Scattering of electrons on screw dislocations Richard Bausch and Rudi Schmitz Institut fu ¨r Theoretische Physik IV, Heinrich-Heine-Universita ¨t Du ¨sseldorf, Universita ¨ tsstrasse 1, D-40225 Du ¨sseldorf, Germany Lukasz A. Turski Center for Theoretical Physics, Polish Academy of Sciences and College of Science, Al. Lotniko ´w 32/46, 02-668 Warszawa, Poland Received 26 August 1998 A previously established general framework for the description of long-wavelength quantum states of elec- trons in a crystal with topological defects is used to discuss the scattering of electrons on a screw dislocation. The corresponding Schro ¨dinger equation contains contributions of the type of a vector potential as well as of a repulsive scalar potential. Together they give rise to modified Aharonov-Bohm interferences in the scattering amplitude, for which the far-field expression is calculated exactly. S0163-18299901121-2 In connection with investigations of residual low- temperature resistivities, the scattering cross section of elec- trons on a screw dislocation has been calculated in Born approximation by Hunter and Nabarro, 1 and in a partial-wave expansion by Stehle and Seeger. 2 Both approaches use the defect-free crystal as a reference system, and the screw dis- location was represented by a deformation potential. 3 In a continuum version, the latter was expected to give an ad- equate description of the long-wavelength scattering waves. More recently it has been pointed out by Kawamura 4 that the topological nature of a screw dislocation invalidates as- sumption of an ideal lattice structure at arbitrary distances from the dislocation core. Correspondingly, an incoming electron cannot be described by a simple plane wave. In the continuum limit of a tight-binding model, defined on a lattice with a built-in screw dislocation, Kawamura derived a Schro ¨ dinger equation with only a kinetic term in the Hamil- tonian. The Fourier transform of this equation with respect to the axial coordinate of the dislocation is reminiscent of that for an electron moving in the vector potential of a magnetic flux line. Accordingly, Kawamura predicted Aharonov- Bohm interferences 5 in the scattering process of an electron on a screw dislocation. However, due to the assumption of constant transfer energies in the tight-binding model, Kawa- mura misses a long-ranged repulsive potential which can compete with the kinetic terms, and which dominates lattice corrections of the Hamiltonian in the core region. 6 In a recent general treatment of the long-range quantum states of an electron in a crystal with topological defects 7 we also used the tight-binding approximation as an intermediate step. For constant transfer energies this led in the continuum limit to a covariant Schro ¨ dinger equation which, in the lan- guage of the continuum theory of defects, 8 lives on a Riemann-Cartan manifold. Additional noncovariant terms show up in this equation when the transfer energies, i.e., the tunneling rates of the particle, are assumed to depend on the local lattice deformations caused by the defects. Applied to the case of a single straight screw dislocation, the covariant part immediately reproduces Kawamura’s form of the Schro ¨ dinger equation. The noncovariant contributions, arising from the most natural and most simple deformation dependence of the transfer energies, include a repulsive po- tential which has the same long-distance behavior as the co- variant kinetic terms. This invalidates any kind of perturba- tion expansion for the scattering amplitude which therefore in the following is calculated without such approximations. Before continuing, we mention an alternative way to de- rive the Schro ¨ dinger equation for a particle moving in a me- dium with topological defects. 9 This approach uses the idea of a gauge-field theory of topological defects, 10 and in prin- ciple allows to avoid the tight-binding approximation. Within our approach 7 we have explicitly derived the Schro ¨ dinger equation for a spinless electron moving in a PHYSICAL REVIEW B CONDENSED MATTER AND MATERIALS PHYSICS THIRD SERIES, VOLUME 59, NUMBER 21 1 JUNE 1999-I BRIEF REPORTS Brief Reports are accounts of completed research which, while meeting the usual Physical Review B standards of scientific quality, do not warrant regular articles. A Brief Report may be no longer than four printed pages and must be accompanied by an abstract. The same publication schedule as for regular articles is followed, and page proofs are sent to authors. PRB 59 0163-1829/99/5921/134913/$15.00 13 491 ©1999 The American Physical Society