Size calibration of strained epitaxial islands due to dipole-monopole interaction
aa r X i v : . [ c ond - m a t . m e s - h a ll ] O c t Size calibration of strained epitaxial islands due todipole-monopole interaction
V. I. Tokar and H. Dreyss´e
IPCMS, Universit´e de Strasbourg–CNRS, UMR 7504, 23 rue du Loess, F-67034Strasbourg, FranceE-mail: [email protected]
E-mail: [email protected]
Abstract.
Irreversible growth of strained epitaxial nanoislands has been studied withthe use of the kinetic Monte Carlo (KMC) technique. It has been shown that thestrain-inducing size misfit between the substrate and the overlayer produces long rangedipole-monopole (d-m) interaction between the mobile adatoms and the islands. Tosimplify the account of the long range interactions in the KMC simulations, use hasbeen made of a modified square island model. Analytic formula for the interactionbetween the point surface monopole and the dipole forces has been derived and usedto obtain a simple expression for the interaction between the mobile adatom and therectangular island. The d-m interaction was found to be longer ranged than the con-ventional dipole-dipole potential. The narrowing of the island size distributions (ISDs)observed in the simulations was shown to be a consequence of a weaker repulsion ofadatoms from small islands than from large ones which led to the preferential growthof the former. Furthermore, similarly to the unstrained case, the power-law behaviorof the average island size and of the island density on the coverage has been found. Incontrast to the unstrained case, the value of the scaling exponent was not universalbut strongly dependent on the strength of the long range interactions. Qualitativeagreement of the simulation results with some previously unexplained behaviors of ex-perimental ISDs in the growth of semiconductor quantum dots was observed.
Keywords: irreversible aggregation phenomena (theory), thin film deposition(theory), heteroepitaxy (theory) molecular beam epitaxy (theory) ize calibration due to dipole-monopole interaction
1. Introduction
The phenomenon of self-assembly of size-calibrated coherent nanoislands taking placein some heteroepitaxial systems during strained epitaxy has been extensively studiedfor more than two decades because of its prospective use in microelectronics [1, 2]. Itseems to be well established that the phenomenon is governed by the elastic strain in theoverlayer caused by the lattice size misfit with the substrate [1, 2]. Usually, it is assumedthat the main role in the size calibration (SC) play the long range forces propagated viathe elastic strain in the substrate [3, 4, 5]. However, explicit growth simulations with theuse of the kinetic Monte Carlo (KMC) technique within the models accounting for suchforces in [6, 7, 8], did not show any narrowing of the island size distributions (ISDs).Moreover, in [6] even some broadening of the ISD was seen. Notably, the broad ISDsobtained in the simulations were very similar to those seen during irreversible growth[9]. So by all evidence the growth in [6, 7, 8] was controlled by kinetics; this is farthersupported by the fact that in thermodynamically controlled strained epitaxy in [10, 11]ISDs narrowing were observed. These results seems to suggest that the SC is implausibleunder conditions of kinetically controlled growth. Such growth, however, usually takesplace at smaller temperatures and at faster deposition rates than the thermodynamicallylimited growth [10] which presents some important practical advantages. For example,smaller substrate-deposit interdiffusion and so better control in heteroepitaxial growth[12]. Therefore, the question of whether the SC may be achieved under conditions ofkinetically controlled growth is of considerable practical interest.The aim of the present paper is to suggest a new mechanism of the SC duringirreversible growth underlain by the repulsive long-range forces induced by the misfitstrain. In contrast to the thermodynamically controlled growth when the conventionaldipole-dipole (d-d) interatomic interactions [3, 4] are sufficient to ensure the SC [10], thed-d forces are too weak to assure SC in the kinetically controlled case, as the explicitsimulations in [6, 7, 8] and our arguments in section 3 below show. However, it is knownthat besides the dipole forces at the strained surfaces there also exist monopole forcesthat are of longer range than the d-d interactions and in the case of the step-boundedsurface structures, such as steps, islands, and pits play a dominant role in their energeticsand kinetics [5, 13, 14, 15, 16, 17, 18].In the present paper we will show that the monopole forces interacting with theforce dipoles induced by the mobile adatoms play similarly important role in the growthkinetics. In particular, they provide a mechanism of the SC during irreversible growth.As will be shown in section 3, in the presence of d-m interactions the strength of therepulsion between the island and the mobile atom may grow with the island size in sucha way that the atoms will avoid larger islands by preferentially attaching to smallerones, thus making the island sizes more homogeneous. The explicit confirmation of thismechanism in explicit KMC simulations will be provided in section 4. But becausethe main difficulty in studying the strained epitaxy with the KMC technique is thenecessity of accounting for the long-range interactions acting between all atoms in the ize calibration due to dipole-monopole interaction
2. The model of strained islands
The main problem in KMC simulation of many-body systems with long-rangeinteractions is that to simulate the change of the position of a single atom, the energyof its interaction with all other atoms in the system need be calculated first. And inthe case of the Metropolis algorithm [21] which is quite appropriate in this case (atsufficiently high temperatures, at least), the calculation need to be performed twice: forthe initial and the final positions of the moving atom. And in the end the move can bediscarded by the algorithm.Because one is usually interested in the thermodynamic limit, the simulated systemshould be reasonably large to mitigate the finite-size effects. For example, in oursimulations we used, following [10], the system consisting of 250 ×
250 sites on a squaresubstrate lattice. With maximum coverage θ = 0 . ize calibration due to dipole-monopole interaction k .The displacements u j of atoms j = 1 , , . . . , l within a chain consisting of l atomscan be calculated as u j = f sinh[ φ (2 j − l − / [ √ α cosh( φl )] , (1)where f is the size misfit between the substrate and the overlayer, α = k/k NN the ratioof the rigidities of the elastic springs binding the atom to the substrate k and to thenearest neighbor atom k NN and φ = ln( p α/ √ α/ . (2)In (1) and everywhere below all lengths are measured in the substrate lattice units (l.u.).So that the numerical value of the misfit parameter f coincides with the relative misfit.In Ge/Si(001), for example, f = 0 .
042 because the relative misfit in the system is 4.2%.We used this value of f together with small value of α in figure 1 to schematicallyillustrate the distribution of the atomic displacements inside atomic chains of differentlength. u i ( l. u . ) i Figure 1.
Symbols: Atomic displacements calculated according to (1) with α = 0 . f = 0 .
042 for the chains consisting of (from left to right) 10, 20, 50 and 80 atoms.Only one half of the displacements are shown; the other half has the same values butthe opposite sign. The solid line shows the displacements of the end 40 atoms of aninfinitely long chain.
The small value of α in the figure was chosen to be close to the rigid-core casecorresponding to k NN → ∞ and α →
0. As can be seen from (1) and (2), in this case u j = f (2 j − l − / , (3)i. e., the atomic displacements depend linearly on the distance from the middle of thechain [11]. As we will show in the next section, the SC may take place in islands ize calibration due to dipole-monopole interaction /k NN ) is finite, though it can be small. From figure 1 it isseen that the displacements follow the linear behavior (3) only in sufficiently smallislands. In large islands the displacements and the monopole forces saturate andphenomenologically can be described by constant monopole force density at the islandedge, as in [5, 13, 14, 15, 16, 17, 18]. In this case the small and large islands repulsemobile monomers with similar force so our SC mechanism became inefficient.The discrete strain can be calculated as (assuming the chain is oriented in the x direction) ε xx = ∆ u/ ∆ x = u i +1 − u i (4)because the x -coordinate difference is equal to 1 l.u.. As can be seen from figure 1,the average strain in small islands has appreciable value which diminishes with growingisland size. In large islands strain remains only in the ends of the chain, so the averagestrain will tend to zero as 1 /l . It is exactly the behavior found in square Co islands on theCu(001) surface in ab initio molecular dynamics simulations in [32]. The authors assessthat the saturation starts in islands of ∼
3. Long range elastic interactions on the surface
In this section we derive expressions for the substrate-propagated elastic interactionsto be used in KMC simulations in section 4. Due to the long-range nature of theinteractions [34], the hopping adatom interacts with all other atoms on the surface. Inorder to calculate the interaction with the large number of atoms at each atomic hop,computationally efficient expressions for the interaction energy are needed.As was explained previously, such expressions can be derived for epitaxial islandsof simple geometry. Thus, following [23, 24, 6, 25, 26, 27] we assume that at lowtemperature the islands on the square substrate lattice acquire rectangular shapes andwill derive the expressions for the interaction of the mobile monomers with such islands.
