 devices reach the same results:

• PN Junction(this chapter)• Schottky barrier or Metal / SC contact (next chapter)

2

The Junction’s formation mechanism

3

Flat Fermi level: No current / thermal equilibrium

•PN Junction at equilibrium1st Step: diffusion mechanism

2nd Step: built in Electric Field appears compensates diffusion forces

E int

e-h pair’srecombination

« built in potential VB i»

4

• Definition : Potential drop between N and P regions

PNbi VVV

0)()()()(

dxxdpDxExpexJ pPP

dxxdp

xpxE

Dp

p )()(

1)(

dxxdp

xpdxxdV

kTe )(

)(1)(

)ln(n

pbi p

pe

kTV

Holes current equation:

or or

Integrating from P to N region:

finally: )nNN(

ekTVV

i

DAbiD 2ln )

nNN(

ekTVV

i

DAbiD 2ln

•Field, potential and Space Charge width(1)• Poisson’s equation:

5

sc

xdx

xVd )()(

2

2

In N and P region:

Dsc

Nedx

xVd

2

2 )(NWx0

Asc

Nedx

xVd

2

2 )( 0 xWP

-WP -WN

6

Electric Field E(x)

)()( Nsc

Dn WxeNxE

)()( P

sc

AP WxeNxE

Continuity of Field on x=0:

PAND WNWN

sc

PA

sc

NDM

WeNWeNE

-WP -WN

•Field, potential and Space Charge width(2)

7

Built in potential V(x)

nNsc

Dn VWxeNxV 2)(

2)(

pPsc

Ap VWxeNxV 2)(

2)(

Depletion layer (ZCE)

sc

pA

sc

nDdpn

WeNWeNVWVWV

22)()(

22

dDAA

Dscdp V

NNNN

eVW

)(2

)(

dDAD

Ascdn V

NNNN

eVW

)(2)(

dDA

ADscd V

NNNN

eVW

2)( -WP WN

•Field, potential and Space Charge width(3)

WARNING: WHEN A VOLTAGE V ISAPPLIED ON P SIDE, VBI HAVE TO

BE REPLACED BY VBI - V

8

BIASED PN JUNCTION

• When a positive voltage is applied on p side, the equilibrium is destroyed and a net current can flow

9

simplifying assumptions : Depletion layer with no free carriers (e‐ and h+) Low injection Boltzmann’s approximation Drop voltage only in depletion layer No generation‐recombination mechanisms present

• Foward Biasing• Positive voltage on P• lowering of built in

potential• Diffusion mechanism

dominates• High current

10

BIASED PN JUNCTION

• Forward biasing• Lowering of built in Field due to

opposite external field• Electrons injected from N to P

regions: minoritary carriers injection

• High current due to full « reservoir »

11

Fdiff e-

Fdiff h+

BIASED PN JUNCTION

Forward Biasing

12

Eext

Jonction PN sous polarisation

• Reverse Biasing• Global Electric Field increases(External Field added to built in Field)

• Injection of electrons from P to N and holes injection from N to P: majority carriers injection

• Low current ( leak current) due to empty « reservoir »

• Reverse Biasing• Global Electric Field increases(External Field added to built in Field)

• Injection of electrons from P to N and holes injection from N to P: majority carriers injection

• Low current ( leak current) due to empty « reservoir »

13

Fconde‐

Fcond h+

PN Junction under biasing

14

•Boltzmann’s Approximation: The Boltzmann approximation is to say that the resulting current being small compared with the components of this current, we consider that we are still in quasi equilibrium and therefore that the current's equation is still valid by replacing Vbi by Vbi ‐VA :

At equilibrium, null current two components compensatebetween it. Taken separately, the magnitude of these components 104 A / cm² (ie 1A for typical diode) and at low injection I is of the order of few mA (max 10mA)

dxxdp

xpdxxdV

kTe )(

)(1)(

Density of carriers injected to the limits of depletionlayer

• If Va=0

• If Va

15

)exp()(kTeV

pp

pWp bi

p

n

p

N

0 )exp())(exp(')('kTeV

pp

kTVVe

pp

pWp A

p

nAbi

p

n

p

N

)exp()exp('2

kTeV

Nn

kTeV

pp A

D

iAnn )exp()exp('

2

kTeV

Nn

kTeVnn A

A

iApp

)exp(** 2''

kTeVnnppn a

innpp

Holes density injected versus bias voltage Va

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7104

105

106

107

108

109

1010

1011

1012

1013

1014

1015

1016

1017

Na= 1E17 cm-3

Vd=0.7 V

P'(W

n) (c

m-3)

Va (V) 16

Minority carriers distribution in neutral region

• Due to gradient concentration, carriers will diffuse and produce diffusion current (no electric field in neutral region!)

