Semiconductor Device Modeling
and Characterization – EE5342
Lecture 2 – Spring 2011
Professor Ronald L. Carter
ronc@uta.edu
http://www.uta.edu/ronc/
Web Pages
* Bring the following to the first class
• R. L. Carter’s web page
– www.uta.edu/ronc/
• EE 5342 web page and syllabus
– http://www.uta.edu/ronc/5342/syllabus.htm
• University and College Ethics Policies
www.uta.edu/studentaffairs/conduct/
www.uta.edu/ee/COE%20Ethics%20Statement%20Fall%2007.pdf
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First Assignment
• e-mail to listserv@listserv.uta.edu
– In the body of the message include
subscribe EE5342
• This will subscribe you to the EE5342
list. Will receive all EE5342 messages
• If you have any questions, send to
ronc@uta.edu, with EE5342 in
subject line.
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A Quick Review of Physics
• Review of
– Semiconductor Quantum Physics
– Semiconductor carrier statistics
– Semiconductor carrier dynamics
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Bohr model H atom
• Electron (-q) rev. around proton (+q)
• Coulomb force, F=q2/4peor2,
q=1.6E-19 Coul,
eo=8.854E-14 Fd/cm
• Quantization L = mvr = nh/2p
• En= -(mq4)/[8eo2h2n2] ~ -13.6 eV/n2
• rn= [n2eoh]/[pmq2] ~ 0.05 nm = 1/2 Ao
for n=1, ground state
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Quantum Concepts
•
•
•
•
Bohr Atom
Light Quanta (particle-like waves)
Wave-like properties of particles
Wave-Particle Duality
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Energy Quanta
for Light
1
2
Tmax  mv  h f  fo   qVstop
2
• Photoelectric Effect:
• Tmax is the energy of the electron
emitted from a material surface when
light of frequency f is incident.
• fo, frequency for zero KE, mat’l spec.
• h is Planck’s (a universal) constant
h = 6.625E-34 J-sec
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Photon: A particle
-like wave
• E = hf, the quantum of energy for
light. (PE effect & black body rad.)
• f = c/l, c = 3E8m/sec, l = wavelength
• From Poynting’s theorem (em waves),
momentum density = energy density/c
• Postulate a Photon “momentum”
p = h/l = hk, h = h/2p
wavenumber, k = 2p /l
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Wave-particle
Duality
• Compton showed Dp = hkinitial - hkfinal,
so an photon (wave) is particle-like
• DeBroglie hypothesized a particle
could be wave-like, l = h/p
• Davisson and Germer demonstrated
wave-like interference phenomena for
electrons to complete the duality
model
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Newtonian Mechanics
• Kinetic energy, KE = mv2/2 = p2/2m
Conservation of Energy Theorem
• Momentum, p = mv
Conservation of Momentum Thm
• Newton’s second Law
F = ma = m dv/dt = m d2x/dt2
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Quantum Mechanics
• Schrodinger’s wave equation
developed to maintain consistence
with wave-particle duality and other
“quantum” effects
• Position, mass, etc. of a particle
replaced by a “wave function”, Y(x,t)
• Prob. density = |Y(x,t)• Y*(x,t)|
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Schrodinger Equation
• Separation of variables gives
Y(x,t) = y(x)• f(t)
• The time-independent part of the
Schrodinger equation for a single
particle with KE = E and PE = V.
2
2
 y x  8 p m
 2 E V ( x )  y x   0
2
x
h
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Solutions for the
Schrodinger Equation
• Solutions of the form of
y(x) = A exp(jKx) + B exp (-jKx)
K = [8p2m(E-V)/h2]1/2
• Subj. to boundary conds. and norm.
y(x) is finite, single-valued, conts.
dy(x)/dx is finite, s-v, and conts.

*
y
 x y x dx  1
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
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Infinite Potential Well
• V = 0, 0 < x < a
• V --> inf. for x < 0 and x > a
• Assume E is finite, so
y(x) = 0 outside of well
2
np x 

y x  
sin 
, n = 1,2,3,...
a
 a 
2 2
2 2
h n
h k
h hk
En 

,p  
2
2
l 2p
8ma
4p
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Step Potential
•
•
•
•
V = 0, x < 0 (region 1)
V = Vo, x > 0 (region 2)
Region 1 has free particle solutions
Region 2 has
free particle soln. for E > Vo , and
evanescent solutions for E < Vo
• A reflection coefficient can be def.
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Finite Potential Barrier
•
•
•
•
Region 1: x < 0, V = 0
Region 1: 0 < x < a, V = Vo
Region 3: x > a, V = 0
Regions 1 and 3 are free particle
solutions
• Region 2 is evanescent for E < Vo
• Reflection and Transmission coeffs.
For all E
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Kronig-Penney Model
A simple one-dimensional model of a
crystalline solid
• V = 0, 0 < x < a, the ionic region
• V = Vo, a < x < (a + b) = L, between ions
• V(x+nL) = V(x), n = 0, +1, +2, +3, …,
representing the symmetry of the
assemblage of ions and requiring that
y(x+L) = y(x) exp(jkL), Bloch’s Thm
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K-P Potential Function*
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K-P Static
Wavefunctions
• Inside the ions, 0 < x < a
y(x) = A exp(jbx) + B exp (-jbx)
b = [8p2mE/h]1/2
• Between ions region, a < x < (a + b) = L
y(x) = C exp(ax) + D exp (-ax)
a = [8p2m(Vo-E)/h2]1/2
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K-P Impulse Solution
• Limiting case of Vo-> inf. and b -> 0,
while a2b = 2P/a is finite
• In this way a2b2 = 2Pb/a < 1, giving
sinh(ab) ~ ab and cosh(ab) ~ 1
• The solution is expressed by
P sin(ba)/(ba) + cos(ba) = cos(ka)
• Allowed values of LHS bounded by +1
• k = free electron wave # = 2p/l
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K-P Solutions*
x
x
P sin(ba)/(ba) + cos(ba) vs. ba
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K-P E(k)
Relationship*
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Analogy: a nearly
-free electr. model
• Solutions can be displaced by ka = 2np
• Allowed and forbidden energies
• Infinite well approximation by
replacing the free electron mass with
an “effective” mass (noting E = p2/2m
= h2k2/2m) of
2  2  1
h
 E
*
m 
2
2
4p  k 
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Generalizations
and Conclusions
• The symm. of the crystal struct.
gives “allowed” and “forbidden”
energies (sim to pass- and stop-band)
• The curvature at band-edge (where k
= (n+1)p) gives an “effective” mass.
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Silicon Covalent
Bond (2D Repr)
• Each Si atom has 4
nearest neighbors
• Si atom: 4 valence
elec and 4+ ion core
• 8 bond sites / atom
• All bond sites filled
• Bonding electrons
shared 50/50
_
= Bonding electron
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Silicon Band
Structure**
• Indirect Bandgap
• Curvature (hence
m*) is function of
direction and band.
[100] is x-dir, [111]
is cube diagonal
• Eg = 1.17-aT2/(T+b)
a = 4.73E-4 eV/K
b = 636K
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References
*Fundamentals of Semiconductor
Theory and Device Physics, by Shyh
Wang, Prentice Hall, 1989.
**Semiconductor Physics & Devices,
by Donald A. Neamen, 2nd ed., Irwin,
Chicago.
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