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surface diffusion is a very important process in many phenomena , in particular in crystal growth .
that is why the diffusion of single adatoms on stepped metal surfaces has been recently widely investigated both experimentally and theoretically .
energy barriers for the moves of an adatom on a surface with steps or islands are not easily accessible by experiment but for many elementary processes they can be calculated on microscopical level by molecular dynamics .
the knowledge of the barriers can then be utilized in the construction of kinetic monte carlo models to study growth processes .
the diffusion energy barriers have already been calculated for various metals and different surface orientations .
for example , in the case of fcc ( 111 ) surface there are calculations for al @xcite , ag @xcite , au @xcite , cu @xcite , ni @xcite , pt @xcite , ir @xcite . in most of these studies
the semi - empirical potentials were used due to their simplicity allowing a systematic study of numerous possible processes .
comparable ab initio calculations demand much more computer power , therefore the number of investigated processes must considerably be reduced .
although the recent first principles calculations @xcite indicates that in the case of pt(111 ) the semi - empirical potentials may be insufficient , in many other studies they lead to reasonable results and their application helped at least qualitative understand diffusion energetics and reveal new processes ( see e.g. @xcite ) .
in this paper we study diffusion on stepped rh(111 ) surface .
the research was motivated by a recent stm experiment on unstable growth of rh(111 ) @xcite where a coarsening due to a step - edge barrier was observed over three orders of magnitude of deposited amount .
more recent observation on pt(111 ) @xcite indicates almost no coarsening over the similar interval of deposited material . whereas the step - edge barriers on pt(111 ) surfaces were extensively studied ( see references in @xcite ) , results for rh(111 ) are not available .
we present here a systematic study of energy barriers for inter - layer transport as well as for diffusion along the step edges .
our simulations were done for finite atomic slabs with a free surface on the top , two atomic layers fixed on the bottom , and periodic boundary conditions in the two directions parallel to the surface .
the slab representing the substrate of ( 111 ) surface was 11 layers thick with 448 atoms per layer .
we used systems of approximately 5000 atoms consisting of 19 to 44 layers , with 110 to 240 atoms per layer for diffusion along channels on the vicinal surfaces ( 311 ) , ( 211 ) , ( 331 ) , ( 221 ) and ( 322 ) .
the semi - empirical many - body rosato  guillope 
legrand ( rgl ) potential @xcite including interactions up to the fifth nearest neighbors @xcite was used . for computational details
see @xcite .
the energy barrier for a particular diffusion process was obtained by testing systematically various possible paths of an adatom .
the path with the lowest diffusion barrier was chosen to be the optimum one , and the diffusion barrier , @xmath0 , was calculated as @xmath1 where @xmath2 and @xmath3 are the total energies of the system with the adatom at the saddle point and at the equilibrium adsorption site , respectively .
we considered both the jump and exchange processes .
the minimum energy path for jump diffusion was determined by moving an adatom in small steps between two equilibrium positions and by allowing the adatom to relax in a plane perpendicular to the line connecting two equilibrium positions .
the rest of the atoms in the system were allowed to relax in all directions .
the energy barrier for exchange process was determined by moving the edge atom , that should be replaced , in small steps toward its final position .
this final position was one of neighboring equilibrium sites .
the moving atom was allowed to relax in the plane perpendicular to the exchange direction at each step , whereas the other atoms , including the adatom , relaxed free in all directions .
in our simulation we obtained the energy barrier @xmath4 ev for self - diffusion on the flat rh(111 ) surface , which is in good agreement with experiments . in the field - ion - microscope ( fim ) experiment @xcite the barrier @xmath5 ev was found and recently the value @xmath6 ev was obtained in the stm experiment @xcite from the temperature dependence of island density .
the results of molecular statics calculations and experimental values are summarized in table 1 .
we calculated also binding energy for the supported dimer .
the value @xmath7  ev is in good agreement with @xmath8  ev obtained in the stm experiment @xcite .
we studied the descent of an adatom to the lower terrace from both types of steps on the ( 111 ) surface , i.e. , step a with \{100 } microfacet and step b with \{111 } microfacet ( see fig .
we performed calculations for several geometries : straight steps , steps with a kink , and also for a small island atoms .
for all considered geometries we systematically investigated all possible adatom jumps and pair exchange processes .
our results for straight steps and steps with a kink are summarized in table 2 .
the energy barrier for a direct jump from the upper to the lower terrace is 0.73 ev for straight step a and 0.74 ev for straight step b. the presence of a kink decreases the barrier for the jump to @xmath9 ev on both steps .
we can see that the energy barriers for the jumps are always larger than for the exchange which are and 0.39 ev for a and b step , respectively .    in more complex geometries the number of competing processes to be energetically compared increases , e.g. in the case of step b with a kink we consider four types of processes according to which step - edge - atom ( denoted by r1 , r2 , r3 , or r4 ) is pushed out ( see fig .
we call them exchange next to corner , exchange over kink i , exchange over kink ii , and exchange next to kink , respectively .
we consider all possible combinations of initial and final positions .
for example , in the case of the exchange next to the corner there are three possible processes : @xmath10 , @xmath11 , @xmath12 . in the process
@xmath12 , e.g. , the adatom starts in the fcc site labeled by 3 and pushes out the edge atom r1 .
two possible directions of moving for pushed atom r1 are shown schematically in fig .
the lowest barrier for the inter - layer transport is the barrier for two exchange processes near the kink on the step b ( @xmath13 ev ) , i.e. the ehrlich - schwoebel barrier is only @xmath14 mev .
the barriers for exchange processes on the step a are significantly higher . for a island ,
the minimal values were obtained for the exchange of the atom in the middle of the edge ( 0.43 ev for a - type edge and 0.24 ev for b - type edge ) .
we found that for rh(111 ) similar as for pt(111 ) @xcite the barriers for the descent at a small island are significantly lower than for the descent at straight long steps .
