VOLUME 76, NUMBER 8 P H Y S I C A L R E V I E W L E T T E R S 19 FEBRUARY 1996

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New Approach for Determination of Diffusion Parameters of Adatoms

Michael Bott, Michael Hohage, Markus Morgenstern, Thomas Michely,* and George ComInstitut für Grenzflächenforschung und Vakuumphysik, Forschungszentrum Jülich, 52425 Jülich, Ge

(Received 13 July 1995)

A new approach for the determination of activation energy and attempt frequency for the diffusof single adatoms on a surface is proposed and demonstrated for Pt adatom diffusion on Pt(The method is based on a combination of low temperature scanning tunneling microscopy (Smeasurements and kinetic Monte Carlo simulations. The method involves only a minimumassumptions and is independent of classical nucleation theory. This independence allows onanalyze the applicability of the latter. In addition, it is found that the measurement of the mesquare displacement of individual adatoms is influenced by the STM tip.

PACS numbers: 81.15.Hi, 61.16.Ch, 68.35.Fx, 68.55.Jk

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Probably the most fundamental diffusion processmolecular beam epitaxy is the migration of the deposiatoms on the substrate, which enables the formation ofposit islands. The basic quantities characterizing adamigration are the activation energyEd and the attemptfrequencyn0. Besides the field emission microscopy aproach [1] there are two methods to determine theserameters: the measurement of the temperature dependof the mean square displacement adatoms by field ioncroscopy (FIM method) [2] and the analysis of the tempature dependence of the island number density by theequation approach of the nucleation theory (nucleation tory method) [3–5]. While the FIM method involves onlbasic diffusion theory, the nucleation theory is rather sphisticated and its application always involves assumptiand approximations of the physical situation. Both meods have been applied successfully. In addition, MoCarlo simulations of simple models were used to test aimprove its formulation [6,7]. However, to the authorbest knowledge it has not yet been tried to test experimtally the applicability of the nucleation theory without presupposing its validity. Such a test of the nucleation themight allow direct insight into the nature and relevancethe approximations to the physical situation involved inapplication.

In this Letter we report a new method for determinatiof Ed andn0 in the temperature range just above the oset of adatom migration, based on scanning tunnelingcroscopy (STM) measurements and kinetic Monte Ca(MC) simulations, involving only a minimum of assumptions. This method is independent of nucleation theoand enables us to perform a test of the nucleation themethod for the Pt adatom migration on Pt(111).

The experiments were performed in a variable tempeture STM system with a base pressure of5 3 10211 mbardescribed elsewhere [8]. The sample crystal was cafully prepared and cleaned before each deposition expiment according to standard procedures [8]. In orderremove gas atoms adsorbed during the initial cool doto 20 K, the sample is flashed prior to the low tempeture deposition. After that, the density of impurities w

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less than1026 Å22 as checked by STM imaging. Pt de-position was performed by electron bombardment heaing of a thoroughly cleaned Pt foil located in front ofthe sample. Immediately after deposition of the desireamount with the precisely controlled deposition rateR ­s6.6 6 0.7d 3 1024 ML s21 (where ML denotes mono-layer) at the specified sample temperature, the sample wcooled to 20 K for STM imaging.

In the MC simulations the deposition with a rate o6.6 3 1024 ML s21 on a triangular lattice with3 3 104

sites is chosen randomly in space and in time [9]. Fotemperatures below 245 K considered here (i) adatom mgration on the Pt(111) surface and (ii) atom jumps fromonefold coordinated to twofold coordinated sites at steedges are the only atomic processes relevant in the simlation. In particular, (iii) the diffusion of twofold coordi-nated atoms along step edges, (iv) dimer dissociation,dimer diffusion, and (vi) transient mobility are either ruledout or irrelevant for the island densities obtained in thMC simulations. Assumptions (ii), (iii), and (iv) followdirectly from a quantitative analysis of the temperaturdependence of the average coordination number of atoin large adatom islands (coverageø0.1 ML) [10] and arein agreement with the available FIM data for Pt and othefcc metals [2,11]. It has to be emphasized that due to tsmall island sizes in the main experiment, which estalishes the new method, the corresponding MC simulatioare not affected by the validity of assumptions (ii) and(iii). The assumption of the irrelevance of dimer diffu-sion up to 245 K [process (v)] needs a closer inspectioIn order to test this approximation we performed MC simulations assuming not only adatoms but also dimers tomobile. As long as the activation energy for dimer diffusion was at least 0.09 eV higher than for adatom diffusiothe island densities obtained from these simulations weidentical within the limits of error to those assuming immobile dimers. Such a difference in activation energy oadatoms and dimers is in agreement with available FIresults [11]: Ir-dimer motion is found to have an activation energy of 0.16 eV higher than Ir-adatom motionFinally a transient mobility of one or two lattice distance

© 1996 The American Physical Society

VOLUME 76, NUMBER 8 P H Y S I C A L R E V I E W L E T T E R S 19 FEBRUARY 1996

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upon accommodation, as suggested by FIM experimeand effective mass theory calculations [12], was agafound to have no measurable influence on the simulatresults, thereby justifying assumption (vi).

