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Density-functional calculation of phonons in adsorbate-covered semiconductor surfaces
Surface Science 427–428 (1999) 58–63
www.elsevier.nl/locate/susc
Density-functional calculation of phonons in adsorbatecovered semiconductor surfaces J. Fritsch *, M. Arnold, C. Eckl, R. Honke, P. Pavone, U. Schro¨der Institut fu¨r Theoretische Physik, Universita¨t Regensburg, D-93040 Regensburg, Germany
Abstract We present the results from density-functional calculations of the phonon dispersion of adsorbate-covered semiconductor surfaces. The plane-wave method is used together with the slab-supercell description for the surfaces. We focus on vibrational states that characterize the chemisorption site of the adsorbed atoms and phonon modes of the interface. By comparing the vibrational states in the presence of the adsorbates with those of the clean surfaces, adsorption-induced changes of the surface geometry and force constants can be identified. We study the chemisorption of hydrogen, antimony, and group-III elements on the (110) surfaces of III–V compounds, as well as on the (001) and (111) surfaces of silicon. © 1999 Elsevier Science B.V. All rights reserved. Keywords: Adatoms; Chemisorption; Density-functional theory; Gallium phosphide; Low-index single crystal surfaces; Silicon; Surface phonons; Surface relaxation and reconstruction
1. Introduction The investigation of surface phonon modes in chemisorption systems is interesting for many reasons. One important aspect is that chemical bonds can be identified by characteristic phonon frequencies appearing in the vibrational spectra. Another interesting aspect is that the basic principles responsible for the formation of stable surface structures, i.e. autocompensation and chemical bonding, can lead to completely different atomic arrangements, when comparing a clean surface with an adsorption configuration. With the help of density-functional theory (DFT ) [1,2] and * Corresponding author. Fax: +49 941-943-4382. E-mail address: [email protected] (J. Fritsch)
its extension to density-functional perturbation theory (DFPT ) [3,4], entire phonon dispersion curves of free and adsorbate-covered surfaces of elemental and compound semiconductors can be obtained on the basis of calculations that are free from any adjustable parameter. By comparing the computed phonon frequencies with experimental results from high-resolution electron energy loss spectroscopy (HREELS) [5], helium-atom scattering [6 ], and other techniques like Raman spectroscopy, detailed insights into the bonding structure of an adsorption system can be obtained. In this paper, we summarize our results for the vibrational states computed for the (110) surfaces of GaP and InAs covered by hydrogen and antimony. We discuss also the adsorption of hydrogen on Si(001) and the (앀3×앀3) reconstruction induced by the chemisorption of gallium on the Si(111) surface.
0039-6028/99/$ – see front matter © 1999 Elsevier Science B.V. All rights reserved. PII: S0 0 39 - 6 0 28 ( 99 ) 0 02 3 3 -2
J. Fritsch et al. / Surface Science 427–428 (1999) 58–63
2. Calculation To describe the adsorbate-covered surfaces, configurations of periodically repeated thin crystal films are used. The computations for hydrogen and antimony chemisorbed on the III–V(110) surfaces are carried out using slabs with seven substrate layers. The films are covered on each side by 1 monolayer (ML) of hydrogen and antimony, respectively. In the case of hydrogen adsorbed on Si(001(2×1), ten substrate layers are used. The two (001) surfaces of each crystal film are terminated by the (2×1) monohydride structure. For the adsorption of gallium on Si(111), we use crystal films with eight substrate layers that are terminated on one surface by hydrogen and by gallium on the opposite surface. In all cases, the vacuum separation between neighboring crystal films is large enough in order to guarantee decoupling. The electronic structure calculations are carried out in the framework of DFT [1] using the local-density approximation [2]. We employ the plane-wave formalism and use norm-conserving pseudopotentials for the electron–ion interaction. The exchange-correlation potential is parametrized in the form proposed by Perdew and Zunger [7]. Brillouin zone integrations are approximated by the summation over six special points.For the adsorption of Ga on Si(111), we use four special points. Prior to the computation of the phonon frequencies, the atomic equilibrium positions are determined by means of the Hellmann–Feynman forces. Complete phonon dispersion curves are obtained by computing all interatomic force constants in the crystal films with the help of DFPT [3,4]. A description of all details of our calculation is given, for example, in Refs. [8,9].
