3.1 Ground State Geometries of Surfaces

Transcription

1 Chapter 3. Semicnductr Surface Studies Chapter 3. Semicnductr Surface Studies Spnsr Jint Services Electrnics Prgram (Cntracts DAAL3-86-K-2 and DAAL3-89-C-1) Academic and Research Staff Prfessr Jhn D. Jannpuls, Dr. E. Kaxiras, Dr. Oscar L. Alerhand Graduate Students Tmas A. Arias, Mark Needels, Andrew M. Rappe, Eugen G. Tarnw, Jing Wang Understanding the prperties f surfaces f slids and the interactins f atms and mlecules with surfaces has been f extreme imprtance bth frm technlgical and academic pints f view. The recent advent f ultrahigh vacuum technlgy has made micrscpic studies f well- characterized surface systems pssible. The way atms mve t reduce the energy f the surface, the number f layers f atms invlved in this reductin, the electrnic and vibratinal states that result frm this mvement, and the final symmetry f the surface layer are all f utmst imprtance in arriving at a fundamental and micrscpic understanding f the nature f clean surfaces, chemisrptin prcesses, and the initial stages f interface frmatin. Actually, ne f the mst difficult and fundamental prblems in surface studies, bth frm the experimental and theretical pints f view, is simply the determinatin f the precise psitins f the atms n a surface. Currently, there are many surface gemetries, even fr elemental surfaces, that remain extremely cntrversial. The theretical prblems assciated with these systems are quite cmplex. We are, hwever, currently in the frefrnt f being able t slve fr the prperties f real surface systems (rather than simple mathematical mdels). In particular, we are cntinuing ur gal f calculating the ttal grund-state energy f a surface system frm "first principles" s that we may be able t prvide accurate theretical predictins f surface gemetries. Our effrts in this prgram have cncentrated in the areas f surface grwth, surface recnstructin gemetries, structural phase transitins, and chemisrptin. 3.1 Grund State Gemetries f Surfaces Using "first principles" ttal energy calculatins, it is pssible t determine n a micrscpic scale hw atms behave when they are n the surface f a slid. This is a fundamental prblem that has plagued bth therists and experimentalists fr decades. The difficulty lies with the very strng interactins that may exist between the surface atms and the hst atms cnstituting the rest f the slid. These interactins can strngly disturb the riginal idealized atmic arrangement at the surface, changing the nature f the bnding and even the riginal stichimetry. At shrt length-scales the atmic rerientatins n the surface typically lead t what is knwn as "recnstructin." At lng lengthscales, it has always been assumed that there exists a perfect state f the surface that is therwise atmically smth and unifrm. This implies that the intrinsic grund-state f a facetted surface f a crystal, which in general includes a recnstructin, is therwise atmically smth and unifrm. T test this hypthesis we have cmbined quantum ttal energy calculatins which can describe accurately the micrscpic prperties f the surface at very shrt length scales with cntinuum mechanical calculatins which can describe cllective phenmena at very lng length-scales. The results f ur theretical analysis lead t the fllwing exciting predictin: surfaces f crystals which recnstruct with brken

