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Prof. Yo-Sep Min Electronic Materials: Semiconductor Physics & Devices Chapt. 5 - Lec 10-1
pn Junction Diodes
• A systematic analysis of pn junction diodes is typically divided
into four segments:
electrostatics (charge density, E-field, and potential): Chapt. 5
steady-state (d.c.) response: Chapt. 6
small-signal (a.c.) response: Chapt. 7 skipped
transient (pulsed) response: Chapt. 8 skipped
Prof. Yo-Sep Min Electronic Materials: Semiconductor Physics & Devices Chapt. 5 - Lec 10-2
Chapter 5. pn Junction Electrostatics
• You will also learn about:
Poisson’s Equation
Built-in Potential
Depletion Approximation
Step-Junction Solution
In this chapter, you will learn about pn junction electrostatics:
charge density, electric field and electrostatic potential existing
inside the diode under equilibrium and steady state conditions.
Prof. Yo-Sep Min Electronic Materials: Semiconductor Physics & Devices Chapt. 5 - Lec 10-3
pn-junction fabrication
pn-Junctions are created by
several processes including:
1. Diffusion
2. Ion-implantation
3. Epitaxial deposition
Each process results in different
doping profiles.
Prof. Yo-Sep Min Electronic Materials: Semiconductor Physics & Devices Chapt. 5 - Lec 10-4
Ideal step-junction doping profile
Step junction Linearly-graded junction
Approximation of an ion-implantation
or shallow diffusion into a lightly
doped wafer
Approximation of deep diffusion into
a heavily doped wafer
Prof. Yo-Sep Min Electronic Materials: Semiconductor Physics & Devices Chapt. 5 - Lec 10-5
Poisson’s Equation
ε
ρ
dx
d
E = charge density
= Ks o
Ks = dielectric constant of semiconductor
• According to the Poisson’s equation, in one dimensional
problem, the charge density (ρ) is proportional to dE/dx.
• Therefore, a diagram of ρ versus x can be deduced from an
E-x plot by simply noting the slope.
• Assuming the dopants to be totally ionized, the charge
density inside a semiconductor is given by
AD NNnpqρ
For a uniformly-doped semiconductor in equilibrium,
neutrality charge the todue 0ρ
Prof. Yo-Sep Min Electronic Materials: Semiconductor Physics & Devices Chapt. 5 - Lec 10-6
Drawing energy band diagram of pn junction in equilibrium
kT
EE
enpFi
i
EF = same everywhere
under equilibrium
Join the two sides of
the bands by a smooth
curve.
kT
EE
enniF
i
p-type n-type
Band bending ???
Prof. Yo-Sep Min Electronic Materials: Semiconductor Physics & Devices Chapt. 5 - Lec 10-7
Electrostatic variables of pn junction in equilibrium
x
E
qx
E
q d
d1
d
d1 iC E
• Potential energy of electrons,
P.E.(electron) = (EC–Eref)
xd
dE
Electrostatic
potential
E-field
Charge
density
Where this charge come from?
• Potential energy difference between
the two sides is EC = –qVbi (< 0).
• There is a built-in potential (Vbi) in
the junction which is equal to
(1/q)(EC).
p-type n-type
Prof. Yo-Sep Min Electronic Materials: Semiconductor Physics & Devices Chapt. 5 - Lec 10-8
Conceptual pn-junction formation
The ionized dopant atoms will build
up an electric field which will prevent
further movement of carriers.
p and n type regions
before junction formation
Majority carriers diffuse to the
opposite side leaving ionized
dopant atoms (immobile charges)
Where this charge come from?
Immobile dopant ions
Space charge region
Depletion region
Prof. Yo-Sep Min Electronic Materials: Semiconductor Physics & Devices Chapt. 5 - Lec 10-9
The built-in potential, Vbi
When the junction is formed, electrons from the n-side and
holes from the p-side will diffuse leaving behind charged
dopant atoms. Remember that the dopant atoms cannot
move! Electrons will leave behind positively-charged donor
atoms and holes will leave behind negatively-charged
acceptor atoms.
The net result is the build up of an electric field from the
positively-charged atoms to the negatively-charged atoms,
i.e., from the n-side to p-side. When steady state condition is reached after the formation of junction, the built-in E-field (or
the built-in potential) will prevent further diffusion of electrons
and holes.
Is there net current across the pn junction?
Prof. Yo-Sep Min Electronic Materials: Semiconductor Physics & Devices Chapt. 5 - Lec 10-10
Equilibrium conditions
Under equilibrium conditions, the net electron current and
hole current will be zero in the pn junction.
electron drift current
opposite to electron flux
electron diffusion current
opposite to electron flux
hole diffusion current
hole drift current net = 0
net = 0
p-type n-type
Prof. Yo-Sep Min Electronic Materials: Semiconductor Physics & Devices Chapt. 5 - Lec 10-11
The built-in potential, Vbi
• The voltage drop across the depletion region under
equilibrium is called built-in potential.
