Homework 5; due Thursday, October 31PY 502, Computational Physics (Fall 2013)Department of Physics, Boston UniversityInstructor: Anders SandvikSCHR¨ODINGER EQUATION IN TWO DIMENSIONSConfining potentials can be realized in semi-conductor nano-structures by combining materials withdifferent band gaps. In a quantum dot, one or several electrons are confined to a finite volume, theshape of which can be taylored using nano-fabrication technologies. Due to the confinement, theelectrons exhibit discrete energy levels and are therefore some times called “artificial atoms”. Inthis assignment you will calculate the lowest energy eigenstates and wave functions for a rectangularquantum dot with an internal structure, as shown in the figure. The dot is assumed to be verythin in the third direction, so that it is effectively two-dimensional. The shaded areas have a lower(negative) potential −V0than the rest of the structure (at potential V = 0), and hence an electronwill be attracted to these regions.LxLyx0y0abFigure 1: Geometry of the semiconductur nano-structure. The potential V = −V0in the shadedareas an V = 0 elsewhere inside the quantum dot. The two low-potential areas are identical andsymmetrically located as shown. Outside the boundaries of the Lx× Lystructure the potential isassumed to be infinite.1We write the Schr¨odinger equation in two dimensions in the form−12∂2∂x2+∂2∂y2Ψ(x, y) + αV (x, y)Ψ(x, y) = αEΨ(x, y), (1)where α = m/¯h2. It will be convenient to measure lengths in units of nm (10−9m), and energiesin eV (1.602 · 10−19J). With the electron mass m = 9.109 · 10−31kg and ¯h = 1.055 · 10−34Js, wethen get−12∂2∂x2+∂2∂y2Ψ(x, y) + βV (x, y)Ψ(x, y) = βEΨ(x, y), (2)whereβ =(10−9m/nm)2(1.602 · 10−19(J/eV) · 9.109 · 10−31kg1.11302 · 10−68J2s2=13.11eVnm2. (3)You should use two different methods to solve the two-dimensional Schr¨odinger equation approx-imately: A) Using a variational method with the particle-in-a-box eigenstates as a basis set, andB) using the Lanczos method to solve the real-space discretized equation. You should write yourprograms so that any values of the parameters defined in the figure can be studied. You shouldthen compare results obtained with the two methods for the four lowest eigenstates of a structureexplained in the figure, with the following parameters:Lx= 5nm, Ly= 10nm, a = 1.0nm (corresponding to x0= 2nm), b = 0.5nm, y0= 2nm, andV0= 0.1eV (and note the sign; V = −V0in the regions of the attractive internal potential).A) Variational calculationThe particle-in-a-box eigenfunctions for a rectangular infinite-wall box wiht x ∈ [0, Lx] and y ∈[0, Ly] areφkx,ky(x, y) = fkx(x)gky(y), kx, ky= 1, 2. . . . , (4)wherefkx(x) =r2Lxsin (kxπx/Lx), gky(y) =s2Lysin (kyπy/Ly). (5)In the variational calculation, you should use the basis set consisting of the N = NxNyfunctionsφkx,kywith kx= 1, . . . , Nxand ky= 1, . . . , Ny, where Nxand Nyshould be given as input to theprogram. Note that in the program the two indices kx, kyshould be combined into a single indexlabeling the states, e.g., k = kx+ (ky− 1)Nx, so that k = 1, 2, . . . , N and the hamiltonian can beconstructed as an N × N matrix.The energies of these wavefunctions are, keeping the constant β as defined in the Schr¨odingerequation (2),βEkx,ky=π22"kxLx2+kyLy2#. (6)The integrals needed for the potential-energy part of the variational Hamiltonian areVpx,pykx,ky=Z Zdxdy φpx,py(x, y)V (x, y)φkx,ky(x, y), (7)2which can be easily calculated analytically since the potential inside the box is non-zero only inrectangular regions. To this end, you will need the following indefinite integrals:Zsin2(ax)dx =12x −14asin(2ax), (8)Zsin(ax)sin(bx)dx =sin[(a − b)x]2(a − b)−sin[(a + b)x]2(a + b), |a| 6= |b|. (9)In addition to the eigenvalues βEnobtained by diagonalizing the Hamiltonian, you should also ex-tract the eigenfunctions corresponding to the four lowest eigenvalues. They are linear combinationsof the basis functions, with weights given by the eigenvectors H that are obtained in the form of atransformation matrix in the diagonalization. Chose a reasonable grid, e.g., 100 × 200 (x, y) points,for writing the wave functions to files.B) Lanczos calculationIn the real-space discretized 2D Schr¨odinger equation the kinetic energy acting on a localized stategivesK|ji =2∆2|ji −121∆2Xδ[j]|δ[j]i, (10)where δ[j] is a nearest-neighbor of element j. For interior elements there are 4 neighbors, but onthe edges there can be no “hopping” out of the dot. The potential energy is diagonal;βV |ji = −βV0|ji, if (xj, yj) inside low−potential region, 0 else. (11)From these expressions the matrix elements hj|K|li and hj|V |li are obtained.The discretization ∆ should be chosen in a such a way that the system dimensions are integermultiples of ∆; Lx= Nx∆, Ly= Ny∆.Use the variant of the lanczos method where the normalized basis states |φmi are generated directly.This avoids problems with large numbers when computing the normalization constants of the un-normalized states |fmi.Your program should write all the Lanczos eigenvalues to disk, and also produce the wave functionscorresponding to the four lowest energies.Specific instructionsYou should test your program, and write a short report on the results (for the structure withparameters specified above). Consider the four lowest energy eigenvalues and the correspondingstates. Give all energies in eV (i.e., divide out the constant β from the calculated eigenvaluesβEn—note that with the units we are using the unit of βE is nm−2).You should check the energy convergence as the size of the basis is increased in the variationalcalculation. You should carry out the Lanczos calculation for different discretizations ∆ (e.g., ∆ =0.2, 0.1, 0.05, 0.025 and smaller if you can with your computer resources) and check the convergenceof the energies as the number of Lanczos iterations is increased. You may not be able to completelyconverge the calculations before the computational effort becomes too large, or instabilities relatedto non-orthogonality (due to build-up of numerical truncation errors) start to plague the Lanczoscalculation. Mention any numerical problems you encounter in the report.3You should produce graphs showing your energy convergence tests. You should also produce plotsshowing the probability densities (square of the wavefunctions) inside