2 Chapter 3. Semicnductr Surface Studies rientatinal symmetry and exhibit an anistrpic intrinsic stress tensr are unstable t a spntaneus frmatin f elastic stress dmains. Thus, the grund-state f such a surface is nt unifrm and will frm an rdered array f dmains f recnstructins in different rientatins. Recently an experiment was perfrmed by F. Men, W. Packard, and M. Webb, Bull. Am. Phys. Sc. 33:472 (1988), n Si(1) that we believe prvides evidence fr ur predictin. Their results are shwn in the tp panel f figure 1. They bserve that upn annealing, the fractin f ne type f dmain grws at the expense f the ther when an external strain is applied t the surface. The dmains fr which the applied cmpressin is alng the dimers are the nes that grw, in agreement with ur calculatins r the surface stress tensr. When the external strain is released, hwever, the surface returns t its "initial" cnfiguratin f equal dmain ppulatins. This surprising result is cnsistent with ur idea f spntaneus frmatin f stress dmains, where the grundstate f the surface crrespnds t a dmain cnfiguratin. Cmparisn f the experimental intensities f dmain ppulatins with the theretical curves (tp and lwer panels in figure 2, respectively) suggests that the experimental surface is nly in quasiequilibrium. This is presumably because f lcal surface miscuts r kinematic cnsideratins which allw fr nly lcal equilibratin. Indeed, the difference in the data between psitive and negative applied strains is indicative f kinematic cnstraints. With ur calculated value f the stress anistrpy, F, and ur fitting the experimental data t determine the strain cnstant s =.3%, we can predict the characteristic size f the dmains, /. We btain a value fr / between 3 and 1 A. The uncertainty cmes frm the fit f E, frm the chice f bulk elastic cnstants, and frm the uncertainty in F. This length scale is cnsistent with the experiment f Men, Packard, and Webb, which puts a lwer bund fr / f apprximately 5 A. We are currently investigating this system in mre detail t understand the effects f different types f steps, defects and miscuts n the final gemetry. All ab-initi theretical calculatins n surface systems have been at zer temperature. It is nw becming pssible, hwever, t begin studying the statistical mechanics and temperature related phase transitins f surfaces f slids. This is a cmpletely new and unexplred area. As an example, the myriad f surface recnstructins that may exist n clean semicnductr surfaces at different temperatures is an extremely interesting and fundamental prblem that needs t be investigated. Mdern studies f phase transitins utilize a pwerful theretical tl which is the renrmalizatin grup scheme. The scheme is based n scaling ideas, and has as input simple spin Hamiltnians which mdel the degrees f freedm f the system. Until nw there has been n way f calculating what these Hamiltnian parameters shuld be fr real surfaces f slids. The ttal energy calculatins described abve, hwever, shuld prvide precisely the kind f infrmatin needed. The exciting pssibility 2 U] - *- -I I I I I STRAIN (%) Figure 1. Tp: Experimental intensity f (1x2) (pen circles) and (2x1) (filled circles) dmains n the Si(1) surface as a functin f applied external strain. Bttm: Thery fr s =.3%. The thick (thin) lines crrespnd t quasi (glbal) equilibrium. 98 RLE Prgress Reprt Number 131

3 Chapter 3. Semicnductr Surface Studies then arises f cupling the results f micrscpic studies f surface systems (at zer temperature) with simple spin Hamiltnians and the renrmalizatin grup apprach t study phase transitins at finite temperatures frm "first principles." (a) In the past, using a simple-empirical ttal energy apprach, we succeeded in develping such a scheme and have applied it t the Si(1) surface, reslving imprtant questins regarding the structure f the Si(1) surface. We are nw investigating the pssible phase transitins that may ccur n the Ge(1) surface. This system, hwever, cannt be described accurately by a semi-empirical apprach s that we are frced t use the mre pwerful and much mre cmplex ab-initi ttal energy methds. T perfrm these studies we have been develping a new scheme fr relaxing a system with many degrees f freedm t its lwest energy cnfiguratin. The scheme is based n a mlecular dynamics apprach t calculating quantum mechanical ttal energies and resembles a simulated quench. Using this apprach we have calculated the ttal energy f fur members f the (2x1) family f buckled dimer recnstructins. These are illustrated in figure 2. The ttal energies are given in table 1. Table 1. Ttal energy f (2x1) family symmetry cnfiguratins. System Ttal energy (ev/dimer) b (2 X 1). p (4 X 1).35 c (4 X 2) -.66 p (2 X 2) -.69 Figure 2. Perspective view f dimer mdels f the Ge(1) surface. The slid atms are the surface layer. (a) Buckled (2x1) symmetry cnfiguratin. (b) Centered (4x2) symmetry cnfiguratin. (c) Primitive (4x1) symmetry cnfiguratin. (d) Primitive (2x2) symmetry cnfiguratin. We expected p(2x2) and c(4x2) t be clse in energy because f ur previus tightbinding wrk in Si. We find that c(4x2) is lwer in energy than b(2x1) because the alternatin f up and dwn dimers alng the dimer rws allws the secnd-layer atms t relax. T keep bnd distances clse t the bulk value, the atms in the secnd layer want t mve twards the up dimer atm t which they are bnded and away frm the dwn dimer atm. This mtin is energetically favrable nly when the "up" and "dwn" dimers alternate alng the dimer

