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An ab initio study on the atomic geometry of reconstructed Ge(110 )16×2 surface
Surface Science 544 (2003) 58–66 www.elsevier.com/locate/susc
An ab initio study on the atomic geometry of reconstructed Ge(1 1 0)16 2 surface Toshihiro Ichikawa
*
Department of Physics, School of Science and Technology, Meiji University, Tama-ku, Kawasaki 214-8572, Japan Received 4 April 2003; accepted for publication 6 August 2003
Abstract In order to understand the atomic geometry of 16 · 2 superstructure reconstructed on clean Ge(1 1 0) surfaces, a 16 · 2 structural model was proposed, based on scanning tunneling microscopic (STM) observations, and relaxed by using an ab initio structure optimization program. STM images calculated from the relaxed model could well explain features of experimental STM images, which indicates that five-membered clusters of Ge adatoms are entities of pentagons observed in the STM images. 2003 Elsevier B.V. All rights reserved. Keywords: Computer simulations; Density functional calculations; Surface structure, morphology, roughness, and topography; Germanium; Low index single crystal surfaces
1. Introduction When Ge(1 1 0) surfaces are cleaned and annealed, surface reconstruction takes place and superstructures form on the surfaces. According to a low energy electron diffraction (LEED) study by Olshanetsky et al. [1], which was pioneerÕs work in structural study of Ge(1 1 0) surface, a c(8 · 10) superstructure appears by annealing above 430 C and below 380 C and (17 15 1)-type faceted structures emerge by annealing in the intermediate temperature range. A subsequent reflection high energy electron diffraction (RHEED) study by Noro and Ichikawa [2] claimed that the structure
*
Tel./fax: +81-44-934-7435. E-mail address: [email protected] (T. Ichikawa).
stable in the intermediate temperature range is not the (17 15 1)-type faceted structures but a 16 · 2 superstructure. A scanning tunneling microscopic (STM) study by Ichikawa et al. [3] revealed that the high-temperature phase above 430 C is not the c(8 · 10) structure but a disordered one and that the structure in the intermediate temperature is certainly the 16 · 2 one. A recent RHEED study [4] clarified that very long annealing lasting two days at temperatures below 380 C converts the c(8 · 10) structure to the 16 · 2 one. This means that the occurrence of the c(8 · 10) structure is due to kinetics, i.e. a sluggish formation of the 16 · 2 structure, and the c(8 · 10) structure is transient and metastable. It can be said from these studies that a stable superstructure on reconstructed Ge-(1 1 0) surfaces is only the 16 · 2 structure and it undergoes an order–disorder phase transition at 430 C.
0039-6028/$ - see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.susc.2003.08.011
T. Ichikawa / Surface Science 544 (2003) 58–66
According to the STM study [3], the 16 · 2 structure consists of a periodic up-and-down sequence of (1 1 0) terraces with the height difference of d220 and Ge adatom clusters of pentagonal shape regularly arrange on the terraces, while the c(8 · 10) structure is a structure in which the same Ge adatom clusters are distributed on a flat surface with the periodicity of c(8 · 10). Gai et al. [5] proposed on the base of their high-resolution Ge(1 1 0) STM images a detailed c(8 · 10) structural model in which Ge surface atoms form twins of pentagons together with Ge adatoms. Takeuchi [6] carried out first principles total energy calculations for Si(1 1 0) and Ge(1 1 0) surface reconstructions, using models with small unit meshes of 2 · 2, 2 · 1 and 1 · 1, and obtained an interesting result that addition of Ge adatoms is favorable energetically rather than bond conserving relaxation. These findings suggest that pentagons of Ge adatoms may play a dominant role in Ge(1 1 0) surface reconstruction. In the 16 · 2 structure the up-and-down sequence of terraces are important building element responsible for the formation of the structure as well as the pentagons. In order to comprehend the Ge(1 1 0)16 · 2 surface reconstruction, the periodic terraces has to be considered together with the pentagons of adatoms. Thus we have performed an ab initio calculation of reconstructed Ge(1 1 0) surfaces, using a new structural model with unit mesh of the full 16 · 2 periodicity in which the two kinds of building elements are present.
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move, while the fourth layer and the H atoms were frozen in order to simulate a bulklike environment. Only the C point was sampled. This corresponds to sampling of 32 k-points in the 1 · 1 surface brillouin zone. The wave function was expanded in plane waves with kinetic energy cutoff Ecut ¼ 10 Ry. We have used norm-conserving pseudopotentials of the Hammann type.
