1. EE 5340 Semiconductor Device Theory Lecture 2 - Fall 2009
Professor Ronald L. Carter

2. Quantum Concepts Bohr Atom Light Quanta (particle-like waves)
Wave-like properties of particles
Wave-Particle Duality
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3. Wave-particle duality
Davisson and Germer demonstrated wave-like interference phenomena for electrons to complete the duality model
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4. 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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5. 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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6. 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 Total E = E and PE = V. The Kinetic Energy, KE = E - V
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7. 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.
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8. 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
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9. 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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10. 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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11. 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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12. K-P Potential Function*
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13. 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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14. 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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15. K-P Solutions*
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16. K-P E(k) Relationship*
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17. 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
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18. Generalizations and 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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19. 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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20. Silicon Band Structure**
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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21. Si Energy Band Structure at 0 K
Every valence site is occupied by an electron
No electrons allowed in band gap
No electrons with enough energy to populate the conduction band
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22. Si Bond Model Above Zero Kelvin
Enough therm energy ~kT(k=8.62E-5eV/K) to break some bonds
Free electron and broken bond separate
One electron for every “hole” (absent electron of broken bond)
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23. Band Model for thermal carriers
Thermal energy ~kT generates electron-hole pairs
At 300K
Eg(Si) = eV
>> kT = meV,
Nc = 2.8E19/cm3
> Nv = 1.04E19/cm3
>> ni = 1.45E10/cm3
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24. Donor: cond. electr. due to phosphorous
P atom: 5 valence elec and 5+ ion core
5th valence electr has no avail bond
Each extra free el, -q, has one +q ion
# P atoms = # free elect, so neutral
H atom-like orbits
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25. Bohr model H atom- like orbits at donor
Electron (-q) rev. around proton (+q)
Coulomb force, F=q2/4peSieo,q=1.6E-19 Coul, eSi=11.7, eo=8.854E-14 Fd/cm
Quantization L = mvr = nh/2p
En= -(Z2m*q4)/[8(eoeSi)2h2n2] ~-40meV
rn= [n2(eoeSi)h2]/[Zpm*q2] ~ 2 nm
for Z=1, m*~mo/2, n=1, ground state
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26. Band Model for donor electrons
Ionization energy of donor Ei = Ec-Ed ~ 40 meV
Since Ec-Ed ~ kT, all donors are ionized, so ND ~ n
Electron “freeze-out” when kT is too small
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27. Acceptor: Hole due to boron
B atom: 3 valence elec and 3+ ion core
4th bond site has no avail el (=> hole)
Each hole, adds --q, has one -q ion
#B atoms = #holes, so neutral
H atom-like orbits
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28. Hole orbits and acceptor states
Similar to free electrons and donor sites, there are hole orbits at acceptor sites
The ionization energy of these states is EA - EV ~ 40 meV, so NA ~ p and there is a hole “freeze-out” at low temperatures
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29. Impurity Levels in Si: EG = 1,124 meV
Phosphorous, P: EC - ED = 44 meV
Arsenic, As: EC - ED = 49 meV
Boron, B: EA - EV = 45 meV
Aluminum, Al: EA - EV = 57 meV
Gallium, Ga: EA - EV = 65meV
Gold, Au: EA - EV = 584 meV EC - ED = 774 meV
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30. 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.
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