3. Results We first discuss the adsorption of hydrogen on the (110) surfaces of GaP and InAs. The chemisorption removes the well-known bond-angle relaxation of the uncovered substrate as a result of the saturation of the dangling bonds. A detailed summary of our results obtained for 1 ML of
59
hydrogen is given in Ref. [10]. Prominent vibrational features introduced by hydrogenation are bond-stretching and bond-bending oscillations, which characterize the adsorption sites of hydrogen and the bond strengths. The upper part of Fig. 1 shows the dispersion of the corresponding phonon modes determined for H:GaP and H:InAs(110). In addition, Fig. 1 illustrates also the dispersion of all other vibrational states obtained by DFPT for the adsorption system. Most of the phonon branches resemble those of the clean surfaces, being characterized by similar dispersion and eigenvectors. An exception, however, is that the surface phonon branches appearing in the gap between the acoustic and optical bulk phonon modes are drastically changed due to the chemisorption. The variations are mainly related to the hydrogen-induced removal of the surface relaxation. This is indicated by the filled circles in the figure, which show the corresponding frequencies obtained for the ideal uncovered surfaces. A vibrational feature of particular interest of the clean and adsorbate covered surfaces is the rocking mode, which is the dynamical version of the bondangle relaxation of the clean substrate. The rocking mode appears at about 12.0 meV in GaP(110) and 8.0 meV in InAs(110). Its frequency is not significantly influenced by hydrogenation. As for the clean surfaces, the rocking mode mixes strongly with bulk states and cannot be resolved for all wavevectors in the surface Brillouin zone (SBZ ). The adsorption of heavy adatoms, however, changes this situation. The chemisorption of antimony on a III–V(110) surface leads to the so-called epitaxial continued layer structure ( ECLS), in which the adatoms form zigzag chains above the valleys between the anion–cation chains of the substrate. Since antimony has five valence electrons, the dangling bonds of all adatoms are doubly occupied. The adsorption layer, therefore, shows a rather small relaxation which is mainly related only to the chemical inequivalence of the substrate anions and cations [11]. Here, we focus on the results obtained for 1 ML of Sb adsorbed on GaP(110). In the relaxed structure computed for Sb:GaP(110), the Ga-bonded Sb atoms are ˚ above the Sb atoms chemisorbed on phos0.12 A phorous. Because of the large mass of the adatoms,
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Fig. 1. Phonon dispersion of the (110) surfaces of GaP and InAs covered by 1 ML of hydrogen. The dotted lines indicate the gap phonon frequencies calculated for the relaxed clean surfaces, and the filled circles show the corresponding phonon frequencies of the ideal clean surfaces.
many vibrational modes of the pristine surface are shifted to smaller frequencies. In particular, the branch related to the rocking mode moves downwards in the phonon dispersion to a frequency range where the bulk density of states is small. Because of the much weaker coupling to bulk
vibrations, the rocking mode is strongly localized in the surface region of Sb:GaP(110). This is clearly seen in Fig. 2. The flat branch denoted as (r), which has a zone-center energy of 10.7 meV, can be associated with the rocking oscillation throughout the entire SBZ. The frequencies of the
J. Fritsch et al. / Surface Science 427–428 (1999) 58–63
Fig. 2. Phonon dispersion of the (110) surface of GaP covered by 1 ML of antimony.
