Contains an introduction, reports on two research projects and a list of publications.Joint Services Electronics Program Contract DAALO3-92-C-000

Arias, Tomas A.

Ariel, Imadiel

Cho, Kyeongjae

Dal Pino, Arnaldo

Joannopoulos, John D.

Meade, Robert D.

Rappe, Andrew M.

Wang, Jing

DSpace@MIT

Chapter 4. Semiconductor Surface StudiesChapter 4. Semiconductor Surface StudiesAcademic and Research StaffProfessor John D. Joannopoulos, Dr. Arnaldo Dal Pino, Dr. Robert D. MeadeGraduate StudentsTomas A. Arias, Kyeongjae Cho, Andrew M. Rappe, Jing WangTechnical and Support StaffImadiel Ariel4.2 Surface ReconstructionSponsorJoint Services Electronics ProgramContract DAALO3-92-C-0001Understanding the properties of surfaces of solidsand the interactions of atoms and molecules withsurfaces has been of extreme importance both from,chnological and academic points of view. Therecent advent of ultrahigh vacuum technology hasmade possible microscopic studies of well-characterized surface systems. The way atomsmove to reduce the energy of the surface, thenumber of layers of atoms involved in this reduc-tion, the electronic and vibrational states that resultfrom this movement, and the final symmetry of thesurface layer are of utmost importance in arriving ata fundamental and microscopic understanding ofthe nature of clean surfaces, chemisorption pro-cesses, and the initial stages of interface formation.The theoretical problems associated with thesesystems are quite complex. However, we are cur-rently at the forefront of solving the properties ofreal surface systems. In particular, we are contin-uing our efforts in developing new techniques forcalculating the total ground-state energy of asurface system from "first principles," so that wecan provide accurate theoretical predictions ofsurface geometries and behavior. Our efforts inthis program have concentrated in the areas ofsurface growth, surface reconstruction geometries,structural phase transitions, and chemisorption.The (7x7) reconstruction of Si( 11) is perhaps themost complex and widely studied surface of a solid.Since its discovery through low-energy electrondiffraction (LEED) more than thirty years ago, anenormous amount of effort has been expended toelucidate the properties of this important surface.Based on this work, it is now generally acceptedthat the geometry of the (7x7) reconstruction isdescribed by the dimer-adatom-stacking-fault (DAS)model as proposed by Takayanagi et al. The com-plexity of this geometry, however, has defied anycomplete and realistic theoretical treatment of itsproperties. The only progress that could be madetheoretically was by isolating and modeling bits andpieces of the surface. The only attempt at a com-plete work used an empirical tight-binding model tostudy the Si(111)-(7x7) reconstruction in a supercellgeometry with 196 atoms.In this work, we have exploited the power of thestate of the art in parallel computation to demon-strate the feasibility of performing ab initio calcu-lations with supercells approching 1000 atoms.Specifically, we have performed the first ab initiocalculation of the Si(111)-(7x7) using a supercellgeometry with 700 effective atoms. These calcu-lations predict the fully relaxed atomic geometry ofthis system, allow construction of theoretical scan-ning tunneling microscope (STM) images as a func-tion of bias voltages, and predict the energydifference between the (7x7) and (2xl) recon-structions.The DAS model for the Si(111)-(7x7) surface recon-struction is shown in figure 1. The unit-cellboundary is outlined by a dashed line. The largestsolid circles denote the twelve atoms. The smallersolid circles denote six rest atoms that lie one layerbelow the surface and are threefold coordinated.There is also a single corner hole in each unit cell.Together, these 19 atoms account for the 19unbonded electrons on the reconstructed surface.4.1 IntroductionChapter 4. Semiconductor Surface Studies(a) Top view of the DAS model for the Si(111)-(7x7) surface