1. The Bandstructure Problem A one-dimensional model (“easily generalized” to 3D!)

2. A One-Dimensional Bandstructure Model 1 e - Hamiltonian: H = (p) 2 /(2m o ) + V(x) p  -iħ(∂/∂x), V(x)  V(x + a) V(x)  Effective Potential. Repeat Distance = a. GOAL: Solve the Schrödinger Equation: Hψ k (x) = E k ψ k (x) k  eigenvalue label First it’s convenient to define The Translation Operator  T. For any function, f(x), T is defined by: T f(x)  f(x + a)

3. Translation Operator  T. For any f(x), T is defined by: T f(x)  f(x + a) For example, if we take f(x) = ψ k (x)  Eigenfunction solution to the Schrödinger Equation: T ψ k (x) = ψ k (x + a) (1) Now, find the eigenvalues of T: T ψ k (x)  λ k ψ k (x) (2) λ k  An eigenvalue of T. Using (1) & (2) together, it is fairly easy to show that: λ k  e ika and ψ k (x)  e ikx u k (x) with u k (x)  u k (x + a) For proof, see the texts by BW or YC, as well as Kittel’s Solid State Physics text & any number of other SS books

4. This shows that the translation operator applied to an eigenfunction of the Schrödinger Equation (of the Hamiltonian H which includes a periodic potential) gives: Tψ k (x) = e ika ψ k (x)  ψ k (x) is also an eigenfunction of the translation operator T! It also shows that the general form of ψ k (x) must be: ψ k (x) = e ikx u k (x) where u k (x) is a periodic function with the same period as the potential! That is u k (x) = u k (x+a)

5. In other words: For a periodic potential V(x), with period a, ψ k (x) is a simultaneous eigenfunction of the translation operator T and the Hamiltonian H From The Commutator Theorem of QM, this is equivalent to [T,H] = 0. The commutator of T & H vanishes; they commute!  They share a set of eigenfunctions. In other words: The eigenfunction (the electron wavefunction!) always has the form: ψ k (x) = e ikx u k (x) with u k (x) = u k (x+a)  “Bloch’s Theorem”

6. Bloch’s Theorem From translational symmetry For a periodic potential V(x), the eigenfunctions of H (wavefunctions of e - ) always have the form: ψ k (x) = e ikx u k (x) with u k (x) = u k (x+a)  “Bloch Functions” Recall, for a free e -, the wavefunctions have the form: ψ f k (x) = e ikx (a plane wave)  A Bloch function is the generalization of a plane wave for an e - in periodic potential. It is a plane wave modulated by a periodic function u k (x) (with the same period as V(x)).

7. Bandstructure A one dimensional model The wavefunctions of the e - have in a periodic crystal Must always have the Bloch function form ψ k (x) = e ikx u k (x), u k (x) = u k (x+a) (1) It is conventional to label the eigenfunctions & eigenvalues (E k ) of H by the wavenumber k: p = ħk  e - “quasi-momentum ” (rigorously, this is the e - momentum for a free e -, not for a Bloch e - ) So, the Schrödinger Equation is Hψ k (x) = E k ψ k (x) where ψ k (x) is given by (1) and E k  The Electronic “Bandstructure”.

8. Bandstructure: E versus k The “Extended Zone scheme”  A plot of E k with no restriction on k But note! ψ k (x) = e ikx u k (x) & u k (x) = u k (x+a) Consider (for integer n): exp[i{k + (2πn/a)}a]  exp[ika]  The label k & the label [k + (2πn/a)] give the same ψ k (x) (& the same energy)! In other words, the translational symmetry in the lattice  Translational symmetry in “k space”! So, we can plot E k vs. k & restrict k to the range -(π/a) < k < (π/a)  “First Brillouin Zone” (BZ ) (k outside this range gives redundant information!)

9. “Reduced Zone sScheme”  A plot of E k with k restricted to the first BZ. For this 1d model, this is -(π/a) < k < (π/a) k outside this range is redundant & gives no new information! Illustration of the Extended & Reduced Zone schemes in 1d with the free electron energy: E k = (ħ 2 k 2 )/(2m o ) Note: These are not really bands! We superimpose the 1d lattice symmetry (period a) onto the free e - parabola. Bandstructure: E versus k

10. Free e - “bandstructure” in the 1d extended zone scheme: E k = (ħ 2 k 2 )/(2m o )

11. Free e - “bandstructure” in the 1d reduced zone scheme: E k = (ħ 2 k 2 )/(2m o ) For k outside the 1 st BZ, take E k & translate it into the 1 st BZ by adding  (πn/a) to k Use the translational symmetry in k-space as just discussed.  (πn/a)  “Reciprocal Lattice Vector”

12. Bandstructure To illustrate these concepts, we now discuss an EXACT solution to a 1d model calculation (BW Ch. 2, S, Ch. 2) The Krönig-Penney Model Developed in the 1930’s & is in MANY SS & QM books Why do this simple model? The process of solving it shares contain MANY features of real, 3d bandstructure calculations. Results of it contain MANY features of real, 3d bandstructure results!

13. The Krönig-Penney Model Developed in the 1930’s & is in MANY SS & QM books Why do this simple model? The results are “easily” understood & the math can be done exactly We won’t do this in class. It is in the books! A 21 st Century Reason to do this simple model! It can be used as a prototype for the understanding of artificial semiconductor structures called SUPERLATTICES! (Later in the course?)

