PPT Semiconductors: A General Introduction

PN Junction Diode: I-V Characteristics

Sung June Kim

[email protected]

http://nanobio.snu.ac.kr

Chapter 6.

Contents

 Qualitative Derivation

 Quantitative Solution Strategy

 Quasineutral Region Considerations

 Depletion Region Considerations

 Boundary Conditions

2

 The Ideal Diode Equation

• Qualitative Derivation

 Equilibrium situation

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

potential hill

high-energy carrier diffusion

drift

balance

E

 Forward bias situation

 a lowering of the potential hill

 The same number of minority carriers are being swept

 More majority carriers can surmount the hill  I_N and I_P

 I (pn)

 The number of carriers that have sufficient energy to

surmount the barrier goes up exponentially with V_A 

exponential increase of the forward current

 The barrier increase

reduces the majority carrier diffusion to a negligible level

 The p-side electrons and n- side holes can wander into the depletion region and be swept to the other side  reverse I (np)

 Reverse bias situation

 an increase of the potential hill

 Being associated with

minority carriers, the reverse bias current is expected to be extremely small

 The minority carrier drift currents are not affected by the height of the hill (The situation is similar to a waterfall)

Reverse current is expected to saturate(bias independent)

 If the reverse bias saturation current is taken to be –I₀, the overall I-V dependence is

I-V characteristic

A/ ref

0

(

1)

I  I e 

Rectification q

V_ref  kT

ohmic ohmic

minority

minority

excess majority carriers  local

excess majority carriers  local

E

E

Excess carriers move to the contact with a relaxation time

 greatly fast recombination

 Current component

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

n-region (far) : electrons

• Quantitative Solution Strategy

 Basic assumptions

(1) Steady state conditions

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

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

N

( )

( )

J  J x  J x

AJ

I 

P p P

J qu n qD dn

dx J qu p qD dp

dx

 

 

E E

and low-level injection

 minority carrier diffusion equations

 0 E

• 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





   

  

 

      

 

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





 

    

 

   

 Since and dn

E  0

p

N N p

P P n n

. . . . . .

J qD d n x x

dx

J qD d p x x

dx

    

    

0 p

0 n

n n n

p p p

  

  



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

• Depletion Region Considerations

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



 



 







 





 









 







1 1

G thermalR N

t n dx

dJ

q 

 

 1 0

G thermalR P

t p dx

dJ

q 

 



 1 0

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

  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

(

)

( )

J  J  x  J x

∴

• Boundary Conditions

 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=



 At the Depletion Region Edges

Under nonequilibrium conditions:

 

  0

0

















x p

x n

n p

 





i kT

E F i

p i i

N

p n e

e n

n 

, 

L

  L

N P

( ) /

2 F F kT

np n e 

A / 2

p n

. . .

qV kT

np n e 

 x   x x

A

Fp F

P N

qV

E E

F

F









If the equal sign is assumed to

hold throughout the depletion region

: law of the junction

 Evaluating the equation at the p-edge

A/ 2

p p p A i

( ) ( ) ( )

n x p x     n x N  n e

A

2 i / p

A

( ) n

n x e

  N

A

2 i /

p p

A

( ) n (

1)

n x e

   N 

 Similarly,

A

2 i /

n n

D

( ) n (

1)

p x e

  N 

• Derivation Proper

 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 

 

   

'

n

( ) 0

 p x   

A

2 /

' i

n

D

( 0) n (

1)

p x e

   N 

 Boundary conditions

 The general solution

P P

'/ '/

'

n 1 2

'

( )

. . . 0

x L x L

p x A e A e x

 

 

   L

 D



 A₂  0 because exp(x’/L_p)   as x’  

 With , A₁=p_n(x’=0)

A P

2

/ '/

' i '

n

D

( ) n (

1)

. . . 0

p x e e x

N

  

 

A P

2 / '/

' n P i '

P P '

P D

( ) d p D n (

1)

. . . 0

J x qD q e e x

dx L N



     

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

N A

2 / "/

" i "

p

A

( ) n (

1)

. . . 0

n x e e x

N

  

 

N A

p 2 / "/

" N i "

N N "

N A

( ) d n D n (

1)

. . . 0

J x qD q e e x

dx L N



     

 The current densities at the depletion region edges,

A

2 /

" N i

N p N

N A

( ) ( 0) D n (

1)

J x x J x q e

     L N 

A

2 /

' P i

P n P

P D

( ) ( 0) D n (

1)

J x x J x q e

    L N 

A

2 2

N i P i /

N A P D

(

1)

D n D n

I AJ qA e

L N L N

 

     

 

A / 0

2 2

N i P i

0

N A P D

(

1) I I e

D n D n I qA

L N L N

  

 

   

 

Ideal diode equation or Shockley equation

• Junction Theory

processes other

G R thermal P

processes other

G R thermal N

t p t

p dx

dJ q t

p

t n t

n dx

dJ q t

n



 



 



 





 



 

 







1 1

•1-D general continuity equation

•In the depletion region, steady state, ignoring last two terms,

dx dJ q 1

0 

dx dJ q 1

0  

and

∴ inside depletion region ,  0 dx

dJ dx

dJ

•In the quasineutral region, E = 0 and diffusion is dominant,

dx n qD d

dx J p qD d

J



 



 ,







P

J J

J Constant through the PN junction

To get continuity equation in quasineutral regions,

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





 

    

 

   

) ( ),

( x n x

p





Solutions.

