the ig nobel prizes are “booby prizes”!

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Ig Nobel Prizes are “Booby Prize

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The Ig Nobel Prizes are  “Booby Prizes”!. Electronic Band Structures BW, Ch. 2; YC, Ch 2; & S, Ch. 2 ~ Follow BW & YC + input from S & outside sources where appropriate. Band Structure  E(k) - PowerPoint PPT Presentation

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Page 1: The Ig Nobel Prizes are   “Booby Prizes”!

The Ig Nobel Prizes are “Booby Prizes”!

Page 2: The Ig Nobel Prizes are   “Booby Prizes”!

Electronic Band StructuresBW, Ch. 2; YC, Ch 2; & S, Ch. 2

~ Follow BW & YC + input from S & outside sources where appropriate

Band Structure E(k)• We are interested in the behavior of electronic energy

levels in crystalline materials (any, not just semiconductors!)

as a function of wavevector k or momentum p = ħk • Group Theory: The math of symmetry that is

useful to simplify calculations of E(k).• A Mathematical Subject! Detailed math coverage &

discussion will be kept to a minimum. Results &

PHYSICS will be emphasized over math!

• Many calculational methods exist:– They are highly sophisticated & computational!

Page 3: The Ig Nobel Prizes are   “Booby Prizes”!

Basic knowledge assumed for this discussion:1. You know that electron energies are quantized.2. You’ve seen the simple Schrödinger Equation

(from quantum mechanics)Note: If you are weak on this or need a review, please get

& read an undergraduate quantum mechanics book!3. You are familiar with the crystal structures of some simple crystals, such as those we recently discussed. Note: If you are weak on this or need a review, please get & read an undergraduate solid state book!4. In a solid, the electronic energy levels form into regions of allowed energy (bands) & forbidden energy (gaps).

See any undergraduate solid state physics book!

Page 4: The Ig Nobel Prizes are   “Booby Prizes”!

• Overview: A qualitative/semi-quantitative discussion now. Later, a detailed & quantitative discussion.

Electronic Energy Bands E(k)These describe the behavior of the electronic energy levels in crystalline materials as function of wavevector k or momentum p = ħk.

• Recall that for FREE electrons, the energies have the simple form: E(k) = (p)2/2mo = (ħk)2/2mo

(mo= free electron mass)

• However, in crystalline solids: E(k) forms into regions of allowed energies (bands) & regions of forbidden energies (gaps).

Page 5: The Ig Nobel Prizes are   “Booby Prizes”!

• As we’ll see soon, Often in solids, a good Approximation is:

E(k) = (ħk)2/2m*

m* is not mo!

m* = m*(k) “effective mass”

However, the true E(k) are complicated!• Accurate calculations of E(k) require

sophisticated quantum mechanics, AND they also require sophisticated computational methods to obtain them numerically.

Page 6: The Ig Nobel Prizes are   “Booby Prizes”!

Qualitative Picture of Bands in “r-Space”(as functions of position in the solid)

One way to distinguish between solid types is byhow the electrons fill the bands & by the band gaps.

Page 7: The Ig Nobel Prizes are   “Booby Prizes”!

Electronic Energy BandsQualitative Picture

Electrons occupying quantized energy states around a nucleus must obey the

Pauli ExclusionPrinciple..

This prevents morethan 2 electrons (ofopposite spin) fromoccupying the samelevel. Number of Atoms

Allowedband

Forbidden gap

Forbiddengap

Allowedband

Allowedband

Page 8: The Ig Nobel Prizes are   “Booby Prizes”!

It can be shown that

Neither full nor empty bands participatein electrical conduction.

Semiconductors, Insulators, ConductorsFull Band

All energy levels are occupied

(contain electrons)

All energy levels are empty

Empty Band

Page 9: The Ig Nobel Prizes are   “Booby Prizes”!

• Consider the case of an Intrinsic Semiconductor (a material with no impurities, so there are no acceptors or donors).

That is, there is no doping. • For such a material, the highest energy band that is

completely filled with electrons (at T = 0 K) is called the highest

Valence Band. • The next highest band is completely empty of electrons

(at T = 0K) & is called the lowest

Conduction Band. • The Energy Difference between the bottom of

the Conduction Band & the top of the Valence Band is called the Fundamental Band Gap.

(Sometimes also called “Band Gap” or “Bandgap”, or “Gap” )

Page 10: The Ig Nobel Prizes are   “Booby Prizes”!

• So, for such an Intrinsic Semiconductor, for energies near the Band Gap, A Qualitative Picture of the Bands in “r-Space” (as functions of position ) looks like the figure below.

• Consider the Conduction Process for this material. Suppose that somehow, some electrons are added to the conduction band & some holes are added to the valence band.

Electron Conduction• In an applied electric field, the electrons in the conduction

band will move almost like free particles.

Hole Conduction • Holes are treated similarly to positively charged particles in the valence

band. In an applied field, they behave almost like free particles.

Page 11: The Ig Nobel Prizes are   “Booby Prizes”!

Intrinsic Semiconductors• Consider a nominally pure

semiconductor at T = 0 K.• There are no electrons in

the conduction band• At T > 0 K a small

fraction of electrons is thermally excited into the conduction band, “leaving” the same number of holes in the valence band.