14. QM Review: The 1d (finite) Rectangular Potential Well In most QM texts!! [-{ħ 2 /(2m o )}(d 2 /dx 2 ) + V]ψ = εψ (ε  E) V = 0, -(b/2) < x < (b/2); V = V o otherwise We want bound states: ε < V o We want to solve the Schrödinger Equation for:

15. Solve the Schrödinger Equation: [-{ħ 2 /(2m o )}(d 2 /dx 2 ) + V]ψ = εψ (ε  E), V = 0, -(b/2) < x < (b/2) V = V o otherwise The bound states are in Region II In Region II: ψ(x) is oscillatory In Regions I & III: ψ(x) is exponentially decaying -(½)b (½)b VoVo V = 0

16. The 1d (finite) rectangular potential well A brief math summary! Define: α 2  (2m o ε)/(ħ 2 ); β 2  [2m o (ε - V o )]/(ħ 2 ) The Schrödinger Equation becomes: (d 2 /dx 2 ) ψ + α 2 ψ = 0, -(½)b < x < (½)b (d 2 /dx 2 ) ψ - β 2 ψ = 0, otherwise. Solutions: ψ = C exp(iαx) + D exp(-iαx), -(½)b < x < (½)b ψ = A exp(βx), x < -(½)b ψ = A exp(-βx), x > (½)b Boundary Conditions:  ψ & dψ/dx are continuous SO:

17. Algebra (2 pages!) leads to: (ε/V o ) = (ħ 2 α 2 )/(2m o V o ) ε, α, β are related to each other by transcendental equations. For example: tan(αb) = (2αβ)/(α 2 - β 2 ) Solve graphically or numerically. Get: Discrete energy levels in the well (a finite number of finite well levels!)

18. Even eigenfunction solutions (a finite number): Circle, ξ 2 + η 2 = ρ 2, crosses η = ξ tan(ξ) VoVo o o b

19. Odd eigenfunction solutions: Circle, ξ 2 + η 2 = ρ 2, crosses η = -ξ cot(ξ) |E 2 | < |E 1 | b b o o VoVo

20. The Krönig-Penney Model Repeat distance a = b + c. Periodic potential V(x) = V(x + na), n = integer Periodically repeated wells & barriers. Schrödinger Equation: [-{ħ 2 /(2m o )}(d 2 /dx 2 ) + V(x)]ψ = εψ V(x) = Periodic potential  The Wavefunction has the Bloch form: ψ k (x) = e ikx u k (x); u k (x) = u k (x+a) Boundary conditions at x = 0, b: ψ, (dψ/dx) are continuous  Periodic Potential Wells (Kronig-Penney Model)

21. Algebra: A MESS! But doable EXACTLY! Instead of an explicit form for the bandstructure ε k or ε(k), we get: k = k( ε ) = (1/a) cos -1 [L( ε /V o )] OR L = L( ε /V o ) = cos(ka) WHERE L = L( ε /V o ) =

22. L = L(ε/V o ) = cos(ka)  -1< L< 1 ε in this range are the allowed energies (BANDS!) But also, L( ε /V o ) = messy function with no limit on L ε in the range where |L| >1  These are regions of forbidden energies (GAPS!) (no solutions exist there for real k; math solutions exist, but k is imaginary) The wavefunctions have the Bloch form for all k (& all L): ψ k (x) = e ikx u k (x)  For imaginary k, ψ k (x) decays instead of propagating!

23. Krönig-Penney Results For particular a, b, c, V o Each band has a finite well level “parent”. L(ε/V o ) = cos(ka)  -1< L< 1 But also L(ε/V o ) = messy function with no limits For ε in the range -1 < L < 1  allowed energies (bands!) For ε in the range |L| > 1  forbidden energies (gaps!) (no solutions exist for real k)  Finite Well Levels   

24. Every band in the Krönig-Penney model has a finite well discrete level as its “parent”!  In its implementation, the Krönig-Penney model is similar to the “almost free” e - approach, but the results are similar to the tightbinding approach! (As we’ll see). Each band is associated with an “atomic” level from the well. Evolution from the finite well to the periodic potential

25. More on Krönig-Penney Solutions L(ε/V o ) = cos(ka)  BANDS & GAPS! The Gap Size: Depends on the c/b ratio Within a band (see previous Figure) a good approximation is that L ~ a linear function of ε. Use this to simplify the results: For (say) the lowest band, let ε  ε 1 (L = -1) & ε  ε 2 (L = 1) use the linear approximation for L(ε/V o ). Invert this & get: ε - (k) = (½) (ε 2 + ε 1 ) - (½)(ε 2 - ε 1 )cos(ka) For next lowest band, ε + (k) = (½) (ε 4 + ε 3 ) + (½)(ε 4 – ε 3 )cos(ka) In this approximation, all bands are cosine functions!!! This is identical, as we’ll see, to some simple tightbinding results.

26. All bands are cos(ka) functions! Plotted in the extended zone scheme. Discontinuities at the BZ edges, at k =  (nπ/a) Because of the periodicity of ε(k), the reduced zone scheme (red) gives the same information as the extended zone scheme (as is true in general). Lowest Krönig-Penney Bands In the linear approximation for L(ε/V o ) ε = (ħ 2 k 2 )/(2m 0 )