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Semiconductor Device Modeling and CharacterizationEE5342, Lecture 1-Spring 2005

Professor Ronald L. Carterronc@uta.edu

http://www.uta.edu/ronc/

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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– www.uta.edu/ronc/5342/syllabus.htm

• University and College Ethics Policies– http://www.uta.edu/studentaffairs/judicialaffairs/

– www.uta.edu/ronc/5342/Ethics.htm

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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=q2/4or2, q=1.6E-19 Coul, o=8.854E-14 Fd/cm

• Quantization L = mvr = nh/2• En= -(mq4)/[8o

2h2n2] ~ -13.6 eV/n2

• rn= [n2oh]/[mq2] ~ 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

• 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) constanth = 6.625E-34 J-sec

stopomax qVffhmvT 2

21

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Photon: A particle-like wave• E = hf, the quantum of energy for

light. (PE effect & black body rad.)• f = c/, c = 3E8m/sec, = wavelength• From Poynting’s theorem (em waves),

momentum density = energy density/c• Postulate a Photon “momentum”

p = h/= hk, h = h/2 wavenumber, k =2/

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Wave-particle Duality• Compton showed p = hkinitial - hkfinal,

so an photon (wave) is particle-like• DeBroglie hypothesized a particle

could be wave-like, = 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”, (x,t)

• Prob. density = |(x,t)• (x,t)|

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Schrodinger Equation

• Separation of variables gives(x,t) = (x)• (t)

• The time-independent part of the Schrodinger equation for a single particle with KE = E and PE = V.

2

2

280

x

x

mE V x x

h2 ( )

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Solutions for the Schrodinger Equation• Solutions of the form of

(x) = A exp(jKx) + B exp (-jKx)K = [82m(E-V)/h2]1/2

• Subj. to boundary conds. and norm.(x) is finite, single-valued, conts.d(x)/dx is finite, s-v, and conts.

1dxxx

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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

(x) = 0 outside of well

248

2

2

22

2

22 hkhp,

kh

ma

nhE

1,2,3,...=n ,axn

sina

x

n

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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 , andevanescent 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 (x+L) = (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

(x) = A exp(jx) + B exp (-jx) = [82mE/h]1/2

• Between ions region, a < x < (a + b) = L (x) = C exp(x) + D exp (-x) = [82m(Vo-E)/h2]1/2

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K-P Impulse Solution• Limiting case of Vo-> inf. and b -> 0,

while 2b = 2P/a is finite• In this way 2b2 = 2Pb/a < 1, giving

sinh(b) ~ b and cosh(b) ~ 1• The solution is expressed by

P sin(a)/(a) + cos(a) = cos(ka)• Allowed values of LHS bounded by +1• k = free electron wave # = 2/

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K-P Solutions*

P sin(a)/(a) + cos(a) vs. a

xx

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K-P E(k) Relationship*

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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.