principles of semiconductor devices-l18
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8/8/2019 Principles of Semiconductor Devices-L18
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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 = +
Alam ECE606 S09 4
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+
Alam ECE606 S09 5
g p
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Continuity Equations via Energy Bands
n x N
N r EC
1nEV
= N N nt q1
Alam ECE606 S09 6
= +
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.
Alam ECE606 S09 11
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
Alam ECE606 S09 14( ) ( )0 00+ + = + = + + =
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.
Alam ECE606 S09 22
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