3. Results and discussion 3.1. 16 2 structural model A characteristic feature of atomic geometry on ideal Ge(1 1 0) and Si(1 1 0) surfaces is zigzag atomic rows running in h1 1 0i direction separated by deep valleys, as shown in Fig. 1. Theoretical study by Menon et al. [8] showed that Si adatoms at special sites above the valleys can make bonds with four Si atoms in neighboring zigzag chains and stabilize the Si(1 1 0) surfaces. Takeuchi [6] obtained similar result that such stabilization is possible on Ge(1 1 0) surfaces, if the Ge adsorption sites have c(2 · 2) periodicity. Two Ge adatoms siting on the special sites are exhibited with closed circles A and B in Fig. 1 as examples. Five-membered clusters C and D are constructed on the ideal Ge(1 1 0) surface by means of
2. Theoretical method Structure relaxation calculations have been performed within local density approximation by means of an ab-initio molecular dynamics program FHI98md [7] based on the density functional theory. We employed a repeated slab geometry, each slab consisting of four (1 1 0) layers with the 16 · 2 lateral periodicity. Dangling bonds on the lower surface were saturated by adsorption of hydrogen atoms. Ge adatoms were placed on the topmost Ge layer. Two consecutive slabs were separated by an wide. The Ge adatoms and unempty space 10 A derlying three layers of the slab had full freedom to
Fig. 1. Stable adsorption sites of Ge adatoms and formation of 5-mem clusters on ideal Ge(1 1 0) surface. Closed, open and gray circles represent adatoms, topmost-layer atoms and second-layer atoms, respectively. For good recognition, closed circles forming 5-mem clusters are linked with lines but the lines do not mean chemical bonds.
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three adatoms of this sort and two adatoms placed near bridge sites, as shown in Fig. 1. Hereafter we abbreviate five-membered clusters to 5-mem clusters. The 5-mem clusters are quite different from pentagons proposed by Gai et al., because the latter pentagons are composed of an adatom sitting at H3 site and four surface atoms in the zigzag atomic rows of the topmost-layer. Ge adatoms in the 5-mem clusters do not show the c(2 · 2) periodicity. However, we conjectured from careful observations of Ge(1 1 0)16 · 2 STM images that the 5-mem clusters are entities of pentagons seen in STM observations. Fig. 2 is a typical constantcurrent filled-state image taken from a Ge(1 1 0)16 · 2 surface, in which zigzag chains of pentagons running in h1 1 2i direction are impressive. A zigzag chain D seen in Fig. 2 is d220 lower than the neighboring zigzag chains C and E, reflecting the periodic up-and-down terrace structure. Next we put four 5-mem clusters on the upper and lower terraces every a 16 · 2 unit mesh,
Fig. 2. Constant-current filled-state STM image of Ge(1 1 0)16 · 2 surface. A unit mesh of the 16 · 2 structure is represented with broken lines.
referring to Ge(1 1 0)16 · 2 STM images. On the occasion, Ge surface atoms under adatoms near the bridge sites were a0 =4 moved in h0 0 1i or h0 0 1i direction, respectively, to make space to accommodate the Ge adatoms, where a0 is lattice constant of bulk Ge crystal. Fig. 3(a) is a top view of a Ge(1 1 0)16 · 2 structural model thus constructed. We employed this model as an initial atomic geometry in the present ab initio calculation. The calculations were executed until force applying to atoms became less than 0.005 Ry/a.u. This convergence criterion is rather loose, but we believe that this will not lead to serious error, because structure difference between calculations by convergence criteria 0.005 and 0.1 Ry/a.u. was minor. Fig. 3(b) shows the atomic geometry of the model which converged. The 5-mem clusters, though they are much distorted from the initial equilateral pentagons, are distinctly visible in the model, which means that the 5-mem clusters are really stable. In addition, large lateral displacement of Ge atoms is seen to take place in layers just under the 5-mem clusters and the root mean square in the displacement amounts to 0.741 and 0.408 A upper and the lower terrace, respectively. The displacement remains, though it becomes smaller, in the inner layers and the root mean square dis in the third-layer adjacent to placements is 0.045 A the frozen layer under the upper terrace. Fig. 4 shows mean heights and root mean square height deviations of the topmost, second, third and fourth layers and 5-mem clusters. The large height deviations represents that the 5-mem clusters and layers just under the 5-mem clusters are appreciably roughened. Layer spacings between the topmost and the second-layer, between the second and the third-layer and between the third and the fourth layer are 1.999, 2.027 and , respectively, which are nearly equal to the 1.952 A in the bulk. The 5-mem clusters are value 2.000 A above the topmost and located 1.463 and 1.220 A second-layers, respectively, which is rather small in comparison with the layer spacings. This may reflect firm bonding between the 5-mem rings and the underlying layers. Total density distribution of valence electrons around adatoms in the 5-mem clusters can be
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Fig. 3. Top view of 16 · 2 model before (a) and after (b) relaxation. Unit meshes of the 16 · 2 structure are represented with dotted lines. Closed, open, light gray and dark gray circles represent adatoms, topmost-layer atoms, second-layer atoms and third-layer atoms, respectively.
computed from output data of the present relaxation calculations. Obtained total density distribution suggests formation of p-bond between neighboring adatoms A and B marked in Fig. 3(b) but does not show any evidence for forming rbonds between adatoms in the 5-mem clusters. In order to examine appropriateness of the present model, constant-current STM images were simulated. In the present simulation, we chose )1.5 and 1.5 V as tunneling bias voltages of filledstate and empty-state images, respectively. Fig. 5(a) and (b) are filled-state and empty-state STM images, respectively. The 5-mem clusters are visi-
ble in both the two images, which indicates that the present model is a good model of the Ge(1 1 0)16 · 2 structure. In the experimental filled-state STM image of Fig. 2, the pentagons seem to have homogeneous intensity around their centers. Experimental empty-state STM images also show well-resolved pentagons composed of five spots of similar intensity, though the images are not represented in this paper. In the empty-state STM image of Fig. 5(b), the 5-mem clusters on the upper and lower terraces are imaged as pentagons consisting of five spots of similar intensity as well as in the
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Fig. 4. Average height and root mean square height deviation of layers in a slab of 16 · 2 model. Figures in parentheses represent root mean square height deviations of the respective layers.
Fig. 5. Simulated constant-current filled-state (a) and emptystate (b) STM images of 5-mem clusters. Pentagonal images originating from 5-mem clusters are illustrated with dotted lines.
experiment, while spots corresponding to adatoms C marked in Fig. 3 look very dark in comparison
to the other four spots in the filled-state image of Fig. 5(a). We suppose that the insufficient thickness of the slab in the present model may be responsible for the discrepancy between the experimental and the simulated STM image, because pentagons on the upper terraces are composed of bright spots of similar intensity. A superstructure with the same periodicity of 16 · 2 appears on reconstructed Si(1 1 0) surfaces [9–13]. STM images [14] of the Si(1 1 0)16 · 2 structure have features very similar to those of the Ge(1 1 0)16 · 2 surface. An et al. [14] proposed four Si(1 1 0)16 · 2 structural models on the ground of their STM images and concluded that tetramerinterstitial model is most appropriate. We assumed that the tetramer-interstitial 16 · 2 structure is formed also on Ge(1 1 0) surfaces. And for reference we relaxed the model with the same manner as the present model and simulated STM images from the relaxed tetramer-interstitial model. Fig. 6 shows the relaxed model in which pentagons are retained in the same fashion as before relaxation and zigzag chains of the pentagons run in h1 1 2i direction as well. A remarkable change by the relaxation is upward and downward movements, as shown with arrows, of neighboring two pentagons, i.e. ‘‘a pair of pentagons’’ named by An et al. Fig. 7(a) and (b) are simulated filled-state and empty-state STM images from the model, respectively, in which features like pentagons are seen. However they seem too small in comparison with pentagons seen in Fig. 2 and look also like tetramers or pairs. The 5-mem clusters in our model are placed on the zigzag atomic rows, while the pentagons in the tetramer-interstitial model lie on the valleys between the zigzag atomic rows. This is an important key for elucidating the 16 · 2 structure. No pentagons are seen in the region extending from A to B in Fig. 2, being due presumably to its narrowness. Instead parallel broad lines like a ladder are visible. It is noted that the pentagons mount on extensions of the broad lines, as indicated with arrows. In order to demonstrate whether the zigzag atomic rows characteristic of the Ge(1 1 0) surface are imaged as the broad lines and the 5-mem clusters mount really on the rows, a 16 · 2 model where the periodic up-and-down sequence of (1 1 0) terraces are missing and only a
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Fig. 6. Top view of tetramer-interstitial 16 · 2 model. A unit mesh of the 16 · 2 structure is represented with dotted lines. Closed circles represent adatoms and raised topmost-layer atoms. Open, light gray and dark gray circles represent topmost-layer atoms, second-layer atoms and third-layer atoms, respectively. Arrows show movement of pairs of pentagons by relaxation.
Fig. 7. Simulated constant-current filled-state (a) and emptystate (b) STM images of pairs of pentagons.
piece of the zigzag chain of the 5-mem clusters exist in its unit mesh was constructed and relaxed. We name the model a 16 · 2 adsorption model. Fig. 8 is a top view of the model after relaxation. The zigzag chains, though distorted, are preserved stable after relaxation. Fig. 9(a) and (b) are simulated filled-state and empty-state STM images of the zigzag chains of the 5-mem clusters and the surrounding area, re-
spectively. The zigzag chains of the 5-mem clusters are surely imaged in the STM images. In addition zigzag chains of intensity maxima originating from the zigzag atomic rows are visible to run in h1 1 0i direction in the region between the zigzag chains of pentagons. It is noteworthy that the 5-mem-cluster images mount on the zigzag chains of intensity maxima. Real STM images are deteriorated by mechanical vibration, thermal drift, circuit noise of the STM instrument, etc. If the present simulated STM images suffer from the deterioration, the zigzag chains of intensity maxima may be imaged as broad lines and the 5-mem-cluster images will mount on the lines. This is consistent with the feature seen in the region extending from A to B in Fig. 2. In order to investigate relationship of the pairs of pentagons in the tetramer-interstitial model with the zigzag atomic rows, a new tetramerinterstitial model without the periodic up-anddown sequences of terraces was constructed and relaxed as well. Fig. 10 is a top view of the relaxed model. Fig. 11(a) and (b) are filled-state and empty-state STM images of the pairs of pentagons and the surrounding area, respectively. The STM images reveal that pentagons lie really over valley between neighboring atomic rows. This is
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Fig. 8. Top view of 16 · 2 adsorption model in which zigzag chains of 5-mem clusters are distributed with a periodicity of 16 · 2. A unit mesh of the 16 · 2 structure is represented with dotted lines. Closed, open and gray circles represent adatoms, topmost-layer atoms and second-layer atoms, respectively.
Fig. 9. Simulated constant-current filled-state (a) and emptystate (b) STM images of 5-mem clusters and surrounding area.
inconsistent with the STM observation. Namely the tetramer-interstitial model is inappropriate as a good model of the Ge(1 1 0)16 · 2 structure. 3.2. Stability of the 16 2 structure The number of Ge atoms constituting models varies, depending on models chosen in the present
study. Therefore chemical potential of Ge atom is required to discuss the stability among the models. We approximated the chemical potential by the total energy of a Ge atom in Ge bulk crystal of diamond structure, neglecting contribution due to entropy. For convenience of discussion, a 16 · 2 model, in which the periodicity of 16 · 2 is fictitiously enforced but the atomic geometry is the same as that of an 1 · 1 model with adsorption of hydrogen atoms on the lower surface, was built and relaxed, and its total energy was calculated. Energy gain and energy loss per 16 · 2 unit mesh for the formation of seven models are exhibited in Fig. 12, taking the formation energy of the fictitious 16 · 2 model as a standard. The figure shows that the formation of the 5-mem clusters or pairs of pentagons stabilizes the surface, while the introduction of the periodic up-and-down sequence of terraces makes the surface much unstable. The tetramer-interstitial model and the present model, which have the above two conflicting structural elements, are much more stable than the periodic up-and-down terrace model without 5mem clusters but is unstable compared with the fictitious 16 · 2 one. It is to be noted that the most stable model is the present model without the periodic up-and-down sequence of terraces. This is
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Fig. 10. Top view of 16 · 2 model composed of only zigzag chains consisting of pairs of pentagons. A unit mesh of the 16 · 2 structure is represented with dotted lines. Closed, open and gray circles represent adatoms, topmost-layer atoms and second-layer atoms, respectively.
Fig. 11. Simulated constant-current filled-state (a) and emptystate (b) STM images of pairs of pentagons and surrounding area.
inconsistent with the experimental finding. Thus the present calculation failed in a full description of the Ge(1 1 0)16 · 2 surface reconstruction. The present calculation has at least two problems. One is the looseness of converging criterion. Another is the thinness of the slab. The appearance of superstructures on clean semiconductor surfaces is generally understood to depend on competition between the energy gain by decrease
of dangling bonds and the energy loss by increase of strain energy associated with the decrease. In the present study we employed 5-mem-cluster layer or pairs of pentagon layer + four sheets of (1 1 0) layers + hydrogen atom layer as the slab. Suppose that a large strain is present in the upper and lower terraces in the present model. The strain is usually relaxed gradually over a long distance. However the fourth layer has to come back to bulk structure in the present slab structure. Since strain energy is proportional to a square of strain, the strain energy calculated in the present model will be overestimated. Especially in the models with the up-and-down sequence of terraces only a sheet of (1 1 0) layer is present to relax strain below the lower terraces and so the overestimation will be larger. Therefore we conjecture that the failure is due to the thinness of the slab in the present models rather than inappropriateness of the geometry of Ge adatoms on the upper and lower terraces.
4. Conclusion A 16 · 2 structural model whose characteristic building blocks are 5-mem clusters of Ge adatoms
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appropriateness of the present model for Ge(1 1 0)16 · 2 structure. However this view was not proved from the energetics, probably because of the small thickness of the present model. A larger scale of calculation employing thicker slab is requested for better understanding of Ge(1 1 0) surface reconstruction.
Acknowledgements The computation in this study has been done using Fujitsu VPP5000 of the Information Science Center, Meiji University. The author thanks staff of the center for the use of the facilities.
References
Fig. 12. Energy gain and energy loss for formation of seven models as compared with fictitious 16 · 2 model. A, B, D, E and F represent energy gain and energy loss of present model, tetramer-interstitial model, 16 · 2 adsorption model, tetramerinterstitial model without periodic up-and-down terraces and present model without periodic up-and-down terraces, respectively. Two pieces of the zigzag chains of the 5-mem clusters are running in a unit mesh of the model F, which are double compared to the model D. C represents energy gain of 16 · 2 model in which a 5-mem cluster per 16 · 2 unit mesh lines up in h1 1 2i direction without forming zigzag chains on a flat Ge(1 1 0) surface. G represents energy loss of 16 · 2 periodic upand-down terrace model without 5-mem clusters.
and a periodic up-and-down sequence of terraces was constructed in the ground of STM observations of reconstructed Ge(1 1 0)16 · 2 surfaces and then relaxed by means of the ab initio molecular dynamics program FHI98md [7]. STM images calculated from the relaxed model well explain pentagonal feature observed in STM images of reconstructed Ge(1 1 0)16 · 2 surfaces. This shows
[1] B.X. Olshanetsky, A.M. Repinsky, A.A. Shklyaev, Surf. Sci. 64 (1977) 224. [2] H. Noro, T. Ichikawa, Jpn. J. Appl. Phys. 24 (1985) 1288. [3] T. Ichikawa, T. Sueyoshi, T. Sato, M. Iwatsuki, F. Udagawa, I. Sumita, Solid State Commun. 93 (1995) 541. [4] T. Ichikawa, H. Fujii, A. Sugimoto, in: Proc. 25th Int. Conf. Phys. Semicond., Osaka 2000, 2001, pp. 296–297. [5] Z. Gai, R.G. Zhao, W.S. Yang, Phys. Rev. B 57 (1998) R6795. [6] N. Takeuchi, Surf. Sci. 494 (2001) 21. [7] M. Bockstedte, A. Kley, J. Neugebauer, M. Scheffler, Comput. Phys. Commun. 107 (1997) 187. [8] M. Menon, N.N. Lathiotakis, A.N. Andriotis, Phys. Rev. B 56 (1997) 1412. [9] Y. Yamamoto, S. Ino, T. Ichikawa, Jpn. J. Appl. Phys. 25 (1986) L331. [10] T. Ichinokawa, H. Ampo, S. Miura, A. Tamaru, Phys. Rev. B 31 (1985) 5183. [11] E.J. van Loenen, D. Dijkkamp, A.H. Hoeven, J. Microsc. 152 (1988) 487. [12] Y. Yamamoto, S. Kitamura, M. Iwatsuki, Jpn. J. Appl. Phys. 5B (1992) L635. [13] Y. Yamamoto, T. Sueyoshi, T. Sato, M. Iwatsuki, Jpn. J. Appl. Phys. A 4 (1993) L532. [14] T. An, M. Yoshimura, I. Ono, K. Ueda, Phys. Rev. B 61 (2000) 3006.
An ab initio study on the atomic geometry of reconstructed Ge(1 1 0)16 2 surface Toshihiro Ichikawa
*
Department of Physics, School of Science and Technology, Meiji University, Tama-ku, Kawasaki 214-8572, Japan Received 4 April 2003; accepted for publication 6 August 2003
Abstract In order to understand the atomic geometry of 16 · 2 superstructure reconstructed on clean Ge(1 1 0) surfaces, a 16 · 2 structural model was proposed, based on scanning tunneling microscopic (STM) observations, and relaxed by using an ab initio structure optimization program. STM images calculated from the relaxed model could well explain features of experimental STM images, which indicates that five-membered clusters of Ge adatoms are entities of pentagons observed in the STM images. 2003 Elsevier B.V. All rights reserved. Keywords: Computer simulations; Density functional calculations; Surface structure, morphology, roughness, and topography; Germanium; Low index single crystal surfaces
1. Introduction When Ge(1 1 0) surfaces are cleaned and annealed, surface reconstruction takes place and superstructures form on the surfaces. According to a low energy electron diffraction (LEED) study by Olshanetsky et al. [1], which was pioneerÕs work in structural study of Ge(1 1 0) surface, a c(8 · 10) superstructure appears by annealing above 430 C and below 380 C and (17 15 1)-type faceted structures emerge by annealing in the intermediate temperature range. A subsequent reflection high energy electron diffraction (RHEED) study by Noro and Ichikawa [2] claimed that the structure
*
Tel./fax: +81-44-934-7435. E-mail address: [email protected] (T. Ichikawa).
stable in the intermediate temperature range is not the (17 15 1)-type faceted structures but a 16 · 2 superstructure. A scanning tunneling microscopic (STM) study by Ichikawa et al. [3] revealed that the high-temperature phase above 430 C is not the c(8 · 10) structure but a disordered one and that the structure in the intermediate temperature is certainly the 16 · 2 one. A recent RHEED study [4] clarified that very long annealing lasting two days at temperatures below 380 C converts the c(8 · 10) structure to the 16 · 2 one. This means that the occurrence of the c(8 · 10) structure is due to kinetics, i.e. a sluggish formation of the 16 · 2 structure, and the c(8 · 10) structure is transient and metastable. It can be said from these studies that a stable superstructure on reconstructed Ge-(1 1 0) surfaces is only the 16 · 2 structure and it undergoes an order–disorder phase transition at 430 C.
0039-6028/$ - see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.susc.2003.08.011
T. Ichikawa / Surface Science 544 (2003) 58–66
According to the STM study [3], the 16 · 2 structure consists of a periodic up-and-down sequence of (1 1 0) terraces with the height difference of d220 and Ge adatom clusters of pentagonal shape regularly arrange on the terraces, while the c(8 · 10) structure is a structure in which the same Ge adatom clusters are distributed on a flat surface with the periodicity of c(8 · 10). Gai et al. [5] proposed on the base of their high-resolution Ge(1 1 0) STM images a detailed c(8 · 10) structural model in which Ge surface atoms form twins of pentagons together with Ge adatoms. Takeuchi [6] carried out first principles total energy calculations for Si(1 1 0) and Ge(1 1 0) surface reconstructions, using models with small unit meshes of 2 · 2, 2 · 1 and 1 · 1, and obtained an interesting result that addition of Ge adatoms is favorable energetically rather than bond conserving relaxation. These findings suggest that pentagons of Ge adatoms may play a dominant role in Ge(1 1 0) surface reconstruction. In the 16 · 2 structure the up-and-down sequence of terraces are important building element responsible for the formation of the structure as well as the pentagons. In order to comprehend the Ge(1 1 0)16 · 2 surface reconstruction, the periodic terraces has to be considered together with the pentagons of adatoms. Thus we have performed an ab initio calculation of reconstructed Ge(1 1 0) surfaces, using a new structural model with unit mesh of the full 16 · 2 periodicity in which the two kinds of building elements are present.
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move, while the fourth layer and the H atoms were frozen in order to simulate a bulklike environment. Only the C point was sampled. This corresponds to sampling of 32 k-points in the 1 · 1 surface brillouin zone. The wave function was expanded in plane waves with kinetic energy cutoff Ecut ¼ 10 Ry. We have used norm-conserving pseudopotentials of the Hammann type.
3. Results and discussion 3.1. 16 2 structural model A characteristic feature of atomic geometry on ideal Ge(1 1 0) and Si(1 1 0) surfaces is zigzag atomic rows running in h1 1 0i direction separated by deep valleys, as shown in Fig. 1. Theoretical study by Menon et al. [8] showed that Si adatoms at special sites above the valleys can make bonds with four Si atoms in neighboring zigzag chains and stabilize the Si(1 1 0) surfaces. Takeuchi [6] obtained similar result that such stabilization is possible on Ge(1 1 0) surfaces, if the Ge adsorption sites have c(2 · 2) periodicity. Two Ge adatoms siting on the special sites are exhibited with closed circles A and B in Fig. 1 as examples. Five-membered clusters C and D are constructed on the ideal Ge(1 1 0) surface by means of
2. Theoretical method Structure relaxation calculations have been performed within local density approximation by means of an ab-initio molecular dynamics program FHI98md [7] based on the density functional theory. We employed a repeated slab geometry, each slab consisting of four (1 1 0) layers with the 16 · 2 lateral periodicity. Dangling bonds on the lower surface were saturated by adsorption of hydrogen atoms. Ge adatoms were placed on the topmost Ge layer. Two consecutive slabs were separated by an wide. The Ge adatoms and unempty space 10 A derlying three layers of the slab had full freedom to
Fig. 1. Stable adsorption sites of Ge adatoms and formation of 5-mem clusters on ideal Ge(1 1 0) surface. Closed, open and gray circles represent adatoms, topmost-layer atoms and second-layer atoms, respectively. For good recognition, closed circles forming 5-mem clusters are linked with lines but the lines do not mean chemical bonds.
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three adatoms of this sort and two adatoms placed near bridge sites, as shown in Fig. 1. Hereafter we abbreviate five-membered clusters to 5-mem clusters. The 5-mem clusters are quite different from pentagons proposed by Gai et al., because the latter pentagons are composed of an adatom sitting at H3 site and four surface atoms in the zigzag atomic rows of the topmost-layer. Ge adatoms in the 5-mem clusters do not show the c(2 · 2) periodicity. However, we conjectured from careful observations of Ge(1 1 0)16 · 2 STM images that the 5-mem clusters are entities of pentagons seen in STM observations. Fig. 2 is a typical constantcurrent filled-state image taken from a Ge(1 1 0)16 · 2 surface, in which zigzag chains of pentagons running in h1 1 2i direction are impressive. A zigzag chain D seen in Fig. 2 is d220 lower than the neighboring zigzag chains C and E, reflecting the periodic up-and-down terrace structure. Next we put four 5-mem clusters on the upper and lower terraces every a 16 · 2 unit mesh,
Fig. 2. Constant-current filled-state STM image of Ge(1 1 0)16 · 2 surface. A unit mesh of the 16 · 2 structure is represented with broken lines.
referring to Ge(1 1 0)16 · 2 STM images. On the occasion, Ge surface atoms under adatoms near the bridge sites were a0 =4 moved in h0 0 1i or h0 0 1i direction, respectively, to make space to accommodate the Ge adatoms, where a0 is lattice constant of bulk Ge crystal. Fig. 3(a) is a top view of a Ge(1 1 0)16 · 2 structural model thus constructed. We employed this model as an initial atomic geometry in the present ab initio calculation. The calculations were executed until force applying to atoms became less than 0.005 Ry/a.u. This convergence criterion is rather loose, but we believe that this will not lead to serious error, because structure difference between calculations by convergence criteria 0.005 and 0.1 Ry/a.u. was minor. Fig. 3(b) shows the atomic geometry of the model which converged. The 5-mem clusters, though they are much distorted from the initial equilateral pentagons, are distinctly visible in the model, which means that the 5-mem clusters are really stable. In addition, large lateral displacement of Ge atoms is seen to take place in layers just under the 5-mem clusters and the root mean square in the displacement amounts to 0.741 and 0.408 A upper and the lower terrace, respectively. The displacement remains, though it becomes smaller, in the inner layers and the root mean square dis in the third-layer adjacent to placements is 0.045 A the frozen layer under the upper terrace. Fig. 4 shows mean heights and root mean square height deviations of the topmost, second, third and fourth layers and 5-mem clusters. The large height deviations represents that the 5-mem clusters and layers just under the 5-mem clusters are appreciably roughened. Layer spacings between the topmost and the second-layer, between the second and the third-layer and between the third and the fourth layer are 1.999, 2.027 and , respectively, which are nearly equal to the 1.952 A in the bulk. The 5-mem clusters are value 2.000 A above the topmost and located 1.463 and 1.220 A second-layers, respectively, which is rather small in comparison with the layer spacings. This may reflect firm bonding between the 5-mem rings and the underlying layers. Total density distribution of valence electrons around adatoms in the 5-mem clusters can be
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Fig. 3. Top view of 16 · 2 model before (a) and after (b) relaxation. Unit meshes of the 16 · 2 structure are represented with dotted lines. Closed, open, light gray and dark gray circles represent adatoms, topmost-layer atoms, second-layer atoms and third-layer atoms, respectively.
computed from output data of the present relaxation calculations. Obtained total density distribution suggests formation of p-bond between neighboring adatoms A and B marked in Fig. 3(b) but does not show any evidence for forming rbonds between adatoms in the 5-mem clusters. In order to examine appropriateness of the present model, constant-current STM images were simulated. In the present simulation, we chose )1.5 and 1.5 V as tunneling bias voltages of filledstate and empty-state images, respectively. Fig. 5(a) and (b) are filled-state and empty-state STM images, respectively. The 5-mem clusters are visi-
ble in both the two images, which indicates that the present model is a good model of the Ge(1 1 0)16 · 2 structure. In the experimental filled-state STM image of Fig. 2, the pentagons seem to have homogeneous intensity around their centers. Experimental empty-state STM images also show well-resolved pentagons composed of five spots of similar intensity, though the images are not represented in this paper. In the empty-state STM image of Fig. 5(b), the 5-mem clusters on the upper and lower terraces are imaged as pentagons consisting of five spots of similar intensity as well as in the
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Fig. 4. Average height and root mean square height deviation of layers in a slab of 16 · 2 model. Figures in parentheses represent root mean square height deviations of the respective layers.
Fig. 5. Simulated constant-current filled-state (a) and emptystate (b) STM images of 5-mem clusters. Pentagonal images originating from 5-mem clusters are illustrated with dotted lines.
experiment, while spots corresponding to adatoms C marked in Fig. 3 look very dark in comparison
to the other four spots in the filled-state image of Fig. 5(a). We suppose that the insufficient thickness of the slab in the present model may be responsible for the discrepancy between the experimental and the simulated STM image, because pentagons on the upper terraces are composed of bright spots of similar intensity. A superstructure with the same periodicity of 16 · 2 appears on reconstructed Si(1 1 0) surfaces [9–13]. STM images [14] of the Si(1 1 0)16 · 2 structure have features very similar to those of the Ge(1 1 0)16 · 2 surface. An et al. [14] proposed four Si(1 1 0)16 · 2 structural models on the ground of their STM images and concluded that tetramerinterstitial model is most appropriate. We assumed that the tetramer-interstitial 16 · 2 structure is formed also on Ge(1 1 0) surfaces. And for reference we relaxed the model with the same manner as the present model and simulated STM images from the relaxed tetramer-interstitial model. Fig. 6 shows the relaxed model in which pentagons are retained in the same fashion as before relaxation and zigzag chains of the pentagons run in h1 1 2i direction as well. A remarkable change by the relaxation is upward and downward movements, as shown with arrows, of neighboring two pentagons, i.e. ‘‘a pair of pentagons’’ named by An et al. Fig. 7(a) and (b) are simulated filled-state and empty-state STM images from the model, respectively, in which features like pentagons are seen. However they seem too small in comparison with pentagons seen in Fig. 2 and look also like tetramers or pairs. The 5-mem clusters in our model are placed on the zigzag atomic rows, while the pentagons in the tetramer-interstitial model lie on the valleys between the zigzag atomic rows. This is an important key for elucidating the 16 · 2 structure. No pentagons are seen in the region extending from A to B in Fig. 2, being due presumably to its narrowness. Instead parallel broad lines like a ladder are visible. It is noted that the pentagons mount on extensions of the broad lines, as indicated with arrows. In order to demonstrate whether the zigzag atomic rows characteristic of the Ge(1 1 0) surface are imaged as the broad lines and the 5-mem clusters mount really on the rows, a 16 · 2 model where the periodic up-and-down sequence of (1 1 0) terraces are missing and only a
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Fig. 6. Top view of tetramer-interstitial 16 · 2 model. A unit mesh of the 16 · 2 structure is represented with dotted lines. Closed circles represent adatoms and raised topmost-layer atoms. Open, light gray and dark gray circles represent topmost-layer atoms, second-layer atoms and third-layer atoms, respectively. Arrows show movement of pairs of pentagons by relaxation.
Fig. 7. Simulated constant-current filled-state (a) and emptystate (b) STM images of pairs of pentagons.
piece of the zigzag chain of the 5-mem clusters exist in its unit mesh was constructed and relaxed. We name the model a 16 · 2 adsorption model. Fig. 8 is a top view of the model after relaxation. The zigzag chains, though distorted, are preserved stable after relaxation. Fig. 9(a) and (b) are simulated filled-state and empty-state STM images of the zigzag chains of the 5-mem clusters and the surrounding area, re-
spectively. The zigzag chains of the 5-mem clusters are surely imaged in the STM images. In addition zigzag chains of intensity maxima originating from the zigzag atomic rows are visible to run in h1 1 0i direction in the region between the zigzag chains of pentagons. It is noteworthy that the 5-mem-cluster images mount on the zigzag chains of intensity maxima. Real STM images are deteriorated by mechanical vibration, thermal drift, circuit noise of the STM instrument, etc. If the present simulated STM images suffer from the deterioration, the zigzag chains of intensity maxima may be imaged as broad lines and the 5-mem-cluster images will mount on the lines. This is consistent with the feature seen in the region extending from A to B in Fig. 2. In order to investigate relationship of the pairs of pentagons in the tetramer-interstitial model with the zigzag atomic rows, a new tetramerinterstitial model without the periodic up-anddown sequences of terraces was constructed and relaxed as well. Fig. 10 is a top view of the relaxed model. Fig. 11(a) and (b) are filled-state and empty-state STM images of the pairs of pentagons and the surrounding area, respectively. The STM images reveal that pentagons lie really over valley between neighboring atomic rows. This is
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Fig. 8. Top view of 16 · 2 adsorption model in which zigzag chains of 5-mem clusters are distributed with a periodicity of 16 · 2. A unit mesh of the 16 · 2 structure is represented with dotted lines. Closed, open and gray circles represent adatoms, topmost-layer atoms and second-layer atoms, respectively.
Fig. 9. Simulated constant-current filled-state (a) and emptystate (b) STM images of 5-mem clusters and surrounding area.
inconsistent with the STM observation. Namely the tetramer-interstitial model is inappropriate as a good model of the Ge(1 1 0)16 · 2 structure. 3.2. Stability of the 16 2 structure The number of Ge atoms constituting models varies, depending on models chosen in the present
study. Therefore chemical potential of Ge atom is required to discuss the stability among the models. We approximated the chemical potential by the total energy of a Ge atom in Ge bulk crystal of diamond structure, neglecting contribution due to entropy. For convenience of discussion, a 16 · 2 model, in which the periodicity of 16 · 2 is fictitiously enforced but the atomic geometry is the same as that of an 1 · 1 model with adsorption of hydrogen atoms on the lower surface, was built and relaxed, and its total energy was calculated. Energy gain and energy loss per 16 · 2 unit mesh for the formation of seven models are exhibited in Fig. 12, taking the formation energy of the fictitious 16 · 2 model as a standard. The figure shows that the formation of the 5-mem clusters or pairs of pentagons stabilizes the surface, while the introduction of the periodic up-and-down sequence of terraces makes the surface much unstable. The tetramer-interstitial model and the present model, which have the above two conflicting structural elements, are much more stable than the periodic up-and-down terrace model without 5mem clusters but is unstable compared with the fictitious 16 · 2 one. It is to be noted that the most stable model is the present model without the periodic up-and-down sequence of terraces. This is
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Fig. 10. Top view of 16 · 2 model composed of only zigzag chains consisting of pairs of pentagons. A unit mesh of the 16 · 2 structure is represented with dotted lines. Closed, open and gray circles represent adatoms, topmost-layer atoms and second-layer atoms, respectively.
Fig. 11. Simulated constant-current filled-state (a) and emptystate (b) STM images of pairs of pentagons and surrounding area.
inconsistent with the experimental finding. Thus the present calculation failed in a full description of the Ge(1 1 0)16 · 2 surface reconstruction. The present calculation has at least two problems. One is the looseness of converging criterion. Another is the thinness of the slab. The appearance of superstructures on clean semiconductor surfaces is generally understood to depend on competition between the energy gain by decrease
of dangling bonds and the energy loss by increase of strain energy associated with the decrease. In the present study we employed 5-mem-cluster layer or pairs of pentagon layer + four sheets of (1 1 0) layers + hydrogen atom layer as the slab. Suppose that a large strain is present in the upper and lower terraces in the present model. The strain is usually relaxed gradually over a long distance. However the fourth layer has to come back to bulk structure in the present slab structure. Since strain energy is proportional to a square of strain, the strain energy calculated in the present model will be overestimated. Especially in the models with the up-and-down sequence of terraces only a sheet of (1 1 0) layer is present to relax strain below the lower terraces and so the overestimation will be larger. Therefore we conjecture that the failure is due to the thinness of the slab in the present models rather than inappropriateness of the geometry of Ge adatoms on the upper and lower terraces.
4. Conclusion A 16 · 2 structural model whose characteristic building blocks are 5-mem clusters of Ge adatoms
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appropriateness of the present model for Ge(1 1 0)16 · 2 structure. However this view was not proved from the energetics, probably because of the small thickness of the present model. A larger scale of calculation employing thicker slab is requested for better understanding of Ge(1 1 0) surface reconstruction.
Acknowledgements The computation in this study has been done using Fujitsu VPP5000 of the Information Science Center, Meiji University. The author thanks staff of the center for the use of the facilities.
References
Fig. 12. Energy gain and energy loss for formation of seven models as compared with fictitious 16 · 2 model. A, B, D, E and F represent energy gain and energy loss of present model, tetramer-interstitial model, 16 · 2 adsorption model, tetramerinterstitial model without periodic up-and-down terraces and present model without periodic up-and-down terraces, respectively. Two pieces of the zigzag chains of the 5-mem clusters are running in a unit mesh of the model F, which are double compared to the model D. C represents energy gain of 16 · 2 model in which a 5-mem cluster per 16 · 2 unit mesh lines up in h1 1 2i direction without forming zigzag chains on a flat Ge(1 1 0) surface. G represents energy loss of 16 · 2 periodic upand-down terrace model without 5-mem clusters.
and a periodic up-and-down sequence of terraces was constructed in the ground of STM observations of reconstructed Ge(1 1 0)16 · 2 surfaces and then relaxed by means of the ab initio molecular dynamics program FHI98md [7]. STM images calculated from the relaxed model well explain pentagonal feature observed in STM images of reconstructed Ge(1 1 0)16 · 2 surfaces. This shows
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