In the main experiment to be described now winvestigated the onset of nucleation due to the onof the thermally activated migration of adatoms. Avarious fixed sample temperatures between 23 and 16an amount ofs4.2 6 0.4d 3 1023 ML Pt was deposited(Fig. 1). This quantity has been chosen small in ordto allow precise counting of the imaged objects. Th“object” density upon deposition at 23 K iss6.4 6 0.7d 3

1024 Å22 and the adatom number density correspondito the deposited amount iss6.3 6 0.6d 3 1024 Å22.Thus the bright spots seen in Fig. 1(a) are single adatoEven with a sharp STM tip as used in Fig. 1(a) thapparent diameter of an adatom is 7–8 Å. Repeaimaging of adatom distributions shows that at 20 K (adatoms are thermally immobile and (2) their locatioare not affected by STM imaging. The decreasethe object density from Figs. 1(a) to 1(d) is apparenFigure 2 shows quantitatively the temperature dependeof the object density as a “nucleation curve.” It iconstant up to 110 K, which demonstrates that undergiven experimental conditions (rate, deposited amouthe adatom mobility at and below 110 K is insufficienfor the adatoms to meet. With increasing temperatuadatoms become mobile and during deposition and6 s before the sample is quenched, the adatoms sto meet, forming stable dimers. With further increasintemperature the chance of adatoms to attach to sta

FIG. 1. STM topographs of Pt(111) surface after deposition4.2 3 1023 ML at sample temperatures of (a) 23 K, (b) 115 K(c) 140 K, and (d) 160 K, respectively; scan size 480 Å animaging at 20 K.

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islands instead of forming new dimers increases, athereby the object density decreases even further.

The MC simulations performed in order to reproducthe object densities involve besides the well-definexperimental conditions only two free parameters:Ed

and n0. These two values were varied to reproducthe onset of nucleation at 110 K and the slope of texperimental object density above 110 K. Our bestis obtained withEd ­ 0.26 eV and n0 ­ 5 3 1012 Hz(dotted line in Fig. 2). The dashed (Ed ­ 0.24 eV, n0 ­6 3 1011 Hz) and dash-dotted (Ed ­ 0.28 eV, n0 ­ 4 3

1013 Hz) curves in Fig. 2 demonstrate the sensitivity othe fitting procedure, from which we estimate the erroof our best fit to beEd ­ 0.26 6 0.01 eV andn0 ­ 5 3

101260.5 Hz. Ed as determined here is identical withinthe error to that determined in a FIM experiment for thsame system and lies in magnitude between embedatom method and first principle calculations [13]. Forn0no other experimental data are available, but its magnituis in the expected range [13].

After having determined the diffusion parameters, idependent experiments to test the new procedure wperformed. First, the dependence of island density on cerage in the range between 0.002 and 0.15 ML was msured at 180 K. Examples of STM topographs are shoin Fig. 3. The island densities obtained from such tpographs are plotted as full squares in Fig. 4. The dotline represents the result of a MC simulation withEd ­0.26 eV andn0 ­ 5 3 1012 Hz as determined above. Thesimulation reproduces the experimental data accurateSecond, the dependence of the island density on tempture at a coverage of 0.1 ML was determined. At 0.1 Mthe island density is saturated (i.e., it deviates less thafew percent from its maximum value) with the entire temperature range. The data (full squares) and the correspo

FIG. 2. Island density after deposition of4.2 3 1023 MLvs 1yT ; experimental data (black squares) and correspondlinear regression (solid line); MC simulation withEd ­0.26 eV and n0 ­ 5 3 1012 Hz (dotted line); MC simulationwith Ed ­ 0.24 eV, n0 ­ 6 3 1011 Hz (dashed) andEd ­0.28 eV, n0 ­ 4 3 1013 Hz (dash-dotted), respectively.

