1. EE 5340 Semiconductor Device Theory Lecture 03 – Spring 2011
Professor Ronald L. Carter

2. Review the Following R. L. Carter’s web page:
EE 5340 web page and syllabus. (Refresh all EE 5340 pages when downloading to assure the latest version.) All links at:
University and College Ethics Policies
Makeup lecture at noon Friday (1/28) in 108 Nedderman Hall. This will be available on the web.
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3. First Assignment Send e-mail to ronc@uta.edu
On the subject line, put “5340 ”
In the body of message include
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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4. Second Assignment Submit a signed copy of the document posted at
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5. 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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6. K-P Potential Function*
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7. 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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8. K-P Solutions*
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9. K-P E(k) Relationship*
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10. 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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11. 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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12. 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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13. 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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14. 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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15. 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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16. 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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17. 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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18. 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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19. 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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20. 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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21. 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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22. 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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23. 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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24. 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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25. 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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