Semiconductor Device Modeling and Characterization EE5342, Lecture 1-Spring 2005 L1 January 18 Professor Ronald L. Carter [email protected]

http://www.uta.edu/ronc/ 1

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 – www.uta.edu/ronc/5342/syllabus.htm

• University and College Ethics Policies – http://www.uta.edu/studentaffairs/judicialaffairs/ – www.uta.edu/ronc/5342/Ethics.htm

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First Assignment

• e-mail to [email protected]

– 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 [email protected], with EE5342 in subject line.

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A Quick Review of Physics

• Review of – Semiconductor Quantum Physics – Semiconductor carrier statistics – Semiconductor carrier dynamics L1 January 18 4

Bohr model H atom

• Electron (-q) rev. around proton (+q) • Coulomb force, F=q 2 /4 pe o r 2 , q=1.6E-19 Coul, e o =8.854E-14 Fd/cm • Quantization L = mvr = nh/2 p • E n = -(mq 4 )/[8 e o 2 h 2 n 2 ] ~ -13.6 eV/n 2 • r n = [n 2 e o h]/[ p mq 2 ] ~ 0.05 nm = 1/2 A o for n=1, ground state L1 January 18 5

Quantum Concepts

• Bohr Atom • Light Quanta (particle-like waves) • Wave-like properties of particles • Wave-Particle Duality L1 January 18 6

Energy Quanta for Light

• Photoelectric Effect: • T • f o

T max

max L1 January 18  1 2

mv

2 

h



f



f o

 

qV stop

is the energy of the electron emitted from a material surface when light of frequency f is incident.

, frequency for zero KE, mat’l spec.

• h is Planck’s (a universal) constant h = 6.625E-34 J-sec 7

Photon: A particle -like wave

• E = hf, the quantum of energy for light. (PE effect & black body rad.) • f = c/ l , c = 3E8m/sec, p = h/ l l = wavelength • From Poynting’s theorem (em waves), momentum density = energy density/c • Postulate a Photon “momentum” = hk, h = h/2 p wavenumber, k = 2 p / l L1 January 18 8

Wave-particle Duality

• Compton showed D p = hk initial - hk final , so an photon (wave) is particle-like • DeBroglie hypothesized a particle could be wave-like, model l = h/p • Davisson and Germer demonstrated wave-like interference phenomena for electrons to complete the duality L1 January 18 9

Newtonian Mechanics

• Kinetic energy, KE = mv • Momentum, p = mv • Newton’s second Law 2 /2 = p F = ma = m dv/dt = m d 2 /2m Conservation of Energy Theorem Conservation of Momentum Thm 2 x/dt 2 L1 January 18 10

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)| L1 January 18 11

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.



x

  2  8 p

h

2

2

 

( )

 y  0 L1 January 18 12

Solutions for the Schrodinger Equation

• Solutions of the form of y y (x) = A exp(jKx) + B exp (-jKx) K = [8 p 2 m(E-V)/h 2 ] 1/2 • Subj. to boundary conds. and norm.

(x) is finite, single-valued, conts.

d y (x)/dx is finite, s-v, and conts.  y *    

dx

 1    L1 January 18 13

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 y    2

a sin

  

n

p

a x

  

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

E n



h

2

n

8

ma

2 2 

h

4 2 p

k

2 2

, p



h

l 

hk

2 p L1 January 18 14

Step Potential

• V = 0, x < 0 (region 1) • V = V o , x > 0 (region 2) • Region 1 has free particle solutions • Region 2 has free particle soln. for E > V o , and evanescent solutions for E < V o • A reflection coefficient can be def.

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Finite Potential Barrier

• Region 1: x < 0, V = 0 • Region 1: 0 < x < a, V = V o • Region 3: x > a, V = 0 • Regions 1 and 3 are free particle solutions • Region 2 is evanescent for E < V For all E o • Reflection and Transmission coeffs. L1 January 18 16

Kronig-Penney Model

A simple one-dimensional model of a crystalline solid • V = 0, 0 < x < a, the ionic region • V = V o , 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 L1 January 18 17

K-P Potential Function*

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K-P Static Wavefunctions

• Inside the ions, 0 < x < a y (x) = A exp(j b x) + B exp (-j b x) y b • Between ions region, a < x < (a + b) = L (x) = C exp( a = [8 = [8 p p 2 2 mE/h] a x) + D exp ( m(V o 1/2 -E)/h 2 ] 1/2 a x) L1 January 18 19

K-P Impulse Solution

• Limiting case of V o -> inf. and b -> 0, while a 2 b = 2P/a is finite • In this way a 2 b 2 = 2Pb/a < 1, giving sinh( a b) ~ P sin( b a b and cosh( a)/( b a) + cos( b a b) ~ 1 • The solution is expressed by a) = cos(ka) • Allowed values of LHS bounded by +1 • k = free electron wave # = 2 p / l L1 January 18 20

K-P Solutions*

x x P sin( b a)/( b a) + cos( b a) vs. L1 January 18 b a 21

K-P E(k) Relationship*

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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.

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