reconstruction. The (7x7) unit cell is outlined.increasing distances from the surface are indicated by circles of decreasing size. The large solid circlese twelve adatoms. the smaller solid circles represent the rest atoms. The faulted half of the unit cell is on thell open circles denote the dimers, while small solid circles and dots represent atoms in the unreconstructed) Side view. Atoms on the lattice plane along the long diagonal of the surface unit cell are shown with largern those behind them. The numbers label the 37 unique atoms in the four-layer slab. All others are related bysymmetry operations.The smallest solid circles denote the fourth layer ofatoms below the surface on the unfaulted half ofthe unit cell. Nine dimers are present two layersbelow the surface. One dimer is labeled by atomswith the number 16 and 17. The side view showsthat the stacking sequence in the right half of theunit is the same as in bulk Si, while the stackingsequence in the left half is faulted.Our model system consists of a supercell with aslab geometry containing vacuum on both sides.Periodic boundary conditions are applied in alldirections. The slab contains the adatom layersand four surface layers shown in figure 1(b) plus amirror image reflection in the vertical direction.Thus, the supercell consists of eight layers of atomswith adatom layers on both slab surfaces and aregion of 10 A of vacuum. The 400 silicon atomsand the vacuum layer make this supercell equiv-alent to a 700-atom system. Unit-cell dimensionsare 22 A x 2 A x 26 A.All atoms were allowed to move freely except theinnermost layer, which was frozen in its bulk posi-tion. Thus, our model of the surface includes therelaxation of three surface layers in addition to theadatom layer. The atoms were assumed to be intheir fully relaxed positions when the forces on theions were converged to 0.15 eV/A.132 RLE Progress Report Number 1353 4Figure 1.Atoms atdenote theleft. Smalayers. (bcircles thaChapter 4. Semiconductor Surface StudiesTable 1 shows the relaxed ionic positions. Thecomputed positions in the surface plane are nearlyidentical with those from the tight-binding calcu-lations. The most significant differences involveexpansion of the surface layers away from the bulklayer. Compared to the tight-binding calculations,the adatoms move - 0.03 A away from the bulklayer. The rest atoms move - 0.02 A toward thebulk layer. Most of the remaining atoms move-0.05 A away from the bulk layer. However,atoms directly below adatoms move only - 0.03 A,Atom X Y ZAdatoms I 1.500 1.500 1.6622 4.492 4.491 1.5943 6.002 6.002 1.5554 9.005 9.005 1.603First-layer 5 1.033 1.033 -0.057atoms 6 1.958 0.993 -0.0587 2.481 2.481 0.3488 4.021 4.022 -0.1559 4.974 4.050 -0.09910 5.980 5.057 -0.135II 6.478 6.478 -0.16312 8.019 8.019 0.34513 9.029 8.059 -0.10414 9.472 9.472 -0.111Second-layer 15 1.504 1.504 -1.512atoms 16 1.162 0.006 -1.05717 1.800 0.007 -1.04218 2.492 1.499 -0.86719 2.969 2.969 -0.86320 4.494 4.495 -1.56121 5.403 4.780 -1.04622 6.004 6.004 -1.56723 7.532 7.532 -0.87124 8.505 7.510 -0.87425 8.998 8.998 -1.537Third-layer 26 0.001 0.001 -3.986atoms 27 1.500 1.500 -4.41628 1.006 -0.002 -3.98929 1.990 0.000 -3.96430 2.499 1.499 -3.94631 2.997 2.997 -3.95132 4.499 4.499 -4.46033 5.496 4.512 -3.97334 5.999 5.999 -4.44035 7.497 7.497 -3.93336 8.505 7.496 -3.93037 9.001 9.001 -4.414Table 1. Relaxed atomic positions for the adatom layerand the first three surface layers for the ab initio (7x7)calculation. The coordinate system is the same as in ref-erence 10. All reduced coordinates (X, Y, Z) are withrespect to the Cartesian system indicated in figure 1,where the x axis is along the cubic [110] direction, andthe y axis is along the [111] direction outward normal tothe surface. The actual atomic coordinates (x, y, z) arerelated _(X, Y, Zty the scaling relations x = aX,y=aY/ 3, z=aZ/ 24, where a3.85 A is the (lxi)surface hexagonal lattice constant.resulting in a more distorted adatom configuration.The atoms two layers directly