1. Professor Ronald L. Carter ronc@uta.edu http://www.uta.edu/ronc/
Semiconductor Device Modeling and Characterization – EE5342 Lecture 2 – Spring 2011
Professor Ronald L. Carter

2. Web Pages Bring the following to the first class
R. L. Carter’s web page
EE 5342 web page and syllabus
University and College Ethics Policies
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3. 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 with EE5342 in subject line.
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4. A Quick Review of Physics
Semiconductor Quantum Physics
Semiconductor carrier statistics
Semiconductor carrier dynamics
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5. 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] ~ eV/n2
rn= [n2eoh]/[pmq2] ~ 0.05 nm = 1/2 Ao
for n=1, ground state
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6. Quantum Concepts Bohr Atom Light Quanta (particle-like waves)
Wave-like properties of particles
Wave-Particle Duality
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7. Energy Quanta for Light
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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8. 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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9. 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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10. 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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11. 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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12. 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.
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13. 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.
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14. 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
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15. 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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16. 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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17. 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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18. K-P Potential Function*
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19. 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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20. 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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21. K-P Solutions* x
P sin(ba)/(ba) + cos(ba) vs. ba
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22. K-P E(k) Relationship*
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23. 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
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24. 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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25. 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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26. 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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27. 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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