PN JUNCTION THEORY

PN JUNCTION THEORY

1

Prof. Philippe LORENZINI Polytech-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 equilibrium

1st Step: diffusion mechanism

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

E int

e-h pair’s recombination

« built in  potential V

»

4

• Definition : Potential drop between N and P regions

P N

bi V V

V  

) 0 ) (

( ) ( )

(  

 

 dx

x D dp

x E x p e

x

J_P _{P p}

dx x dp x x p

D_p E

p ( )

) ( ) 1

( 



dx x dp x p dx

x dV kT

e ( )

) ( 1 )

( 



) ln(

n p

bi p

p e

V  kT Holes current equation:

or or

Integrating from P to N region:

finally: )

n N ( N

e V kT

V

i D A bi

D   ln 2 )

n N ( N

e V kT

V

i D A bi

D   ln 2

•Field, potential and Space Charge width(1)

• Poisson’s equation:

5 sc

x dx

x V d



( ) )

(

2

2  

 In N and P region

:

D sc

e N dx

x V d



2 

2 ( )

WN

x 

 0

A sc

e N dx

x V d



2 

2 ( )

0



W_P x

-W_P -W_N

6

 Electric Field E(x)

) (

)

( _N

sc D

n eN x W

x

E   

 ^{( )} _sc ^{( P)}

A

P eN x W

x

E  



 Continuity of Field on x=0

:

P A N

D

W N W

N 

sc P A sc

N D

M

W eN W

E eN



 ^{ }





-W_P -W_N

•Field, potential and Space Charge width(2)

7

 Built in potential V(x)

n N

sc D

n eN x W V

x

V   (  )²  ) 2

( 

p P

sc A

p eN x W V

x

V  (  )²  ) 2

( 

 Depletion layer (ZCE)

sc p A sc

n D d

p n

W W eN

V eN W

V W

V( ) ( ) 2 2

2 2











d D A

A sc D d

p V

N N

N N V e

W ( )

) 2

(   

d D A

D

A sc

d

n V

N N

N

N V e

W ( )

) 2

(   

d D

A

A D

sc

d V

N N

N N

V e

W 

 2 )

( ^-W_P ^WN

•Field, potential and Space Charge width(3)

WARNING: WHEN A VOLTAGE V IS APPLIED ON P SIDE, V

HAVE TO

BE REPLACED BY V

- 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

F_diff e-

F_diff h+

BIASED  PN   JUNCTION

Forward Biasing

12

E_ext

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

F_conde‐

F_cond 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 V_bi by V_bi‐V_A :

At equilibrium, null current  two components compensate between 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)

dx x dp x

p dx

x dV kT

e ( )

) ( 1 )

( 



Density of  carriers injected to   the  limits of  depletion layer

•

If Va=0

•

If Va

15

) ) exp(

(

kT eV p

p p

W

p _bi

p n p

N  

 0

kT eV p

p kT

V V

e p

p p

W

p _A

p n A

bi p

n p

N  







) exp(

) exp(

'

2

kT eV N

n kT

p eV

p ^A

D i A

n

n   ' exp( ) exp( )

2

kT eV N

n kT

n eV

n ^A

A i A

p

p  

) exp(

*

*

'

kT n eV

n p

p

n





Holes density injected versus bias voltage V

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7

10⁴ 10⁵ 10⁶ 10⁷ 10⁸ 10⁹ 10¹⁰ 10¹¹ 10¹² 10¹³ 10¹⁴ 10¹⁵ 10¹⁶ 10¹⁷

Na= 1E17 cm^-3 Vd=0.7 V

P'(Wn) (cm-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 geometry dependant

• Discrimatory parameter : length diffusion L_Dn,p of electrons and holes and neutral region widths d_n,p

17 -W_P 0 W_N

Minority carriers distribution in neutral region

• Long regions ( )

18

n p p

n L

d _,  _,

p N

a

L x kT W

eV n

n p e e

p x

p'( )   ( ^kTa 1) ^(WNx)/Lp

eV n

n p e e

p x

p'( )   ( 1) ^{(  )/}

n a

L Wp kT x

eV p

p n e e

n x

n'( )   ( ^kTa 1) ^(xWp)/Ln

eV p

p n e e

n x

n'( )   ( 1) ^{(  )/}

 Short (narrow) regions ( )d_n,p  L_p,n ) )(

1 (

) (

' e x x

d p p

x

p ^kT _c

eV

n n n

a  



 ( 1)( )

) (

' e x x

d p p

x

p ^kT _c

eV

n n n

a  





) '

)(

1 (

) (

' e x x

d n n

x

n ^kT _c

eV

p p p

a  



 ( 1)( ' )

) (

' e x x

d n n

x

n ^kT _c

eV

p p p

a  





 General case  











 







p kT c

eV

p n n

n L

x sh x

e L

sh d p p

x p

a

) 1 (

) ( )

