ECE606: Solid State Devices
Lecture 17
Schottky Diode
Gerhard Klimeck
gekco@purdue.edu
1
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Presentation Outline
1) Importance of metal-semiconductor junctions
2) Equilibrium band-diagrams
3) DC Thermionic current (simple derivation)
4) Intermediate Summary
5) DC Thermionic current (detailed derivation)
6) AC small signal and large-signal response
7) Additional information
8) Conclusions
Reference: Semiconductor Device Fundamentals, Chapter 14, p. 477
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
2
Metal-semiconductor Diode
Symbols
M
N
ND
Metal
Wire
N
3
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Applications of M-S Diode ….
STM(scanning tunneling microscope)
Detectors
on semiconductor
CNT(carbon nanotube)
Original Bipolar
Transistors
www.fz-juelich.de/ibn/index.php?index=674
Originally, Gelena (PbS), Si as semiconductor and
Phospor Bronze for metal (cat’s whisker)
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
4
Presentation Outline
1) Importance of metal-semiconductor junctions
2) Equilibrium band-diagrams
3) DC Thermionic current (simple derivation)
4) Intermediate Summary
5) DC Thermionic current (detailed derivation)
6) AC small signal and large-signal response
7) Additional information
8) Conclusions
5
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Topic Map
Equilibrium DC Small
signal
Large
Signal
Circuits
Diode
Schottky
BJT/HBT
MOS
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
6
Drawing Band Diagram in Equilibrium…
∇ • D = q ( p − n + N D+ − N A− )
Equilibrium
∂n 1
= ∇ • J N − rN + g N
∂t q
J N = qnµ N E + qDN ∇n
DC dn/dt=0
Small signal dn/dt ~ jωtn
Transient --- full sol.
∂p
1
= − ∇ • J P − rP + g P
∂t
q
J P = qp µ P E − qDP ∇p
7
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Band-Diagram
1. EF flat in equilibrium
2. EC/EV in Metal
3. Vacuum level in Metal
4. Ec/Ev in semiconductor
5. Vacuum level in semiconductor
ND
6. Connect vacuum levels
Vacuum level
ΦM
7. duplicate the connections
down to Ec/Ev
χ
EC
EV
EF
Since NA is large,
xp is negligible ..
N A x p = N D xn
Charge(metal) = charge(semiconductor)
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
8
Built-in Potential: bc @Infinity
neglect
qVbi
ΦM
χ
ΦB
∆
qVbi = ( Φ M − χ ) − ∆ ≡ Φ B − ∆
∆ + χ + qVbi = Φ M
= Φ B − k BT ln
ND
nondegenerate
NC
9
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Analytical Solution (Simple Approach)
Depletion Approximation
ρ
qND
Actual
Xn
QM =-QD
E
integrate
Xn
x
x
Q
qN x
Emax = D = − D n
K
ε
K sε 0
Area negligible s 0
QD =
NDXn
integrate
φ
qφB
Φmax=qVbi
EC
x
Xn
EV
Xn
x
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
φmax = −
Emax xn
2
Notice how we neglect xp here, is
10
there a proof?
Complete Analytical Solution
Charg
e
x
+
-
n
Position
x
E ( 0+ ) =
qN D xn
k sε 0
E ( 0− ) =
qN M x p
ksε 0
?
⇒ N D xn = N M x p
p
E-field
Position
Potentia
l
qVbi =
=
E ( 0− ) xn
2
2
+
E ( 0+ ) x p
2
qN M x p 2
qN D xn
+
2k s ε 0
2k sε 0
Position
11
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Depletion Regions
ND
xp xn
N D xn = N M x p
qVbi =
2
qN D xn 2 qN M x p
+
2k s ε 0
2k s ε 0
xn =
2k sε 0
NM
2k s ε 0 1
Vbi →
Vbi
