Charge Control Model

Suppose that I (2)

_p

(x

_n

=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

_p

(x

_n

=0) – I

_n

(x

_p

=0)

No recombination in W

( I

p

(x

n

=0) and I

n

(x

p

=0) are constant )

I

_p

(x

_n

) is diffusion current decreasing exponentially ( I

p

(x

n

) is proportioned to δ

p

(x

n

) )

I

_n

(x

_n

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

I

_n

(x

_p

) = I – I

_p

(x

_n

)

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

_r

>> 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 

_(N

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

0

M  I

^m

BR A

V M V

 

 



 

 1

1

 

^1/2

0 0

) 2 0

( 



 



  



 





 







_{bi A}

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

_CR

V

_bi

 V

_A

 V

_bi

 V

_BR

 V

_BR

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

^-6

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

^{qV nkT}

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,

p→0)

qAn W I

_{R G} ⁱ

2 

0





 

n ^{E E kT}



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

_n

: electron capture rate

 g

_n

: electron emission rate

 r

_p

: hole capture rate

 g

_p

: 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 )

  

p i ^{E E kT}



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_{( )/ ( )/}

/ ) (













 

  

p i ^{E E kT}



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 (



  



 











) (

)

(

_{1 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

_R-G

is expected to vary roughly as exp(qV

_A

/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

_DIFF

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

_A

 V

_bi

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

 



c p



^c

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

 



c p



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,

 



c p



p c

n

n x L

L x p x

x

p exp ' /

/ ) ' ' ( )exp 0 ( )

'

( 







D i p

p

kT qV DIFF

N n L qA D I

e I

I ^A

2 0

/ 0

'

) 1 (

'







If x’_c →0 _, or x’_c /L_P <<1,

 



c p



p c

n

n x L

L x p x

x

p sinh ' /

/ ) ' ' ( )sinh 0 ( )

'

( 





 



 



 







c n

n x

p x x

p '

1 ' ) 0 ( )

' (

•The perturbed carrier concentration becomes a linear function of position.

This is a direct consequence of negligible thermal R-G in a region much shorter than a diffusion length.

•This observation is justification for henceforth neglecting the thermal R-G term in the minority carrier diffusion equation when the quasineutral width is small compared to a diffusion length

Punch-Through

This involves the reverse-bias current

x’_c /L_P <<1,

The width of the quasineutral region x’_c decreases with

increasing reverse bias because of the growing depletion width

_IDIFF (V_A<0) does not saturate ( )

x_c is sufficiently small, x’_c →0

The situation where an entire device region becomes depleted is reffered to as punch through

D i c

p

N n x qAD I

2

0' '

' 0' 1/x_c I 



Piecewise-linear approximations of junction diode

characteristics

large E

_g

small n

_i

, small I

_o

, large E

_o

, large V

_br

Low doping large V

_br

In this time, watch out for punchthrough

(In Punchthrough, entire device region is depleted.) guard ring to prevent premature breakdown

Contact Resistance decreases (From terminal to n

⁺

doping)

p

⁺

-n-n

⁺

structure (forward resistance decreases)

Wise choices

Beveled edge and guard ring to prevent edge breakdown under reverse bias:

(a) diode with beveled edge; (b) closeup view of edge, showing reduction of depletion region near the bevel; (c) guard ring

문서에서 PPT Semiconductors: A General Introduction (페이지 32-67)