ECE606: Solid State Devices
Lecture 14
Electrostatics of p-n junctions
Gerhard Klimeck
gekco@purdue.edu
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Outline
1) Introduction to p-n junctions
2) Drawing band-diagrams
3) Accurate solution in equilibrium
4) Band-diagram with applied bias
Ref. Semiconductor Device Fundamentals, Chapter 5
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
2
What is a Diode good for ….
GaAs lasers
solar cells
Organic LED
Avalanche Photodiode
GaN lasers
Image.google.com
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
3
p-n Junction Devices …
Schematic of a p-n Diode
Symbol
Contact
P-doped
pocket
NA
N
P
ND
N-doped
substrate
Point-contact diode
Contact
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
4
Topic Map
Topic Map
(Today : Diode in Equilibrium)
Equilibrium
DC
Small
signal
Large
Signal
Circuits
Diode
Schottky
Diode in Equilibrium.
(No external voltage applied)
BJT/HBT
MOS
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
5
Drawing Band Diagram in Equilibrium…
Previously constant in homogeneous
semiconductors. But for pn diode: f(x) !!
∇ • D = q ( p − n + N D+ − N A− )
∂n 1
= ∇ • J N − rN + g N
∂t q
J N = qnµ N E + qDN ∇n
∂p −1
= ∇ • J P − rP + g P
∂t
q
J P = qp µ P E − qDP ∇p
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Equilibrium
(Start here)
In equilibrium J=0 (no current flow).
But, Electric fields or diffusion might still be
present. Detailed balance
Non-Equilibrium (refine later)
DC dn/dt=0
Small signal dn/dt ~ jωt x n
Transient --- full solution
6
P and N doped Material Side by Side …
N-doped
E
P-doped
Vacuum level (determined by material)
Conduction Band
Ec
EF
EF
EV
Valence Band
Valence Band
For P-doped
EF close to EV
k
-k
Conduction Band
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
7
Forming a Junction
Space charge (fixed)
Mobile carriers
NA
ND
Donor-side (N-side)
Squares are fixed donor atoms.
Every donor atom has given away
one electron (blue circle)
Acceptor-side (P-side)
Squares are fixed acceptor atoms.
Every acceptor atom has captured one
electron (from the valence band).
Every acceptor atom leaves behind one
hole in the valence band. (red circle)
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Forming a Junction
Space charge (fixed)
Mobile carriers
NA
ND
p = NA
n = ND
n0 = ni
2
NA
Before joining the
p and the n side
x
Junction
Valid only in equilibrium
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Forming a Junction
Diffusion
E-field
Space charge (fixed)
Mobile carriers
NA
ND
p = NA
n = ND
n0 = ni
2
Depletion approximation
ln(n),
ln(p)
Actual Carrier
Concentrations
n-side
Before joining the
p and the n side
x
Junction
n
NA
After joining the
p and the n side
p
p-side
n
p
Depletion approximation
Bulk Region
Depleted
Region
Bulk Region
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
x
10
Formation of a Junction
+
NA
ND
ln(ND)
ln(n)
ln(p)
ln(NA)
Net charge on N-side: qND
(subtract green from blue line)
Space charge region: fixed
donor atoms
Neglect electrons in p-side
Xn: depletion region in N-side
Net charge on P-side: -qNA
(subtract brown from red line)
Space charge region: fixed
acceptor atoms
Xp: depletion region in P-side
Q=p-n+ND-NA
Depleted Region
qND
red and blue are of equal size
charge balance
Depletion Region means
depleted of “mobile” charges
Bulk Region
Bulk Region
xn
xp
-qNA
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
11
Sketch of Electrostatics
Vbi
Potential
position
Integrate
Emax =
E-field
q
q
xp N A =
xn N D
K sε o
K sε o
position
Integrate
qND
p-n+ND-NA
Depleted
Region
position
Bulk Region
xn
xp
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Bulk Region
-qNA
12
Sketch of Electrostatics
Potential
xn
xp
Invert to go
from potential
to energy scale
position
Band Diagram
position
In equilibrium Fermi-level
must be flat
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
13
Outline
1) Introduction to p-n junctions
2) Drawing band-diagrams
3) Analytical solution in equilibrium
4) Band-diagram with applied bias
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
14
Short-cut to Band-diagram
Neutral
Space
Charge
Drawing Recipe
Neutral
ND N A
Vacuum level
χ2
1)
2)
3)
4)
5)
6)
7)
Start with EF
Ec/Ev in bulk n-side
Ec/Ev in bulk p-side
Vacuum level in N
Vacuum level in P
Join vacuum levels
“Transfer” vacuum
slopes to join Ec/Ev
EC
χ1
EV
EF
… is equivalent to solving the Poisson equation
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
15
Built-in Potential: boundary conditions @infinity
Always true in equilibrium
qVbi
∆1 + χ1 + qVbi = χ 2 + E g ,2 − ∆ 2
χ2
χ1
Eg,2
∆1
∆2
delta1,2 determined via
doping concentrations
X1,2 material parameters
Built-in potential
Vbi unknown!
qVbi = Eg ,2 − ∆ 2 − ∆1 + χ 2 − χ1