q ND ( NM + ND )
q ND
NM ∞
2k sε 0
ND
xp =
Vbi → 0
q NM ( NM + ND )
This is why we can neglect xp
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
12
Presentation Outline
1) Importance of metal-semiconductor junctions
2) Equilibrium band-diagrams
3) DC Thermionic current (simple derivation)
4) Intermediate Summary
5) DC Thermionic current (detailed derivation)
6) AC small signal and large-signal response
7) Additional information
8) Conclusions
13
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Topic Map
Equilibrium DC
Small
signal
Large
Signal
Circuits
Diode
Schottky
BJT/HBT
MOS
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
14
Band Diagram with Applied Bias…
∇ • D = q ( p − n + N D+ − N A− )
∂n 1
= ∇ • J N − rN + g N
∂t q
J N = qnµ N E + qDN ∇n
Band diagram …
This works for doping-modulated
Semiconductors.
∂p
1
= − ∇ • J P − rP + g P
∂t
q
Does not work for heterostructures
(when the conduction band is not
continuous)
J P = qp µ P E − qDP ∇p
Metal-Semiconductor is a HS
Need theory of thermionic
Klimeck – ECE606 Fall 2012 – notes adopted from Alamemission
15
Depletion Regions with Bias
ND
xp
xn
xn =
2k sε 0
NM
2 k sε 0 1
Vbi →
(Vbi − VA )
q ND ( NM + ND )
q ND
xp =
2 k sε 0
ND
Vbi → 0
q NM ( NM + ND )
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Forward bias: xn decrease
Reverse bias: xn increase
16
Band-diagram with Bias
Forward Bias
qVbi-V
EC-EF
VA
qVbi
EC-EF
Reverse Bias
VA
EC-EF
17
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
I-V Characteristics
Forward Bias
Ec
EF
VA
I
Metal
EF
ESF
EV
M
N-type
Semi.
1,2,3
4 – generation current
5 – impact ionization
I
6,7
Reverse Bias
EF
4
M
Ec
EF
ESF
EV
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
5
1 – diffusion control
2 – ambipolar
3 – resistance drop
6 – defects – trap assist
7 – Esaki
18
Current Flow Concept
Barrier
The same!
M=>S current
unchanged!
Partition problem in
2 currents
M=>S S<=M
Barrier smaller
M<=S increased
ΦB
qVbi-VA
EC-EF
Forward Bias
VA
qVbi
ΦB
EC-EF
Barrier
The same!
M=>S current
unchanged!
ΦB
Barrier bigger
M<=S decreased
q(Vbi+VA)
VA
In Equilibrium:
and
are equal
M=>S current
Independent
of bias!
EC-EF
Reverse Bias
19
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Left Boundary Condition
We use thermionic emission
because:
1. EC is not continuous here
2. There’s no minority carriers!
J T (VA = 0) = 0 = J m → s (0) − J s → m (0)
⇒ J m → s (0) = J s → m (0)
J T (VA ) = J m→ s (VA ) − J s →m (VA )
= J m→ s (0) − J s → m (VA )
q(Vbi-VA) E
C
VA
(detailed balance)
M=>S current
independent of bias
J T (VA ) = J s → m (0) − J s → m (VA )
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
20
Semiconductor to Metal Flux
V
ns − q kTbi
J s →m (0) = − q e υth = J m→ s (0)
J m→ s (0) =
2
V −V
qΦ
ns − q bikT A
nm − kT
υth
J m→ s (VA ) = − q e
υth M J s → m (VA ) = − q e
2
2
qV
qVA
nsυth − kTbi
Only the ones
= −q
e
× e kT
Only half of above the
2
them goes to barrier goes to
qV
B
the right
= J s → m (0) e
the right
q(Vbi-VA)
EC-EF
VA
= J m → s (0) e
qVA
kT
nυ −
= −q m th Me
2
qnmυthM − qkTΦΦm
J T = J s → m (0) − J s →m (VA ) =
e
2
B
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
A
kT
qΦ B
kT
e
qVA
kT
 qVkTA 
e − 121