) (

) 0 ( )

(

) (

) 0 ( )

(

N N N

L x p

p

P P P

L x n

n

D L

e n

x n

D L

e p

x p

N P

































kT qV

i

e

n p

n  





Under non-equilibrium condition

p

n

J

p 





n

p

J

n 

  J

A

2 i /

p p

A

( ) n (

1)

n x e

   N 

A

2 i /

n n

D

( ) n (

1)

p x e

  N 

0

0

A/ 0

2 2

N i P i

0

N A P D

(

1) I I e

D n D n I qA

L N L N

  

 

   

 

) 1 (

) 0 ''

( )

0 '

(    



 J

x J

x I e

J

A P

2 / '/

' n P i '

P P '

P D

( ) d p D n (

1)

. . . 0

J x qD q e e x

dx L N



     

A N

p 2 / "/

" N i "

N N "

N A

( ) d n D n (

1)

. . . 0

J x qD q e e x

dx L N



     

With slight shift of coordinates,

(recap)diode I-V characteristic (The Diode Equation)

• Diode current is determined by the diffusion current of the minority carriers

• Both hole current and electron current should be considered

• They then are added up

This is the Diode

Equation

(recap)Reverse Saturation Current

• After defining Io(reverse

saturation current), the well

known J-V or I-V relation is

established

• Examination of Results

Ideal I-V

1. For forward biasing greater than a few kT/q, 2. For reverse biases greater than a few kT/q,

q few kT V

if kTV

I q

I)ln( ) _A ... _A 

ln( ₀

) /

0 exp(qV kT I

I  _A

I0

I  

I0

I  I  I₀exp(qV_A /kT)

 The Saturation Current

 The current depends on doping of the LIGHTLY doped region.

diodes n

N p n L qA D I

D i P

P 

  ^

2 0

diodes p

N n n L qA D I

A i N

N 

  ^

2 0



 



 



D i P

P A

i N

N

N n L D N

n L qA D I

2 2

0

 Carrier currents

 The total current density is constant

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

 Carrier concentrations

 Forward biasing increases the concentration

Reverse decreases

 Under the low-level injection, the majority carrier

concentrations in these regions are everywhere approximately equal to their equilibrium values

 Under reverse biasing the depletion

region acts like a “sink” for minority carriers

 Larger reverse biases have little effect

N_A > N_D

new

Trends of diffusion and drift currents

• Separation of the energy band

• Diffusion current is strongly dependent on the potential barrier changing to bias.

• Drift current of minority carriers do not change much because they are limited in number.

• Therefore the total current is mostly diffusion current in forward bias, and mostly drift current of minority carriers (called generation current) in

reverse direction.

Charge Control Model(1)

Charge Control Model

Suppose that I (2)

(x

=0) is a supplying current to maintain the condition for

every

Result is same as charge control model (1)

(from slope of minority carrier distribution)

can be calculated in the same way

(recap) Total current (1)

Fig 5-17

Electron and hole components of current in a forward-biased p-n junction. In this example, we have a higher injected minority hole current on the n-side than electron current on the p side because we have a lower n doping than p doping.

Diffusion of minority carrier

Drift of majority carrier

“The summation of current is

constant while current component is changing”

Total current(2)

I = I

(x

=0) – I

(x

=0)

No recombination in W

( I

(x

=0) and I

(x

=0) are constant )

I

(x

) is diffusion current decreasing exponentially ( I

(x

) is proportioned to δ

(x

) )

I

(x

) is drift current which supplies hole for p-area and electron for n=area by recombination

I

(x

) = I – I

(x

)

the electric field of neutral region is very small

compared with the field of pn junction area

(recap)Reverse Biased pn junction(1)

Fig 5-18

Reverse-biased p-n junction: (a) minority carrier distribution near the reverse-biased junction; (b) variation of the quasi-Fermi levels

(recap)Reverse Biased pn junction(2)

For V

>> kT/q

Minority carrier extraction

Quasi Fermi Level widens

 Deviations from the Ideal

• Ideal Theory versus Experiment

 I-V characteristic derived from a Si diode

A large reverse-bias current flows

when the reverse voltage exceeds a certain value

Breakdown

Reverse Bias Breakdown

 V_BR tends to increase with band gap of the semiconductor and the doping on the lightly doped side of the junction

75 . 0

1

B

BR

N

V 

B is the doping on the lightly doped side of the junction )

Avalanche Breakdown

Impact ionization.