N 
N
=  Eg ,2 + k BT ln A  + k BT ln D + ( χ 2 − χ1 )
NV ,2 
N C ,1

N A ND
= k BT ln
+ ( χ 2 − χ1 )
−E / k T
NV ,2 NC ,1e g ,2 B
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
16
Interface Boundary Conditions
Homo - Junction
D
xp
Hetero - Junction
xn
E = (D/kεo)
xn
xp
Field not continuous
across junction
position
position
D1 = K1ε 0 E (0− ) = K 2ε 0 E (0− ) = D2
E (0− ) =
K2
E (0+ )
K1
Displacement is continuous across the interface, field need not be ..
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
17
Built-in voltage for Homo-junctions
Space
Charge
Neutral
Neutral
ND NA
qVbi
Drawing Recipe
1)
2)
3)
4)
5)
6)
7)
Start with EF
Ec/Ev in bulk n-side
Ec/Ev in bulk p-side
Vacuum level in N
Vacuum level in P
Join vacuum levels
“Transfer” vacuum
slopes to join Ec/Ev
Vacuum level
χ1
χ1
EC
EF
EV
Zero for homo-junctions
qVbi = k BT ln
N AND
NV ,2 N C ,1e
− Eg ,2 / k BT
+ ( χ 2 − χ1 ) = k BT ln
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
N AND
N N
= k BT ln A 2 D
− E g / k BT
ni
NV N C e
18
Analytical Solution of Poisson Equation
Depleted Region
qND
Q= p-n+ND-NA
-qNA
position
xp
xn
KSε 0
d 2V
= − q ( p − n + N D+ − N A− )
2
dx
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
19
Analytical Solution for Homojunctions
(Charge)
p-n+ND-NA
qND
-qNA
E-field
position
Integrate
xn
xp
qN D xn
E ( 0− ) =
k sε 0
E ( 0+ ) =
E-field
qN A x p
k sε 0
⇒ N D xn = N A x p
Integrate
position
Potential
Integrate
Vbi
position
Small !
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
qVbi =
E (0
−
)x
n
Potential
+
E ( 0+ ) x p
2
2
qN D xn 2 qN A x p
=
+
2ks ε 0
2ks ε 0
2
20
Depletion Regions in Homojunctions
Space
Charge
Neutral
Neutral
ND N A
xp
xn
Solve for xn, xp
N D xn = N A x p
qVbi =
2
qN D xn 2 qN A x p
+
2k s ε 0
2k s ε 0
xn =
2k s ε 0
NA
Vbi
q ND ( N A + ND )
xp =
2k sε 0
ND
Vbi
q N A ( N A + ND )
Small Project: Solve the same problem for a hetero-junction
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
21
Complete Analytical Solution
ρ =0
ρ = q ( ND − NA)
N
ρ =0
P
x
-xp 0 xn
ρ
V
E
Integrate
Integrate
qND
-xp
-xp 0
x
xn
x
x
xn
-qNA
If you need to
calculate electric
field at specific
points…
∫
E ( x)
0
qN A
dε ′ = −∫
dx′
− xp K ε
S 0
x
E ( x) = −
-xp
 −KqNε oA .......... − xp ≤ x ≤ 0
s
d E  qND
=  ................0 ≤ x ≤ xn
dx  K sε o
0............. x ≤ − xp , x ≥ xn
qN A
( x p + x).......... − xp ≤ x ≤ 0
KSε0
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
E ( x) = −
xn
0
∫ε
( x)
dε ′ = ∫
xn
x
qN D
dx′
KSε 0
qN D
( xn − x)..........0 ≤ x ≤ xn
KSε0
22
Outline
1) Introduction to p-n junction transistors
2) Drawing band-diagrams
3) Analytical solution in equilibrium
4) Band-diagram with applied bias
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
23
Topic Map
Equilibriu DC
m
Small
signal
Large
Signal
Circuit
s
Diode
Schottk
y
BJT/HB
T
MOS
Diode in Non-Equilibrium
(External DC voltage applied)
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
24
Applying Bias to p-n Junction
IV characteristics of a Diode
To be discussed in detail
ln(I)
3
2
1
6,7
Reverse Bias
Forward Bias
VA
4
1.
Diffusion limited
2.
Ambipolar transport
3.
High injection
4.
R-G in depletion
5.
Breakdown
6.
Trap-assisted R-G
7.
Esaki Tunneling
5
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
25
Forward and Reverse Bias
ND
NA
Current flows
easily
Forward Bias
ND
NA
Current does
not flow easily
Reverse Bias
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
26
Band Diagram with Applied Bias…
∇ • D = q ( p − n + N D+ − N A− )
Band diagram (this segment)
∂n 1
= ∇ • J N − rN + g N
∂t q
J N = qnµ N E + qDN ∇n
Next segment / lecture …
∂p
1
= − ∇ • J P − rP + g P
∂t
q
J P = qp µ P E − qDP ∇p
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
27
Applying a Bias: Poisson Equation
qVbi
No bias
Question: Max value of Vbi?
EC-EF
EF-EV
Applied
bias
-qV
Answer: for degenerate s.c.,
if EC-EF=0, EF-EV=0 Eg
n( x ) = ni e ( Fn −Ei ) β
q(Vbi-V)
p( x ) = ni e
EC-Fn
Fp-EV
− ( Fp − Ei ) β
n × p = ni 2 e
( Fn −Fp ) β
P-side grounded
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
28
Depletion Widths
ND
-V
xn
NA
GND
xp
From previous lecture
(homo-junction)
ND xn = NA x p
