Parallel Computation for Electronic Waves in Quantum Corrals

Recent scanning tunneling microscopy (STM) studies on the (111) faces of noble metals have directly imaged electronic surface-confined states and dramatic standing-wave patterns have been observed 1,2]. We solve for the local density of electronic states in these "leaky" quantum corral confinement structures using a coherent elastic scattering theory. We seek solutions of the two-dimensional SchrOdinger equation compatible with non-reflecting boundary conditions which asymptotically satisfy the Sommerfeld radiation condition [11,14]. The large matrices generated by the discretization of realistic quantum corral structures require the use of sparse matrix methods. In addition, a parallel finite element solution was undertaken using the message passing interface standard (MPI) and the Portable, Extensible, Toolkit for Scientific Computation (PETSc) [5] for an efficient computational solution on both distributed and shared memory architectures. Our calculations reveal excellent agreement with the reported experimental dl/dV STM data.


INTRODUCTION
Recent scanning tunneling microscopy (STM) differ- ential conductance studies in the vicinity of defects and step edges on Cu(111) at T--4.2 K [1-3] and on Au(111) and Ag(111) at room temperature [4,5] have directly imaged the dramatic standing wave patterns of the electronic surface-confined states [6].We solve for the local density of electronic states (LDOS) in quantum corral structures using a coherent elastic scattering theory.We seek solutions of the two- dimensional Schr6dinger equation compatible with Sommerfeld radiation conditions [3,4] in these "leaky" confinement structures.We obtain excellent agreement with the reported experimental STM dif- ferential conductance studies.Our results are consist- ent with the STM junction current model of Tersoff and Hamann [7] and Lang [8], as extended by Davis et al. [9] to the case of the 2DEG formed by surface- state confinement, in which the differential conduct- ance, dl/dV, is a measure of the surface LDOS.
The LDOS is evaluated from the analytic Green function obtained from the system Hamiltonian com- patible with an outward compatible with standard finite element methods (FEM) [10,11].We discuss our parallel FEM approach using the message passing interface stand- ard (MPI) [12] and the Portable, Extensible, Toolkit for Scientific Computation (PETSc) [13].The proces- sor decomposition is designed to reduce the commu- nication required for the assembly of the sparse global matrix for an efficient solution on both distributed and shared memory architectures.
MODEL SYSTEM An electronic scattering state, VE, with energy E must satisfy the Schr6dinger equation for the system Hamiltonian, Hxl/'E= EE where ---V.--Ve(r) +V(r)e(r) Eve(r). (1) The potential term, V(r), encompasses the elastic scatter- ing centers, which are modeled as finite-height potential barriers.The local density of states (LDOS) is obtained from the diagonal of the Green function, Following the work of Bayliss and coworkers [14], we formulate approximate non-reflecting boundary conditions on an artificial boundary at a radius R 0 which encloses the scattering domain; the potential is assumed constant outside the artificial boundary and is taken as the zero reference.A boundary constraint is imposed which restricts the solutions to the class of wavefunctions which are compatible with an outward Sommerfeld radiation condition 15,16], limr_ r(d'l)/2(rikv) O, where d is the system dimensionality.In the two-dimensional case, the scat- tered part of the wavefunction may be expanded in the form qscatt a e ikr ..o (O)/ (kr) j+.A set of lin- ear operators, B m, are constructed such that each m th higher order term is eliminated by the B m operation on / 4m+ 1, the far-field expansion, BmtscattlF O[R;''-'].
The boundary constraint is exact in the limit of R 0and has a radially decaying truncation error on the boundary.The set of operators are therefore defined as B 0 1, B m (O r ik + am/r)Bm_ 1, where a m is a constant chosen to eliminate the m th term of the expansion.The subsequent formulation of the Green function using a finite element basis and recast into matrix notation has the form: where T and V contain the kinetic and potential energy matrix elements, respectively, C contains the matrix elements which result from the NRBC 10,11], and the matrix elements of M result from the non- orthogonal finite element basis set.
The large systems generated by the FEM discreti- zation of realistic quantum corral structures can bene- fit from parallel sparse matrix storage and parallel computation techniques.Using triangular mesh ele- ments and standard node number reordering methods, narrowly banded systems are generated.By distribut- ing the independent local element calculations across processors and likewise distributing the sparse global system matrix, assembly of the linear system can be performed with minimal interprocess communication.Schematically shown in Fig.
is the processor domain decomposition of the finite element mesh.Each box superimposed over the schematic finite ele- ment domain on the left encloses the local elements assigned to each processor.Since all the processors simultaneously initialize with the element data required for their respective subdomain computations, the only required interprocessor communication is for global matrix assembly.As is schematically shown in the figure, the storage of the sparse global matrix is distributed across the processors such that each com- putes most of its locally stored global matrix ele- ments.The required communication is therefore due to the assembly of the processor boundary contribu- tions.This communication penalty can be further reduced through the use of non-blocking sends to overlap the FEM integral computations with the glo- bal matrix assembly.The local NRBC matrix ele- ments, C, do not disturb the sparse matrix structure and require no additional communication overhead.
There has also been much recent effort to develop efficient parallel Krylov subspace (KSP) type iterative (MPI) standard [12], provides a simple programmer interface to parallel sparse vector and matrix datastruc- tures and many powerful KSP linear and non-linear solvers.While insulating the programmer from the details of the underlying communication, the PETSc library provides access to the full richness of the MPI standard while maintaining as much as possible the "single system view" of traditional sequential codes.The portable library is available for most common hardware platforms from workstations to supercomput- ers.Since it is built using installed MPI libraries, it takes advantage of any site dependent MPI optimiza- tions for its underlying hardware communications.