acoustic modes below the bulk continuum are also lower than those of the pristine surface. Moreover, additional surface acoustic branches appear as a result of the adsorption of Sb, whereas the number and the frequencies of the surface acoustic modes essentially do not change in the case of hydrogen chemisorption. One significant feature of Sb adsorbed on GaP(110) and the (110) surfaces of other III–V compounds is a shear vibration that is restricted to the zigzag chains of the adatoms. The frequency of this mode, denoted as (sh) in the figure, is mainly predetermined by the strength of the Sb–Sb bonds and lies at about 20 meV irrespective of the substrate [11]. The vibrational energies calculated for Sb:GaP(110) are 20.9, 21.2, 21.0 meV at the C9 , X 9 and X∞ point. This is in good agreement with 20.7 meV measured by Raman spectroscopy (see experimental data summarized in Ref. [11]). Another example of an adsorption-induced surface derelaxation is the chemisorption of hydrogen on Si(001) (2×1). In the monohydride structure, each of the two dangling bonds per dimer of the surface is saturated by one hydrogen atom. This removes the characteristic dimer buckling of the uncovered system. In their relaxed positions, the hydrogen atoms are bonded to the substrate atoms
61
˚ , the computed dimer bond at a distance of 1.49 A ˚ length is 2.38 A. Characteristic modes of the monohydride complex are the stretching and bending vibrations of the adsorbed hydrogen atoms as well as hindered translations and rotations of the monohydride unit, which is bonded to the second substrate layer. By means of DFPT, the following vibrational frequencies were obtained at the C9 point of the surface Brillouin zone (SBZ ): the individual stretching oscillations of the two hydrogen atoms are coupled to an antisymmetric vibration at 269.7 meV and a symmetric stretching mode at 270.7 meV. The computed frequencies are in good agreement with 259.1 and 260.6 meV, as measured by infra-red spectroscopy [12]. The splitting of 1.5 meV is slightly underestimated by theory. For the bending oscillations in H:Si(001) (2×1), DFPT yields four states with C9 point energies of about 71.4, 72.6, 73.1, and 74.9 meV. The modes at 73.1 and 74.9 meV involve atomic motions of the adsorbed hydrogen atoms parallel to the dimer rows. The two other eigenstates are polarized in the plane perpendicular to the dimer rows. In contrast with the stretching modes, the lower vibrational energy is associated here with the symmetric bending mode. With the help of electron-energy loss spectroscopy, a bondbending oscillation was resolved at about 78.5 meV [13]. In the Si(001) (2×1) monohydride geometry, tilted dimers are no longer favored over symmetric dimers. As a result of the derelaxation, the frequency of the dimer-rocking mode is increased. In the dispersion along the dimer rows, it appears at about 26–27 meV. The respective frequencies computed for the clean Si(001) surface lie in the range 20–23 meV [8]. A detailed discussion of all vibrational features in H:Si(001) (2×1) is given in Ref. [14]. The deposition of group-III elements on Si(111) leads to the formation of (앀3×앀3)R30° superlattices. The particular order of the reconstruction is related to the fact that the adsorbed group-III atoms can saturate three dangling bonds of the free surface. With the help of refined experiments [15–17] and total energy calculations [17,18], the T position was identified as the adsorption site. 4 In the (앀3×앀3)R30° adsorption geometry, the
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J. Fritsch et al. / Surface Science 427–428 (1999) 58–63
atoms relax under the constraint of preserving the C symmetry. Hence, the atoms in the second and 3v third substrate layer are allowed to relax only perpendicular to the surface, while relaxations in the first substrate layer are allowed along and radially to the symmetry axis. Here, we discuss our results obtained for the chemisorption of gallium [19]. The first substrate layer atoms are ˚ and relax towards the shifted upwards by 0.03 A ˚ adatom by 0.14 A. The atoms in the second substrate layer that are directly below the adatoms ˚ , while all other are shifted downwards by 0.33 A ˚ . Fig. 3 atoms in this layer relax upwards by 0.17 A illustrates the phonon dispersion computed by DFPT for the Ga:Si(111) (앀3×앀3) R30° adsorption configuration [10,19] and compares the results with the data from HREELS [5] and HAS [6 ]. Most of the features recorded by means of inelastically scattered electrons and helium atoms can be attributed to localized surface states identified in the density-functional calculation. For the explanation of the flat branch at about 40 meV and the other vibrational features seen by HREELS, the projected density of states
vector, j, was chosen to lie in the sagittal plane at an angle of 30° with respect to the surface normal. As can be seen from the right-hand side of Fig. 3, all of the branches resolved in the HREELS measurements follow maxima of the projected density of states. A prominent feature in the measured and calculated phonon dispersion of Ga:Si(111) ( 앀3×앀3) R30° is a flat branch above the continuum of the bulk states, with a computed C9 point frequency of about 67.7 meV, which is in good agreement with the experimental value of about 68.0 meV. The corresponding eigenvector is dominated by a bond-stretching oscillation of the atoms, which are in the second and third substrate layers and directly below the adatoms. The gallium atoms and the atoms in the first substrate layer are essentially at rest. A detailed analysis of all vibrational modes obtained for the adsorption of gallium and other group-III elements on Si(111) is given in [19].
