Equilibrium Carrier Transport

Equilibrium Carrier Transport

Equilibrium Carrier Flows

Electron : diffusion flow n  p, drift flow p n

At equilibrium,

diffusion flux = drift flux for electrons & holes

Band diagram explanation

Carrier distribution at equilibrium : Boltzmann distribution

Electrons of energy E-E_c < qV_b cannot overcome the barrier

energy, only the els E-E_c > qV_b can diffuse to p-side

While small concentration of n in p- side drift to n-side

Similar explanation for holes flow

qV_b E_c

E_F

E_v

E

n drift n diffusion

p drift p diffusion

Applied bias

By convention : apply voltage (V_a) on p-side with respect to n-side

Positive (Forward, V_f) or negative (Reverse, V_r) bias

Built-in potential

– V_b increases for reverse bias and decreases for forward bias

pn Junction with Bias

Depletion width

Since

w increases for reverse bias and decreases for forward bias

p n

+ _

V

V

   V

 V

Forward bias : Reverse bias :

V V V V

V V V V

b a b f

b a b r

  

  

V V q N N

N N w

b a

s

D A

D A

 

 1

2

2



 

x V V

q

N

N N N

n

s o b a A

D D A

 





 

 2

 

( )

 

x V V

q

N

N N N

p

s o b a D

A D A

 





 

 2

 

( )

qN_A

pn Junction with Bias

Band diagram

p-side goes up for reverse bias and goes down for forward bias

Electric field

-x_po x_no

r

qN_D

x

E

E_c

E_v

E

x

E _o

o s

n D o

x qN



 



E

Transport with Bias

Biased Junctions

Steady state response of pn junction

Bias = external applied voltage. By convention the bias is applied on p- side of the junction.

Most of the bias voltage drops (is consumed) accross the space charge region (Neutral regions have a lot of free carriers thus voltage drop is small)

Forward bias = (+) voltage on p- side barrier energy V_b reduce to V_b-V_f

Reverse bias = (-) voltage on p- side barrier energy V_b increase to V_b+V_r

Bias voltages control the diffusion barrier hill height and thus control the diffusion fluxes

p n

+

V

q(V_b-V_a) qV_a

E_v n drift n diffusion

Transport with Bias

Diffusion current

Drift current?

Drift current is not limited by how fast, but is limited by how often

In any bias condition, the elect field is strong enough for free carriers in the space charge region to sweep down

Only the carriers within diffusion length L_n or L_p from the edge of the space charge region contribute

These carriers are thermally

generated minority carriers within L_n or L_p, thus small number : call GENERATION CURRENT

Therefore drift current in junction is small and practically independent of the bias condition constant At equilibrium

n n qV

diff o

kT

  a

  exp 

e drift p

n L_n

G

I I qV

n

kT

diff

o

n a

 

  exp 

I

 I

 I

 0

I V I V

I I

n diff

a n

drift a n

gen

o n

(  )  (  )

 

0 0

Diode Equation

Total current equation

with

: ideal diode equation

Assymetric I-V curve enables the rectifying function of diodes

Diffusion current is important near the pn junction

I I qV

tot o

kT

 

 

 







exp 1  I

V

0

0

-I

exponential diffusion current

constant generation current

forward reverse bias

E  0 E ~0 bias E ~0

diffusion

drift drift

p n

I

 I

 I

Diode Currents

Assumptions

Steady state One dimensional Low-level injection

Processes of diffusion, drift, and thermal G-R only

Currents

Total current density is constant throughout the diode

Quasi-neutral region

E ~ 0 and dn=dDn, thus

Steady state continuity eq. applies

Injected carrier distributin

Above solution for x>x_n or x<-x_p  do not know about J_tot

dx qD dp p

q J

dx qD dn n

q J

x J x J J

AJ I

p p

p

n n

n

p n















E E





) ( )

(

0 0

2 2

2 2

   

  

D n

x

n x x

D p

x

p x x

n

p p

n

p

p

n n

p

n



 



 

