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EE 5340 Semiconductor Device Theory Lecture 02 – Spring 2011. Professor Ronald L. Carter [email protected] http://www.uta.edu/ronc. 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 - PowerPoint PPT Presentation
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EE 5340Semiconductor Device TheoryLecture 02 – Spring 2011
Professor Ronald L. [email protected]
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 [email protected]– 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.