Chapter 6. - PN Junction Diode: IV Characteristics

PN Junction Diode: I-V Characteristics

Sung June Kim

kimsj@snu.ac.kr

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 Derivation

ü Equilibrium situation

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

potential hill

high-energy carrier diffusion

drift

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 I_N and I_P à I

ü The number of carriers that have sufficient energy to

surmount the barrier goes up exponentially with V_A à

exponential increase of the forward current

ü The barrier increase

reduces the majority carrier diffusion to a negligible level ü The p-side electrons and n-side holes can wander into the depletion region and be swept to the other side à reverse 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-V dependence is

I-V characteristic

A/ ref

0

(

^{V V}

1)

I = I e -

Rectification

q

V

_ref

= kT

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 ohmic

minority

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 component

Depletion region : electrons and holes p-region (far) : holes

n-region (far) : electrons

• Quantitative Solution Strategy ü Basic assumptions

(1) Steady state conditions

(2) A nondegenerately doped step junction (3) One-dimensional

(4) Low-level injection (5) G_L=0

N

( )

P

( )

J = J x + J x

N n N

P p P

J qu n qD dn

dx J qu p qD dp

dx

= +

= +

E E

AJ

I =

• Quasineutral Region Considerations

2

p p p

N 2 L

n 2

n n n

P 2 L

p

n n n

D G

t x

p p p

D G

t x

t t

¶D ¶ D D

= - +

¶ ¶

¶D ¶ D D

= - +

¶ ¶

2

p p

N 2 p

n 2

n n

P 2 n

p

0 . . .

0 . . .

d n n

D x x

dx

d p p

D x x

dx

t t

D D

= - £ -

D D

= - ³

and low-level injection à minority carrier diffusion equations

0

E @

ü Since and dn

E @ 0

₀/dx=dp₀/dx=0 in the quasineutral regions

p

N N p

n

P P n

. . . . . .

J qD d n x x

dx

J qD d p x x

dx

= D £ -

= - D ³

0 p

0 n

n n n

p p p

= + D

= + D

Q

ü We can only determine J_N(x) in the quasineutral p-region and J_P(x) in the quasineutral n-region

• Depletion Region Considerations

N P

thermal thermal

R G R G

1 1

0 dJ n , 0 dJ p

q dx t q dx t

- -

¶ ¶

= + = - +

¶ ¶

processes other

G R thermal P

processes other

G R thermal N

t p t

J p q

t p

t n t

J n q

t n

¶ + ¶

¶ + ¶

× Ñ -

¶ =

¶

¶ + ¶

¶ + ¶

× Ñ

¶ =

¶

- -

1

1

ü Suppose that thermal recombination-generation is negligible throughout the depletion region;

ü à J_N and J_P are constants inside the

depletion region

thermal R-G thermal R-G

/ | / | 0

n t p t

¶ ¶ = ¶ ¶ =

/ 0 and / 0

N P

dJ dx = dJ dx =

N p n N p

P p n P n

( ) ( )

( ) ( )

J x x x J x

J x x x J x

- £ £ = -

- £ £ =

N

(

p

)

P

(

n

)

J = J - x + J x

• Boundary Conditions

ü At the Ohmic Contacts

The ideal diode is usually taken to be a “wide-base” diode

The contacts may effectively be viewed as being positioned at x=

±¥

ü At the Depletion Region Edges

Under nonequilibrium conditions:

Equilibrium conditions Nonequilibrium conditions

( )

( ) 0

0

= +¥

® D

= -¥

® D

x p

x n

n p

( )

(

^{E F}

)

^kT

i kT

E F i

p i i

N

p n e

e n

n =

^{- /}

, =

^{- /}

L

n

� � L

p

N P

( ) /

2 F F kT

np = n e

i ^-

A / 2

p n

. . .

qV kT

np = n e

i

- x £ x £ x

A

Fp F

P N

qV

E E

F

F

N

=

-

£ -

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

: law of junction

ü Evaluating the equation at the p-edge

A/ 2

p p p A i

( ) ( ) ( )