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FIG. 3. STM topographs after deposition at 180 K of (a2 3 1023 ML, (b) 5 3 1023 ML, (c) 1 3 1022 ML, and (d)1 3 1021 ML; scan size 960 Å and imaging at 20 K.

ing linear regression (solid line) are shown as an Arrhenplot in Fig. 5. The result of the MC simulations obtaineagain withEd ­ 0.26 eV andn0 ­ 5 3 1012 Hz as deter-mined above is represented by the dotted line. While tMC simulation tends to give slightly higher island densitiethan experimentally observed (10% higher), the slopesthe experiment and the simulation line are practically idetical. Note also that the clear deviation of the experimenisland density at 265 Ks3.77 3 1023 K21d indicates thebreakdown of assumptions underlying the MC simulatioat this high temperature. Probably either assumption (

FIG. 4. Island density vs coverage, after deposition at 180experimental data (black squares); MC simulation withEd ­0.26 eV and n0 ­ 5 3 1012 Hz (dotted line); numerical solu-tion of the rate equations [14] withEd ­ 0.26 eV and n0 ­5 3 1012 Hz and with capture numberssfr (dashed ) andslatt

(dash-dotted), respectively. The dash–double-dotted curveinferred from Fig. 2 of Ref. [7].

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(stability of dimers) or (v) (dimer diffusion) or both are novalid any more.

In conclusion, the two test experiments and the agrement of our results with independent FIM experimen[13] demonstrate that the take up of a nucleation curand its fit by MC simulations is an efficient and accuraprocedure for determining both diffusion parametersadatom migration. The procedure is applicable for sytems where adatoms are the diffusing species withhighest mobility. Its clue is that at the onset of nucleatioall other activated atomic processes are either frozenor irrelevant (all processes affecting cluster shape, clusdissociation, etc.) such that the only two free parametin the MC simulation aren0 andEd , which thereby can beunequivocally determined. Compared to the FIM methit has the advantage of being applicable not only to marials of large mechanical strength.

The new method also avoids some of the complicatioof the nucleation theory method as we will see now. Tsimplest way to use nucleation theory for the determintion of Ed andn0 is with the help of the well-known scal-ing relation for the total island densityN ­ fsRyDdg withD ~ n0e2EdykT andg ­ 1

3 for the physical situation underconsideration here [3,7]. The application of this relationpossible only for larger coverages where the island densis saturated. Therefore the functionf ­ fssd dependson the ability of the islands to incorporate free adatom(capture numbers); it is not straightforward to determines precisely, since it principally depends on island sha(specifically its dimensionality), as well as the lateral anthe size distribution of islands [6,7]. Thus the nucleatiotheory method necessitates additional knowledge (e.g.,atomic processes at the island ledge) compared to thecleation curve method (where, for instance, atomic prcesses at the island ledge are irrelevant). As an exaple of what is meant with these remarks we consider t

FIG. 5. Saturated island density at1 3 1021 ML coverage vs1yT : experimental data (black squares) and correspondingear regression (solid line); MC simulations withEd ­ 0.26 eVandn0 ­ 5 3 1012 Hz (dotted line); numerical solutions of therate equation with the same parameters for the diffusion coficient D with sfr (dashed) andslatt (dash-dotted) as capturenumbers (see text).

VOLUME 76, NUMBER 8 P H Y S I C A L R E V I E W L E T T E R S 19 FEBRUARY 1996

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case of Pt-adatom migration on Pt(111) by using simplifirate equations [14]. These rate equations were numericintegrated with two different standard expressions forcapture number [14]slatt (lattice approximation) andsfr

(geometrical concept for fractals) and withEd ­ 0.26 eVandn0 ­ 5 3 1012 Hz as an input. From Fig. 4 it is apparent that both capture number concepts show deviatfrom the experimental dependence of island densitycoverage. However, both curves can befittedto the exper-imental values by adjusting the attempt frequency. FrFig. 5 it can be seen that both solutions exhibit in facpower law dependence onRyD. The use ofslatt repro-duces the correct slope in the Arrhenius representatioFig. 5 and therefore enables a correct determination ofEd.The use ofsfr leads in contrast to a wrong value forEd

becausesfr and thusfssfr d is strongly temperature dependent. Note that the determination ofEd assuming atemperatureindependentfunction fssd and the determi-nation of n0 by fitting with the coverage dependencethe island density withsfr used in the rate equation—adone in Ref. [5]—is inconsistent. In the mean field aproximation of nucleation theory better agreement with texperimental data for the coverage dependence of theland density may be achieved, if the island size, densand structure dependence of the capture numbers isplicitly taken into account. Such a solution, as inferrfrom Fig. 2 of Ref. [7] for our experimental conditionsDyR ­ 0.95 3 108d, is represented by the dash–doubldotted line in Fig. 4.

The most direct method for determination ofEd andn0

by STM would be the monitoring of the displacementsindividual adatoms in successive STM images at variofixed temperatures similar to the FIM method. Howevwe found, similar to the case of semiconductors [15,1that the STM tip influences the adatom diffusion oPt(111). The mean square displacement was too la(e.g., for U ­ 30 mV and I ­ 0.2 nA we determinedEd ­ 0.17 eV and n0 ­ 1.3 3 1011 Hz) and increasedwith decreasing tip-sample distance. Minimization of tinteraction of time of the migrating adatom with thSTM tip and carefully chosen tunneling parameters miglead to useful results with this approach. However,improved knowledge of the details of the interaction of ttip with the migrating adatom is necessary, which neemore detailed experiments [17].