4 Chapter 3. Semicnductr Surface Studies rws as in the p(2x2) and c(4x2) recnstructins. When the secnd-layer atms are either bnded t tw "up" r tw "dwn" dimer atms, they cannt mve twards r away frm a dimer withut further stretching r cmpressing an already stretched r cmpressed bnd, as the case may be. If this were the sle mechanism differentiating the energies f the varius dimer recnstructins, p(2x2) wuld be degenerate with c(4x2) and p(4x1) wuld be degenerate with b(2x1). Since p(4x1) is significantly higher in energy than b(2x1), this is strng evidence that an additinal mechanism is at wrk. The magnitudes f the displacements f the atms are extremely similar between p(2x2) and c(4x2) dimers. In fact, if ne takes the psitin f the tw independent dimers in the p(2x2) gemetry and maps them int the ther tw dimers accrding t c(4x2) rules, ne gets an identical energy with the riginal minimum energy c(4x2). In cntrast t previus wrk, there was negligible breaking f the reflectin symmetry f the (1) plane. This symmetry breaking did nt affect the ttal energy and prbably arse frm difficulties in perfrming thse calculatins. We find a dimer tilt f 14 degrees fr all fur minimum-energy cnfiguratins. T predict a phase-transitin ne maps the dimer prblem dimensinal Ising-spin-prblem: temperature t a tw- -H = V si, si,j+ 1 + H -si,j si+l,j I,j + D Esij Si +l,j+ + UYsi,j Si,j+2 ij i ij (1) ) 1+" Figure 3. Effective cuplings between adjacent dimers. Table 2. Effective cupling cnstants between adjacent dimers. Cupling cnstant E (mev) V - 43 H 1 D 4 As can be seen by the relative magnitude f V t H and D, the strngest cupling between dimers is alng the rws. The terms invlving U and F interactins cntribute equally t the ttal energies fr all fur symmetries and are initially set equal t zer. A psitin-space renrmalizatin-grup-thery calculatin is perfrmed with a finite cluster f fur cells f five sites each and the flws ccur in the parameter space f V, H, D, and F. These values f the parameters lead t a phase-transitin temperature f K. These calculatins als predict that the critical expnents assciated with diffractin spt intensities and widths shuld be in the Ising universality class. + F si, j si,j+ si+l,j si+l,j+ 1 This Hamiltnian includes all interactins up t tw surface-atm spacings as shwn in figure 3. The T= values f V, H, and D are derived frm the ttal-energy differences f the fur cnfiguratins and are given in table Hydrgenatin The interactin f atmic hydrgen with cleaned semicnductr surfaces has been extensively studied fr ver a decade. Hydrgen atms appear t saturate surface dangling bnds resulting in a nearly ideal, bulk-terminated plane f expsed surface atms. It is interesting that in cases where the surface des nt have the gemetry and 1 RLE Prgress Reprt Number 131