First we show that the conventional dipole forces are not sufficiently strong for the aboveSC mechanism to be operative. The potential acting between two adatoms at distance r apart in this case is [3, 4] V d − d = γr , (5)where γ > r , so r is dimensionless and the closestatoms are 1 l.u. apart. ize calibration due to dipole-monopole interaction a and b along the x and y directions, respectively, and summing contributions of the form of (5) over all atomswithin the island in the continuum approximation one gets: V d − d ab ( x, y ) = γ hh ¯ r ¯ x ¯ y i ¯ x + ¯ x − i ¯ y + ¯ y − . (6)Here two-dimensional radius vector r = ( x, y ) points from the island center which weplaced for simplicity at the coordinate origin (0 ,
0) to the position of the mobile atom;¯ x ± = x ± a/
2, ¯ y ± = x ± b/
2, and ¯ r = (¯ x + ¯ y ) / . The square brackets in (6) denote thesubstitution of four possible combinations of ¯ x ± , ¯ y ± for ¯ x and ¯ y with necessary signs.Rectangular islands were grown in the KMC simulations in [6] but no narrowing ofthe ISDs was found. Equation (6) allows us to understand this. According to [35] theisland capture numbers in the rate equations that define the rate at which the islandscapture the mobile adatoms is proportional to the density of the adatoms at the sitessituated one hopping step away from the island. They may be called the island nearestneighbors (NN). At high values of the diffusion to deposition rates ratio that we aregoing to simulate in section 4 the growth is in the thermodynamically controlled regimethe distribution of mobile atoms is very close to equilibrium [10, 11] so their density canbe assessed as N NN ∝ exp[ − E ( i NN ) /k B T ] , (7)where E ( i NN ) is the energy of the adatom at some point i NN NN to the island. Let usfor definiteness consider the point i NN = ( a/ , a of an island centered at the coordinate origin (0 , a, b E ( i NN ) = V d − d ab ( a/ ,
0) saturates to a constant value.Thus, the mobile atoms can reach both large and small islands with equal ease, so thed-d repulsion does not cause SC during irreversible growth.
The dipole-dipole interactions correspond to the situation when each adatom ispositioned in the geometric center of the substrate lattice unit cell so all atomicdisplacements in the substrate are also symmetric and the resulting force distributioncorresponds to the force dipole. Such a behavior would describe rather non-strainedsituation because the presence of a positive misfit f means that the adatoms are toobig to fit into the substrate cells without pushing their neighbors. This situation can bedescribed within the model of the rigid-core adatoms [11] which should be adequate forsituations where f > f , then the atomwith the coordinates r = ( x, y ) inside the island centered at the origin will be displacedby its neighbors from the center of the lattice cell it belongs to in the direction r as∆ r = f r . (8) ize calibration due to dipole-monopole interaction k . So the displaced adatom at position r within theisland will exert on the substrate a shear force in the x − y plane with the density F m ( r ′ ) = k ∆ r δ ( r − r ′ ) = kf r δ ( r − r ′ ) , (9)where for simplicity we again resorted to the continuum approximation. We suppliedthe force density with the superscript “ m ” to stress its monopole character. Of course,there always exists an atom at r = − r with the opposite force density so that at largedistances from the island the potential will have the dipole-dipole asymptotic of thetype of (5). In large islands, however, the distance 2 r between the atoms can be largeso an adatom approaching the island boundary will experience effectively the monopoleforces.The energy of interaction of an adatom with a force monopole is the work performedby the dipole force distribution F d along the field of displacements u : V d − m ( r ) = − Z F d ( r − r ′ ) · u ( r ′ ) dx ′ dy ′ . (10)The dipole force distribution produced by an adatom at r is [4] F d ( r − r ′ ) = A ∇ δ ( r − r ′ ) , (11)where A is a constant. The displacement field due to the monopole force F appliedat the coordinate origin for the isotropic case was calculated in [34]. So substitutingexpressions (8.19) for u ( r ) from that reference together with (11) into (10) one findsafter some algebra V d − m ( r ) = A (1 − ν )2 πµ F · r r , (12)where ν is the Poisson ratio and µ the shear modulus. As is seen, the interaction in(12) is longer-ranged than in (5). We remind that everywhere in the present study we,following [3, 4], consider only 2D in-plane forces and other vectors, so the componentof the monopole force in the direction perpendicular to the surface was set to zero inexpressions (8.19) from [34]. In more sophisticated models of strain in epitaxial islands(see, e. g., [36]) this component is non-vanishing and should be included in (12) alongthe line of derivation presented above.Now substituting (9) into (12) and integrating over the rectangular island a × b centered at the origin (0 ,
0) one gets the potential of interaction with the adatom placedat the point ( x, y ) external to the island as V d − m ab ( x, y ) = C hh ( x + ¯ x ) ln(¯ y + ¯ r ) + ( y + ¯ y ) ln(¯ x + ¯ r ) i ¯ x + ¯ x − i ¯ y + ¯ y − , (13)where C = Akf (1 − ν ) / πµ and other notation is the same as in (6).To farther simplify the simulations, below we will assume that the islands are ofsquare shape [23, 24, 25] because in the case of weak elastic forces and small islands weare going to study the aspect ratios of the rectangular islands are known to be close tounity [6, 18]. ize calibration due to dipole-monopole interaction E ( i NN ) = V d − m aa ( a/ , | a →∞ ∼ Ca ln a (14)which means that in contrast to the potential (6) derived from the dipole-dipoleinteraction, the potential based on the dipole-monopole forces does differentiate betweenmobile adatom capture by large and small islands, as can be seen from (7),—thusproviding a mechanism for kinetically controlled SC.
