Last Time The# of allowed k states (dots) is equal to the number of primitive cells in the crystal

Learning Objectives for Today After today’s class you should be able to:  Explain the meaning and origin of energy bands and “forbidden band gaps”  Understand difference between metals, semiconductors and insulators!  (If time) Relate DOS to energy bands Another source on today’s topics, see Ch. 7 of Kittel or search Kronig-Penney model

Transmission III Incoming from left ReflectedTransmitted “Conservation of probability”: must either reflect or transmit

Trans./Refl. Probabilities 2 equation, 3 unknowns!? Transmission and Reflection coefficients:

Concept Test An electron approaches the end of a long wire

Wavefunction III

Coefficients Incoming from left ReflectedExponential growth Exponential decay III Note: no transmitted wave in the solution. 0 (Non-physical) Boundary conditions:

Using Bloch’s Theorem: The Krönig-Penney Model Bloch’s theorem allows us to calculate the energy bands of electrons in a crystal if we know the potential energy function. Each atom is represented by a finite square well of width a and depth V 0. The atomic spacing is a+b. We can solve the SE in each region of space: 0 < x < a -b < x < 0 V x 0a a+b 2a+b2(a+b) V0V0 -b

Boundary Conditions and Bloch’s Theorem x = 0 The solutions of the SE require that the wavefunction and its derivative be continuous across the potential boundaries. Thus, at the two boundaries (which are infinitely repeated): Now using Bloch’s theorem for a periodic potential with period a+b: x = a (1) (2) k = Bloch wavevector Now we can write the boundary conditions at x = a: (3) (4) The four simultaneous equations (1-4) can be written compactly in matrix form  Let’s start it!

Results of the Krönig-Penney Model Since the values of a and b are inputs to the model, and  depends on V 0 and the energy E, we can solve this system of equations to find the energy E at any specified value of the Bloch wavevector k. What is the easiest way to do this? Taking the determinant, setting it equal to zero and lots of algebra gives: By reducing the barrier width b (small b), this can be simplified to:

Graphical Approach Right hand side cannot exceed 1, so values exceeding will mean that there is no wavelike solutions of the Schrodinger eq. (forbidden band gap) Ka Plotting left side of equation Problems occur at Ka=N or K=N /a small b

Turning the last graph on it’s side ka/ Energy in terms of E 0 This equation determines the energy bands. For values of K where the left side of the equation has a magnitude < 1, then k is real and energy bands are allowed. BAND 1 BAND 2 Forbidden band gap

Greek Theater Analogy: Energy Gaps

Energy Levels of Single vs Multiple Atoms Single Atom Multiple Atoms

15 Ideal Double Quantum Wells How do we start?

The two solutions have different energies  Symmetric (Bonding) and Antisymmetric (Antibonding)

Energy vs. Barrier Width What would 3 wells look like? Spins not coupled What happens as make b go to 0?

18 Triple Quantum Wells Which has the lowest energy? Any relation between nodes and energy?

19 Quadruple Quantum Wells

20 Five Quantum Wells

Figure 1.7: Coupled Well Energies How would the energy levels look for multiple wells? What happens to these levels as the atoms get closer (b smaller)?

Band Overlap  Often the higher energy bands become so wide that they overlap with the lower bands  Many materials are conductors (metals) due to the “band overlap” phenomenon

23 Energy Band Overlap 14 Si: 3s 2 3p 2 Out of 8 possible n=3 electrons (2s and 6p) Valence Band Typically the last filled energy band Conduction Band The bottom unfilled energy band Mixing of bands known as hybridization (Si=sp 3 )

24 Energy Band Formation Valence Band last filled Conduction Band bottom unfilled Metal No gap Semiconductor Small gap (<~1eV) Insulator Big gap (>~1eV) Diagram (flat or with momentum k) showing energy levels is a band diagram. This is at T=0. What happens at higher T?

Semiconductor Flat Band Diagram (Quantum Well) In What do I mean by flat? 1.Before any movement of charge, could cause bands to bend 2.At a single point in the crystal (changes with momentum) 1.43 eV

Insulators vs High Temp (Flat Band Diagrams) A small fraction of the electrons is thermally excited into the conduction band. These electrons carry current just as in metals (holes too) The smaller the gap the more electrons in the conduction band at a given temperature Resistivity decreases with temperature due to higher concentration of electrons in the conduction band Insulator low temp high

What happens as you approach the gap? ka/ Energy in terms of E 0 Classically E = ½ m v 2 What happens to v as k gets close to Brillouin zone edge? BAND 1 BAND 2 Forbidden band gap Find v for the free electron energy.

Compare to the free-electron model Free electron dispersion Let’s slowly turn on the periodic potential –/a /a

Electron Wavefunctions in a Periodic Potential (Another way to understand the energy gap) Consider the following cases: Electrons wavelengths much larger than atomic spacing a, so wavefunctions and energy bands are nearly the same as above Wavefunctions are plane waves and energy bands are parabolic: –/a /a V x 0a a+b 2a+b2(a+b) V1V1 -b

Wavelength much greater than atomic spacing Similar to how radio waves pass through us without affecting Energy of wave What happens as I lower this energy?

Electron Wavefunctions in a Periodic Potential U=barrier potential Consider the following cases: Electrons wavelengths much larger than a, so wavefunctions and energy bands are nearly the same as above Wavefunctions are plane waves and energy bands are parabolic: Electrons waves are strongly back-scattered (Bragg scattering) so standing waves are formed: Electrons wavelengths approach a, so waves begin to be strongly back-scattered by the potential: –/a /a

The nearly-free-electron model (Standing Waves) Either: Nodes at ions Or: Nodes midway between ions a Due to the ±, there are two such standing waves possible: These two approximate solutions to the S. E. at have very different potential energies. has its peaks at x = a, 2a, 3a, …at the positions of the atoms, where V is at its minimum (low energy wavefunction). The other solution, has its peaks at x = a/2, 3a/2, 5a/2,… at positions in between atoms, where V is at its maximum (high energy wavefunction).

The nearly-free-electron model Strictly speaking we should have looked at the probabilities before coming to this conclusion: a Different energies for electron standing waves Symmetric and Antisymmetric Solutions

34 Summary: The nearly-free-electron model BAND GAPS APPEAR AT EACH BRILLOUIN ZONE EDGE -2π/a –π/a π/a 2π/a In between the two energies there are no allowed energies; i.e., wavelike solutions of the Schrodinger equation do not exist. Forbidden energy bands form called band gaps. The periodic potential V(x) splits the free- electron E(k) into “energy bands” separated by gaps at each BZ boundary. E-E- E+E+ EgEg

Approximating the Band Gap BAND GAPS APPEAR AT EACH BRILLOUIN ZONE EDGE -2π/a –π/a π/a 2π/a E-E- E+E+ EgEg For square potential: V(x) =V o for specific values of x (changes integration limits)