Semiconductor Device Modeling and

Characterization – EE5342 Lecture 3 – Spring 2011

Professor Ronald L. Carterronc@uta.edu

http://www.uta.edu/ronc/

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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

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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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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 ( )

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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 values of LHS bounded by +1

• k = free electron wave # = 2p/l

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K-P Solutions*

P sin(ba)/(ba) + cos(ba) vs. ba

xx

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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 = 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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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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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 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 BandStructure 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 ModelAbove 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 forthermal 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 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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Band Model fordonor 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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Acceptor: Holedue 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 andacceptor 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: 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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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.