Semiconductor Device Modeling and

Characterization – EE5342 Lecture 2 – Spring 2011

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– http://www.uta.edu/ronc/5342/

syllabus.htm• University and College Ethics Policieswww.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)/[8eo

2h2n2] ~ -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• 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 221

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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 = mvConservation 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.

y

py

22

28 0xx

m E V x x h2 ( )

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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.

* 1dxxx yy

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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

plp

py

248

2

222

222 hkhp,kh

manhE

1,2,3,...=n ,axnsin

ax

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 , 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 ModelA 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 byP 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*

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

xx

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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 1

22

22

4

*

kEhm

p

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Generalizationsand 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 BandStructure**• 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.