EE 5340Semiconductor Device TheoryLecture 02 – Spring 2011

Professor Ronald L. Carterronc@uta.edu

http://www.uta.edu/ronc

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Web Pages

* Review the following• R. L. Carter’s web page

– www.uta.edu/ronc/• EE 5340 web page and syllabus

– www.uta.edu/ronc/5340/syllabus.htm• University and College Ethics Policies

– www.uta.edu/studentaffairs/conduct/– www.uta.edu/ee/COE%20Ethics%20Statement%20Fall%2007.p

df

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First Assignment

• Send e-mail to ronc@uta.edu– On the subject line, put “5340 e-mail”– In the body of message include

• email address: ______________________• Your Name*: _______________________• Last four digits of your Student ID: _____

* Your name as it appears in the UTA Record - no more, no less

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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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Bohr model for the H atom (cont.)En= -

(mq4)/[8eo2h2n2] ~

-13.6 eV/n2 *

rn= [n2eoh2]/[pmq2] ~ 0.05 nm = 1/2 Ao *

*for n=1, ground state

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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/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 = 2 p / l

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Wave-particle duality

• Compton showed Dp = hkinitial - hkfinal, so an photon (wave) is particle-like

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Wave-particle duality

• DeBroglie hypothesized a particle could be wave-like, l = h/p

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Wave-particle duality

• 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 givesY(x,t) = y(x)• f(t)

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

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

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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,

88E

1,2,3,...=n ,sin2

2

22

2

22

nhkh

pmkh

manh

axn

ax

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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 valued of LHS bounded by +1

• k = free electron wave # = 2p/l

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K-P Solutions*

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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

2

2

2

2

4

k

Ehm

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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 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.

M&K = Device Electronics for Integrated Circuits, 3rd ed., by Richard S. Muller, Theodore I. Kamins, and Mansun Chan, John Wiley and Sons, New York, 2003.