4. KMC simulations of the growth of the square islands
To assess the efficiency of the proposed mechanism, kinetic Monte Carlo (KMC)simulations were carried out with the use of a variant of the square-island modeldeveloped in [23, 24, 25]. As explained in the previous section, to simplify the calculationof the elastic forces we slightly modified the model by applying the approach of [26, 27]to the square islands instead of the circular ones. Namely, we assumed that the sidelength of the island a = √ s , where s is the island size. Then the square capture zone [9]with the side length a + 2 is formed by surrounding the island with a strip of width 1.Any atom that enters the capture zone either by direct impingement or via the hoppingdiffusion is irreversibly caught by the island whose size becomes s + 1. To validate our KMC setup we first carried out the simulations without the long-rangeinteractions. Two points could arose concern in connection with our approach. First,because the capture of the monomers by the islands is different from the conventionalsquare island model [23, 24, 25], the question arises on whether the physics of the growthremains qualitatively the same. Second, because of the difficulties with accounting forthe interaction of the diffusing monomer with all atoms in the system, the size of thesimulated lattice was chosen to be 250 ×
250 sites which is smaller than typically usedfor the simulations of the growth without long-range interactions. Because our maininterest in the present study are the ISDs, we compared the total number of atomsin two-dimensional square islands obtained in our approach with corresponding resultsfrom [6, 25]. The diffusion constant was calculated according to the standard expression D = ν att exp( − E d /k B T )with typical values for the attempt frequency ν att = 1 THz, the diffusion barrier E d = 0 . ize calibration due to dipole-monopole interaction N s s v / θ s/s av θ = 0 . Figure 2.
Simulated ISD for irreversible growth at 400 ◦ C in the absence of misfitstrain in our model (histogram); for comparison are shown simulation data from [6](filled circles) and [25] (crosses) N d s = v / θ d/ √ s av θ = 0 . Figure 3.
Shaded histogram—distribution of the diameters (defined as d = √ s )of the islands grown in the KMC simulations at 400 ◦ C with the elastic interactioncorresponding to γ = 20 meV. The dashed histogram is the diameter distribution for γ = 0. compact islands. The data shown in figure 2 are plotted in the scaling variables [9], asis conventional in the precoalescence regime at coverage θ not exceeding ∼ . N s = θf ( s/s av ) /s , (15)where N s is the density of islands of size s , s av is the average island size, and f auniversal scaling function. As is seen, the agreement is very good; most importantly, noISD narrowing is seen in our data which means that the SC obtained in the simulationsshown on figure 3 is due to the strain and not because of the approximations made. To simulate the growth with realistic strain in our model we need to chose the valueof the constant C in (13). For consistency we chose it in such a way that islands with a = 1 corresponding to isolated adatoms asymptotically reproduced the dipole-dipolepotential (5). After some algebra the asymptotic of (13) was found to be C/ (12 r ), ize calibration due to dipole-monopole interaction (a) (b) (c) . θ d a v σσ / d a v Figure 4.
Log–log plots of the coverage dependences of the mean islands diameters(a), of the diameter distributions dispersions (b), and of the dispersion to the diameterratios (c) for five values of the interaction parameter γ : 0 (+), 2.5 meV ( • ), 5 meV( ◦ ), 10 meV ( × ), and 20 meV (∆). Solid lines are linear fit to the data; dashed linesare guides to the eye. Note that only γ = 0 data exhibit the scaling behavior. so C = 12 γ would assure that at distances larger than 1 l.u. the interaction of anisland consisting and of an adatom will have the same strength as the adatom-adatominteraction (5).The range of numerical values of γ used in the simulations was chosen according toestimates made in [7, 6, 3]. Because the estimated values vary in a rather broad rangeand ab initio estimates of non-local interactions are known to be unreliable [37], thesimulations were carried out for five values of γ = 0, 2.5, 5, 10 and 20 meV.The main results of our KMC simulations are shown in figures 4 and 5 Twoconclusions can be drawn from the data. The value of index ω introduced in [38] fromthe power-law dependence s av ∝ θ − ω (16)can be found from our data on d av ≃ √ s av ∝ θ − ω/ . This index is convenient forexperimental measurement because it is directly connected to the index characterizingthe total island density N = θ/s av ∝ θ ω (17)Our first conclusion is that the index strongly depends on the strength of the elastic ize calibration due to dipole-monopole interaction (a)
200 300 400Temperature ( ◦ C) (b) . . d a v σ / d a v Figure 5.