5 Chapter 3. Semicnductr Surface Studies peridicity f the bulk-terminated plane, the interactin f hydrgen with surface atms is strng enugh t unrecnstruct the cmplicated recnstructin patterns. This prcess takes place fr example n the (2x1) Si(111) surface, which exhibits a lw-energy 9=bnded-chain recnstructin. Upn hydrgenatin this chain f Si atms with (2x1) peridicity reverts t the (1xl) pattern f the bulk-terminated plane. Similar phenmena have been bserved n the Ge(111) surface. Theretically, this prcess is nt very well understd and a realistic, firstprinciples study with adequate accuracy t define precise lw energy psitins f atms, crrespnding ttal energies, and vibratinal excitatins abve the grund-state has been cmpletely lacking. Presently, we have undertaken precisely such a study. Using ab-initi quantum mechanical ttal energy calculatins, we find that the atmic psitins f the hydrgenated Si and Ge(l11) surfaces differ significantly frm thse f an ideal bulk terminated plane. In particular, the Si-H and Ge-H bnds are fund t be cnsiderably larger than the sum f cvalent radii. The substrate relaxatins are small and their physical rigin can be explained in terms f electrnic charge transfer, which eliminates the surface diple mment by shifting charge frm the hydrgen bnd t the back-bnds. T determine vibratinal excitatins we need t investigate the change in ttal energy with varius cnfiguratin crdinates. There are three vibratinal mdes that are f interest t study. Tw are easily identified in terms f simple atmic mtins and invlve hydrgen atms mving relative t the substrate. The large difference in mass between H, Si and Ge effectively decuples the mtin f H frm cllective mdes f the lattice. These are the stretching (whstretch) and bending (cwhwag) mdes f the Si-H and Ge-H bnds. The third mde (wl) invlves a vibratin f the upper bilayer f the system. The calculated ttal energies and crrespnding fitted curves t quadratic plynmials are shwn in figures 4 thrugh 6. Higher rder plynmial fits give frequencies that differ by less than 1 percent frm the quadratic-fit values fr all cases cnsidered. The nearly perfect harmnic character 5 > I C Si-H bnd length A) (b) Ge-H bnd length (A) Figure 4. Calculated ttal energies and quadratic plynmial fits fr the hydrgen bnd stretching mde in (a) Si( 11):H and (b) Ge( 11):H. Energies are given in mev per (1xl) surface unit cell, with respect t the fully relaxed cnfiguratin. f the energy curves is a reflectin f the high degree f cnvergence f the calculatins, which gives the energy differences with a relative accuracy f 1-4 ev. The frequencies fr the three mdes (whstretch, chwag, and wl) and the experimental values are cllected in table 3. There is n experimental measurement fr wl because it lies in the regin f ther vibratinal mdes f the crystal and cuples strngly t them. There is, hwever, a similar vibratinal mde in the amrphus materials fr which experimental measurements exist. Vibratinal frequencies assciated with H mtin in amrphus hydrgenated Si and Ge have als been determined experimentally; their values are included in table 3. 11

6 Chapter 3. Semicnductr Surface Studies f E 5 ) 'p Si-H wag angle (degrees).68 Si first bilayer spacing (A) t cd t,3) Ge-H wag angle (degrees) Figure 5. Same as in figure 4 fr the hydrgen bnd wagging (bending) mde. The calculated values fr whstretch and whwag are in excellent agreement with the experimentally determined frequencies. By cntrast, the calculated values fr l differ by apprximately 35 percent frm the experimental nes. Given the gd agreement we can btain between thery and experiment fr the hydrgen-related vibratins, we cnclude that the discrepancy in the values f wl is nt related t sme theretical shrtcming. In particular, the systematic errrs in the hydrgen relaxatins and the surface Ge first bilayer spacing A) Figure 6. Same as in figure 4 fr the tp layer vibratin mde. layer relaxatins are the same and the expected accuracy fr the cl mde is therefre 5-1 percent. The large discrepancy in l indicates a significant difference between the mtin f atms in the hydrgenated surfaces and in micrvids f the amrphus material. The cmparisn predicts that internal surfaces (micrvids) in the amrphus netwrk are lcally much sfter than the crrespnding crystalline surface cnfiguratin. 12 RLE Prgress Reprt Number 131

7 Chapter 3. Semicnductr Surface Studies Table 3. Cmparisn f calculated vibratinal frequencies with experiment and with crrespnding values in amrphus materials. Frequencies are in mev. The crrespnding wave number in cm- 1 is als given after each frequency in brackets. (foretch is the stretching mde f the hydrgen bnd, U)yag is the wagging mde f the same bnd, and w, the frequency fr vibratin f the tp layer f hydrgenated atms. Si Thery Expt.a LSi:Hb Thery Expt.a asi:hb (Wretch 245 (1978) 257 (285) 248 (2) 231 (186) 245 (1975) 235 (1895) w a g 71 (569) 77 (63) 79 (64) 63 (55) 66 (53) 7 (565) w1 35 (283) 26 (21) 2 (162) 15 (12) Nte: afrm: Frzheim et al., Phys. Rev. B 27:2278 (1983); Richter et al., J. Nn-crystalline Slids 59-6:181 (1983). bfrm: Papagn et al., Phys. Rev. 34:7188 (1986); Shen at al., Phys. Rev. B 22:2913 (198). Ge 13

8 Prfessr Marc A. Kastner 14 RLE Prgress Reprt Number 131