banerjee foils ch1 2

8/13/2019 Banerjee Foils Ch1 2

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Lecture notes for:Solid State Electronic Devices

by S. Banerjeewith modifications by M. Nadeem Akram and Per Ohlckers

The most important topics in thesyllabus:

Crystal structure of semiconductors Quantum physics

Bandstructure, carrier drift

Optical Generation-Recombination,Diffusion

Bipolar (PN) junctions

Field Effect Transistors: MOS ICs

Bipolar Junction Transistors

Optoelectronics

Chapter 1 and 2 slides in this file


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Moores Law:Exponential increase oftransistor count, and

decrease of sizes with time

Chapter 1

2


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Slide 3

Semiconductors


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Periodic Table

l = 1 (p)

l = 0 (s)

l = 2 (d)

l = 3 (f)

1

2

3

4

5

6

7

NobleGas

Halogen

GroupVI

GroupIV

GroupV

GroupIII

n

4


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5


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6

In addtion; Ternary and quartenarysemiconductors are used in specific applications

Examples of ternary semiconductors: AlxGa1-xAs, GaAsyP1-y

Example of quartenary semiconductors: AlxGa1-xAsyP1-y

Silicon (Si) (Elementary semiconductor) is by far the mostused material for integrated circuits combining excellent

technical perforance with low cost.

Gallium Arsenide (GaAs) (Binary semiconductor) is importantin optoelectronics, favourably combining electronic and

optical features.


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A LATTICE is an infinite,periodic arrangement of points. The symmetry

determines the lattice, notthe magnitudesof the distances.At each lattice point if we put a physical entity or BASIS which is one atom (ora group of atoms in the same arrangement), we get a CRYSTAL.

A crystal has long rangeorder, polycrystalline material only short rangeorderand an amorphous material no order. 7

Crystal Lattice


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PRIMITIVE cell ODEF is the smallest cell that can be translated by integernumber of primitive vectors a and b to replicate the lattice. It has latticepoints ONLY at corners of cell. It is not unique but convention is to choose

smallest and most orthogonal primitive vectors.

UNIT cell (rectangle) allows lattice points at corners, face center and bodycenter in 3-D. It is sometimes used instead of the primitive cell if it canrepresent the symmetry of the lattice better. (in this example centered

rectangular 2-D lattice). It replicates the lattice by integer translations ofbasis vectors. 8

Primitive Cell/Unit Cell


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3-D lattices are called SPACE lattices or Bravais lattices.

These are examples of cubicUNIT cells because they obeyPOINT symmetry operations (reflection, rotation, inversion) of a

cube.The PRIMITIVE cells are different: It is cubic for the SC case,but non-cubic for FCC and BCC. The primitive cell has only ONElattice point per cell, and a volume of 1/n if n is the number

density of points. 9

Space lattices or Bravais lattices


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1. Choose coordinate system with origin at anylattice point and orient axes

along edges of cube.2. Determine intercepts of planes with axes in multiplesof unit cell edges.3. Take reciprocals of intercepts and reduce to lowest set of integers,unless

intercept is fraction of unit cell edge4. Equivalentsets of (100) planes: ({100} planes) by rotation of the unit cell

within the cubic lattice.

Miller Indices for Planes

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1. Fix coordinate system and choose vector along direction in the lattice.2. Determine components of vector along axes.3. Reduce to lowest set of integers.4. Equivalent [100] directions indicated as .

Miller Indices for Directions

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If all the atoms are the same as in Si or Ge, it is a d iamondstructure. If the

nearest neighbor atoms are different as in GaAs,it is zinc blende. The 4nearest neighbors are shown in a symmetric, tetrahedral arrangement.

Diamond latticestructure:(a) a unit cell of the

diamond latticeconstructed by placingatoms from each atom inan fcc; (b) top view(along any 100 direction)

of an extended diamondlattice. The coloredcircles indicate one fccsublattice and the blackcircles indicate the

interpenetrating fcc.

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13

Packing fraction:Si Unit Cell shown


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14

Number density:Si Unit Cell shown


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15--->

The 3 most importantSilicon Crystal Planes


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Pull crystal SLOWLY (~ mm/minute) while rotating (for uniformity). Diameterincreases as pull rate reduced. Industry grows large 300 mm diameter boulesoringots. Few stress-related defects because crystal unconstrained.Molten Si at 1412 C. Add dopants to melt, but incorporation governed bydistribution coefficient or segregation coefficient, k

d= C

S/C

L. Common

impurities are C and O from crucible.

Czochralski Bulk Crystal Growth

Steps in (100) Si wafer fabrication:

1. Grind boule into cylinder and putnotch on {110}orientation.

2. Saw into wafers, and grinding/polishing of damage.

3. Chamfer edges and chemical-mechanical polish front.

Ref: Plummer

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Atmospheric pressure CVD systems. The Si wafers are held in slots cut on thesides of a SiC-coated graphite susceptor that flares out near the base topromote gas flow patterns conducive to uniform epitaxy. Typical precursors anddeposition temperatures are SiCl

4(1200 C) and SiH

2Cl

2(1000 C).