below move the mostcompared to the tight-binding calculations, relaxing-0.10 A away from the bulk layer. The relativeheights of adatoms 1, 2, 3, and 4 are 0.085, 0.031,0.000, and 0.038 A, respectively. ExperimentalLEED values are 0.120, 0.080, 0.0, and 0.040.Having generated the electronic wave functions inour pseudopotential calculation, we proceded tomake theoretical STM figures. In figure 2, wecompare the theoretical results (top) with corre-sponding experiments at a tip voltage of 2 V(bottom) for both occupied and unoccupied states.The theoretical STM probe was placed 3.8 A abovethe surface adatom layer. The probe height islimited by the presence of a second surface layerabove the probe due to the periodic replication ofthe unit cell. The theoretical grey scale wasmatched to the experimental grey scale at the topsof the adatoms and bottoms of the corner holes,and linearly interpolated in between. Experimentsshow a variation of 2-3 A corresponding to thisscale. The corresponding theoretical variation is 5A. Depths greater than 3 A are set to minimumintensity in order to match the experimental greyscale.The most fundamentally intereresting part of thiscalculation is the comparison of formation energiesof the (7x7) reconstruction and the (2x1) surface.Cleavage of Si to create the (111) surface results ina (2x1) metastable structure. This surface mustthen be annealed to generate the stable (7x7)structure. We find that the (7x7) reconstruction isenergetically favorable over the metastable (2x1)surface by 60 meV per (141) cell. Our absolutevalues are 1.179 eV for the (7x7) surface and 1.239eV for the (2x1).The calculations discussed in this section weremade possible because of the considerable compu-tational power associated with massively parallelarchitectures. The calculations elevate ab initioinvestigations to a new echelon and demonstratethat studies of complex material systems withsupercells containing nearly an order of magnitudemore atoms than previously possible are now fea-sible.4.3 Computation at Large LengthScalesAs we discussed in the previous section, withrecent advances in highly parallel computer archi-tecture, calculations on systems with hundreds ofatoms are now tractable. This not only permitsmore comfortable extraction of the thermodynamic133Chapter 4. Semiconductor Surface StudiesFigure 2. STM images of the Si(111)-(7x7) surface reconstruction based on the results of this calculation (top panels)and the experimental results of Avouris and Wolkow (bottom panels). Shown are unoccupied (left panel) and occupied(right panel) state contributions. The determination of the grey scale in the top panels is described in the text.limit but also extends the qualitative level of com-plexity in accessible problems. However, withgrowing interest in calculations involving largersystems, it has been demonstrated that as thelongest physical length sale in a calculationincreases, integration of the fictitious electronicequations of motion becomes dominated by fluctu-ations in the electronic Hartree energy ("chargesloshing"). These fluctuations force the time stepfor integration of the equations of motion toapproach zero as the system size increases. Whilethe onset of this instability depends on the physicsof the system under study, the effect has beendemonstrated dramatically in a silicon system of sixprimitive cells (-46 A long) at 8 Ry. To fullybenefit from the next generation of computer tech-nologies, an approach free of charge sloshing isneeded.134 RLE Progress Report Number 135Chapter 4. Semiconductor Surface StudiesRecently, conjugate-gradient procedures introducedby Teter, Payne, Allan, and Gillan for relaxing theelectronic wave functions at fixed atomic arrange-ments have been shown not to exhibit the afore-mentioned instability and to consume an order ofmagnitude less CPU time than the best fictitiouselectronic dymamics schemes. Despite thisimprovement, performing a dynamics calculationwith conjugate-gradient techniques is very costlybecause