Temperature dependences of the mean island diameter d av (a) and ofthe ratio of the diameter distribution dispersion σ to d av (b) at coverage θ = 0 . γ (meV) ω Table 1.
Dependence of the scaling exponent ω as defined in (16-17) on the strengthof the long-range interaction parameter γ (5) forces, especially at small γ , as can be seen from table 1. Experimentally the smallervalues of ω may look as the growth saturation, as seems to be the case in [39]. Thesecond conclusion is that in the presence of strain the dispersion σ of the island diameterdistribution (IDD) exhibits a saturated behavior, as can be seen from figure 4(b). Incombination with monotonous growth of d av the dispersion to mean diameter ratiodiminishes with coverage in qualitative agreement with experimental data [39]. The dispersion to mean diameter ratios in figures 4(c) and 5(b) are not as small as insome experimental data that exhibit the best cases of SC. One of reasons is that inour calculations we used the standard formulas of statistics and took into account allavailable IDD data from the smallest to the largest island sizes. Because they contain atail in the IDD curves at small diameters (see figure 3), the values of d av are smaller thandiameter values at the IDD maxima. This augments both σ and σ/d av values. Hereit is pertinent to note that similarly asymmetric IDDs are rather commonly observedexperimentally (see, e. g., [39, 20, 19, 40]). But because experimental data as a rulecontain more different structures than exists in our simple model, size distributions for ize calibration due to dipole-monopole interaction d av andthe IDDs widths are calculated only in its vicinity [39, 20, 19]. In [39], for example,the relative FWHM with respect to d av taken to be equal to d at the IDD maximumwas found to be quite small (15%). But as can be seen from our figure 3, in our case ithas similar value ∼ ≈ . σ so the effective value of ( σ/d av ) eff withsuch data processing can be as small as 0.06. Thus, the SC mechanism proposed mayunderly even the best cases of SC seen experimentally.
5. Conclusion
To conclude, in this paper we suggested a kinetic mechanism of the SC in strainedepitaxy. It differs from the kinetic mechanisms in the presence of the Ostwaldripening [39, 40] in that there is no need for atomic detachments. This means thatthe mechanism can be operative at smaller temperatures which may have importantpractical advantages. But because of similar growth behavior, the mechanism can alsocontribute to the phenomena attributed to the ripening. Furthermore, it may underlythe unusual narrowing of IDD with temperature observed in the QDs growth in [20, 19](cf. our figure 5). Such behavior is qualitatively different from that observed both inthermodynamically controlled growth and in the kinetically controlled growth in theabsence of strain.The proposed mechanism of SC heavily relies on the island size dependent monopoleforces due to the misfit shear strain in the substrate, so the strength and spatial extentof the forces are of crucial importance. The simple rigid-core model [11] we used toillustrate our mechanism is presumably good for small islands simulated in the presentpaper and for those grown experimentally in [12]. For very large islands, however,saturation toward the constant monopole density similar to that on the surface stepsshould be expected. Calculations in [41] revealed the shear strain that linearly variesacross the QD/substrate interface in capped InAs/GaAs pyramids of at least 12 nmin diameter. Experimentally large interface shear strain of considerable spatial extentcaused by Ge/Si(001) QDs of an order of magnitude larger diameter was observed in[42]. This may mean that the mechanism described in the present paper contributesto the SC of QDs of all sizes. But even restricted to islands a few nanometer indiameter, the mechanism would still be of considerable practical interest. The maximalcatalytic efficiency is achieved in size calibrated metallic islands of small diameters [33].In semiconductor heteroepitaxy the quantum size effect that allows for variation ofthe QD photoluminescence wavelength is operative only in small QDs. Finally, themodel of monolayer-high islands that we studied in the present paper can be taken asa starting point for modeling the growth of the submonolayer QDs [28]. Their small ize calibration due to dipole-monopole interaction
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