Metal Organic CVD (MOCVD) for III-V semiconductors.

Barrel-type reactor for Si Vapor Phase Epitaxy

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Crystal growth bymolecular beam epitaxy

(MBE):(a) evaporation cells inside ahigh -vacuum chamber(10-11Torr) directingbeams of Al, Ga, As, anddopants onto a GaAssubstrate at ~600C;

(b) (b) scanning electronmicrograph of the crosssection of superlattice of

GaAs (dark lines) andAlGaAs (light lines).Each layer is fourmonolayers thick.

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1-D: Heteroepitaxy and misfitdislocations due to lattice mismatchbetween SiGe and Si leading tocompressive strain in the SiGe layer.(a) For less than critical layerthickness, tc, pseudomorphic growthoccurs; (b) above tc, misfitdislocations form at the interface.

Crystal Defects

Ref: Plummer

0-D: Vacancy(missing atom);Interstitial (extra atom in void);

Impurity-substitutional/interstitial1-D Dislocation

2-D: Surface, stacking fault

3-D: Precipitate or cluster19

Ch 2 Q Ph i


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Chapter 2:Quantum Physics

Diffraction can only be explained by

treating light as a wave instead of aparticle, supporting Huygens over Newton.

(l)(n) = c = 3 x 108m/s

= speed of light (constant)

Characterized by: Wavelength or Frequency;Amplitude (A) or Intensity (=A2), and Phase


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Planck/Einstein- Light Quanta Photons

Planck found that in order to

model black body radiation,one has to envision thatenergy (in the form of light)is lost (or gained) in integervalues according to:

DE = nhn

Energy Change

n = 1, 2, 3 (integers)

frequency

h = Plancks constant = 6.626 x 10-34J.s

Ephoton= hn

21Slide


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Photoelectric Effect- Einstein

212

e photonm h qn n

Energy needed to release electron =

workfunction of metal (measured in eV or J)

Light apparently behaves both as a particle and a wave: wave-particle duality. Perhaps so does matter?? Leads to quantum mechanics

0n0

Frequency ( n)

metal

One needs a minimum amount of photonenergy to see electrons.

Also for n no, number of electrons increaseslinearly with light intensity. As frequency of incident light is increased,

kinetic energy of emitted e-increases linearly.


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Continuous spectrum from solids Quantized spectrum from atoms

E

E

23

Only certain DE are allowedAny DE is possible

Slide


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Rydbergs Model

Rydberg extends the Balmer model (n1=2) for visible spectra to otherspectral series.

222

1

11

nnRyn

n1= 1, 2, 3, ..

n2= n1+1, n1+2,

Ry= 3.29 x 10151/s

This suggests that the energy levels of the H atom areproportional to 1/n2

Ritz discovers the combinatorial principle betweenphoton frequencies.


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Hydrogen Atom ModelsRutherford proposed a planetary model

where the light, negative electrons orbit

around a heavy nucleus made up of

positive protons and neutral neutrons.

He placed no restrictions on electron

orbits, which was a problem because

according to Maxwell such orbiting

electrons should lose energy andcombine with nucleus.

1. Bohr proposed that electrons can

only have certain allowed orbits.

2. In these orbits, the angularmomentum is quantized as:

p = mvr = nh/2

3. Electrons can lose or gain energy

onlyduring transitions between these

orbits.

DE = Ei-Ef= hn

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De Broglie Waves


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De Broglie Waves Based on wave-particle duality of matter, de Broglie suggested that particles have a

wavelength given by:

mv

h

p

h l

not allowed

khh

p l

l

2

2mv

h

p

h l

The Bohr orbits satisfied the standing wave condition

l nmv

hnr 2

p = mvr = nh/2

27Slide

H i b U i P i i l


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Heisenberg Uncertainty Principle Another limitation of classical physics was that it assumed we could know both

the position and momentum of an electron exactly. Heisenberg development of quantum mechanics led him to the observation that

there is a fundamental limit to how well one can know boththe position andmomentum of a particle simultaneously.

4

htE DD

4

hpx xDD

Uncertainty in position Uncertainty in momentumSimilar uncertainties in y,z.

4thuncertainty relation

What is the uncertainty in momentum for an electron in a 1 radiusorbital in which the positional uncertainty is 1% of the radius.?

Dp h

4Dx 6.626x1034J.s 41x1012m

5.27x1023kg.m /s

28Slide


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Quantum Mechanics: Schrodinger equation

Schrodinger developed a mathematical formalism

that incorporated the wave nature of matter:( - greek letter Psi pronounced (p)sigh)( H - greek letter Etapronounced ay-tay)

HE

HThe Hamiltonian: p2

2m (PE)Kinetic Energy

operating on

the Wavefunction:

gives us the

total energy

x

E = energy

What is anoperator?

e.g. derivatives

d/dt or d2/dx2

Schrodinger equation:


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Wavefunction

What is a wavefunction?= a probability amplitude (complex)

Consider a wave:

n tkxjAey 2

222*2 AAeeAy tkxjtkxj nnIntensity =

Probability of finding a particle in space:

*

Probability =

With the normalizedwavefunction, we can

describe spatial distributions.