Carrier multiplication.

Electron-hole pairs created by impact ionization:

(a) band diagram of a p-n junction in reverse bias

showing(primary) electron gaining kinetic energy in the field of the depletion region, and creating a (secondary) electron-hole pair by impact ionization, the primary electron losing most of its kinetic energy in the process;

(b) a single ionizing collision by an incoming electron in the depletion region of the junction;

(c) primary, secondary and tertiary collisions.

and the doping on the lightly doped side of the junction

Carrier multiplication model(mutiplication factor M)

I

M  I

BR A

V M V

 

 



 

 1

1

 

0 0

) 2 0

( 



 



  



 





 







D A

D A S

n S

D

V V

N N

N N K

x q K

qN

 E 

BR D

A

D A S

CR

V

N N

N N K

q  

 





 

0

2

2

E 

D A

D BR A

N N

N V  N 

B

BR

N

V  1

empirical fit to experimental data,

M is used to correct the ideal diode equation to account for avalanching and carrier multiplication

In other words, breakdown occurs when the electric field in the depletion region reaches some critical value

Then, when,

E(0)  E

V

 V

 V

 V

 V

Electric field is independent of doping. So,

 Zener Process

• Tunneling

– The particle energy remains constant during the process.

 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.(d < 10

cm)

Reverse bias↑ # of filled valence electrons placed opposite empty ⇒

conduction-band states↑ current↑ ⇒

Zener Breakdown

By Tunneling.(decrease of d in reverse bias) highly doped p⁺n⁺ junction.

as voltage regulator

EE_cc

EE_ff EE_vv

EE_cc EE_ff

EE_vv

VV_RR II_RR

V_R = 0 V (Equilibrium)

EE_cc

EE_ff EE_vv

EE_cc EE_ff

EE_vv

VV_RR II_RR

hh⁺⁺

V_R < 0 V V_R = 0 V ee^--

VV_RR II_RR

EE_cc

EE_ff EE_vv

EE_cc EE_ff

EE_vv

ee^-- ee^-- ee^--

ee^-- ee^--

V_R << 0 V (Zener Breakdown, Tunneling)

Ideality factor n

 The recombination current is complicated by the fact that recombination rate, which depends on the carrier concentrations, varies with position within depletion region.

 The diode equation can be modified by including the parameter n :

) 1 (

'

0



 I e

I

n varies between 1 and 2, depending on the material , temperature and voltage

theory

Forward and reverse current-voltage

characteristics plotted on semi-log scales, with current normalized with respect to saturation

current Io; (a) the ideal forward characteristic is an exponential with an ideality factor n=1 (dashed straight line on log-linear plot). The actual forward characteristics of a typical diode(colored line) have four regimes of operation; (b) ideal reverse

characteristic (dashed line) is a voltage- independent current = -Io. Actual leakage characteristics(colored line) are higher due to generation in the depletion region, and show breakdown at high voltages.

small forward bias and all reverse biases.

← thermal recombination-generation in the depletion region

 Reverse biasing  carrier concentration in depletion region are reduced below their equilibrium values

 lead to the thermal generation

 Forward biasing  carrier concentration increase above their equilibrium values  carrier recombination

 In steady state, net R-G rate is the same for electrons and holes



 

 

 ⁿ

p

x

x thermalR G G

R dx

t qA n

I

 In depletion region, the general R-G relationship is used

) (

)

( _{1 1}

2

p p n

n

n np t

n

n p

i G

thermalR   

 

 



  



   

 ⁿ 

p

x

x p n

G i

R dx

p p n

n

n qA np

I ( ₁) ( ₁)

2





 For reverse biases greater than a few kT/q, carrier concentration is negligible(n→0,

qAn W I

2 





 



kT E E p n

p e ^{T i} e ^{i T}

p p n

n _{( )/ ( )/}

0 1 0

0 1

2 1 2

1 _{ }



 



 



 

    



 Recombination mechanism(optional)

E_c E_t E_v

r_n

E_c E_t E_v

g_n

E_c E_t

E_v r_p

E_c E_t

E_v g_n

 r

: electron capture rate

 g

: electron emission rate

 r

: hole capture rate

 g

: hole emission rate

) 1

( _t

t

nnN f

C 



t t nN f

 e

t t ppN f

C

) 1

( _t

t

nN f

e 



0 )

1 (

0 ) 1 (

























t t p t

t p p p

t t

n t t n n n

f pN C f

N e r dt g

dp

f nN

C f N e r dt g

dn

(steady state)

t n t

n f

nC f e  1

t p t

p f

pC f

e  

1

kT E

t E_{t i}

f e_{( )/} 1

1

 



