PN Junction Diode: I-V Characteristicsocw.snu.ac.kr/sites/default/files/NOTE/7012.pdf ·...

Chapter 6. PN Junction Diode : I-V Characteristics

PN Junction Diode: I-V Characteristics

Sung June [email protected]

http://helios.snu.ac.kr

Chapter 6.


Contents

q Qualitative Derivation

q Quantitative Solution Strategy

q Quasineutral Region Considerations

q Depletion Region Considerations

q Boundary Conditions

2


q The Ideal Diode Equation

• Qualitative Derivationü Equilibrium situation

ü The I-V characteristics of the ideal diode are modeled by the ideal diode equation à qualitative and quantitative derivation

potentialhill

high-energy carrierdiffusiondrift

balance

E


ü Forward bias situation à a lowering of the potential hill

ü The same number of minority carriers are being swept

ü More majority carriers can surmount the hill à IN and IPà Iü The number of carriers that have sufficient energy to surmount the barrier goes up exponentially with VA à

exponential increase of the forward current


ü The barrier increase reduces the majority carrier diffusion to a negligible levelü The p-side electrons and n-side holes can wander into the depletion region and be swept to the other side àreverse I (nàp)

ü Reverse bias situation à an increase of the potential hill

ü Being associated with minority carriers, the reverse bias current is expected to be extremely small


ü The minority carrier drift currents are not affected by the height of the hill (The situation is similar to a waterfall)ü If the reverse bias saturation current is taken to be –I0, the overall I-Vdependence is

I-V characteristic

A ref/0 ( 1)V VI I e= -

Rectification

qkTVref =


ü Whenever an electron on the p-side moves to the n-side, it is replaced by an electron generated through one of the R-G centers

ohmic

ohmicminority

minority

excess majority carriers à local

excess majority carriers à local E

E

Excess carriers move to the contact with a relaxation time à greatly fast

recombination

ü Current componentDepletion region : electrons and holesp-region (far) : holesn-region (far) : electrons


• Quantitative Solution Strategyü Basic assumptions

(1) Steady state conditions(2) A nondegenerately doped step junction(3) One-dimensional(4) Low-level injection(5) GL=0

N P( ) ( )J J x J x= +

N n N

P p P

dnJ qu n qDdxdpJ qu p qDdx

= +

= +

E

E

AJI =


• Quasineutral Region Considerations

2p p p

N L2n

2n n n

P L2p

n n nD G

t xp p pD Gt x

t

t

¶D ¶ D D= - +

¶ ¶

¶D ¶ D D= - +

¶ ¶

2p p

N p2n

2n n

P n2p

0 . . .

0 . . .

d n nD x x

dxd p pD x xdx

t

t

D D= - £ -

D D= - ³

and low-level injection à minority carrier diffusion equations0@E


ü Since and dn0/dx=dp0/dx=0 in the quasineutral regions0@E

pN N p

nP P n

. . .

. . .

d nJ qD x x

dxd pJ qD x xdx

D= £ -

D= - ³

0 p

0 n

n n np p p

= + D

= + D

Q

ü We can only determine JN(x) in the quasineutral p-region and JP(x) in the quasineutral n-region• Depletion Region Considerations

N P

thermal thermalR G R G

1 10 , 0dJ dJn pq dx t q dx t

- -

¶ ¶= + = - +

¶ ¶

processesotherGRthermalP

processesotherGRthermalN

tp

tpJ

qtp

tn

tnJ

qtn

¶¶

+¶¶

+×Ñ-=¶¶

¶¶

+¶¶

+×Ñ=¶¶

-

-

1

1


ü Suppose that thermal recombination-generation is negligible throughout the depletion region;ü à JN and JP are constants inside the depletion region

thermal R-G thermal R-G/ | / | 0n t p t¶ ¶ = ¶ ¶ =/ 0 and / 0N PdJ dx dJ dx= =

N p n N p

P p n P n

( ) ( )

( ) ( )

J x x x J xJ x x x J x

- £ £ = -

- £ £ =

N p P n( ) ( )J J x J x= - +


• Boundary Conditionsü At the Ohmic Contacts

The ideal diode is usually taken to be a “wide-base” diodeThe contacts may effectively be viewed as being positioned at x= ±¥

ü At the Depletion Region EdgesUnder nonequilibrium conditions:

Equilibrium conditions Nonequilibrium conditions

( )( ) 0

0

=+¥®D

=-¥®D

xpxn

n

p

( ) ( ) kTFEi

kTEFi

piiN enpenn // , -- ==


nL� pL�N P( ) /2 F F kT

inp n e -=

A /2p n. . .qV kT

inp n e x x x= - £ £

A

FpFPN

qV

EEFFN

=

-£-

If the equal signal is assumed to hold throughout the depletion region

: law of junction


ü Evaluating the equation at the p-edge

A /2p p p A i( ) ( ) ( ) qV kTn x p x n x N n e- - = - =

A

2/i

pA

( ) qV kTnn x eN

- =

A

2/i

p pA

( ) ( 1)qV kTnn x eN

D - = -

ü Similarly,


A

2/i

n nD

( ) ( 1)qV kTnp x eN

D = -


• Derivation Properü The origin of coordinates is shifted to the n-edge of the depletion region

2'n n

P '2p

0 . . . 0d p pD xdx tD D

= - ³

'n ( ) 0p xD ® ¥ =

A

2/' i

nD

( 0) ( 1)qV kTnp x eN

D = = -

ü Boundary conditions

ü The general solution

P P'/ '/'n 1 2

'

( )

. . . 0

x L x Lp x Ae A ex

-D = +

³ P P pL D t=Q


ü A2 à 0 because exp(x’/Lp) à ¥ as x’ à ¥ü With , A1=Dpn(x’=0)

A P

2/ '/' 'i

nD

( ) ( 1) . . . 0qV kT x Lnp x e e xN

-D = - ³

A P

2/ '/' 'n P i

P P 'P D

( ) ( 1) . . . 0qV kT x Ld p D nJ x qD q e e xdx L N

-D= - = - ³

ü On the p-side of the junction with the x’’-coordinate.

NA

2"//" "i

pA

( ) ( 1) . . . 0x LqV kTnn x e e xN

-D = - ³

NA

2p "//" "N i

N N "N A

( ) ( 1) . . . 0x LqV kTd n D nJ x qD q e e xdx L N

-D= - = - ³


ü The current densities at the depletion region edges,

A

2/" N i

N p NN A

( ) ( 0) ( 1)qV kTD nJ x x J x q eL N

= - = = = -

A

2/' P i

P n PP D

( ) ( 0) ( 1)qV kTD nJ x x J x q eL N

= = = = -

A

2 2/N i P i

N A P D

( 1)qV kTD n D nI AJ qA eL N L N

æ ö= = + -ç ÷

è ø

A /0

2 2N i P i

0N A P D

( 1)qV kTI I e

D n D nI qAL N L N

= -

æ öº +ç ÷

è ø

Ideal diode equation or Shockley equation


• Examination of Results


ü Carrier currentsü The total current density is constantü The majority-carrier current densities are obtained by graphically subtracting the minority-carrier current densities from the total current density


ü Carrier concentrationsü Forward biasing increases the concentration

Reverse decreases

ü Under the low-level injection, the majority carrier concentrations in these regions are everywhere approximately equal to their equilibrium values


ü Under reverse biasing the depletion region acts like a “sink” for minority carriersü Larger reverse biases have little effect

NA > ND


6.2.2 Reverse-Bias Breakdown


q Zener Process• Tunneling

– The particle energy remains constant during the process.ü The particle and the barrier are not damaged.

(1) There must be filled states on one side and empty states on the other side at the same energy.

(2) d must be very thin.


Reverse bias↑ ⇒ # of filled valence electrons placed opposite empty conduction-band states↑ ⇒ current↑

6.2.3 The R-G Currentü A current far in excess of that predicted by the ideal theory exists

at small forward bias and all reverse biases.← thermal recombination-generation in the depletion region


Ec

Ef

Ev

Ec

Ef

Ev

VR

IR

VR = 0 V (Equilibrium)


Ec

Ef

Ev

Ec

Ef

Ev

VR

IR

h+

VR < 0 VVR = 0 V

e-


VR

IR

Ec

Ef

Ev

Ec

Ef

Ev

e-

e-e-

e-e-

VR << 0 V (Zener Breakdown, Tunneling)VR = 0 V


ü Reverse bias ⇒ , ⇒ thermal generation ü Forward bias ⇒ , ⇒ recombination(1) The net R-G rate is the same for electrons and holes.(2) For every electron-hole pair created or destroyed per second, one

electron per second flows into or out of the diode contacts.

0nn < 0pp <0nn > 0pp >


GRDIFF

kTEEn

kTEEp

in

ip

iGR

x

xnp

iGR

np

i

GRthermal

GRthermal

x

xGR

III

eenp

nn

WqAnI

pn

dxppnn

nnpqAI

ppnnnnp

tn

dxtnqAI

TiiT

n

p

n

p

-

--

-

--

-

---

+=

+=+º

-=

®®

+++-

=

+++-

-=¶¶

¶¶

-=

ò

ò

)(21)(

21

2

0,0

)()(

)()(

/)(/)(110

0

11

2

11

2

ttttt

t

tt

tt kTEEi

kTEEi

Ti

iT

enp

enn/)(

1

/)(1

-

-

º

º


Summary

31


Summary

32

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