D D

D D

J qD d n

dx x x

J qD d p

dx x x

n n p

p p n

  

  

D D

D D

D D

n x n e x x

p x p e x x

p p

x L

p

n n

x L

n n

p

( ) ( )

/ /

  





Diode Currents

Depletion region

In the depletion region

If thermal G-R is negligible, dJ_n/dx=0 and J_n=constant

Therefore From



Since



Therefore for the injected holes



Similarly for the injected electrons

* Refer the book derivation

p x

p x

qV kT

po p

no n

( ) b

( ) exp

 

 



 

p x p x

q V V kT

p x

p x

qV kT

p p

n n

b a

po p

no n

a

( )

( ) exp

( )

( ) exp

   

 



 



 



p_p(x_p) p_po(x_p)Dp_p(x_p) p_po(x_p) p x

p x

p x

p x

qV kT p x

p x

qV kT

p p

n n

po p

no n

a

p p

no n

a

( )

( )

( )

( ) exp

( )

( ) exp

  



 



 



 



p x p x qV

n n no n kT

( )  ( ) exp a

 



Dp x

p x p x

p x qV

kT

n n n n no n

no n

a

( ) ( ) ( )

( ) exp

 

 

 

 





 1







 n

t q

dJ dx

n t

p n p

G R

  



1 0

J x x x J x

J x x x J x

n p n n p

p p n p n

( ) ( )

( ) ( )

    

   

J

J

x

J

x

Dn n x qV

p po p

kT

  

 

 







( ) exp 1 

Diode Currents

Injected carriers behavior

Redefine the x coordinate for p and n x' = x-x_n and x" = -(x+x_p)

with boundary conditions of

Therefore

and

With similar expressions for els, the total current density

 

J x qD d p x dx

qD

L p x qD

L p e e

p p

n p

p n

p p

no

qV_a kT x L_p

( ) ( )

( )

/ /

   

  

  ^{ }

D D

1

D p

( x   ) D p

( ) 0 e

Dp

( x     ) 0

 

D D

D

p x p e e

p e

n no

qV kT x L

n

x L

a p

p

( )

( )

/ /

/

  



 

 

1 0

x_n -x_p

x^' x''

p n

 

 

I A J x J x

qA D

L p D

L n e

I qA D

L p D

L n

n p

p p

no

n n

po

qV kT

o

p p

no

n n

po

a

     

  

 

 

  

 



( ) ( )

/

0 0

1

with

Diode Currents

Ideal I-V curve

for V_a > few kT/q

Saturation current I

Since p_no=n_i²/N_D, etc, different

semiconductors show different I_o 

smaller gap shows larger saturation current for the same doping

For high-low junctions, highly doped side term is negligible for p⁺n diode

In general, heavily doped side can be ignored in the electrical characteristics of the junction (as in most real diodes)

I

V

0

0

-I

I_oexp(qV_a/kT)

ln( ) I ln( I ) q kT V

o a

 

V

0

0

slope=q/kT ln(I_o)

ln(I)

I qA D

L p D L n

o

p p

no n n

   po

 



I qA D L p

o

p p



Figure assumed N_A > N_D

Total current density constant everywhere in the diode

The injected minority carrier current decays by recombining with the majority carriers

Thus the same amount of majority carrier current decreases making overall current becomes I_tot

In the depletion region the currents J_n and J_p are constant

In the neutral region far from the junction, only the majority carriers make the current by drift

Even we approximated that E ~ 0 in the neutral region, there are small E with large carrier concentration

 drift current of I_tot

Carrier Currents

x

J_n

x_n

J

J_p

-x_p

p n

majority

minority

J

Carriers Concentration

Forward bias

Note that the log scale, Dp > Dn If N_A >> N_D , Dp >> Dn  Hole injection only J J_p



Reverse bias

 Zero minority carriers at x_n=0 and x_p=0. Minority Carrier Extraction

lo g(n ,p )

x

x_n N_D

-x_p

p n

N_A

p_no n_po

n_no p_po

Dp p qV

kT p

n no

r

  no

 