Page 12: The Ig Nobel Prizes are   “Booby Prizes”!

Calculated Si Bandstructure in k Space

Eg

GOALSGOALSAfter this chapter, you should:

1. Understand the underlying Physics behind the existence of bands & gaps.

2. Understand how to interpret this figure.

3. Have a rough, general idea about how realistic bands are calculated.

4. Be able to calculate energy bands forsome simple models of a solid.

NoteSi has an indirect band gap!

Page 13: The Ig Nobel Prizes are   “Booby Prizes”!

Brief Quantum Mechanics (QM) & Solid State (SS) Review

• QM results that I must assume that you know!

The Schrödinger Equation(time independent; see next slide) describes electrons

– The solutions to the Schrödinger Equation result in quantized (discrete) energy levels for electrons.

• SS results that I must assume that you know!– Solutions to the Schrödinger Equation in a

periodic crystal give bands (allowed energy regions) & gaps (forbidden energy regions).

Page 14: The Ig Nobel Prizes are   “Booby Prizes”!

Quantum Mechanics (QM)• The Schrödinger Equation: (time independent)

Hψ = EψThis is a differential eigenvalue equation.H Hamiltonian operator for the system

(energy operator)

E Energy eigenvalue, ψ wavefunction

Particles are QM waves!|ψ|2 probability density; ψ is a function of ALL

coordinates of ALL particles in the problem!

Page 15: The Ig Nobel Prizes are   “Booby Prizes”!

The Physics Behind E(k) E(k) Solutions to the Schrödinger

Equation for an electron in a solid.

QUESTIONSWhy (qualitatively) are there bands?

Why (qualitatively) are there gaps?

Page 16: The Ig Nobel Prizes are   “Booby Prizes”!

Bands & Gaps • These can be understood from two

(very different) qualitative pictures!• The two pictures are models (& Opposite

Limiting Cases) of the true situation.• An electron in a perfectly periodic crystalline solid:

– The potential seen by this electron is perfectly periodic

The existence of this periodicpotential is the cause of the

bands & the gaps!

Page 17: The Ig Nobel Prizes are   “Booby Prizes”!

Qualitative Picture #1“A Physicist’s viewpoint”- The solid is looked at “collectively”

Almost Free Electrons (done in detail in Kittel’s Ch. 7!)

For free electrons: E(k) = (p)2/2mo = (ħk)2/2mo

Almost Free Electrons:• Start with the free electron E(k), add small

(weakly perturbing) periodic potential V.• This breaks up E(k) into bands (allowed

energies) & gaps (forbidden energy regions).• Gaps: Occur at the k where the electron waves

(incident on atoms & scattered from atoms) undergo constructive interference (Bragg reflections!)

Page 18: The Ig Nobel Prizes are   “Booby Prizes”!

Qualitative Picture #1 Forms the basis for REALISTIC bandstructure computational methods!

• Starting from the almost free electron viewpoint & adding a high degree of sophistication & theoretical + computational rigor:

Results in a method that works VERY WELL forcalculating E(k) for metals & semiconductors!• An “alphabet soup” of computational techniques:

– OPW: Orthogonalized Plane Wave method– APW: Augmented Plane Wave method– ASW: Antisymmetric Spherical Wave method– Many, many others

The Pseudopotential Method (the modern method of choice!)

Page 19: The Ig Nobel Prizes are   “Booby Prizes”!

Atomic / Molecular Electrons• Atoms (with discrete energy levels) come

together to form the solid.• Interactions between the electrons on

neighboring atoms cause the atomic energy levels to split, hybridize, & broaden.

Quantum Chemistry!• First approximation: Small interaction V! Occurs

in a periodic fashion (the interaction V is periodic).

Qualitative Picture #2 “A Chemist’s viewpoint”

The solid is looked at as a collection of atoms & molecules.

Page 20: The Ig Nobel Prizes are   “Booby Prizes”!

Quantum Chemistry!• First approximation: Small interaction V! Occurs

in a periodic fashion (interaction V is periodic). • Groups of levels come together to form bands (& gaps).• The bands E(k) retain much of the character of

their “parent” atomic levels (s-like & p-like bands, etc.)

Gaps:• Also occur at the k where the electron

waves (incident on atoms & scattered from atoms) undergo constructive interference (Bragg reflections!)

Qualitative Picture #2

Page 21: The Ig Nobel Prizes are   “Booby Prizes”!

• Starting from the atomic / molecular electron viewpoint & adding a high degree of sophistication & theoretical & computational rigor Results in a method that works VERY WELL for

calculating E(k) (mainly the valence bands) for

insulators & semiconductors!(Materials with covalent bonding!)

• An “alphabet soup” of computational techniques:– LCAO: Linear Combination of Atomic Orbitals method– LCMO: Linear Combination of Molecular Orbitals method

– The “Tightbinding” method & many others.

The Pseudopotential Method (the modern method of choice!)

Qualitative Picture #2

Page 22: The Ig Nobel Prizes are   “Booby Prizes”!

Bandstructure Theories in Crystalline Solids

Pseudopotential Method Tightbinding (LCAO) Method

Semiconductors, Insulators

Electronic Interaction

Metals

Almost Free Electrons

Molecular Electrons

Free Electrons

Isolated Atom, Atomic Electrons