Embed Size (px)
Citation preview
Semiconductor Device Modeling and
Characterization – EE5342 Lecture 2 – Spring 2011
Professor Ronald L. [email protected]
http://www.uta.edu/ronc/
©rlc L02 21Jan2011
2
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
©rlc L02 21Jan2011
3
First Assignment
• e-mail to [email protected]– 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 [email protected], with EE5342 in subject line.
©rlc L02 21Jan2011
4
A Quick Review of Physics
• Review of –Semiconductor Quantum
Physics–Semiconductor carrier statistics–Semiconductor carrier dynamics
©rlc L02 21Jan2011
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)/[8eo
2h2n2] ~ -13.6 eV/n2
• rn= [n2eoh]/[pmq2] ~ 0.05 nm = 1/2 Ao
for n=1, ground state
©rlc L02 21Jan2011
6
Quantum Concepts
• Bohr Atom• Light Quanta (particle-like waves)• Wave-like properties of particles• Wave-Particle Duality
©rlc L02 21Jan2011
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) constanth = 6.625E-34 J-
sec
stopomax qVffhmvT 2
21
©rlc L02 21Jan2011
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 = 2 p / l
©rlc L02 21Jan2011
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
©rlc L02 21Jan2011
10
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
©rlc L02 21Jan2011
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)|
©rlc L02 21Jan2011
12
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 KE = E and PE = V.
2
2
280
x
x
mE V x x
h2 ( )
©rlc L02 21Jan2011
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.
1dxxx
©rlc L02 21Jan2011
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
248
2
2
22
2
22 hkhp,
kh
ma
nhE
1,2,3,...=n ,axn
sina
x
n
©rlc L02 21Jan2011
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.
©rlc L02 21Jan2011
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
©rlc L02 21Jan2011
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
©rlc L02 21Jan2011
18
K-P Potential Function*
©rlc L02 21Jan2011
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
©rlc L02 21Jan2011
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
©rlc L02 21Jan2011
21
K-P Solutions*
P sin(ba)/(ba) + cos(ba) vs. ba
xx
©rlc L02 21Jan2011
22
K-P E(k) Relationship*
©rlc L02 21Jan2011
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
1
2
2
2
2
4
k
Ehm
©rlc L02 21Jan2011
24
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.
©rlc L02 21Jan2011
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
©rlc L02 21Jan2011
26
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
©rlc L02 21Jan2011
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.