 







  

exp 1

Dn_p  

n

lo g(n ,p )

x

x_n N_D

-x_p

p n

N_A

p_no n_po

n_no p_po

Dp Dn

Non-equilibrium carrier concentrations



in the depletion region : Law of the Junction

In the far neutral region F_n and F_p is for the equilibrium state

The variation in the depletion region would be monotonic though exact level cannot be determined point-by- point

From the law of the junction flat Fermi energies are generally assumed

Fermi Levels of Biased Junction

qV_f E_v

F_n E_c

F_p Injection

region

n n F E

kT

p n E F

kT

i

n i

i

i p

  

 



  

 

 exp

exp

np n F F

kT n qV

i i kT

n p a

  

 

  

 

2 2 

exp exp

qV_r E_v

F_n E_c

F_p

Deviation from Ideal

Experimental I-V

Breakdown at high reverse bias Slope over at high forward bias (V_a

> ~0.7 V)

Slope of q/2kT for V_a < ~0.35 V Large reverse biasing of pn junction causes sudden large increase of reverse current : breakdown

lo g(I )

V

0

0

q/kT

q/2kT reverse bias

breakdown

G-R current in the depletion region

diffusion term

slope-over region

Breakdown

Breakdown voltage (V_BR) depends on the doping concentrations

For high-low junctions

with N_B = doping conc of lightly doped side

Junction Breakdown is recoverable process, not permanent

Breakdown processes are Avalanche and Zener BD

Reverse Bias Breakdown

V

 1 / N

V

N

N

1000

Avalanche region 100

1 10

10¹⁴ 10¹⁵ 10¹⁶ 10¹⁷ 10¹⁸ slope=-0.75

GaAs Si

Ge

Reverse Bias Breakdown

Avalanche BD

Lightly doped pn jct

For small reverse bias, electron mean free path (~10^-6 cm)<< depletion width (~

10^-4 cm)

Carriers lose energy by collision with atoms and it is dissipated by heat

At large reverse bias, the energy transfer is sufficient to ionize atoms and create EHPs : Impact Ionization

The generated carriers make additional impact ionization resulting in carrier multiplication  avalanche process

V_BD > ~6.7 V for Si

Sloping rather than sharp BD

Larger V_BD with T increase due to smaller collision distance

E_v E_c

E

Lattice collision &

heat dissipation

V N N

N N E q

BR

A D

A D

   6 g /

E_v E_c

carrier

multiplication

Zener BD

Highly doped pn jct with small depletion width w

E is large due to small w, ~ 10⁶ V/cm

w increases with reverse bias, but the thickness of the potential

energy barrier d decreases

Tunneling through d (d < ~10^-6 cm for tunneling) : require > ~10¹⁷ cm^-3 doping for Si

V_BD > ~4.5 V for Si

Smaller V_BD with T increase

Reverse Bias Breakdown

E_v E_c

tunneling

filled states

empty states w

d

V

E

q

G-R Current

In ideal diode, G-R term is neglected in the depletion region

For rev bias where carrier conc

decrease below equil value thermal G-R component is important

I increases with bias since w increases with rev bias

For forw bias, large carriers recombine in the depletion region

I_G-R at forward bias

with

h

G-R term is dominant at small forward

bias E_v

E_c

G-R current

E_v E_c

ideal diode current

R-current

I qAn

G R w

i o

   2

 

I_{G R} exp qV_A /hkT

High Current Phenomena

High Level Injection

As V_a > V_b (negligible barrier height), Dp > n_no ~N_D (minority carrier conc approaches to doping conc in the lightly doped side)

Accordingly increased majority carrier conc for charge neutrality

Slope-over region exists

Junction resistance reduces, thus jct voltage drop (V_J) becomes smaller

The voltage drop in the neutral regions is now comparatively large (Series resistance)

lo g(n ,p )

x

x_n

N_D

-x_p

p n

N_A

p_no n_po

n_no p_po

Dp Dn

 

I

qV

kT

V_a V_J IR_s