The Kronig-Penney model extended to arbitrary potentials via - PowerPoint PPT Presentation

The Kronig-Penney model extended to arbitrary potentials via numerical matrix mechanics Robert Pavelich, Frank Marsiglio University of Alberta June 15, 2015

Marsiglio (2009) introduced a matrix diagonalization method for solving 1D potentials like the harmonic oscillator in a new third way 1. 2. 3.

Marsiglio (2009) introduced a (pedagogical) matrix diagonalization method for solving 1D potentials like the harmonic oscillator in a new third way 1. Solving the ODE with Hermite polynomials 2. 3.

Marsiglio (2009) introduced a (pedagogical) matrix diagonalization method for solving 1D potentials like the harmonic oscillator in a new third way 1. Solving the ODE with Hermite polynomials 2. Using Dirac’s raising and lowering operators 3.

Marsiglio (2009) introduced a (pedagogical) matrix diagonalization method for solving 1D potentials like the harmonic oscillator in a new third way 1. Solving the ODE with Hermite polynomials 2. Using Dirac’s raising and lowering operators 3. Matrix diagonalization with an infinite square well basis

Hamiltonian matrix elements are divided into kinetic diagonal terms and potential-dependent terms H nm = � ψ n | ( H 0 + V ) | ψ m � = δ nm E (0) + H V n nm

And the potential terms are computed in the usual way H V nm = � ψ n | V ( x ) | ψ m � � a = 2 � n π x � � m π x � d x sin V ( x ) sin a a a 0

This method gives excellent agreement with analytical solutions at low energies 5000 4000 v Ho 3000 (0) E n /E 1 2000 exact n 2 (0) h � (n − 1/2)/E 1 1000 n 2 + v 0 0 0 10 20 30 40 50 60 70 n

But what about another potential? We try periodic boundary conditions ( ` a la a particle on a ring) φ ( x + a ) = φ ( x )

This gives plane wave basis states... � 1 φ (0) a e i 2 π n a x n ( x ) =

... with similar energies to the infinite square well ka = 2 π n or � � n 2 π 2 � 2 = 4 n 2 E (0) = (2 n ) 2 E (0) E n = 4 2 ma 2 1 1

We now compute our matrix elements in the new basis H V nm = � φ (0) n | V | φ (0) m � � a = 1 d x e − i 2 π nx / a V ( x ) e i 2 π mx / a a 0

We can recreate the results in the 2008 paper in the new basis easily! 1800 n 2 + v off 1600 (0) h ω (n−1/2)/E 1 1400 Infinite square well 1200 Periodic boundary (0) 1000 E n /E 1 800 600 400 200 0 0 10 20 30 40 n

But we want to go beyond a single unit cell, so the boundary condition we actually want is the Bloch condition φ ( x + a ) = e iKa φ ( x )

Given our plane wave basis states, we are essentially just multiplying exponentials so the Bloch contribution to the energy is merely additive ka = 2 π n + Ka

Further, this only affects the main diagonal kinetic energy terms as the potential terms in the Hamiltonian are unaffected � a 1 e − iKx e − i 2 π nx / a V ( x ) e i 2 π mx / a ✟✟✟ ✟ ✟ d x ✟✟✟ e + iKx a 0

Thus our procedure is to first populate our Hamiltonian matrix ignoring the Bloch contribution (2 · 0) 2 + h V   h V h V . . . 00 01 02 (2 · 1) 2 + h V h V h V . . . H nm   10 11 12 = (2 · 2) 2 + h V   h V h V . . . E (0)   20 21 22 . . .  ...  1 . . . . . .

Then we iteratively introduce Bloch terms to the main diagonal, ranging from Ka ∈ ( − π, π ), diagonalizing each to get a new set of eigenvalues (0 + Ka /π ) 2 + h V   h V h V . . . 00 01 02 (2 + Ka /π ) 2 + h V h V h V . . .   10 11 12 (4 + Ka /π ) 2 + h V   h V . . . h 21   20 22 . . .  ...  . . . . . .

With no potential, these solutions recapitulate the expected free electron parabolic bands 25 20 15 (0) E/E 1 10 5 0 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Ka/ π

As we introduce the actual potential, there is a lifting of degeneracies and band gaps appear... 30 25 20 (0) E/E 1 15 10 5 0 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Ka/ π

... and these solutions exactly match up with the known analytic solutions to the Kronig-Penney model 30 25 20 (0) E/E 1 15 10 5 0 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Ka/ π

But the method is general enough to handle any repeating 1D potential; here we investigated several for which we could compute analytical matrix elements (a) Kronig−Penney, ρ = 0.5 (b) Kronig−Penney, ρ = 0.8 V(x) (c) Simple harmonic oscillator (d) Inverted harmonic oscillator (e) Linear well

For example periodic harmonic oscillators... 30 25 20 (0) E/E 1 15 10 5 0 −1 −0.5 0 0.5 1 Ka/ π

... or the so-called linear well 40 35 30 25 (0) E/E 1 20 15 10 5 0 −1 −0.5 0 0.5 1 Ka/ π

We can compare the bandstructure by making the third band in each case “similarly bounded” 0 −0.5 K−P ( ρ = 0.5) −1 Inv. HO (0) E/E 1 −1.5 K−P ( ρ = 0.8) SHO −2 −2.5 Linear −3 −1 −0.5 0 0.5 1 Ka/ π

Further, we can use the second derivatives of these bands as a measure of the effective mass of the electrons/holes d 2 E ( K ) ≡ E (0) � 1 ≡ 1 � 2 e ′′ 1 � � ele m ∗ � 2 d K 2 � K min hol ele K max hol

We find that potentials with more realistic “cusp-like” potentials have lower hole effective masses (there’s less of a potential “seen” by the higher energy band states) Potential e ′′ e ′′ e ′′ ele / e ′′ ele hol hol = m hol / m ele K-P ( ρ = 0 . 5) 13.83 − 25 . 35 − 0 . 55 K-P ( ρ = 0 . 8) 39.09 − 70 . 61 − 0 . 55 Simple HO 37.84 − 121 . 80 − 0 . 31 Inverted HO 19.83 − 55 . 96 − 0 . 35 Linear 31.63 − 102 . 23 − 0 . 31

Work in progress: 2D bandstructures 5 4 0 −2 3 −4 v o (0) E/E 1 2 −6 −8 1 −10 1 0.8 1 0 0.6 0.8 0.6 0.4 0.4 0.2 0.2 −1 Γ X 01 M Γ 0 0 y x

Thanks for your attention! This research was supported in part by by a University of Alberta Teaching and Learning Enhancement Fund (TLEF) grant