EE 5340Semiconductor Device TheoryLecture 2 - Fall 2009

Professor Ronald L. Carterronc@uta.edu

http://www.uta.edu/ronc

L02 Aug 27 2

Quantum Concepts

• Bohr Atom

• Light Quanta (particle-like waves)

• Wave-like properties of particles

• Wave-Particle Duality

L02 Aug 27 3

Wave-particle duality

• Davisson and Germer demonstrated wave-like interference phenomena for electrons to complete the duality model

L02 Aug 27 4

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

L02 Aug 27 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”, (x,t)

• Prob. density = |(x,t)• (x,t)|

L02 Aug 27 6

Schrodinger Equation

• Separation of variables gives(x,t) = (x)• (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

2

2

280

x

x

mE V x x

h2 ( )

L02 Aug 27 7

Solutions for the Schrodinger Equation• Solutions of the form of (x) =

A exp(jKx) + B exp (-jKx) K = [82m(E-V)/h2]1/2

• Subj. to boundary conds. and norm.(x) is finite, single-valued, conts.d(x)/dx is finite, s-v, and conts.

1dxxx

©L02 Aug 28 8

Infinite Potential Well• V = 0, 0 < x < a• V --> inf. for x < 0 and x > a• Assume E is finite, so

(x) = 0 outside of well

2,

88E

1,2,3,...=n ,sin2

2

22

2

22

nhkh

pmkh

manh

axn

ax

©L02 Aug 28 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 , andevanescent solutions for E <

Vo

• A reflection coefficient can be def.

©L02 Aug 28 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

©L02 Aug 28 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 (x+L) = (x) exp(jkL), Bloch’s Thm

©L02 Aug 28 12

K-P Potential Function*

©L02 Aug 28 13

K-P Static Wavefunctions• Inside the ions, 0 < x < a

(x) = A exp(jx) + B exp (-jx) = [82mE/h]1/2

• Between ions region, a < x < (a + b) = L (x) = C exp(x) + D exp (-x) = [82m(Vo-E)/h2]1/2

©L02 Aug 28 14

K-P Impulse Solution• Limiting case of Vo-> inf. and b -> 0,

while 2b = 2P/a is finite• In this way 2b2 = 2Pb/a < 1, giving

sinh(b) ~ b and cosh(b) ~ 1• The solution is expressed by

P sin(a)/(a) + cos(a) = cos(ka)• Allowed valued of LHS bounded by +1• k = free electron wave # = 2/

©L02 Aug 28 15

K-P Solutions*

©L02 Aug 28 16

K-P E(k) Relationship*

©L02 Aug 28 17

Analogy: a nearly-free electr. 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

1

2

2

2

2

4

k

Ehm

©L02 Aug 28 18

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)) gives an “effective” mass.

©L02 Aug 28 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

©L02 Aug 28 20

Silicon BandStructure**• Indirect Bandgap• Curvature (hence

m*) is function of direction and band. [100] is x-dir, [111] is cube diagonal

• Eg = 1.17-T2/(T+) = 4.73E-4 eV/K = 636K

©L02 Aug 28 21

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

©L02 Aug 28 22

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)

©L03 Sept 02 23

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

©L03 Sept 02 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

©L03 Sept 02 25

Bohr model H atom-like orbits at donor• Electron (-q) rev. around proton (+q)

• Coulomb force, F=q2/4Sio,q=1.6E-19 Coul, Si=11.7, o=8.854E-14 Fd/cm

• Quantization L = mvr = nh/2• En= -(Z2m*q4)/[8(oSi)2h2n2] ~-40meV

• rn= [n2(oSi)h2]/[Zm*q2] ~ 2 nm

for Z=1, m*~mo/2, n=1, ground state

©L03 Sept 02 26

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

©L03 Sept 02 27

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

©L03 Sept 02 28

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

©L03 Sept 02 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 meVEC - ED = 774 meV

L02 Aug 27 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.