- Applications Using Band Diagrams and Fermi Energy Level
Applications to Devices – Physics Physics
MSE 310-ECE 340
Electrical Properties
of Materials
Homojunctions
Heterojunctions
p-n
junction
diodes
metal-s/c
junctions
p-n-p
Bipolar
transistors
p-n-p junction
Metal-oxide-semiconductor
junction
&
Light
Emitting
Devices
MOS capacitors
MOS transistors
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MSE 310-ECE 340
Part 1: Metal-Metal Contacts
Electrical Properties
of Materials
(a)
Pt
Mo
Vacuum
Vacuum
(Mo) = 4.20 eV
Flat band
(Pt) = 5.36 eV
- Workfunction Differences -
Fermi level
Fermi level
Electrons
Evac’s aligned
Electrons
(b)
Vacuum
Fermi level
4.20 eV
Equilibrium
5.36 eV
(Pt) - (Mo) = 1.16 eV = eV
Vacuum
Ef’s aligned
Fig. 4.28: When two metals are brought together, there is a contact
potential, V. (a) Electrons are more energetic in Mo so they tunnel
to the surface of Pt. (b) Equilibrium is reached when the Fermi levels
are lined up.
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
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1
Workfunctions of Various Metals
MSE 310-ECE 340
Electrical Properties
of Materials
Workfunction Equation as
Determined by
Mehrotra & Mahanty
ao = Bohr radius
p = plasmon frequency
= numerical value for integral
 = (1/3)0.5 vf
vf = Fermi velocity
ro = radius of equilibrium density
distribution of free electrons
Mehrotra & Mahanty, Free electron contribution to the workfunction of metals, J. Phys. C: Solid State Phys., Vol. 11, 1978.
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MSE 310-ECE 340
Image Potential = Schottky Effect
Electrical Properties
of Materials
 work  Evacuum  EF
Applied PE
Image PE
EF + 
Image Force
Potential Energy:
EF + 
EF + eff
Evacuum
Vimage (r )  Evacuum 
Ef 0
x
e2
 er
16 r
x
x
Vapplied (r )  er
  electric field
2
e
16 r
VTotal (r )  Evacuum 
e2
16 r
an e- a distance r from a
metal surface that has a
potential energy, Vimage.
Vimage (r )  
Net PE
(a)
(b)
VTotal (r )
0
r
r  rmin
(c)
To find eff:
Need to find maximum:
•Take derivative and set = 0
•Find rmin.
•Substitute rmin back into
equation and solve for eff.
Fig. 4.36: (a) PE of the electron near the surface of a conductor, (b)
Electron PE due to an applied field e.g. between cathode and anode
(c) The overall PE is the sum.
Vmax  eeff
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
p. 287-288
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Ng, p. 608-609
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2
MSE 310-ECE 340
Field Emission & Image Force
Electrical Properties
of Materials
PE(x)
EF + eff
Vo
e-
Vmax  eeff
EF
EF
(a)
0
Metal
0
xF
x
x = 0 x = xF
(b)
Vacuum
Field-Assisted Thermionic Emission
Grid or Anode
J e
Cathode
e
E
(c)


Vmax
kT
eeff
kT
where:
HV V
 e3 
eeff  e  
 
 4 o 
  electric field
Fig. 4.37 (a) Field emission is the tunneling of an electron at an
energy EF through the narrow PE barrier induced by a large applied
field. (b) For simplicity we take the barrier to be rectangular. (c) A
sharp point cathode has the maximumfield at the tip where the fieldemission of electrons occurs.
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
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MSE 310-ECE 340
Metal-Metal Contacts – Seebeck Effect
Seebeck effect (thermoelectric power)
Built-in potential difference, V, across a material due to a
temperature difference, T, across it
Electrical Properties
of Materials
S
Knowlton
V
T
Sign of S: potential of the cold side
with respect to the hot side; neg. if
e-’s have accumulated in the cold side.
Kasap, Electronic Materials & Devices (McGraw-Hill, 2006) Ch. 4
6
3
Seebeck Effect
MSE 310-ECE 340
Density of States = Low at Ef
Phonon Scattering will have a
greater effect on electrons
Lhot < Lcold (L = e- mean free path)
Electrical Properties
of Materials
+
+
+
+
-
e.g.: Cu, Li, Au
Density of States = High at Ef
Phonon Scattering will have a
lesser effect on electrons
Lhot > Lcold (L = e- mean free path)
e.g.: Ni, Pt, Al, Pd
Fig 4.61
Knowlton
-
+
+
+
+
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
MSE 310-ECE 340
7
Metal-Metal Contacts – Seebeck Effect
Application = Thermocouple Metal
Electrical Properties
of Materials
Metal
type A
Hot
Cold
Hot
o
o
100 C
Cold
o
100 C
0 C
0
Metal
V
o
0 C
0
Metal
Metal
type B
V
Metal
type B
(b)
(a)
Number of Carriers Diffusing to Hot
Region will differ in each metal, thus
voltage difference occurs
Fig 4.32
(a) If same metal wires are used to measure the Seebeck voltage across the
metal rod, then the net emf is zero. (b)The thermocouple from two different
metals, type A and B. The cold end is maintained at 0 ¡C which is the
reference temperature. The other junction is used to sense the temperature. In
this example it is heated to 100 ¡C.
T
T
VAB   SA  SB dT   SABdT
To
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To
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
4
MSE 310-ECE 340
Electrical Properties
of Materials
Metal-S/C Contacts: Schottky & Ohmic Contacts
Flat band
R.F. Pierret, Semiconductor Device Fundamentals (Addison-Wesley, 1996) Ch. 14
Knowlton
MSE 310-ECE 340
Electrical Properties
of Materials
9
Metal-S/C Contacts: Schottky & Ohmic Contacts
Flat band
Flat band
Equilibrium
Equilibrium
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R.F. Pierret, Semiconductor Device Fundamentals (Addison-Wesley, 1996) Ch. 14
10
5
MSE 310-ECE 340
Electrical Properties
of Materials
Metal-S/C Contacts:
Schottky & Ohmic
Contacts
Band-Bending
Where does it come
from?
  q  p  N D    n  N A  

d



dx
where
   o R
d 2V
d




dx 2
dx
Poisson's Equation
R.F. Pierret, Semiconductor Device Fundamentals
(Addison-Wesley, 1996) Ch. 14
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MSE 310-ECE 340
Electrical Properties
of Materials
Metal-S/C Contacts:
Schottky & Ohmic Contacts
Biasing Effects
R.F. Pierret, Semiconductor Device Fundamentals
(Addison-Wesley, 1996) Ch. 14
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6
MSE 310-ECE 340
Electrical Properties
of Materials
Metal-S/C Contacts: Schottky & Ohmic Contacts
Doping Effects
Equilibrium
R.F. Pierret, Semiconductor Device Fundamentals (Addison-Wesley, 1996) Ch. 14
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MSE 310-ECE 340
Electrical Properties
of Materials
13
Metal-S/C Contacts: Schottky & Ohmic Contacts
Overview
Note: Blocking = Schottky
Equilibrium
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Muller & Kamins, Device Electronics for Integrated Circuits, 3rd Ed. (Wiley, 1996) Ch. 3
14
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