( '

 











 







n kT c

eV

n p p

p L

x sh x

e L

sh d n n

x n

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

dx x eD dp

x

J _{p p} ( )

)

(  

dx x eD dn

x

J_{n n} ( ) )

( 

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

) (

) (

) (

)

(V J _p W_p J_n W_p J _p W_n J_n W_p

J      

 We get the classical and well known diode equation:

) 1 (

)

( V  J

e

 J

Js is the theorical saturation current or reverse current

20 -W_P 0 W_N

p A

n i n

D P i

S N d

D en d

N D J en

2

2 





Long region

n A

n i

P D

P i

S N L

D en L

N D J en

2 2







General case

) ) (

(

2 2

n p n

A

n i

P n P

D

P i S

L th d L N

D en L

th d L N

D

J  en 

Minority currents in neutral region



Short (Narrow) region

• The model is refinedwe take into account G‐R process in  depletion layer

• Well understood mechanism (Shockley‐Read)

The  real diode:  genération‐ recombinaison   mechanism in  depletion layer

21

n p

n

n r pn

i

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 can be rewritten

) exp(

) (

) (

) (

)

( ²

kT n eVa

W n W p W

n W

p _{N N}  _{P P}  _i

2

n

 

 



 

kT eV

r n

exp 2

max

2 

•

Generation‐Recombinaison current in depletion layer can be expressed as:

22









 ^N

P

W GR W

p n

n

n W J W J e rdx

J ( ) ( )



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

2

ni

pn 

2  0



 ⁱ r n



For  forward bias, r

=cte>0 and the current is due to recombinations.

The  real diode:  genération‐ recombinaison  

mechanism in  depletion layer

• Finally GR current present in depletion layer can be expressed as:

23

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



 



 

 ) 1

exp(2

0

kT J eV

J_{GR GR} ^a



 



 

 



 



 

 ) 1

exp(2 1

) exp(

)

( ⁰

kT J eV

kT J eV

V

J _{a S} ^a _GR ^a

T i

GR en W

J 2

0 

 The expression above can be generalized by introducing Ideality factor:



 



 

 exp( ) 1

)

( ₀

nkT J eV

V

J ^a

The  real diode:  genération‐ recombinaison  

mechanism in  depletion layer

Reverse biasing:

Junction breakdown

• Thermal Effect (Narrow bandgap)

• Zener Effect:

• direct flowing from VB to CB by tunnel effect (0), if electric field above critical Field E_c

• Avalanche Effect:

• before« tunneling », hot electrons

(accelerated electrons) excite by impact ionisation electrons from VB to CB

(1,2,3) etc….

• « Punchtrough »

24

B C

BD eN

V E

2 . ²

 

tun nel

Tunnel

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

dV C dQ

C



 Q  eAN

W

 eAN

W

T D

A

D A

A D

j

T

W

A N

N

N N

V V

e C A

C  

 

 

 ( ) ( )

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 P

P

Sp J

Q 



n n

Sn J

Q 

 ^



N

X

W n

Sp

A e p x p dx

Q ( ' ( ) )

Holes density in excess present in N region























) (

) 1 coth(

) )

0 ( ' (

P P n

n P

n Sp

L sh d L

L d p

p e Q

28

Charge Storage (diffusion) capacitance

• We can transform the previous expression by:

29

) (

P

Sp

J W

Q  





















) (

1 1

P n P

L ch d





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

 Narrow diode:        transit time

 Long diode:        lifetime

P n

t

D

d 2







 

Charge Storage (diffusion) capacitance

• The previous expression, valid in N region, can be

generalized in P region and we obtain for the whole diode

:

30

) (

)

( _{( )}

)

(n n P p p N

Sp Sn

S Q Q J W J W

Q   





If we use:

dV C dQ

C_S  _d  ^S

) (

Sp Sn

d

S

K J J

kT C e

C C

C       

K : Geometry dependant Factor  (2/3  narrow)

(1/2  long)

Charge Storage (diffusion) capacitance

Equivalent circuit of foward diode

31

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

r_s : serie resistance of neutral region n and p

I e r_d kT 1



(from Neamen)

Large signal switching of diode

32

As long as the stored charge is positive 

 forward bias diode  voltage across diode is small (few 10 mV)

) 1 (

'   ^kT 

eV n n

n

a

e p p

p



W_n

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

• Storage time:

• Rise (or fall) time :

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

33













 





 ln(1 ) ln(1 )

m f

f m

f p

sd I I

I I

 I



m f

f j

F

f I I

I RC

 



 







  





  avec

3 1 . 2

Large signal switching of diode

Diode Tunnel – diode Backward

34













 3

2 exp 4

* b t

e m

T a 











 











 

pe a pe

a pe

t V

V V

I V

I exp 1

(a)

(e) (d) (c) (b)