3 shows the energy profile for the diffusion along two edges of a large island .
the structure in the middle corresponds to a diffusion around the corner formed by two edges .
the angle contained by the edges is 120@xmath15 .
there is a small minimum just at the corner positions .
the transport between two edges is asymmetric .
we found that the diffusion along the straight step of type a is faster ( the barrier is 0.40 ev ) than along the step of type b ( the barrier is 0.81 ev ) .
this could be attributed to a purely geometrical effect due to different local geometries along the steps .
the adatom diffusing along the step b has to pass closer to the topmost atoms of the lower terrace than when it is diffusing along the step a ( see fig.1 ) .
there are no available experimental data for diffusion along the steps on rh(111 ) surface .
only one measurement on the ( 311 ) and ( 331 ) surface has been published @xcite . in order to have some comparison , we calculated the energy barriers for the diffusion along steps on vicinal surfaces with terraces : ( 211 ) , ( 311 ) - terraces with step edges of type a , and ( 332 ) , ( 221 ) and ( 331 ) - step edges of type b. results are summarized in table 3 .
the vicinal surfaces are ordered according to the distance between terraces .
we can see that there is a clear tendency with the decreasing distance between steps . in the case of a - step
the barrier along the step is increasing with the step distance increasing , whereas for b - step it is decreasing .
we obtained the barriers @xmath16  ev and @xmath17 ev for the diffusion along steps ( 311 ) and ( 331 ) surfaces , respectively .
experimental results of fim measurements are the energy barriers @xmath18 ev and @xmath19 ev @xcite .
there is qualitative agreement between experimental and calculated data : @xmath20 .
using the rgl potential we calculated the energy barrier for self - diffusion on the flat rh(111 ) surface and binding energy of supported dimer which are in good agreement with the experimental data . with the same potential , we systematically studied energy barriers for the descent at straight as well as rough steps on rh(111 ) .
we found that the lowest energy barriers for descent to the lower terrace is for the exchange process near a kink on step b. we also calculated barriers for the diffusion along step edges on rh(111 ) surface and along step edges on several vicinal surfaces .
we found that the diffusion along step a is faster than along step b , which is in qualitative agreement with the fim experiment .
we observed that these barriers are slightly affected by the step - step interaction .
we expect that due to rather large barriers for the diffusion along steps , both steps will be rough during the growth at lower temperatures and the interlayer transport will prefer step b. at a higher temperature the diffusion along step a starts to be active and the descent on both steps will be possible . however , step b will remain rough and the descent on this step will be easier . in island growth
this would imply that b - edges of an island will grow faster than a - edges , therefore , b - edges will become shorter .
however , the number of kinks for the easy descent on a shorter b - edge will be lower .
hence we expect that for a certain interval of temperatures the shape of the growing island will be asymmetric with longer a steps .
this picture seems to be in agreement with the morphologies presented in @xcite .
+ * figure captions * +   + fig
. 1 . two types of step edges , a and b , for a large island .
solid line shows the diffusion path along the island edge .
the atoms of different layers from the surface to the bulk are shown as large filled circles , large open circles , small open circles , and tiny open circles .
[ fig : along ] +   + fig .
2 : different exchange processes near a kink site on step b on rh(111 ) surface . the edge atoms undergoing exchange diffusion ( r1, ... ,r4 ) and starting positions of an adatom ( 1, ... ,9 ) are shown .
the four topmost atomic layers from the surface to the bulk are distinguished by different circle radii ( cf .
+   + fig .
3 : dependence of the adatom energy for the path along the edge of a large island .
the path is composed from three sections : along edge a , around the corner and along edge b. +
energy barriers for different moves of a single rh adatom in the vicinity of steps on rh(111 ) surface are studied with molecular statics . 
 interatomic interactions are modeled by the semi - empirical many - body rosato  guillope  
 legrand potential . 
 we calculate systematically barriers for the descent at straight steps , steps with the kink and small islands as well as barriers for diffusion along the step edges . 
 the descent is more probable on steps with a \{111 } microfacet and near kinks . 
 diffusion along a step with a \{100 } microfacet is faster than along a step with a \{111 } microfacet . 
 we also calculate barriers for diffusion on several surfaces vicinal to rh(111 ) .    _ 
 ecoss-18 , vienna , september 21 - 24 , 1999 , submitted to surface science _ 
 +   +   +     +   + f. mca , m. kotrla , _ institute of physics , ascr , praha , czech republic _ + o. s. trushin , _ institute of microelectronics , rasc , yaroslavl , russia _ +     + keywords : semi - empirical models and model calculations , construction and use of effective interatomic interactions , stepped single crystal surfaces , adatoms , rhodium , surface diffusion