2
qND xn 2 qNA x p
q (Vbi − V ) =
+
2k s ε 0
2k s ε 0
Applied bias
xn =
2ks ε 0
NA
(Vbi − V )
q ND ( NA + ND )
xp =
2ks ε 0
ND
(Vbi − V )
q NA ( NA + ND )
What about heterojunctions?
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
29
Fields and Depletion at Forward/Reverse Biases
VA<0
VA=0
VA>0
Charge
P
N
VA<0
VA=0
VA>0
Position
Barrier height is reduced
at forward biases
Electric Field
Significant increase of
peak field at reverse
bias
VA<0
VA=0
VA>0
Position
Potential
VA<0
VA=0
VA>0
Position
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Summary
PN-Junction Electrostatics
1) Learning to draw band-diagrams is one of the most
important topics you learn in this course. Band-diagrams are
a graphical way of quickly solving the Poisson equation.
2) If you consistently follow the rules of drawing band-diagrams,
you will always get correct results. Try to follow the rules, not
guess the final result.
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
ECE606: Solid State Devices
p-n diode I-V characteristics
Gerhard Klimeck
gekco@purdue.edu
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
31
Outline
1) Derivation of the forward bias formula
2) Solution in the nonlinear regime
3) I-V in the ambipolar regime
4) Conclusion
Ref. SDF, Chapter 6
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
33
Topic Map
Equilibrium
DC
Small
signal
Large
Signal
Circuits
Diode
Schottky
Diode in Non-Equilibrium
(External DC voltage applied)
BJT/HBT
MOSFET
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
34
Continuity Equations for p-n junction Diode
∇ • E = q ( p − n + N D+ − N A− )
∂n 1
= ∇ • J N − rN + g N
∂t q
J N = qnµ N E + qDN ∇n
This section
∂p 1
= ∇ • J P − rP + g P
∂t q
J P = qp µ P E − qDP ∇p
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
35
Applying a Bias: Poisson Equation
Review
qVbi
No bias
EC-EF
EF-EV
q(Vbi-V)
Applied
bias
-qV
EC-Fn
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Fp-EV
36
Depletion Widths
Review
ND
-V
xn
ND xn = NA x p
2
qND xn 2 qNA x p
q (Vbi − V ) =
+
2k s ε 0
2k s ε 0
NA
GND
xp
xn =
2ks ε 0
NA
(Vbi − V )
q ND ( NA + ND )
xp =
2ks ε 0
ND
(Vbi − V )
q NA ( NA + ND )
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
37
Flat Quasi-Fermi Level up to Junction
Equilibrium: Many electrons Nside. Few electrons P-side.
EC
EV
Detailed balance of drift and
diffusion forces. No net current
Jn
Forward Bias: Electron Jn and
hole Jp currents across junction.
EC
EV
Jp
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Diffusion force unchanged.
(because doping did not change)
But, drift force increased due to
applied bias Net current
38
Various Regions of I-V Characteristics
q
ln( I ) ~
VA
k BT
ln(I)
3
1. Diffusion limited
2
2. Ambipolar transport
1
6,7
3. High injection
VA
1
4. R-G in depletion
5. Breakdown
6. Trap-assisted R-G
4
7. Esaki Tunneling
5
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
39
Recall: One Sided Minority Diffusion
Steady state
Acceptor doped
If current is continuous, one can calculate
for the current anywhere. Position doesn’t
matter! Calculate at “easiest” position.
Minority carrier current on P-side.
We know the solution from earlier
Assume steady state, no rn, gn
q(Vbi-V)
-V
Fp-EV
∂n 1 dJn
=
− rn + gn
∂t q dx
Jn = qnµn E + qDn
dn
dx
d 2n
0 = Dn 2
dx
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
40
Boundary Conditions
∆n(0+ ) = n(0+ )VG − n(0+ )VG = 0
Boundary Conditions
n( x = 0+ ) = ni e (Fn −Ei ) β
p( x = 0+ ) = ni e
=
− ( Fp −E i ) β
ni2 qVA β
( e − 1)
NA
Solution profile
Difference of Quasi-Fermi-levels equals
applied voltage
np = ni2 e
( Fn − Fp ) β
p(0+ ) = NA
n(0+ ) =
q(Vbi-VA)
= ni2 e qVA β
-VA
Fp
ni2 qVA β
e
NA
Fn
X=0+
NA
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
41
Right Boundary Condition
n( x = W p ) ≈
ni2
NA
∆n( x = W p ) = 0
EC
metal
V
Infinite surface recombination
velocity at metal excess carrier
concentration = 0
EV
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
42
Example: One Sided Minority Diffusion
DN
d 2n
=0
dx 2
Ansatz
V
∆n( x, t ) = C + Dx
Plug in B.C.
x = W p , ∆n( x = W p ) = 0 ⇒ C = − DW p
x = 0 ', ∆n( x = 0) =
ni2 qVA β
( e − 1) = C
NA

n2
x 
∆n( x, t ) = i ( e qVA β − 1) 1 −
 W 
NA
p 

Final result: Excess electron
carrier concentration (P-side)
as function of position
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
43
Electron & Hole Fluxes
∆n ( x ) =

ni2 qVA β
x
e
− 1)  1 −
(

NA
 Wp



J N = qnµ N E + qDN ∇n
Current (electrons)
J n = qDn
dn
dx
∆n
=−
x=0
2
i
qDn n
e qVA β − 1)
(
Wp N A
Fn
Fp
∆p
Current (holes)
J p = − qD p
dp
dx
=−
x = 0'
qD p ni2 qVA β
( e − 1)
Wn N D
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
44
Total Current
Forward Bias
ln J T ≈ qVA k BT + ln(const.)
To get total current, add electron
and hole current values at
identical x-points.
(assume no recombination)
Reverse Bias
J T ≈ const .
 D n2 D n2 
J T = − q  n i + p i  ( e qVA β − 1)
 W p N A Wn N D 
∆n
Diffusion current
Fn
ln(I)
(4)
(3)
(2)
(1)
VA
Fp
∆p
(5)
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
45
Outline
1) Derivation of the forward bias formula
2) Solution in the nonlinear regime
3) I-V in the ambipolar regime
4) Conclusion
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
48
Nonlinear Regime (3) …
) (
(
)
 D n 2 D n 2  ( qV −∆F −∆F ) β
q V − aJ −bJ β
JT = −q  n i + p i  e A n p − 1 = I 0 e ( A n p ) − 1
W
N
W
N
 p A
n
D 

