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This section discusses the concept of energy bands and band gaps in metals, the formation of energy bands in quantum wells, and the role of periodic potentials in determining band structure. Topics include Bloch's Theorem, band overlap, and the Kronig-Penney model.
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Previously: Simple models of metals. For improvements, we need the band structure. By the end of this section you should be able to: Use/define Bloch’s Theorem Conceptually explain why energy bands and band gaps develop (two different approaches) Coming up soon: Calculating energy bands Today’s Objectives
kz Fermi surface kF ky kx Where do energy bands and gaps come from? GaAs Real materials have gaps in the available levels Metals have available states just above the Fermi level Texts for Today’s Discussion: Kittel: pages 163-176 Ashcroft & Mermin: pages 152-161 and 176-188 Snoke: pages 1-14
Quantum Well for Each Atom Rather than a Giant Quantum Well for the Metal Single Atom Quantum/Atomic Physics Simplest Periodic Potential Multiple Atoms (Condensed Matter Physics)
Approach 1: Get atoms close (conceptual)Ideal Double Quantum Wells How would it change if non-ideal well? How do we start? Which more likely to find electron in middle?
A Small Barrier Adjusts the Energies and Wavefunctions Non-ideal double well Non-ideal single well
Increasing the Barrier Moves the Symmetric and Antisymmetric Closer What would 3 wells look like? Larger barrier Smaller barrier What happens as make b go to 0?
Triple Quantum Wells Which has the lowest energy? Any relation between nodes and energy?
How Energy Bands Form What happens to these levels as the atoms get closer (b smaller)? How would the energy levels look for multiple wells?
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
U 0 L Approach 2: Include U(x) in Sch. Eq.“Realistic” Atomic Potentials in Solids • Multi-electron atomic potentials are complex • Even for hydrogen atom with a “simple” Coulomb potential solutions are quite complex U=0
Bringing Atoms Close Together in a Periodic Fashion For one dimensional case where atoms (ions) are separated by distance a, we can write the condition of periodicity as Even in a more complicated multi-atomic chain, we can use the same formula Note: Band gaps occur even without periodicity, but periodic examples are simpler! Bloch electronsare electrons that obey the Schrodinger equation with a periodic potential. They reduce to free electrons if you take U(x) = 0 = U(x+an)
Bloch’s Theorem This theorem gives the electron wavefunction in the presence of a periodic potential energy. We will prove 1-D version, AKA Floquet’s theorem. (3D proof on page 134 of Ashcroft) When using either theorem, we use the time-indep. Schrodinger equation for an electron in a periodic potential where the potential energy is invariant under a lattice translation of a In 3D (vector):
Bloch Wavefunctions a • Bloch’s Theorem states thatfor a particle moving in the periodic potential, the wavefunctionsψnk(x) are of the form • unk(x) has the periodicity of the atomic potential • The exact form of u(x) depends on the potential associated with atoms (ions) that form the solid
Or just: 1 2 N 3 Main points in the proof of Bloch’s Theorem in 1-D 1. First notice that Bloch’s theorem implies (3D version): Can show that this formally implies Bloch’s theorem, so if we can prove itwe will have proven Bloch’s theorem. 2. To prove the statement shown above in 1-D: Consider N identical lattice points around a circular ring, each separated by a distance a. Our task is to prove: Built into the ring model is the periodic boundary condition:
And has the most general solution: This requires Where we define the Bloch wavevector: Or: 1 2 N 3 Proof of Bloch’s Theorem in 1-D: Conclusion The symmetry of the ring implies that we can find a solution to the wave equation (QM reason too): If we apply this translation N times we will return to the initial atom position: Compare to past discussion of how many states in a band Now that we know C we can rewrite
Kittel’s Restatement • It looks a little different because he rewrites the potential energy as a Fourier series (on board) Discussed in both Kittel and Ashcroft This method used in Kittel for next example I will do it a different way because confusing.
U(x) Ion core a x What periodic potential do we want to try? To better understand how band gaps form, let’s model a 1D crystal, i.e. a lattice with a periodic potential. The exact shape of the periodic potential will not matter, and these potentials could be complicated.
One Common Approach But the exact shape doesn’t matter, so let’s try something easier! What’s Easy? A Physicist Thinks Quantum Wells are Easy (Kroniq-Penney Model or Nearly Free Electron Approx.) U(x) x
U(x) U0 x -b 0 a a+b 2a+b 2(a+b) We can solve the SE in each region of space: 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. First done for a chain of finite square well potentials model by Krönig and Penney in 1931 with E<U0 Each atom is represented by a finite square well of width a and depth U0. The atomic spacing is a+b. 0 < x < a -b < 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): (2) (1) Boundary Conditions and Bloch’s Theorem x = 0 Use Bloch’s Theorem otherwise wavefunction not in terms of k. What else can we do?
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): (2) (1) (3) (4) Boundary Conditions and Bloch’s Theorem x = 0 x = a Now using Bloch’s theorem for R= -(a+b): k = Bloch wavevector Now we can write the boundary conditions at x = a: 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 U0 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 small b Right hand side cannot exceed 1, so values exceeding will mean that there is no wavelike solutions of the Schrodinger eq. (forbidden band gap) Problems occur at Ka=N or K=N/a Ka Plotting left side of equation
Turning the last graph on it’s side 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 2 Forbidden band gap E(k) (a.u.) Ka/ BAND 1 ka/
Note: Notation same as Ashcroft, use Kittel version in lecture notes The potential U(r)=U(r+Na) obeys the Born-von Karman condition UK are the Fourier coefficients of U UK= U-K if the crystal has inverse symmetry Plugging this form of into Schrodinger’s Equation gives a kinetic energy term of:
Understanding the notation of the second proof of Bloch’s theorem Combining sums and then defining q=q’-K Some more manipulation (subbing q=k-K) gives an alternate form of S.E.
It looks more complicated than it is. Schrodinger’s equation in momentum space simplified by condition of periodicity In the free electron case, all UK are 0, so this simplifies:
