I. INTRODUCTION
The epitaxial growth of elemental semiconductors (Si and Ge) is of considerable scientific and technological significance, because Si and Ge are base materials used in electronic devices and serve also as ideal model systems for studying semiconductor surfaces and growth. Misfit strain is inherently present in heteroepitaxial growth of SiGe thin films due to the 4.2% lattice mismatch between the two materials. Consequently, the effect of strain on growth of SiGe films has been extensively studied, both experimentally and theoretically. 1,2 So far, a majority of studies have been devoted to the strain effect on surface and growth thermodynamics, in particular for the growth of SiGe film on a Si͑001͒ substrate, 1,2 while less attention has been paid to the strain effect on growth kinetics, with limited theoretical studies on the effect of strain on surface diffusion for Si and Ge systems. [3] [4] [5] Previously, we have carried out first-principles calculations to investigate the effect of strain on surface selfdiffusion on Si͑001͒. We showed that the effect of strain on surface diffusion is inherently correlated with the intrinsic surface stress induced by the adatom along its diffusion pathways. The diffusion barrier depends linearly on the external strain. Furthermore, we proposed a simple generic method to predict quantitatively the change of surface diffusion under a given external strain by first-principles calculations of adatom-induced surface stress on an unstrained surface. Recently, Walle et al. have carried out first-principles calculations to study the effect of strain on diffusion of a Ge adatom on Si͑001͒ and Ge͑001͒ surfaces. 4 The dependence of the diffusion barrier on external strain from their calculations for a Ge adatom on Si͑001͒ and Ge͑001͒ is noticeably less linear than what we found for a Si adatom on Si͑001͒. 3 To further confirm the physical (linear) dependence of the surface diffusion barrier on external strain that we discovered earlier for Si self-diffusion on Si͑001͒ (Ref.
3) and to provide a complete set of first-principles values of adatom binding energies and diffusion barriers, as well as their strain dependence for homo-and heteroepitaxial growth on Si͑001͒ and Ge͑001͒ surfaces, we present here comprehensive firstprinciples calculations of the strain dependence of adatom binding energies and diffusion barriers for Ge on Si͑001͒, Si on Ge͑001͒, and Ge on Ge͑001͒ surfaces. In general, our calculations show that the surface binding energies of a Si adatom is higher than those of a Ge adatom, and both adatoms bind more strongly to Si͑001͒ than to Ge͑001͒. The diffusion barriers of Si and Ge adatoms on Si͑001͒ are significantly higher than those on Ge͑001͒, and the diffusion barriers for a Si adatom are higher than those for a Ge adatom on the same type of surface. These results are in good agreement with the common intuition of decreasing bond strength from Si-Si to Si-Ge and to a Ge-Ge bond, and they agree with available experimental data. [6] [7] [8] [9] The surface diffusion anisotropy in all the systems studied is comparable. Furthermore, the linear dependence of the diffusion barriers on the external strain is reproduced in all the cases. To determine the range of validity of such linearity, calculations are extended to larger strain to see where the linearity breaks down. Surprisingly, the linear dependence is found to sustain up to rather large strain in some cases. Strain is found to have a much smaller influence on the saddle-point energy than on the minimum-point (binding) energy.
II. COMPUTATIONAL DETAILS
All the calculations are carried out using the ultrasoft pseudopotential total-energy method within the local-density approximation (LDA), following the procedure used in our previous studies. 3 To ensure a consistent set of convergence criteria for systems containing both Si and Ge, the KohnSham orbitals are expanded in plane waves with an energy cutoff of 11 Ry. We use a supercell consisting of a ten-atomic-layer slab with a p͑ ͱ 8 ϫ ͱ 8͒R45°unit cell in lateral directions (eight atoms per layer) and a seven-atomic-layer vacuum to model the Si͑001͒ and Ge͑001͒ surfaces. The atoms in the surface layer form a p͑2 ϫ 2͒ reconstruction, as shown in Fig. 1 . A Si ͑Ge͒ adatom is placed on both the top and bottom surfaces of the slab to retain the inversion symmetry of the supercell. Coordinates of all the atoms are optimized during the structural relaxation, except the two innermost layers that are kept fixed. The potential-energy surface (PES) of the adatom on the unstrained and strained surfaces is mapped out by conjugate gradient minimization, up to a precision of 10 −4 eV in total-energy difference and with forces on the remaining atoms converged to 0.01 eV/ Å. The unstrained Si͑001͒ and Ge͑001͒ surfaces correspond to the calculated bulk lattice constant of 5.39 Å and 5.63 Å, respectively. Two special K points are used for the Brillouinzone sampling. Tests have been done to make sure that all the results are converged with respect to energy cutoff, system size, and k-points sampling.
A common practice, when addressing adatom diffusion, is to map out the PES via placing an adatom laterally over a set of equidistant grid points. At each point, the z coordinate of the adatom is optimized along with the full coordinates of all other atoms. To accurately locate all the minima and saddle points in the complex PES and hence accurately determine the diffusion barriers for different diffusion pathways, the exact location and energy of a (local) minimum site are subsequently determined by fully relaxing all the degrees of freedom of all the atoms (including the adatom), starting from the nearby minimum grid point (i.e., the grid point at which the adatom binding energy is the lowest locally). The saddle point (the transition state) between any two minima and hence the diffusion barrier are then identified using the nudged elastic band (NEB) method. 10 
III. RESULTS AND DISCUSSION

A. Adatom binding and diffusion on unstrained surfaces
As expected, the resulting PES's for Si/ Ge͑001͒, Ge/ Si͑001͒, and Ge/ Ge͑001͒ systems exhibit qualitatively similar features to that of a Si adatom on Si͑001͒. There exist four minimum sites (M, H, Q, and P) and five saddle points (F, C, B, A, and D), as indicated in Fig. 1 . The absolute minimum is located at the M site. The binding energies are −5.24 eV for Si/ Si͑001͒, −4.81 eV for Ge/ Si͑001͒, −4.94 eV for Si/ Ge͑001͒, and −4.47 eV for Ge/ Ge͑001͒, respectively. The relative energies of other local minima, H, Q, and P, with respect to M, are listed in Table I . It must be noted that the LDA methods are known to overestimate the binding energies and the results have a systematic error. The general trend, however, is more reliable. We found that the binding energies of a Si adatom is higher than those of a Ge adatom on the same surface, and both adatoms bind more strongly to Si͑001͒ than to Ge͑001͒. The binding energy of a Si adatom on Ge͑001͒ is about the same as that of a Ge adatom on Si͑001͒. These findings follow the common intuition of decreasing bond strength from Si-Si to Si-Ge and to a Ge-Ge bond. In fact, our calculated ratio of adatom binding energies of Si/ Si͑001͒ to Ge/ Ge͑001͒ and of Ge/ Si͑001͒ [or Si/ Ge ͑001͒] to Ge/ Ge͑001͒ is 1.17 and 1.08 (or 1.11), respectively, which are in good quantitative agreement with the ratio of experimental cohesive energies of Si to Ge and of SiGe (average of the two) to Ge, which is 1.20 and 1.10, respectively. 11 The same trend was also obtained by Mae 12 using the embedded-atom potential.
The complex PES's lead to multiple diffusion pathways. For each combination of the system, we particularly find two possible low-barrier paths for diffusions parallel to the dimer rows, P1͑MFHCHFM͒ and P2͑MBQBM͒, and one possible path perpendicular to the dimer rows, P3͑MAPDQBM͒. Although the binding sites, adsorption geometries, and diffusion pathways of Si and Ge adatoms on the reconstructed Si͑001͒ and Ge͑001͒ surfaces resemble each other, the ener- (1) In all three systems, i.e., Ge/ Si͑001͒, Si/Ge͑001͒, and Ge/ Ge͑001͒, the adatom diffusion is anisotropic, with the fast diffusion direction being along the dimer row, the same as in the case of Si adatom diffusion on Si͑001͒.
3,6,13
The differences of barriers between path P1 ͑P2͒ and P3 in each of the systems are cnash@example.org eV or so, and the overall anisotropy is thus comparable with each other (about three orders of magnitude at room temperature). These are in good agreement with previous calculations 3, 13, 14 and experiments. 6-9 (2) The microscopic details of the adatom motion along the dimer row on Ge͑001͒, however, turn out to be somewhat different from that on Si͑001͒. The Si ͑Ge͒ adatom seems to have a good chance to diffuse along the trough edge between the dimer rows ͑P2͒ on Ge͑001͒, in contrast with the behavior on Si͑001͒, where diffusion is predominantly over the top of the dimer row ͑P1͒.
3,13 For Si/ Si͑001͒ and Ge/ Si͑001͒, the barriers of P1 and P2 are nearly degenerate, with P1 slightly lower than P2, whereas for Si/ Ge͑001͒ and Ge/ Ge͑001͒, the barriers of P1 and P2 differ substantially, with P2 much lower than P1. (3) The diffusion barriers on Ge͑001͒ are significantly lower than on Si͑001͒, and Ge adatom diffusion has slightly lower barriers than Si adatom diffusion on the same surfaces, again reflecting the lower cohesive energy of Ge as compared to Si. Our calculated diffusion barriers of 0.65 eV (Ref.
3) for Si on Si͑001͒ and 0.37 eV for Ge on Ge͑001͒ agree well with available experiments, which are, respectively, 0.67 eV for Si on Si͑001͒ (Ref. 6) and 0.47± 0.12 eV for Ge on Ge͑001͒, 9 and with previous calculations. 13 Furthermore, we note that the same qualitative trend has also been observed experimentally for addimer diffusion. 15, 16 (4) For Ge adatom diffusion on Si͑001͒, the diffusion barrier along the top of the dimer rows is estimated to be 0.617 eV, in excellent agreement with the theoretical 14 (Fig. 3 in Ref. 4) . It is worth mentioning that different starting configurations of c͑4 ϫ 2͒ and p͑2 ϫ 2͒ reconstruction have been used in Ref. 4 and the present work, respectively. As has been discussed earlier, 17, 18 the diffusion barriers might be sensitive to the local tilting of dimers. (5) Finally, we observe that when a Si ͑Ge͒ adatom diffuses into the dimer bridge site (C site in Fig. 1) , it spontaneously opens the otherwise closed dimers on Ge͑001͒, which is necessary for a new layer to form. 19 But in contrast, the presence of an adatom directly above the surface dimer on Si͑001͒ does not lead to the dimer opening. 3,13
B. Adatom binding and diffusion on strained surfaces
To investigate how strain affects adatom binding and surface diffusion, the changes of binding energy and diffusion barrier are examined under different strain conditions for Si and Ge surfaces. We have applied compressive and tensile strain up to 2%, uniaxially or biaxially, ⑀ bi , to the adatomadsorbed p͑2 ϫ 2͒ Si͑001͒ and Ge͑001͒ surfaces. The uniaxial strain is applied in the direction either along the surface dimer bond, ⑀ xx , or perpendicular, ⑀ yy . On every strained surface, diffusion barriers are again determined by energy minimization and by the NEB method. 10 Figure 2 shows the calculated binding energies for an adatom at the global minimum site M as a function of external strain. The tensile stress is positive. It is interesting to note that there is a linear dependence between the binding energy and external strain, the same as the behavior of molecular adsorption energy on a strained metal surface. 20 The linear dependence can be understood within the scheme of linear elastic theory. The binding energy with external strain ⑀ ext can be written as E ad = E ad 0 + A⑀ ext , where E ad 0 is the binding energy on an unstrained surface, A is the surface area, and is the surface stress tensor induced by the adatom. 20 Qualitatively, adatom binding energy depends linearly on external strain; quantitatively, it depends on the magnitude of the surface stress that the adatom induces at the adsorption site. It is rather substantial in all the systems we studied, here up to ϳ0.05 eV for a ±2% strain. A compressive (or tensile) strain may either increase or decrease the adatom binding to the substrate, as reflected by the opposite effects of ⑀ xx versus ⑀ yy in Fig. 2 . If an adatom induces a tensile surface stress, the binding energy increases with increasing external strain; the reverse is true if it induces a compressive surface stress. The dependence of the diffusion barriers along the path P1, P2, and P3 on external strain of ⑀ xx , ⑀ yy , and ⑀ bi is plotted, respectively, in Fig. 3 for the Ge/ Si͑001͒ system. Second, the effect of strain on diffusion is quantitatively rather significant in all these systems. A 2% strain (compressive or tensile) can change the diffusion barrier as large as 170 meV [see, e.g., the change of the barrier for path P3 by ⑀ yy in Fig. 4(b) ], which translates to an increase or decrease of diffusion rate by more than a few times at the typical growth temperature of 500°C. Third, the effect of a given external strain is highly pathway-and system-dependent. The strength and sign of the dependence are determined by A⌬, which are listed in Table II . Fourth, the effect of strain is additive. Figures 3(c) , 4(c), and 5(c), show clearly that the diffusion barriers under biaxial strain (solid circles) agree very well with the additions of the effects of uniaxial strains (open circles). The additive property of the strain effects allows us to achieve the full control of surface diffusion by applying and manipulating only the uniaxial strain, which might be easier to engineer experimentally. Finally, the surface-diffusion anisotropy on both Si͑001͒ and Ge͑001͒ can be enhanced because of the strain-induced change of diffusion pathways.
One can expect that the linear approximation is applicable as long as the multiatomic distortions involved in the diffusion processes remain within the linear regime. It is important to determine the range of linearity, because it must fail beyond a critical large strain limit. Higher-order terms in ⑀ must then be included to adequately describe the diffusion barrier. We have done some calculations to extend the strain to a much larger amount to determine where the linearity breaks down. Consider the fact that the linear dependence of the diffusion barrier on strain results from the strain dependence of adatom adsorption energies at both the minimum and the saddle points. We show in Fig. 6 the variation of the adsorption energy of a Ge adatom on Si͑001͒ at the minimum site E M together with that at the saddle point ͑B͒ E S as a function of the external strain applied perpendicular to the dimer row. Interestingly, the two behave quite differently. Particularly in this case, the linearity breaks down theoretically much sooner at the saddle point (at ϳ5% strain) than at the minimum site (sustaining up to as large as ϳ8% strain). This indicates that a significant change in the multiatomic configurations involved in the adatom adsorption at the saddle point occurs at ϳ5% strain. Quantitatively, the strain has a much smaller influence on the saddle-point energy ͑E S ͒ than on the minimum-point energy ͑E M ͒, in agreement with earlier predictions for metal surfaces. 22, 23 In general, one can expect that the range of linearity can be valid up to a few percent of strain. It depends on the system, the diffusion pathway, and the type of strain (uniaxial versus biaxial) applied. We note that the maximum misfit strain in heteroepitaxial growth of Si and Ge systems is 4%, within the theoretical limit of the linear regime, but in the case of Ge island growth on Si, the compressive strain around the Ge islands may exceed 4%.
IV. SUMMARY
In summary we have carried out a comprehensive systematic first-principles study of the strain dependence of adatom binding and diffusion for Ge/ Si͑001͒, Si/Ge͑001͒, and Ge/ Ge͑001͒ systems. In general, our calculations show that the binding energies of a Si adatom are higher than those of a Ge adatom and both adatoms bind more strongly to Si͑001͒ than to Ge͑001͒. The diffusion barriers of Si and Ge adatoms on Si͑001͒ are significantly higher than those on Ge͑001͒, and diffusion barriers for a Si adatom are higher than those for a Ge adatom on the same type of surface. These results are in good agreement with the common intuition of decreasing bond strength from Si-Si to Si-Ge and to a Ge-Ge bond. The surface diffusion anisotropy in all the systems studied is comparable. Furthermore, the linear dependence of the binding energies and diffusion barriers on the external strain is reproduced in all the cases, giving strong evidence for a priori quantitative prediction of the effect of external strain on surface binding and diffusion. The linear dependence is found to sustain theoretically up to rather large strain in some cases. Strain is found to have a much smaller influence on the saddle-point energy than on the minimum-point (binding) energy. These results, together with our earlier study on the Si/ Si͑001͒ system, 3 provide a complete set of first-principles values of adatom binding energies and diffusion barriers as well as their strain dependence for homo-and heteroepitaxial growth on Si͑001͒ and Ge͑001͒ surfaces. 

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