∑ |j · Vs (q: )|2 (1) Z(v, q: ) dv=∑ ∑ a s vse(v,v+dv) a was calculated. Here, Vs (q: ) is the displacement of a the ath atom given by the normalized eigenvector of the sth phonon branch at the wavevector q: . The summation over a is restricted to the outermost four atomic layers. To account for atomic displacements normal and parallel to the surface, the
We thank S. Baroni and P. Giannozzi for providing numerical support. Our calculations have been accomplished using the Cray-YMP supercomputers of the HLRZ of the KFA in Ju¨lich under Contract No. K2710000 and the Leibniz Rechenzentrum in Mu¨nchen. This work has been supported by a grant of the Deutsche Forschungsgemeinschaft through the Graduiertenkolleg ‘Komplexita¨t in Festko¨rpern: Phononen, Elektronen und Strukturen’.
Acknowledgements
References
Fig. 3. Phonon dispersion of the Ga:Si(111) (앀3×앀3) R30° surface computed by means of DFPT [10,19] in comparison with the results from high-resolution electron-energy loss spectroscopy (circles) [5] and helium-atom scattering (triangles) [6 ]. The gray scale on the right-hand side of the figure illustrates the projected density of states as defined in Eq. (1).
[1] P. Hohenberg, W. Kohn, Phys. Rev. B 136 (1964) 864. [2] W. Kohn, L.J. Sham, Phys. Rev. A 140 (1965) 1133. [3] S. Baroni, P. Giannozzi, A. Testa, Phys. Rev. Lett. 58 (1987) 1861. [4] P. Giannozzi, S. de Gironcoli, P. Pavone, S. Baroni, Phys. Rev. B 43 (1991) 7231. [5] J. Schmid, H. Ibach, J.E. Mu¨ller, Phys. Rev. B 51 (1995) 5233. [6 ] R.B. Doak, J. Vac. Sci. Technol. B 7 (1989) 1252. [7] J.P. Perdew, A. Zunger, Phys. Rev. B 23 (1981) 5048. [8] J. Fritsch, P. Pavone, Surf. Sci. 344 (1995) 159.
J. Fritsch et al. / Surface Science 427–428 (1999) 58–63 [9] J. Fritsch, U. Schro¨der, Phys. Rep. 309 (1999) 209. [10] C. Eckl, R. Honke, J. Fritsch, P. Pavone, U. Schro¨der, Z. Phys. B 104 (1997) 715. [11] W.G. Schmidt, G.P. Srivastava, Surf. Sci. 331 (1995) 333–540. [12] Y.J. Chabal, Surf. Sci. 168 (1986) 594. [13] F. Stucki, J.A. Schaefer, J.R. Anderson, G.J. Lapeyre, W. Go¨pel, Solid State Commun. 47 (1983) 795. [14] M. Arnold, Ph.D. thesis, University of Regensburg, Germany, 1999.
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[15] R.J. Hamers, Phys. Rev. B 40 (1989) 1657. [16 ] J.C. Woicik, T. Kendelewicz, A. Herrera-Gomez, K.E. Miyano, P.L. Cowan, C.E. Bouldin, P. Pianetta, W.E. Spicer, Phys. Rev. Lett. 71 (1993) 1204. [17] J.E. Northrup, Phys. Rev. Lett. 53 (1984) 683. [18] J. Zegenhagen, J.R. Patel, P. Freeland, D.M. Chen, J.A. Golovchenko, P. Bedrossian, J.E. Northrup, Phys. Rev. B 39 (1989) 1298. [19] R. Honke, Ph.D. thesis, University of Regensburg, Germany, 1997.