• By the definition of electric field, dx
dVE
Integrating across the depletion region gives
bipn
xV
xV
x
xVxVxVdVdx
n
p
n
p
)()(
)(
)(E
Under equilibrium condition,
0dx
dnqDnqJ nnn E
Solving for E,
n
dxdn
q
kT
n
dxdnD
n
n //
E
p-type n-type
Prof. Yo-Sep Min Electronic Materials: Semiconductor Physics & Devices Chapt. 5 - Lec 10-12
The built-in potential, Vbi
n
dxdn
q
kT
n
dxdnD
n
n //
E
bipn
xV
xV
x
xVxVxVdV
n
p
n
p
)()(
)(
)(E
Combining two equations,
n
p
n
p
x
x
x
xbi dx
n
dxdn
q
kTdxV E
/
)(
)(ln
p
nx
x xn
xn
q
kT
n
dn
q
kTn
p
For a non-degenerately doped step junction (total ionization),
Dn Nxn )(A
ip
N
nxn
2
)( and
2ln
i
DAbi
n
NN
q
kTVTherefore
Prof. Yo-Sep Min Electronic Materials: Semiconductor Physics & Devices Chapt. 5 - Lec 10-13
The built-in potential, Vbi
2ln
i
DAbi
n
NN
q
kTV
Ex) For a pn junction of Si (NA = ND = 1015/cm3) at 300 K
Vq
eVVbi
6.0
10
1010ln
026.020
1515
• In pn junctions of non-degenerately doped semiconductors,
q
EV G
bi generally
• The electrostatic potential (V) in Volts is numerically equal to
the energy (E) in eV.
Prof. Yo-Sep Min Electronic Materials: Semiconductor Physics & Devices Chapt. 5 - Lec 10-14
The built-in potential, Vbi
iFi ln
n
pkTEE
iiF ln
n
nkTEE
q Vbi = (Ei EF)p-side + (EF Ei)n-side
kT
EE
i
iF
enn
kT
EE
i
Fi
enp
p-type n-type
Alternative derivation of Vbi by using the energy band diagram
Prof. Yo-Sep Min Electronic Materials: Semiconductor Physics & Devices Chapt. 5 - Lec 10-15
The built-in potential, Vbi
Vbi = (1/q) {(Ei EF)p-side + (EF Ei)n-side }
22lnln
lnln
i
DA
i
np
ii
n
NN
q
kT
n
np
q
kT
n
n
q
kT
n
p
q
kT
kT
biVq
n
n
p
pexp
p
n
n
pProve this relation!!
q Vbi = (Ei EF)p-side + (EF Ei)n-side
sidenonionconcentratelectron nand
sideponionconcentrat holepwhere
n
p
Prof. Yo-Sep Min Electronic Materials: Semiconductor Physics & Devices Chapt. 5 - Lec 10-16
Built-in potential in a p+n junction
Ex) Derive an equation for the Vbi for a p+n junction,
Because EF = EV in the p+ side,
Vbi = (1/q) {(Ei EF)p-side + (EF Ei)n-side }
(Ei EF)p-side = EG/2q
i
DGbi
n
N
q
kT
q
EV ln
2
Similarly, for n+p junction,
i
AGbi
n
N
q
kT
q
EV ln
2
Prof. Yo-Sep Min Electronic Materials: Semiconductor Physics & Devices Chapt. 5 - Lec 10-17
Depletion Approximation
To obtain a quantitative solution for the electrostatic variables (potential, E-field and charge density), the Poisson’s equation
should be solved.
ε
ρ
dx
d
E
dx
dVE
AD
ss
NNnpεK
q
εK
ρ
ε
ρ
dx
d
00
E
In order to write down ρ as a function of x, we must know
explicit expression for n and p as a function of x.
Unfortunately, the n and p in E ≠ 0 regions, like the depletion
region, are not specified prior to solving the Poisson’s equation.
We need an approximation method.
Prof. Yo-Sep Min Electronic Materials: Semiconductor Physics & Devices Chapt. 5 - Lec 10-18
Depletion Approximation
The carrier depletion tends to be greatest in the immediate
vicinity of the metallurgical junction and then tails off as one
proceeds away from the junction.
However, in the depletion approximation, we assume as
below:
(1) In the depletion region (–xp ≤ x ≤ xn), the
n and p are negligible compared to the net
doping concentration (ND – NA).
(2) The charge density outside the depletion
region (x ≤ –xp and x ≥ xn) is identically zero.
AD
s
NNnpεK
q
dx
d
0
E
Prof. Yo-Sep Min Electronic Materials: Semiconductor Physics & Devices Chapt. 5 - Lec 10-19
Depletion approximation
Poisson equation
np xxx
dx
dE AD
s
NNεK
q
0
0 else everywhere
Note that the charge density has
precisely the same functional
form as ND-NA within the
depletion region.
Prof. Yo-Sep Min Electronic Materials: Semiconductor Physics & Devices Chapt. 5 - Lec 10-20
Prof. Yo-Sep Min Electronic Materials: Semiconductor Physics & Devices Chapt. 5 - Lec 10-21
Announcements
• Next lecture: p. 209 ~ 226
• Mid-term EXAM
Chapters 1, 2, 3, and 5.1
13:00 ~ 15:00; October 20,
별232
Supplementary sheet and calculator