Section 1.2-3 Homework from this section: 1.5 (We will do a similar problem in class today) Bloch’s Theorem and Krönig

1. For review/introduction of Schrodinger equation: http://web.monroecc.edu/manila/webfiles/spiral/6schrodingereqn.pdf Section 1.2-3Homework from this section: 1.5(We will do a similar problem in class today)Bloch’s Theorem and Krönig-Penney Model

2. Learning Objectives for Today After today’s class you should be able to: • Apply Bloch’s theorem to the Kronig-Penney model or any other periodic potential • Explain the meaning and origin of “forbidden band gaps” • Begin to understand the Brillouin zone • For another source on today’s topics, see Ch. 7 of Kittel’s Intro to Solid State Physics. • Crystal basics to prepare us for next class

3. “Realistic” Potential in Solids • Multi-electron atomic potentials are complex • Even for hydrogen atom with a “simple” Coulomb potential solutions are quite complex • So we use a model infinite one-dimensional periodic potential to get insight into the problem (last time, looked at 1-6 atoms)

4. Periodic Potential • For one dimensional case where atoms (ions) are separated by distance a, we can write condition of periodicity as a

5. Section 1.3: 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 in the book) When using this theorem, we still 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):

6. Bloch Wavefunctions a • Bloch’s Theorem states thatfor a particle moving in the periodic potential, the wavefunctionsψ(x) are of the form • uk(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

7. 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: 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:

8. 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 (and last lecture) implies that we can find a solution to the wave equation: If we apply this translation N times we will return to the initial atom position: Now that we know C we can rewrite

9. Consequence of Bloch’s TheoremProbability of finding the electron • Each electron in a crystalline solid “belongs” to each and every atom forming the solid • Very accurate for metals where electrons are free to move around the crystal! • Makes sense to talk about a specific x (± n a)

10. V V0 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<V0 Each atom is represented by a finite square well of width a and depth V0. The atomic spacing is a+b. 0 < x < a I wish the book had selected different letters than K and , but staying consistent -b < x < 0

11. 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, deriv.) Boundary Conditions and Bloch’s Theorem x = 0 x = a Now using Bloch’s theorem for a periodic potential with period 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 

12. Results of the Krönig-Penney Model Since the values of a and b are inputs to the model, and  depends on V0 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:

13. 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) Gap occurs at Ka=N or K=N/a Ka Plotting left side of equation

14. Not really much different Single Atom Multiple Atoms

15. Greek Theater Analogy: Energy Gaps

16. What Else Can We Learn From This Model?

17. Exercise 1.4

18. Different Ways to Plot ItExtended Zone Scheme Note that the larger the energy, the larger the band/gap is (until some limit). y=coska Different x axis k = Bloch wavevector Ka The range -<ka< is called the first Brillouin zone. Atoms

19. Different Representations of E(k) Reduced zone scheme All states with |k| > /a are translated back into 1st BZ Frequently only one side is shown as they are degenerate. In 3D, often show one side along with dispersion along two other directions (e.g. 100, 110, 111)

20. Band diagrams can refer to either E vs. real space or E vs momentum space k Momentum space example Real space examples

21. ...with first Brillouin zone: –/a ®/a (a the lattice constant) –/a /a Compare to the free-electron model Free electron dispersion Let’s draw it in 3D! Let’s slowly turn on the periodic potential

22. V V1 x -b 0 a a+b 2a+b 2(a+b) Electron Wavefunctions in a Periodic Potential(Another way to understand the energy gap) Consider the following cases: Wavefunctions are plane waves and energy bands are parabolic: –/a /a Electrons wavelengths much larger than atomic spacing a, so wavefunctions and energy bands are nearly the same as above

23. How do X-rays Work? The soft tissue in your body is composed of smaller atoms, and so does not absorb X-ray photons particularly well. The calcium atoms that make up your bones are much larger, so they are better at absorbing X-ray photons.

24. Consequence of Bloch’s Theorem Similar to how radio waves pass through us without affecting

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

26. 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). Either: Or: a Due to the ±, there are two such standing waves possible: The nearly-free-electron model (Standing Waves) Nodes at ions Nodes midway between ions

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

28. -2π/a –π/a π/a 2π/a BAND GAPS APPEAR AT EACH BRILLOUIN ZONE EDGE Summary: The nearly-free-electron model The periodic potential V(x) splits the free-electron E(k) into “energy bands” separated by gaps at each BZ boundary. 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. E- Eg E+

29. -2π/a –π/a π/a 2π/a BAND GAPS APPEAR AT EACH BRILLOUIN ZONE EDGE Approximating the Band Gap E- Eg E+ For square potential: V(x) =Vo for specific values of x (changes integration limits)

30. Crystal Directions • Choose one lattice point on the line as an origin (point O). Choice of origin is completely arbitrary, since every lattice point is identical. • Then choose the lattice vector joining O to any point on the line, say point T. This vector can be written as; R = n1 a + n2 b + n3 c • To distinguish a lattice direction from a lattice point, the triplet is enclosed in square brackets [ ...]. Example: [n1n2n3] • [n1n2n3] is the smallest integer of the same relative ratios. Example: [222] would not be used instead of [111]. • Negative directions can be written as Figureshows [111] direction Also sometimes [-1-1-1]