Diffusion vs. Thermionic Emission
∆n
Fp
Fn
Check that both gives the
same result for a diode…
Fn
Fp
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Thermionic Emission theory
is a more general approach,
can be used for device so
small that no scattering
occur (can’t use diffusion!).
22
Intermediate Summary
Schottky barrier diode is a majority carrier device of great
historical importance.
There are similarities and differences with p-n junction
diode: for electrostatics, it behaves like a one-sided
diode, but current, the drift-diffusion approach requires
modification.
The trap-assisted current, avalanche breakdown, Zener
tunneling all could be calculated in a manner very similar
to junction diode.
23
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Presentation Outline
1) Importance of metal-semiconductor junctions
2) Equilibrium band-diagrams
3) DC Thermionic current (simple derivation)
4) Intermediate Summary
5) DC Thermionic current (detailed derivation)
6) AC small signal and large-signal response
7) Additional information
8) Conclusions
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
24
Energy Resolved Thermionic Flux
Energy <
1 * 2
m υ min
2
will be bounced back
∞
∞ −υ min
J s→m = −q ∫
∫ ∫
−∞
−∞
−∞
Occupation
(non-denegerate)
~ DOS
Ω
dk x dk y dk z e −( E − EF ) β υ x
3
4π
υ
Only movement in the
x direction will contribute
to the current flux
min
q(Vbi-VA)
VA
EC-EF
x
25
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Thermionic Flux from Semi to Metal ..
∞
∞
−υmin
−∞
−∞
−∞
J s →m = −q ∫
∫ ∫
E
E − E F = ( E − EC ) + ( EC − E F ) =
= qe
− ( Ec − EF ) β
∞
∞
−υ min
−∞
−∞
−∞
∫ ∫ ∫
qΩ ( m* )
4π 3 ℏ3
e
1 * 2 1 * 2 1 * 2
m υ x + m υ y + m υ z + ( EC − E F )
2
2
2
*
*
*
( mυx + mυ y + mυ z ) β
Ω d ( m υx ) d ( m υ y ) d ( m υz ) −
2
e
υx
4π 3
ℏ
ℏ
ℏ
3
J s→m =
EC
EF
Ω
− E−E β
dk x dk y dk z e ( F ) υ x
3
4π
− ( Ec − EF ) β
2
2
2
 m*υ x2 
 ∞ − m*υ y2  β
  ∞ − m*υ z2  β