of the tight tolerances to which the elec-tronic problem must be solved (typcially 10#tV/atom) to yield stable atomic trajectories.Very recently we have overcome these difficultiesand introduced a new technique for performingfinite-temperature ab initio total-energy calculationsat long length scales. The method involves theintroduction of a new class of energy functionalsand an efficient scheme for performing interativesolutions.To demonstrate the practicality of the new tech-nique for long-length-scale systems, we have per-formed tests using a system that is well over 50 Along. In particular, we have determined the phononspectrum of Si through the velocity-velocity autocor-relation function of the ions in our calculation. Afterprojecting the ionic displacements from our calcula-tion onto the allowed k states and the known purelylongitudinal and transverse polarization vectors, wethen determine the frequencies present in the timeof autocorrelation functions of these spatial Fouriercoefficients with the maximum entropy method.Taking the frequencies of the peak values of theresulting spectra, we compare our parameter-freeresults with the experimentally measured phononfrequencies in figure 3 and find excellent agree-ment. Our spectra clearly and accurately reproducenot only the celebrated flattening of the loweracoustic (TA) modes as one moves away from theF point but also the delicate splitting of the opticmodes along A. Note that the calculation alsoaccurately reproduces the lowest frequencyacoustic mode despite its period of nearly 1 ps andmanifests the delicate splitting of the optic modes,which beat against each other with even longerperiods. This illustrates that our dynamics tech-nique produces reliable ionic trajectories even overperiods of picoseconds and is thus very precise.4.4 PublicationsArias, T., M. Payne, and J.D. Joannopoulos. "Ab-Initio Molecular Dynamics at Large Length-Scales." Phys. Rev. B45: 1538 (1992).Arias, T., M. Payne, and J.D. Joannopoulos, "Ab-Initio Molecular Dynamics: Analytically Con-Figure 3. Phonon spectrum as determined frommaximum peak values of MEM fits. These values arecompletely ab initio with no free parameters. The left-hand panel displays experimental data (13, 14) and theright hand panel displays frequencies from our dynam-ically determined trajectories.tinued Energy Functionals and Insights intoIterative Solutions." Phys. Rev. Lett. 69: 1077(1992).Brommer, K., B. Larson, M. Needels, and J.D.Joannopoulos. "Implementation of the Car-Parrimello Algorithm for Ab-Initio Total EnergyCalculations on a Massively Parallel Computer."In Computers in Physics.Brommer, K., M. Needels, B. Larson, and J.D.Joannopoulos. "Ab-Initio Theory of the S:(lll)(7x7) Surface Reconstruction - A Challenge forMassively Parallel Computation." Phys. Rev.Lett. 68: 1355 (1992).Kaxiras, E., O. Alerhand, J. Wang, and J.D.Joannopoulos. "Theoretical Modelling of Het-eroepitaxial Growth Initiation." Mat. Sci. Eng.B14: 245 (1992).Payne, M., G. Francis, M. Needels, E. Tarnow, P.Dallot, P. Bristowe, and J.D. Joannopoulos."Ab-Initio Investigations of Surfaces and GrainBoundaries in Germanium." In Ordering at Sur-Chapter 4. Semiconductor Surface Studiesfaces and Interfaces. New York: SpringerVerlag, 1992, p. 169.Rappe, A., A. Dal Pino, M. Needels, and J.D.Joannopoulos. "Mixed Basis PseudopotentialMethod for Iterative Diagonalization Tech-niques." Phys. Rev. B 46: 7353 (1992).Rappe, A., J.D. Joannopoulos, and P. Bash. "ATest of the Utility of Planewaves for the Study ofMolecules from First Principles." J. Am. Chem.Soc. 114: 6466 (1992).136 RLE Progress Report Number 135

Semiconductor Surface Studies

Contains an introduction, reports on two research projects and a list of publications.Joint Services Electronics Program Contract DAAL03-89-C-0001Joint Services Electronics Program Contract DAAL03-92-C-000

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Semiconductor Surface Studies

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