The authors thank Georg Rosenfeld, PavelSmilauer,Zhenyu Zhang, and Andy Zangwill for illuminating discussions, and PavelSmilauer also for critically reading themanuscript. M. H. acknowledges support by the KonraAdenauer-Stiftung and M. M. by the Studienstiftung ddeutschen Volkes.

*To whom correspondence should be addressed.[1] M. Trinigides and R. Gomer, Surf. Sci.155, 254 (1985).[2] G. Ehrlich and F. G. Hudda, J. Chem. Phys.44, 1039

(1966); for a recent review, see G. L. Kellog, Surf. ScRep.21, 1 (1994).

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[3] J. A. Venables, G. D. T. Spiller, and M. Hanbücken, ReProg. Phys.47, 399 (1984).

[4] Y. W. Mo, J. Kleiner, M. B. Webb, and M. G. Lagally,Phys. Rev. Lett.66, 1998 (1991); J. A. Stroscio, D. T.Pierce, and R. A. Dragoset, Phys. Rev. Lett.73, 1955(1994).

[5] H. Brune, H. Röder, C. Boragno, and K. Kern, Phys. ReLett. 73, 1955 (1994).

[6] For example, M. C. Bartelt and J. W. Evans, Phys. Rev.46, 12 675 (1992); C. Ratsch, A. Zangwill, P.Smilauer,and D. D. Vvedensky, Phys. Rev. Lett.72, 3194 (1994);J. G. Amar, F. Family, and P. M. Lam, Phys. Rev. B50,8781 (1994).

[7] G. S. Bales and D. C. Chrzan, Phys. Rev. B50, 6057(1994).

[8] M. Bott, Th. Michely, and G. Comsa, Rev. Sci. Instrum. 66, 4135 (1995); Th. Michely, Jül-Bericht 2569(Forschungszentrum Jülich, Jülich, 1991).

[9] Random deposition also in time turned out to be crcial in order to avoid finite size effects of the simulatiofield. The procedure applied is comparable to the one dscribed in P. A. Maksym, Semicond. Sci. Technol.3, 594(1988).

[10] M. Hohage, M. Bott, M. Morgenstern, Z. Zhang, ThMichely, and G. Comsa (to be published).

[11] D. W. Basset and P. R. Webber, Surf. Sci.70, 520 (1978);G. L. Kellogg, J. Phys. (France) IV47, C2-331 (1986);S. C. Wang and G. Ehrlich, Surf. Sci.239, 301 (1990).

[12] S. C. Wang and G. Ehrlich, Phys. Rev. Lett.62, 2297(1989); P. Stoltze and J. K. Nørskov, Phys. Rev. B48,5607 (1993).

[13] P. J. Feibelman, R. S. Nelson, and G. L. Kellog, Phys. ReB 49, 10 548 (1994).

[14] From the Eqs. 2.3, 2.5, 2.6, 2.8, and 2.11 in Ref. [3] thfollowing two rate equations are derived under the condtions of our experiment (i ­ 1, immobility of stable clus-ters, no desorption):dn1ydt ­ Rs1 2 Zd 2 2s1n2

1D 2

sxn1nxD and dnxydt ­ s1n21D 2 2nx dZydt, wheren1

and nx are the density of adatoms and stable clusterespectively,s1 and sx the corresponding capture number, andZ the coverage of the atoms contained in stble islands or in the present case the total coveragethe atoms minusn1yn0 where n0 is the density of sites.For sx two different expressions were used:slatt

x ­4ps1 2 Zdyflns1yZd 2 s3 2 Zd s1 2 Zdy2g with s1 ­3, obtained by Stowell [18] by solving the diffusionequation in the lattice approximation andsfr

x ­ 2 1

sZn0ynxd1y1.7, which is an expression corresponding to thgeometrical concept [19] applied to fractal islands [5].

[15] L. J. Whitman, J. A. Stroscio, R. A. Dragoset, and R.Cellotta, Science251, 1206 (1991); P. Ebert, M. G.Lagally, and K. Urban, Phys. Rev. Lett.70, 1437 (1993).

[16] Y. Mo, Phys. Rev. Lett.71, 2923 (1993).[17] One might also consider intermediate quenching of t

adatoms during imaging (as done in Ref. [18]) to avotip effects. While this is principally possible, for low temperatures as generally necessary for metals this procedis experimentally demanding with respect to sample cotamination, STM drift, and data acquisition time.

[18] M. J. Stowell, Philos. Mag.26, 349 (1972).[19] G. Zinsmeister, Thin Solid Films7, 51 (1971).

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