www.elsevier.nl/locate/susc
Density-functional calculation of phonons in adsorbatecovered semiconductor surfaces J. Fritsch *, M. Arnold, C. Eckl, R. Honke, P. Pavone, U. Schro¨der Institut fu¨r Theoretische Physik, Universita¨t Regensburg, D-93040 Regensburg, Germany
Abstract We present the results from density-functional calculations of the phonon dispersion of adsorbate-covered semiconductor surfaces. The plane-wave method is used together with the slab-supercell description for the surfaces. We focus on vibrational states that characterize the chemisorption site of the adsorbed atoms and phonon modes of the interface. By comparing the vibrational states in the presence of the adsorbates with those of the clean surfaces, adsorption-induced changes of the surface geometry and force constants can be identified. We study the chemisorption of hydrogen, antimony, and group-III elements on the (110) surfaces of III–V compounds, as well as on the (001) and (111) surfaces of silicon. © 1999 Elsevier Science B.V. All rights reserved. Keywords: Adatoms; Chemisorption; Density-functional theory; Gallium phosphide; Low-index single crystal surfaces; Silicon; Surface phonons; Surface relaxation and reconstruction
1. Introduction The investigation of surface phonon modes in chemisorption systems is interesting for many reasons. One important aspect is that chemical bonds can be identified by characteristic phonon frequencies appearing in the vibrational spectra. Another interesting aspect is that the basic principles responsible for the formation of stable surface structures, i.e. autocompensation and chemical bonding, can lead to completely different atomic arrangements, when comparing a clean surface with an adsorption configuration. With the help of density-functional theory (DFT ) [1,2] and * Corresponding author. Fax: +49 941-943-4382. E-mail address: [email protected] (J. Fritsch)
its extension to density-functional perturbation theory (DFPT ) [3,4], entire phonon dispersion curves of free and adsorbate-covered surfaces of elemental and compound semiconductors can be obtained on the basis of calculations that are free from any adjustable parameter. By comparing the computed phonon frequencies with experimental results from high-resolution electron energy loss spectroscopy (HREELS) [5], helium-atom scattering [6 ], and other techniques like Raman spectroscopy, detailed insights into the bonding structure of an adsorption system can be obtained. In this paper, we summarize our results for the vibrational states computed for the (110) surfaces of GaP and InAs covered by hydrogen and antimony. We discuss also the adsorption of hydrogen on Si(001) and the (앀3×앀3) reconstruction induced by the chemisorption of gallium on the Si(111) surface.
0039-6028/99/$ – see front matter © 1999 Elsevier Science B.V. All rights reserved. PII: S0 0 39 - 6 0 28 ( 99 ) 0 02 3 3 -2
J. Fritsch et al. / Surface Science 427–428 (1999) 58–63
2. Calculation To describe the adsorbate-covered surfaces, configurations of periodically repeated thin crystal films are used. The computations for hydrogen and antimony chemisorbed on the III–V(110) surfaces are carried out using slabs with seven substrate layers. The films are covered on each side by 1 monolayer (ML) of hydrogen and antimony, respectively. In the case of hydrogen adsorbed on Si(001(2×1), ten substrate layers are used. The two (001) surfaces of each crystal film are terminated by the (2×1) monohydride structure. For the adsorption of gallium on Si(111), we use crystal films with eight substrate layers that are terminated on one surface by hydrogen and by gallium on the opposite surface. In all cases, the vacuum separation between neighboring crystal films is large enough in order to guarantee decoupling. The electronic structure calculations are carried out in the framework of DFT [1] using the local-density approximation [2]. We employ the plane-wave formalism and use norm-conserving pseudopotentials for the electron–ion interaction. The exchange-correlation potential is parametrized in the form proposed by Perdew and Zunger [7]. Brillouin zone integrations are approximated by the summation over six special points.For the adsorption of Ga on Si(111), we use four special points. Prior to the computation of the phonon frequencies, the atomic equilibrium positions are determined by means of the Hellmann–Feynman forces. Complete phonon dispersion curves are obtained by computing all interatomic force constants in the crystal films with the help of DFPT [3,4]. A description of all details of our calculation is given, for example, in Refs. [8,9].