  −υmin
−
 β

 e  2  dυ   e  2  dυ  
 2 

d
υ
e
υ
y
z
x
x
∫
∫
∫



−∞
−∞
  −∞


 
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
26
Thermionic Current …
υmin =
J s →m =
q Ω ( m* )
4π 3 ℏ3
3
e
− ( Ec − E F ) β
 m*υ x2 
 ∞ − m*υ y2  β
  ∞ − m*υz2  β

  −υmin
−
β
 2 
 e  2  dυ   e  2  dυ  


υ
υx 
d
e
y
z
x
∫
∫
∫

  −∞

 −∞
  −∞

π
J s →m
2q
(Vbi − VA )
m*
π
1 − q (Vbi −VA ) β
e
2
4π qm*k 2 2 ( EF − EC − qVbi ) β qVA β
=
T e
e
= A0 e qVA β
3
h
J T = J s →m − J m → s = A0 ( e qVA β − 1)
27
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Some insight of the Thermionic Current …
Compare to the current of p-n diodes…
J T = J s →m − J m→ s = A0 ( e qVA β − 1)
Schottky diode
 D n2 D n2 
J T = − q  n i + p i  ( e qVA β − 1)
 W p N A Wn N D 
p-n diode
Both of them depends exponentially on VA, however current of p-n diodes depends
more on temperature (since ni depends strongly on Eg), where the Schottky diode
doesn’t have that dependence.
However, Eg hides in…
J s →m
4π qm* k 2 2 ( EF − EC − qVbi ) β qVA β
=
T e
e
= A0 e qVA β
3
h
The information of material hides in m*
The information of doping concentration hides in EF-EC
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
28
Recombination/Generation/Impact-ionization
EF
M
EF
M
Ec
EF
ESF
EV
SAME technique as in p-n junction except integrate to xp only
29
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Presentation Outline
1) Importance of metal-semiconductor junctions
2) Equilibrium band-diagrams
3) DC Thermionic current (simple derivation)
4) Intermediate Summary
5) DC Thermionic current (detailed derivation)
6) AC small signal and large-signal response
7) Additional information
8) Conclusions
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
30
Topic Map
Equilibrium
DC Small
signal
Large
Signal
Circuits
Diode
Schottky
BJT/HBT
MOS
31
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
AC response
Conductance
Junction
Capacitance
Diffusion
Capacitance
Series
Resistanc
e
We will not have Diffusion Capacitance here…
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
32
Forward Bias Conductance
m depends on which
operation regime you are
(
I = Io e
ln
q (V A − RS I ) β / m
gFB
)
−1
RS
I + Io
= q (VA − RS I ) β / m
I0
1
m
= RS +
g FB
qβ ( I + I 0 )
m
dV
= A − RS
qβ ( I + I o ) dI
33
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Junction Capacitance (Majority Carriers)
Junction
Capacitance
CJ =
CJ =
Response time – dielectric
Very fast propagation
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
κ sε 0 A
W
κ sε 0 A
2κ s ε 0
qN D
(Vbi − V A )
34
No Diffusion Capacitance in Schottky Diode
p-n Diode
n
Schottky diode
When electrons reach the metal side, they
quickly scatter and join the majority
carriers there. Nothing is diffusing…
x
VA
No minority carrier transport and
therefore no diffusion capacitance ..
35
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Reducing diffusion capacitance in p-n diode
p-n Diode
Schottky diode
n
x
Provide
lots of
traps
VA
Short minority carrier lifetime in p-n junction diode
equivalent to rapid energy relaxation in SB diode.
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
36
Presentation Outline
1) Importance of metal-semiconductor junctions
2) Equilibrium band-diagrams
3) DC Thermionic current (simple derivation)
4) Intermediate Summary
5) DC Thermionic current (detailed derivation)
6) AC small signal and large-signal response
7) Additional information
8) Conclusions
37
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Ohmic Contact vs. Schottky contacts ..
What if we just want the
p-n junction? How to
reduce the barrier of
Metal-Semiconductor?
Metal
n+
n-Si
Low Doping
Moderate
Doping
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Oxide
For majority carriers:
there’s not barrier
For minority carriers:
high doping allows them
to tunnel through
Metal will just act like
Ohmic Contact
High Doping
38
Lowering of Schottky Barrier
E
qφ1
qφB
Metal
qφ1
qφB
Semi
Metal
Semi
SurfaceCharge -
Image
Charge
Electron
The fields on the
metal must be
perpendicular
(otherwise there
will be tangential
current flow
which is
impossible)
as if there’s an
image charge
(positive) in the
metal, results in
lowering the
barrier height
x
x
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
Ref. Sze/Ng., p. 143 39
Fermi-level Pinning
Density of
States
E
c
E
Full
f
Ev
Metal
Full
Semiconduct
or
f
Full
Full
Ev
A
Metal
Shift in the
Band
edge
E
c
E
Metal
Why sometimes we don’t
get conductance at all?
B
surface states bends EC/EV so
that the Fermi level is at the
center of the bandgap, they
exchange carriers with metals,
blocking the bulk
semiconductor inside
no modulation in potential
barrier even you connected to
different metals.
Regardless the workfunction, no
modulation in potential.
(e.g. Modern high-k dielectrics)
Metal
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
40
Conclusion
1) Schottky diodes have wide range of applications in
practical devices.
2) The key distinguishing feature of Schottky diode is that
it is a majority carrier device.
3) We use a different technique to calculate the current in
a majority carrier device.
=> thermionic emission.
4) Elimination of diffusion capacitance make the response
of the diodes very fast.
Klimeck – ECE606 Fall 2012 – notes adopted from Alam
41