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31


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)(

kxj

Aex

tjEtxV

xmH

,

2 2

22

Time evolution of wavefunction in 1-D given by Schrodinger equation, whereHamiltonian is H = KE+ PE=T +V

tj

Etj

eetf

tEft

tf

j

)(

)()(

Use separation of variables, for static potential V(x):

)()()()(

)()()V(

2 2

22

tfxE

t

tfx

j

tfxx

xm

Gettime-independent Schrodingereigenvalue equation for spatial part:

For free particle, V=0 putting:

mkE

2

22

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k lh

)(, tkxjAetx

m

p

m

k

tj

E

22

222

For a free particle, multiplying the spatial and temporal parts of thesolution we get a plane wave function of position, x, and time, t,

with normalization constantA,wavevector k and frequency :

(Planck)

(de Broglie)p

Expectationvalues of quantum mechanical

operatorscorrespond to physical quantities.

xj

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Particle in a potential well and Quantization


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Particle in a potential well and Quantization

L

xn

Lx

sin

2

L

xn

Lx

sin

2

kxAx sin Energy is quantizedin terms of quantumnumber n due to

confinement of particle. E

2

22

8mL

hnE

n = 1, 2,

x

0

inf.

0 L

34Slide

Q t h i l t li


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(a) Potential barrier of height V 0and thickness

W;

(b) (b) probability density for an electron with

energy E< V 0, indicating a non-zero value of

the wave function beyond the barrier.

Quantum mechanical tunneling

35

Q ti ti i S h i ll S t i P t ti l


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Quantization in Spherically Symmetric Potentials

In the case of the hydrogen atom, energybecomes quantized due to the presence of a

constraining potential.

r0

SchrodingerEquation

r

ZerV

2

)(

0

Recovers the Bohr model36Slide


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H-atom wavefunctions

If we solve the Schrodinger equation using thispotential, we find that the energy levels arequantized:

En Z2

n2me

4

802h2

2.178x1018J Z

2

n2

n is theprincipal quantum number, and rangesfrom 1 to infinity.

These correspond to the Bohr orbits,

and is due to constraining Coulomb potential.

218 18

22.178 10 2.178 10 13.6 1( )n

ZE x J x J eV for n Rydberg

n

H atom a ef nctions orbitals


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H-atom wavefunctions- orbitals In solving the Schrodinger equation, other quantum numbers become evident:

l, the orbital angular momentum quantum number. Gives ellipticity of orbit.

m,the z component of orbital angular momentum.Ranges in value from -l to 0 to l. Lines up along external magnetic field.

We can then characterize the wavefunctions based onthe quantum numbers (n, l, m, s).

Also have a spin quantum numbers

of

l(0 to (n-1)) is given a letter value as follows: 0 = s (sharp)

1 = p (principal)

2 = d (diffuse)

3 = f (fundamental)


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n l Orbital ml # of Orb.

1 0 1s 0 12 0 2s 0 1

1 2p -1, 0, 1 33 0 3s 0 1

1 3p -1, 0, 1 32 3d -2, -1, 0, 1, 2 5

For each of these, have spin s = .

The total number of electrons in the shells are 2, 8, 18,..

Orbital energies increase as 1/n2

Orbitals of same n, but differentl are considered to be of (almost)

equal energy (degenerate).

39Slide

S (l=0) Orbital Shapes


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S (l=0) Orbital Shapes

r dependence only

as n increases, orbitalsdemonstrate n-1 nodes.

1s

1

Z

ao

32

e

Z

a0r

1

Z

ao

32

e

a0 is the Bohr radius = 0.53

2 (l 1) bit l h


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2p (l = 1) orbital shapes

not spherical, but lobed.

labeled with respect to orientation along x, y, and z.

2pz

1

4 2

Z

ao

32

e

2 cos

Electronic Structure of Elements and Periodic Table


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1A 2A 3A 4A 5A 6A 7A 8A

1

H1s1

2

He1s2

3Li

1s22s1

4Be

1s22s2

5B

1s22s22p1

6C

1s22s22p2

7N

1s22s22p3

8O

1s22s22p4

9F

1s22s22p5

10Ne

1s22s22p6

11Na

[Ne]3s1

12Mg[Ne]3s2

13Al

[Ne]3s23p1

14Si

[Ne]3s23p2

15P

[Ne]3s23p3

16S

[Ne]3s23p4

17Cl

[Ne]3s23p5

18Ar

[Ne]3s23p6

Lowest energy orbitals occupied first: also Pauli exclusion principle 42Slide


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Hybridized Orbitals in a Si Atom


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The spherically symmetric s type wave functions or orbitals arepositive everywhere, while the three mutually perpendicular p

type orbitals ( p x, p y, p z) are dumbbell shaped and have a

positive lobe and a negative lobe. The four sp3hybridized

orbitals, only one of which is shown here, point symmetrically in

space and lead to the diamond lattice in Si.

y

44


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End of Chapter 1 and 2 slides

Most important topics: To be included later

banerjee foils ch1 2

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