RESULTS
Using our elastic scattering theory, we calculate the LDOS for quantum corral structures based on the geometries reported in Refs.
[1] and [3].Each numer- ical scattering domain was discretized by a non-uni- form triangular mesh extending beyond each Fe adatom corral.The Fe adatoms, discretized and mod- eled as hard-wall finite potential barriers, EB,of 1.52 , diameter, are positioned equidistant from the 2.55 spaced nearest-neighbor hexagonal Cu(111) sites.We found that one set of parameters, m* 0.361m 0 and E B 2.5eV, was able to reproduce the available experimental data, and it was used for all our elastic scattering calculations.The effective mass is comparable to the m*/m 0 0.38 + 0.02 reported by Crommie et al. [2].We use an approximate band edge energy of 0.43 eV below E F, a value comparable to the 0.44 + 0.01 eV band edge reported in Ref. [2].Shown in Fig. 2 is a typical two-dimenstional image of the LDOS for the 76 Fe Adatom oval quantum cor- ral.The standing wave patterns are clearly visible and compare well to the reported experimental STM dif- ferential conductance data [3].I00 L O0 -50 0 50 O0 150 FIGURE 2 Image of the theoretical elastic scattering calculation of the LDOS for the 76 Fe adatom oval "stadium" corral for E 0.45 eV.
Excellent agreement is achieved with the reported STM dl/dV image in Ref. [3] CONCLUSION In summary, we have presented an elastic scattering theory for the electronic states in quantum corrals and our approach to a parallel FEM numerical solution using the PETSc 13] libraries.The parallel processor decomposition of the FEM mesh and the global solu- tion matrix are distributed such that little interprocessor communication is necessary during the assembly stage.The overall excellent agreement of our results with available STM data on Cu(111) leads us to con- clude that the Fe adatoms act as elastic scattering centers to the surface-confined electrons, and that there is no significant coupling between the 2DEG and bulk states.

FIGURE
FIGURE Schematic diagram for the parallel domain decomposition of the finite element mesh domain.The element contributions from each processor region on the left assembles into the distributed sparse global matrix on the right.Communication is only required to assemble the processor boundary contributions Department of Electrical Engineering, 268 Fitzpatrick, College of Engineering, University of Notre Dame, Notre Dame, IN 46556.
* Director of Science Computing Facilities, 202-A Nieuwland, College of Science, University of Notre Dame, Notre Dame, IN 46556.+