(The probability that an electron fills trap)

p n e e ,

(The emission coefficient)

kT E

t E_{t i}

f e_{( )/} 1

1

 



t t n

n f

nC f e  1

t t p

p f

pC f

e  

1

1 /

) (

1 /

) (

n C e

n C e

n C e

n C e

p kT

E E i p p

n kT

E E i n n

t i

i t













(steady state)

t t p t

t p t

t n t t

nN f C nN f e N f C pN f

e  (1 )  (1 )

  



kT E E i n

kT E E i p n

t _{t i i t}

t i

e n p C e

n n C

e n C n

f C_{( )/ ( )/}

/ ) (













 

  



kT E E n i

i t

p n

t t

n t t n n n

t i i

t C p n e

e n n C

n np N C C

f nN

C f N e r dt g

dn

/ ) ( /

) (

2) (

) 1 (



  



 











) (

)

(

2

p p

n n

n np t

n

n p

i G

thermalR

  

 

 





 

Then,

p p n

n r g r

dt g dn dt

dp     

 0

 For forward biases, the carrier concentrations cannot be neglected.

 We merely note that I

is expected to vary roughly as exp(qV

/ηkT). Typically η is expected close to 2.

 Then combined forward and reverse bias dependence is approximately described by below











  

 



kT p qV

A n bi

kT qV i

G R

A A

q e kT

V V

W e I qAn

2 / 0

/

0

2 1 /

) 1 (

2





 

) 1

( ^/

2

2  

 



 

 ^{qV kT}

D i P

P A

i N

N

DIFF e ^A

N n L D N

n L qA D I

G R DIFF I I

I   _

 Diffusion current by ideal diode equation,

 And total current is

 In room temperature,

and IR-G current dominates at reverse and small forward biases.

0

2 0

/ I

W

qAn_i  

 With increasing forward biases, I

increases more rapidly

 Because while , the relative weight of the two component varies from

semiconductor to semiconductor

 Also, the reverse bias

diffusion component of the current will increase at a faster rate with increasing temperature

2 i

DIFF n

I  I_RG  n_i

A bi

 High level Injection

Minority and majority carrier concentrations adjacent to the depletion region are perturbed

 The majority carrier concentration must increase to maintain approximate charge neutrality

 An analysis of high-level injection leads to a predicted current varying roughly as exp(q/2kT)

• V

 V

high-current phenomena

•Large current :

voltage drop in quasineutral region and high level injection

 Series Resistance

Quaseneutral region have an inherent resistance R_S

S A

J V IR

V  

 We can rewrite I-V_A relationship

(when I is small, we can ignore IR_A, then V_J = V_A)

bi A

kT IR V q kT

qV I e V V

e I

I  ^J  0 ^{( A S)/}  

/ 0

S J

A V IR

V

V   



• Narrow-Base Diode

 Current Derivation

 x’_c = x_c - x_n and L_P > x’_c

 minority carrier concentration at a contact a finite distance from the depletion region edge depends on the R-G rate at the contact

 Ohmic contact, R-G rate is high and the minority carrier concentration is maintained near its equilibrium value

 Paralleling the derivation of the ideal diode equation,

c p

n n

P p x x

dx p

D d 0 ' '

0 '₂

2   

   _{( ' ' ) 0}

) 1 (

) 0 '

( ^/

2















c n

kT qV D i n

x x p

N e x n

p ^A

c L

x L

x

n x Ae A e x x

p ( ') ₁ ^{/' p}  ₂ ^{/' p} 0 ' '

 ^ 

2

) 1

0

( A A

p_n  

 0 A₁e^x'c/Lp  A₂e^x'c/Lp

 





p c

n

n x x

L x

L x p x

x

p 0 ' '

/ ' sinh

/ ) ' ' ( )sinh 0 ( )

'

(   





 

0

' '

) 0 '

(    _

 _{p P} ⁿ _x

DIFF dx

p qAD d

x AJ I

) / ' sinh(

) / ' cosh(

'

) 1 (

'

2 0

/ 0

p c

p c D

i p

p

kT qV DIFF

L x

L x N

n L qA D I

e I

I ^A







 General solution is

 Applying boundary condition,

Then,

Finally,

• Narrow-Base Diode

 Limiting Cases

 





p c

n

n x L

L x p x

x

p sinh ' /

/ ) ' ' ( )sinh 0 ( )

'

( 







) / ' sinh(

) / ' cosh(

'

) 1 (

'

2 0

/ 0

p c

p c D

i p

p

kT qV DIFF

L x

L x N

n L qAD I

e I

I ^A







If x’_c →∞ _, or x’_c /L_P >>1,

 





p c

n

n x L

L x p x

x

p exp ' /

/ ) ' ' ( )exp 0 ( )

'

( 