^{qV kT}

n - x p - x = n - x N = n e

A

2 i / p

A

( ) n

^{qV kT}

n x e

- = N

A

2 i /

p p

A

( ) n (

^{qV kT}

1)

n x e

D - = N -

ü Similarly,

A

2 i /

n n

D

( ) n (

^{qV kT}

1)

p x e

D = N -

• Derivation Proper

ü The origin of coordinates is shifted to the n-edge of the depletion region

2

n n '

P '2

p

0 d p p . . . 0

D x

dx t

D D

= - ³

'

n

( ) 0

p x

D ® ¥ =

A

2

' i /

n

D

( 0) n (

^{qV kT}

1)

p x e

D = = N -

ü Boundary conditions

ü The general solution

P P

'/ '/

'

n 1 2

'

( )

. . . 0

x L x L

p x A e A e x

D =

-

+

³ Q L

^P

= D

^{P p}

t

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

A P

2

/ '/

' i '

n

D

( ) n (

^{qV kT}

1)

^{x L}

. . . 0

p x e e x

N

D = -

-

³

A P

2

/ '/

' n P i '

P P '

P D

( ) d p D n (

^{qV kT}

1)

^{x L}

. . . 0

J x qD q e e x

dx L N

D

-

= - = - ³

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

N A

2

"/

" i / "

p

A

( ) n (

^{qV kT}

1)

^{x L}

. . . 0

n x e e x

N

D = -

-

³

N A

2

p / "/

" N i "

N N "

N A

( ) d n D n (

^{qV kT}

1)

^{x L}

. . . 0

J x qD q e e x

dx L N

D

-

= - = - ³

ü The current densities at the depletion region edges,

A

2

" N i /

N p N

N A

( ) ( 0) D n (

^{qV kT}

1)

J x x J x q e

L N

= - = = = -

A

2

/

' P i

P n P

P D

( ) ( 0) D n (

^{qV kT}

1)

J x x J x q e

= = = = L N -

A

2 2

N i P i /

N A P D

(

^{qV kT}

1)

D n D n

I AJ qA e

L N L N

æ ö

= = ç + ÷ -

è ø

A/ 0

2 2

N i P i

0

N A P D

(

^{qV kT}

1) I I e

D n D n I qA

L N L N

= -

æ ö

º ç + ÷

è ø

Ideal diode equation or Shockley equation

• Examination of Results

ü Carrier currents

ü The total current density is constant

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

ü Carrier concentrations

ü 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 carriers ü Larger reverse biases have little effect

N_A > N_D

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 Current

ü 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

E_c

E_f E_v

E_c E_f

E_v

V_R I_R

V_R = 0 V (Equilibrium)

E_c

E_f E_v

E_c E_f

E_v

V_R I_R

h⁺

V_R < 0 V V_R = 0 V e^-

V_R I_R

E_c

E_f E_v

E_c E_f

E_v e^-

e^- e^-

e^- e^-

V_R << 0 V (Zener Breakdown, Tunneling)V_R = 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.

n

0

n < p < p

⁰

n

0

n > p > p

₀

G R DIFF

kT E E n kT

E E p i

n i

p i G

R

x

x p n

i G

R

n p

i G

R thermal

G R thermal x

G x R

I I

I

e n e

p n

n qAn W I

p n

p dx p

n n

n qA np

I

p p

n n

n np t

n

t dx qA n

I

T i i

T n

p n

p

-

- -

- - -

-

- - -

+

=

+

= +

º

-

=

®

®

+ +

+

= -

+ +

+ - -

¶ =

¶

¶ - ¶

=

ò ò

) 2 (

) 1 2 (

1

2 0 ,

0

) (

) (

) (

) (

/ ) (

/ ) 1 (

1 0

0

1 1

2

1 1

2

t t

t t

t

t

t t

t

t

_{E E kT}

i

kT E E i

T i

i T

e n p

e n n

/ ) (

1

/ ) (

1

- -

º

º

Summary

31

Summary

32