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Lecture 6
Semiconductor Device Physics
http://zitompul.wordpress.com
2 0 1 3
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Chapter 5pn Junction Electrostatics
Semiconductor Device Physics
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Step junctionidealization
Metallurgical Junction
Chapter 5 pn Junction Electrostatics
Doping profile
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Poisson’s Equation
Chapter 5 pn Junction Electrostatics
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
Chapter 5 pn Junction Electrostatics
pn-Junction diode
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Band diagram
c ref
1( )V E E
q
Qualitative Electrostatics
Chapter 5 pn Junction Electrostatics
Equilibrium condition
Electrostatic potential
( )V x dx E
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Electric field
Qualitative Electrostatics
Chapter 5 pn Junction Electrostatics
Equilibrium condition
Charge density
dV
dx E
S 0K
E
x
S 0
( )x xK
E
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Formation of pn Junction and Charge Distribution
Chapter 5 pn Junction Electrostatics
D A( )q p n N N qNA– qND
+
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Formation of pn Junction and Charge Distribution
Chapter 5 pn Junction Electrostatics
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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
Chapter 5 pn Junction Electrostatics
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
Chapter 5 pn Junction Electrostatics
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
Chapter 5 pn Junction Electrostatics
Solution for ρ
Solution for E
Solution for V
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Relation between ρ(x), E(x), and V(x)
Chapter 5 pn Junction Electrostatics
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
Chapter 5 pn Junction Electrostatics
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
Chapter 5 pn Junction Electrostatics
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
Chapter 5 pn Junction Electrostatics
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
Chapter 5 pn Junction Electrostatics
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
Chapter 5 pn Junction Electrostatics
• To ensure low-level injection conditions, reasonable current levels must be maintained VA should be small
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Step Junction with VA 0
Chapter 5 pn Junction Electrostatics
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
Step Junction with VA 0
Chapter 5 pn Junction Electrostatics
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
Chapter 5 pn Junction Electrostatics
• If voltage drop , then depletion width
• If voltage drop , then depletion width
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Linearly-Graded Junction
Chapter 5 pn Junction Electrostatics
S
1dx
E V dx E
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