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Outline

1) Continuity Equation

2) Example problems

3 Conclusion

Ref. Advanced Semiconductor Fundamentals , pp. 205 210

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Now the Continuity Equations

1 N N N n r gt q = +J

1 p

= + J N N N qn E qD n

= P P P

r gt q

= J P P Pqp E qD p

V

( )+ = + D A D q p n N N

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Now the Continuity Equations N g

x A=1

( ) J x n x x

r

( ) )( () =

+n n J x A x n x A dx A J + N N A Ag r x x

( ) ( )n nn J x J x dx g r += + 1 n N N J g r = +

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t q x q

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Continuity Equations for Electron/HolesContinuity Equations for Electron/Holes

( ) J x N g

1 n N N n J g r t q = + ( ) J x dx+n

N r

( ) p J xr p

= + P P P J g r t qp

( ) p J x dx+

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g p

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Continuity Equations via Energy Bands

n x N

N r EC

1nEV

= N N nt q1

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= +

P PP

t

g r

q

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Continuity Equation: A Good Analogy

Wabash

River

water level in lake = (in flow outflow) + rain evaporation

1n

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= +

N N n

t g r

q

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Equations in place, learning to solve them

1 N N N n r gt q = +J

1 p

J = + N N N qn E qD n V

= P P P

r gt q

P P Pqp E qD p = J Two methods of solution:

( )+ = + D A E q p n N N Numerical and Analytical

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Analytical Solutions

( )+

= + D A D q p n N N 1n

Band diagram N N N t q

J = + N N N qn E qD n1 = +

J P P P

pr g

t q

Diffusion approximation,Minority carrier transport,

P P Pqp E qD p = J Ambipolar transport

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Outline

1 Continuit E uation

2) Example problems

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Consider a Complex System Unpassivated

x

sur aceMetal contact

1 2 3

Acce tor do ed

Light turned on in the middle section.

The right region is full of mid gap trapsInterface traps at the end of the right region.The left region is trap free.

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The left/right regions contacted by metal electrode.

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Recall: Analytical Solution of Schrodinger Equation

d 2 2

+k 2 =01) 2N unknowns

( ) 0 x = =

2)( ) 0 x = + =

B B x x x x += ==

3)

B B x x x x

d d dx dx

+= ==

2N2 unknowns (for continuous U)

Det(coefficient matix)=0And find E by graphical

4) 2( , ) 1 x E dx

= 5)

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or numer ca so ut on or wave unction

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Recall: Bound levels in Finite well

E

U(x)

sin cos A kx B kx = +

x x De Ne ++=

( ) 0 x = = oun ary on ons x x

e Ce

+=

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Example: Transient, Uniform Illumination

1nr

= +J J = + n E D n(uniform)

t q1

( )n n n + nt

= ccep or

ope

uniform1 = +

J p P p

pr g

t q = J p p Pqp E qD p

0( )

+ = + p

p p p Gt

Majority carrier

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D A D A D q p n N N q p n N N pn

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Example: Transient, Uniform Illumination

( )n nG

= +

1nt

nt , ccep or ope

, ,, ( , ) nt n x G A = =

( , ) 1 nt nn x t G e

= time

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Example: One sided Minority Diffusion

1

Acceptor doped Trap free

1 nn dJ N N t q dx 220 = N

d n D

dx

N N N nqn E qD dx = +J

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Example: One sided Minority Diffusion

2d n

Metal contact

2 N

dx=

a 0x

=m m

J q n

,n x t x= +

, ( ' ) 0= = = = x a n x a C Da0 ', ( ' 0 ')= = = x n x C

'' = = x

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, a x

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Example: Minority Diffusion with RG

1 = +n

N N dJ n gt r

q dx

= +J N N N

dnqDqn E

dx

Steady stateAcceptor doped

2 + n nd n

F ux

2 = n

N D dx

2

rap

e

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20

= N n

dx D

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Diffusion with Recombination

2

2 0 = N

d n n D

Metal contact

n n x L x Ln x t Ee Fe = +

n

3

b

2 = nb LF Ee, ( ) 0= = = x b n x b

0 X

n0, ( 0) ( 0)= = = + = = x n x E F n x

2(0) n n n x L b L x Ln

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2, (1 )

nb Le

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Combining them all .n

2 ( ) = =nn x G b0x 0

1 2 322(00 ')( )

=n n

1 ( 0)'

( ') 1

= = x

n x xn'

1n x

G

=

a c oun ary con on

a

2(0')( ) ( ) = n n n x L b L x Ln x e e ena

2( )n n n x L b L x LnG e e e=

n

en

edn

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a cu a ng curren N N N dx

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Why is the Efield for minority carriers negligible ?

( ) D A D q p n N N + = + N N N qD dx

qn = +J E 0 0 0 0 D A D q p n N N + = + =

( )0 0 0 D Aq p n N n N +

= + + 0n D n N N + = + +

n,p

0 A

n =n 2 N

n=n 0+n(x)

x

volts / m

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Conclusion

1) Continuity Equations form the basis of analysis of all

the devices

we

will

study

in

this

course.

2) Full numerical solution of the equations are possible

and many commercial software are available to do so.

3) Analytical solutions however provide a great deal of

insight into the key physical mechanism involved in the operation of a device.

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