Semiconductor Device Physicseng.staff.alexu.edu.eg/~mmorsy/Courses/Undergraduate/EE...Formation of pn Junction and Charge Distribution Chapter 5 pn Junction Electrostatics 10 D e ii

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Lecture 6

Semiconductor Device Physics

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2 0 1 3

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Chapter 5pn Junction Electrostatics

Semiconductor Device Physics

3

Step junctionidealization

Metallurgical Junction

Chapter 5 pn Junction Electrostatics

Doping profile

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Poisson’s Equation


S 0K

E v D

D E

S 0K

S 0x K

E

Poisson’s equation is a well-known relationship in electricity and magnetism.

It is now used because it often contains the starting point in obtaining quantitative solutions for the electrostatic variables.

In one-dimensional problems, Poisson’s equation simplifies to:

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Equilibrium Energy Band Diagram


pn-Junction diode

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Band diagram

c ref

1( )V E E

q

Qualitative Electrostatics


Equilibrium condition

Electrostatic potential

( )V x dx E

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Electric field

Qualitative Electrostatics


Equilibrium condition

Charge density

dV

dx E

S 0K

E

x

S 0

( )x xK

E

8

Formation of pn Junction and Charge Distribution


D A( )q p n N N qNA– qND

+

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Formation of pn Junction and Charge Distribution


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DF i n-side

i i

( ) ln lnNn

E E kT kTn n

bi F i n side i F p side( ) ( )qV E E E E

Ai F p-side

i i

( ) ln lnNp

E E kT kTn n

A Dbi 2

i

lnN N

qV kTn

Built-In Potential Vbi


For non-degenerately doped material,

• Vbi for several materials:

Ge ≤ 0.66 V

Si ≤ 1.12 V

GeAs ≤ 1.42 V

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A1

S

( )qN

x x c

E

The Depletion Approximation

A

S

qNd

dx E

Dn

S

( ) ( )qN

x x x

E

Ap

S

( ) ( )qN

x x x

E

with boundary E(–xp) 0

with boundary E(xn) 0


On the p-side, ρ = –qNA

On the n-side, ρ = qND

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A p

D n

, 0

, 0

0,

qN x x

qN x x

otherwise

Ap p

S

Dn n

S

( ), 0

( )

( ), 0

qNx x x x

xqN

x x x x

E

2Ap p

S

2Dbi n n

S

( ) , 02

( )

( ) , 02

qNx x x x

V xqN

V x x x x

Step Junction with VA0


Solution for ρ

Solution for E

Solution for V

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Relation between ρ(x), E(x), and V(x)


1.Find the profile of the built-in potential Vbi

2.Use the depletion approximation ρ(x)

With depletion-layer widths xp, xn unknown

3.Integrate ρ(x) to find E(x)

Boundary conditions E(–xp) 0, E(xn)0

4.Integrate E(x) to obtain V(x)

Boundary conditions V(–xp) 0, V(xn) Vbi

5.For E(x) to be continuous at x 0, NAxp NDxn

Solve for xp, xn

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Bonus Question

• Answer the following questions in a report.

• Can the built-in potential of a pn junction in

equilibrium be measured directly. Justify your

answer.

• What happens if VA is increased beyond the built

in potential Vbi in the forward biased region.

– Keep all bonus reports and submit them upon

requesting that.

– Only non-cheated reports will be rewarded while

students who will submit copied reports will be

punished.

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2 2A Dp bi n

S S

( ) ( )2 2

qN qNx V x

A p D nN x N x

Step Junction with VA0


At x = 0, expressions for p-side and n-side for the solutions of E and V must be equal:

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S An bi

D A D

2

( )

Nx V

q N N N

S Dp bi

A A D

2

( )

Nx V

q N N N

n px x W

Dn

A

Nx

N

Depletion Layer Width


Eliminating xp,

Eliminating xn,

Summing Sbi

A D

2 1 1V

q N N

Exact solution, try to derive

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S bi Dn p n

D A

2 , 0

V NW x x x

q N N

S bi2 VW

q N

S bi Ap n p

A D

2 , 0

V NW x x x

q N N

One-Sided Junctions


If NA >> ND as in a p+n junction,

If ND >> NA as in a n+p junction,

Simplifying,

where N denotes the lighter dopant density

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D Abi 2

i

lnN NkT

Vq n

S bi

D

2 VW

qN

n 0.115 mx W

Dp n

A

Nx x

N

Example: Depletion Layer Width


A p+n junction has NA 1020 cm–3 and ND 1017cm–3, at 300 K.

a) What isVbi?

b) What is W?

c) What is xn?

d) What is xp?

17 20

10 2

10 1025.86 mV ln 1.012 V

(10 )

1/ 214

19 17

2 11.9 8.854 10 1.0120.115 m

1.602 10 10

30.115 m 10 1.15 Å

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Step Junction with VA 0


• To ensure low-level injection conditions, reasonable current levels must be maintained VA should be small

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In the quasineutral, regions extending from the contacts to the edges of the depletion region, minority carrier diffusion equations can be applied since E ≈ 0.

In the depletion region, the continuity equations are applied.

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A Dbi

i i

ln lnN NkT kT

Vq n q n

Sp n bi A

A D

2 1 1W x x V V

q N N

S D

p bi A

A A D

2,

Nx V V

q N N N

S A

n bi A

D A D

2 Nx V V

q N N N



Built-in potential Vbi (non-degenerate doping):

A D

2

i

lnN NkT

q n

Depletion width W :

,Dp

A D

Nx W

N N

W

NN

Nx

DA

An

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Effect of Bias on Electrostatics


• If voltage drop , then depletion width

• If voltage drop , then depletion width

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Linearly-Graded Junction


S

1dx

E V dx E

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Semiconductor Device Physicseng.staff.alexu.edu.eg/~mmorsy/Courses/Undergraduate/EE...Formation of pn Junction and Charge Distribution Chapter 5 pn Junction Electrostatics 10 D e ii