Today’s lecture: Nonlinear Regime (2,3)
ln(I)
3
Assumption of flat QuasiFermi levels invalid here
2
1
6,7
VA
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
49
Flat Quasi-Fermi Level up to Junction ?
J N = qnµ N E + qDN
dn
dx
J n = nµ n
dFn
dx
⇒ ∆Fn =
Drop of Quasi-Fermi
level across the junction
proportional to current!
Rewrite n into non-equilibrium form, re-arrange Jn equation
n = ni e β ( Fn − Ei )
qDN
J nWn
µn N D
dn
 dF

= qDN β  n − E   ni e β ( Fn − Ei ) 
dx
dx


New diffusion component: Plug this
into original Jn equation
dn
 dF

= qDN nβ  n − E 
dx
 dx

dF
D
kT


= qµN n  n − E  ∵ N = B
µn
q
 dx

Fn
qDN
Fp
Wn
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
50
Forward Bias: Nonlinear Regime …
n(0+ ) =
ni2 ( Fn −Fp )β
e
NA
=
junction
)
(
ni2 ( qVA − ∆Fn − ∆Fp ) β
n 2 ( qV − ∆F − ∆F ) β
e
⇒ ∆n(0 + ) = i e A n p − 1
NA
NA
)
(
 D n 2 D p ni2  ( qVA −∆Fn − ∆Fp ) β
−1
JT = −q  n i +
 e
W p N A Wn N D 
∆n
∆Fn =
J nWn
µn N D
∆Fp =
J pWn
µn N D
VA
∆Fn
∆Fp
Still diffusion dominated transport? Since Quasi-Fermi
levels are not flat in nonlinear regime (drift), this approximation
becomes worse.
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
51
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
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
52
Region (2): Ambipolar Transport
 D D p  ( qVA −∆Fn −∆Fp ) β / 2
JT ≈ −q  n +
 ni e
Wp Wn 
ln( J T ) ≈
Today’s lecture: Ambipolar Transport regime (2)
qVA
2 k BT
Question: Where does the 2
come from?
ln(I)
3
2
1
6,7
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
VA
53
Nonlinear Regime: Ambipolar Transport
np = ni2 e
(
Here not negligibly small.
Ambipolar transport !
( Fn − Fp ) β
)
(
ni2
q (V −∆F −∆F ) β
+ ∆n)( N A + ∆p ) = ni2 e A n p − 1
NA
Excess carrier concentrations >> NA Thus…
∆n ≈ ∆p = ni
≈ ni e
(
(e (
)
q VA −∆Fn −∆Fp β
)
−1
)
q VA −∆Fn −∆Fp β / 2
Currents
J n = − qDn
∆n qDn ni ( qVA −∆Fn −∆Fp ) β / 2
=
e
Wp
Wp
J p = −qD p
∆n qDp ni ( qVA −∆Fn −∆Fp ) β / 2
=
e
Wn
Wn
Note: junction never disappears,
even for large forward bias!
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
54
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
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
55
Forward Bias Nonlinearity (7): Esaki Diode
Esaki-Diode: Heavily doped diode
ln(I)
Fn
3
2
Fp
1
6,7
empty
Fn
VA
Fp
Heavy doping
I
X
Fn
No states!
Fp
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
1
VA
Tunneling in
diodes.
Nobel Prize
(Esaki)
Reverse Bias (5): Zener Tunneling
+VA
Tunneling
(triangular barrier)
empty
Zener tunneling
occurs in every
diode. (reverse
bias)
Fp
4
Fn
5
empty
Remember: Tunneling through a triangular barrier
I = qpT υ
4
α k 
4 cosh αd +  −  sinh 2 αd
k α
(p.49 ADF)
T=
2
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
57
Conclusion
1)
I-V characteristics of a p-n junction is defined by many
interesting phenomena including diffusion, ambipolar transport,
tunneling etc.
2) The separate regions are identified by specific features. Once
we learn to identify them, we can see if one or the other
mechanism is dominated for a given technology.
3) In the next class, we will discuss a few more non-ideal effects.
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
58