• Distribution is geometrydependant

• Discrimatory parameter : length diffusion LDn,p of electrons and holes and neutral region widths dn,p

17

-WP 0 WN

Minority carriers distribution in neutral region

• Long regions ( )

18

nppn Ld ,,

pNa

LxWkTeV

nn eeppxp /)()1()(' pNa

LxWkTeV

nn eeppxp /)()1()('

n

aLWpxkT

eV

pp eennxn /)()1()(' n

aLWpxkT

eV

pp eennxn /)()1()('

Short (narrow) regions ( )nppn Ld ,,

))(1()(' xxedp

pxp ckTeV

n

nn

a

))(1()(' xxedp

pxp ckTeV

n

nn

a

)')(1()(' xxedn

nxn ckTeV

p

pp

a

)')(1()(' xxedn

nxn ckTeV

p

pp

a

General case

p

ckTeV

p

n

nn L

xxshe

Ld

sh

ppxp

a

)1()(

)('

n

ckTeV

n

p

pp L

xxshe

Ld

sh

nnxn

a '

)1()(

)('

Minority currents in neutral region

Knowing minority distribution we are in position to calculate the current wich is a diffusion current (very low field in neutral region):

19

dxxdpeDxJ pp)()( dx

xdneDxJ nn)()(

Hypothesis : no G‐R process in depletion layer (ZCE)

)()()()()( pnnppnpp WJWJWJWJVJ

We get the classical and well known diode equation:

)1()( / kTeVS eJVJ

Js is the theorical saturation current or reverse current

20

-WP 0 WN

pA

ni

nD

PiS dN

DendNDen

J22

Long region

nA

ni

PD

PiS LN

DenLNDen

J22

General case

)()(

22

n

pnA

ni

P

nPD

PiS

Ld

thLN

Den

Ld

thLN

DenJ

Minority currents in neutral region

Short (Narrow) region

• The model is refinedwe take into account G‐R process in depletion layer

• Well understood mechanism (Shockley‐Read)

The real diode: genération‐recombinaison mechanism in depletion layer

21

npnnpn

ri

i

2

1 2

We know that If we suppose np constant in depleted region and np >>

(in forward bias) , the rate r is max when n=p, and it canbe rewritten

)exp()()()()( 2

kTeVanWnWpWnWp iPPNN

2in

kTeVnr ai

2exp

2max

• Generation‐Recombinaison current in depletionlayer can be expressed as:

22

N

P

W

WGRpnnn rdxeJWJWJ )()(

For reverse biasing ( ), we have a negative rate ( ) . It means dans wehave a net generation process

2inpn

02

in

r

For forward bias, rmax=cte>0 and the current isdue to recombinations.


• Finally GR current present in depletion layer can beexpressed as:

23

If we take into account the diffusion current, we get:

1)

2exp(0

kTeV

JJ aGRGR

1)

2exp(1)exp()( 0

kTeV

JkTeV

JVJ aGR

aSa

Ti

GR Wen

J2

0

The expression above can be generalized by introducingIdeality factor:

1)exp()( 0 nkT

eVJVJ a


Reverse biasing: Junction breakdown

• Thermal Effect (Narrow bandgap)• Zener Effect:

• direct flowing from VB to CB by tunnel effect (0), if electric fieldabove critical Field Ec

• Avalanche Effect:• before« tunneling », hot electrons

(accelerated electrons) excite by impact ionisation electrons from VB to CB (1,2,3) etc….

• « Punchtrough »

24

B

CBD eN

EV2

. 2

tunnelTunnel

Avalanche

Small signal model of the diode: capacitances

• Capacitance associated to charges• 2 types of charges present in the junction

• Fixed charges (ionised dopants) in depletion layer• Mobiles (e- et h+) injected when forward biasing

• 2 types of capacitance• Junction (or Transition) Capacitance • Charge Storage (or diffusion) Capacitance

25

Junction capacitance

26

Simply associated to charges present in depletion layer

dVdQCC jT NDPA WeANWeANQ

TDA

DA

ADjT W

ANN

NNVV

eACC

)()(2

2

or:

Charge Storage (diffusion) capacitance

• Reflects the delay between the voltage and current

• Associated with charges injected into the neutral regionsTraduit le retard entre la tension et le courant

27

PPSp JQ

nnSn JQ

C

N

X

W nSp dxpxpeAQ ))('(

Holes density in excess presentin N region

)(

1)coth())0('(

P

nP

nPnSp

Ld

shLd

LppeQ

28


• We can transform the previous expression by:

29

)( NPSp WJQ avec

)(

11

P

nP

Ld

ch

Time expression can be simplified, depending of the neutral geometry:

Narrow diode: transit time

Long diode: lifetimeP

nt D

d2

2

P


• The previous expression, valid in N region, can begeneralized in P region and we obtain for the whole diode:

30

)()( )()( NppPnnSpSnS WJWJQQQ

If we use: dVdQCC S

dS

)( )()( ppnnSpSndS JJKkTeCCCC

K : Geometry dependant Factor (2/3 narrow)(1/2 long)


Equivalent circuit of foward diode

31

rd : diode resistance (dynamic resistance) given by the differential slope of the I‐V characteristics

rs : serie resistance of neutral region n and pIe

kTrd1

(from Neamen)

Large signal switching of diode

32

As long as the stored charge is positive forward bias diode voltage acrossdiode is small (few 10 mV)

)1(' kTeV

nnn

a

eppp

sd Storage time ie nN pWp )('

Wn

• Storage time :The main problem in minoritary carriers devices:• Storage time:

• Rise (or fall) time :

• Cj: mean value of capacitance between zero and –V2

33

)1ln()1ln(

mf

f

m

fpsd II

III

mf

fjFf II

IRC

avec 1

3.2

Large signal switching of diode

Diode Tunnel – diode Backward

34

324

exp*

bt

emaT

pe

a

pe

apet V

VVV

II 1exp

(a)

(e)

(d)

(c)

(b)

PN JUNCTION THEORY - Polytech Nice Sophia ...users.polytech.unice.fr/~lorenz/PN_junction_eng.pdfPN Homojunction • Non linear device • rectifier devices (composants redresseur)

Documents