principles of semiconductor devices-l21

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Lecture 21: p‐n Diode I‐V CharacteristicsMuhammad Ashraful Alam

.

Alam ECE‐606 S09 1

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Outline

1) Derivation of

the

forward

bias

formula

2) Solution in the nonlinear regime

3) I‐V in the ambipolar regime

4 Conclusion

.

,

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Topic Map

E uilibrium DC Small Lar e Circuits

signal SignalDiode

Schottky

BJT/HBT

MOSFET

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Continuity Equations for p‐n junction Diode

+ −• = − − D A

1n∂ N N N t q

= −∂

J μ = + ∇qn E qD n

1P P P

pr g

∂ = ∇ • − +J

Will focus on this part today …

t q

P P Pqp E qD pμ = − ∇J

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Applying a Bias: Poisson Equation

qV bi

E F -E V

E C -E F

q(V bi - V )

C - n

F p -E V

-q V

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Depletion Widths

GND-V DN AN

xn xp

02 ( )s Abi n k N V V x ε = −

22 AD n pqNqN x x − =

D A pn D A D

0 02 2s sk k ε ε ( )0 ( )s D bi

A A D p V q N N N V = −+

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Flat Quasi ‐Fermi Level up to Junction

E C

E V

E C

Jn

E V

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p

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Various Regions of I‐V Characteristics

ln I

n ~ A Bk T

. us on m e

2. Ambipolar transport 2

3. High injection

- 6,7

5. Breakdown A

1

6. Trap-assisted R-G

7. Esaki Tunneling

4

5

8Alam ECE‐606 S09

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Recall: One Sided Minority DiffusionSteady

stateAcceptor doped

Can calculate current q V bi

‐V anywhere, let us solvethe problem where it is the easiest …

1∂ d n J

F p‐E V

‐

= − +∂ n nr g

t q dx

μ = +n nn

dnqn qD Jdx

E 2

20 =n

d nDdx

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Boundary Conditions0 0 0

+ + +

Δ = −n n n

( )( 0 ) β −+= = n i F E i n x n e ( )

2

1 β

=

= −

G G

AqV i ne

N( )

( 0 )β − −+

= =p i F E

i p x n e

( )2 2 β β −= = pn AqV i i

F F n e n en p q(V bi -V A )

F p F n -V A(0 )+ =

A p N

2

(0 ) β + = Aqi

A

V n eN

n AN

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Right Boundary Condition

2

( ) inn x W = ≈

( ) 0 A

pn x W Δ = =

C

V

E V

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Example: One Sided Minority Diffusion

2

20

N D

dx=

( , )n x t C DxΔ = +

2

, ( ) 0 p p p x W n x W C DW = Δ = = ⇒ = −

0 ', ( 0) 1 AqV i

A

x n x e C N

β = Δ = = − =2 ⎛ ⎞

12

( , ) 1 1 Aqi

A p

n x t e N W

Δ = − −⎜ ⎟

⎝ ⎠

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Electron & Hole Fluxes

2 AqV in x⎛ ⎞

− − β n D n= + ∇E

A p N W

⎝ ⎠

( )

2

0 1 AqV n i

n n x A

qD ndn

J qD edx W N β

=

= = − −

Δn

F p

n

Δp

2

Alam ECE‐606 S09 13

( )0'

1 A p qV i p p

x n D

n p J qD e

dx W N

β

=

= − = − −

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Total Current

( )22

1 An i

p

qV T

p i

D A n

D nW N

D nW N

J q e β ⎡ ⎤

= − + −⎢ ⎥⎢ ⎥⎣ ⎦ln ln . J V k T const ≈ +

Forward Bias

Reverse BiasΔn .T J const ≈

ln(I)(2)

(3)F p

n

VA(4)Δp

Alam ECE‐606 S09 14

(5)

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Outline

1 Derivation of the forward bias formula

2) Solution in the nonlinear regime

‐

4) Conclusion

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Nonlinear Regime (3) …

( )2 2

n p AqV pn i i F F D D n n β −Δ Δ−⎡ ⎤= − − ( ) A pn J q V a b J β − −= −

p A n DW N W N ⎢ ⎥

⎣ ⎦

0

2

1

6,7

16

A

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Flat Quasi ‐Fermi Level up to Junction ?

N N N

dnqn qDμ = +J E n n n

n n n

dF J W J F nμ = ⇒ Δ =

dF dn− −

n D

n n iii N i N n n e q q

dx dxn e= = −

⎢ ⎥⎣ ⎦⎣ ⎦

F F n

n N N

dF dnqD qD

dx dxn β ⎡ ⎤= −⎢ ⎥⎣ ⎦

E

n N B N

n

dF D T q ndx q

μ μ

= − =⎢ ⎥⎣ ⎦

∵E

Alam ECE‐606 S09 17

Wn

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Forward Bias: Nonlinear Regime …

2( )(0 ) n pi

junctio

F F

n

nn e β −+ = ( ) ( )( )

2 2

(0 ) 1 p A An pnqV qV i F F F i F n ne n e

N N β β Δ Δ− − − −Δ + Δ= ⇒ Δ = −

( )2 2

n p AqV pn i i F F D D n n β −Δ Δ−⎡ ⎤− −

T p A n DW N W N ⎢ ⎥⎣ ⎦

Δn

n n

n Dn

J W N

F μ

Δ =ΔF

ΔFnV

p p n

n D

F N

J W μ

Δ =

Alam ECE‐606 S09 18

pprox: t us on om nate transport

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Outline

1) Derivation of the forward bias formula

2) Solution in the nonlinear regime

3) I‐V in the ambipolar regime

4) Tunneling and I‐V characteristics

5 Conclusion

Alam ECE‐606 S09 19

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Region (2): Ambipolar Transport

D − −⎡ ⎤

ln(2

) A n pnT i T

p n B

J q n e J W W k T

≈ − + ≈⎢ ⎥⎣ ⎦

2

1

6,7

20

A

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Nonlinear Regime: Ambipolar Transport

( )2

2

n pF F i

q V F F

np n e

n

β

β

−

−Δ −Δ

=

A i

A

n

N

pn e+ =+ −

( ) / 2

1 A n p

A n p

i

q V F F

n e pn

β

− −

−Δ −Δ≈

Δ = −Δ≈

( ) / 2 A n pqV F F n in n

qD nn J qD e

W W β −Δ −ΔΔ= − =

( ) / 2 A n pqV F F p i p p

n n

qD nn J qD e

W W β −Δ −ΔΔ= − =

Alam ECE‐606 S09 21

Note: junction never disappears!

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Outline

1) Derivation of the forward bias formula

2) Solution in the nonlinear regime

3) I‐V in the ambipolar regime4) Tunneling and I‐V characteristics

5) Conclusion

Alam ECE‐606 S09 22

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Forward Bias Nonlinearity (7): Esaki Diode

3ln(I)

F n F p

1

F n

VA

6,7

emptyp

Heavy doping

F n F

X1No states!

Alam ECE‐606 S09

VA

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Reverse Bias (5): Zener Tunneling

+VA

F n

p

4empty

5

4

= υ I qpT empty

2 2

4cosh sinhα⎛ ⎞α + − α⎜ ⎟α⎝ ⎠

k d d k

Alam ECE‐606 S09 24

.

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Conclusion

1) I‐V characteristics of a p‐n junction is defined by many

interesting phenomena including diffusion, ambipolar

2) The separate regions are identified by specific features.

w y , w

other mechanism is dominated for a given technology.

3) In the next class, we will discuss a few more non ‐ideal effects.

Alam ECE‐606 S09 25