Semiconductor Device Modeling
and Characterization – EE5342
Lecture 3 – Spring 2011
Professor Ronald L. Carter
ronc@uta.edu
http://www.uta.edu/ronc/
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 Policies
www.uta.edu/studentaffairs/conduct/
www.uta.edu/ee/COE%20Ethics%20Statement%20Fall%2007.pdf
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First Assignment
• e-mail to listserv@listserv.uta.edu
– 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
ronc@uta.edu, with EE5342 in
subject line.
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Second Assignment
• e-mail to listserv@listserv.uta.edu
– 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
ronc@uta.edu, with EE5342 in
subject line.
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4
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 KE = E and PE = V.
2
2
 y x  8  m
 2 E V ( x )  y x   0
2
x
h
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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 = [82mE/h]1/2
• Between ions region, a < x < (a + b) = L
y(x) = C exp(ax) + D exp (-ax)
a = [82m(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 values of LHS bounded by +1
• k = free electron wave # = 2/l
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K-P Solutions*
x
x
P sin(ba)/(ba) + cos(ba) vs. ba
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K-P E(k)
Relationship*
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Analogy: a nearly-free X
electron model
• Solutions can be displaced by ka = 2n
• Allowed and forbidden energies
• Infinite well approximation by
replacing the free electron mass with
an “effective” mass (noting E = p2/2m
= h2k2/2m) of
2  2  1
h
 E

m 
2
2
4  k 
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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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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)) gives an “effective” mass.
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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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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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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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Band Model for
thermal carriers
• Thermal energy
~kT generates
electron-hole pairs
• At 300K
Eg(Si) = 1.124 eV
>> kT = 25.86 meV,
Nc = 2.8E19/cm3
> Nv = 1.04E19/cm3
>> ni = 1.45E10/cm3
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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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Bohr model H atomlike orbits at donor
• Electron (-q) rev. around proton (+q)
• Coulomb force, F=q2/4eSieo,q=1.6E-19
Coul, eSi=11.7, eo=8.854E-14 Fd/cm
• Quantization L = mvr = nh/2
• En= -(Z2m*q4)/[8(eoeSi)2h2n2] ~-40meV
• rn= [n2(eoeSi)h2]/[Zm*q2] ~ 2 nm
for Z=1, m*~mo/2, n=1, ground state
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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 “freezeout” when kT is too
small
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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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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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Impurity Levels in
Si: EG = 1,124 meV
•
•
•
•
•
•
Phosphorous, P:
Arsenic, As:
Boron, B:
Aluminum, Al:
Gallium, Ga:
Gold, Au:
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EC - ED = 44 meV
EC - ED = 49 meV
EA - EV = 45 meV
EA - EV = 57 meV
EA - EV = 65meV
EA - EV = 584 meV
EC - ED = 774 meV
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Semiconductor Electronics concepts thus far
• Conduction and Valence states due to
symmetry of lattice
• “Free-elec.” dynamics near band edge
• Band Gap
– direct or indirect
– effective mass in curvature
• Thermal carrier generation
• Chemical carrier gen (donors/accept)
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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.
• 1Device Electronics for Integrated Circuits, 2 ed.,
by Muller and Kamins, Wiley, New York, 1986.
• 2Physics of Semiconductor Devices, by S. M. Sze,
Wiley, New York, 1981.
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