3. Results We first discuss the adsorption of hydrogen on the (110) surfaces of GaP and InAs. The chemisorption removes the well-known bond-angle relaxation of the uncovered substrate as a result of the saturation of the dangling bonds. A detailed summary of our results obtained for 1 ML of
59
hydrogen is given in Ref. [10]. Prominent vibrational features introduced by hydrogenation are bond-stretching and bond-bending oscillations, which characterize the adsorption sites of hydrogen and the bond strengths. The upper part of Fig. 1 shows the dispersion of the corresponding phonon modes determined for H:GaP and H:InAs(110). In addition, Fig. 1 illustrates also the dispersion of all other vibrational states obtained by DFPT for the adsorption system. Most of the phonon branches resemble those of the clean surfaces, being characterized by similar dispersion and eigenvectors. An exception, however, is that the surface phonon branches appearing in the gap between the acoustic and optical bulk phonon modes are drastically changed due to the chemisorption. The variations are mainly related to the hydrogen-induced removal of the surface relaxation. This is indicated by the filled circles in the figure, which show the corresponding frequencies obtained for the ideal uncovered surfaces. A vibrational feature of particular interest of the clean and adsorbate covered surfaces is the rocking mode, which is the dynamical version of the bondangle relaxation of the clean substrate. The rocking mode appears at about 12.0 meV in GaP(110) and 8.0 meV in InAs(110). Its frequency is not significantly influenced by hydrogenation. As for the clean surfaces, the rocking mode mixes strongly with bulk states and cannot be resolved for all wavevectors in the surface Brillouin zone (SBZ ). The adsorption of heavy adatoms, however, changes this situation. The chemisorption of antimony on a III–V(110) surface leads to the so-called epitaxial continued layer structure ( ECLS), in which the adatoms form zigzag chains above the valleys between the anion–cation chains of the substrate. Since antimony has five valence electrons, the dangling bonds of all adatoms are doubly occupied. The adsorption layer, therefore, shows a rather small relaxation which is mainly related only to the chemical inequivalence of the substrate anions and cations [11]. Here, we focus on the results obtained for 1 ML of Sb adsorbed on GaP(110). In the relaxed structure computed for Sb:GaP(110), the Ga-bonded Sb atoms are ˚ above the Sb atoms chemisorbed on phos0.12 A phorous. Because of the large mass of the adatoms,
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J. Fritsch et al. / Surface Science 427–428 (1999) 58–63
Fig. 1. Phonon dispersion of the (110) surfaces of GaP and InAs covered by 1 ML of hydrogen. The dotted lines indicate the gap phonon frequencies calculated for the relaxed clean surfaces, and the filled circles show the corresponding phonon frequencies of the ideal clean surfaces.
many vibrational modes of the pristine surface are shifted to smaller frequencies. In particular, the branch related to the rocking mode moves downwards in the phonon dispersion to a frequency range where the bulk density of states is small. Because of the much weaker coupling to bulk
vibrations, the rocking mode is strongly localized in the surface region of Sb:GaP(110). This is clearly seen in Fig. 2. The flat branch denoted as (r), which has a zone-center energy of 10.7 meV, can be associated with the rocking oscillation throughout the entire SBZ. The frequencies of the
J. Fritsch et al. / Surface Science 427–428 (1999) 58–63
Fig. 2. Phonon dispersion of the (110) surface of GaP covered by 1 ML of antimony.
acoustic modes below the bulk continuum are also lower than those of the pristine surface. Moreover, additional surface acoustic branches appear as a result of the adsorption of Sb, whereas the number and the frequencies of the surface acoustic modes essentially do not change in the case of hydrogen chemisorption. One significant feature of Sb adsorbed on GaP(110) and the (110) surfaces of other III–V compounds is a shear vibration that is restricted to the zigzag chains of the adatoms. The frequency of this mode, denoted as (sh) in the figure, is mainly predetermined by the strength of the Sb–Sb bonds and lies at about 20 meV irrespective of the substrate [11]. The vibrational energies calculated for Sb:GaP(110) are 20.9, 21.2, 21.0 meV at the C9 , X 9 and X∞ point. This is in good agreement with 20.7 meV measured by Raman spectroscopy (see experimental data summarized in Ref. [11]). Another example of an adsorption-induced surface derelaxation is the chemisorption of hydrogen on Si(001) (2×1). In the monohydride structure, each of the two dangling bonds per dimer of the surface is saturated by one hydrogen atom. This removes the characteristic dimer buckling of the uncovered system. In their relaxed positions, the hydrogen atoms are bonded to the substrate atoms
61
˚ , the computed dimer bond at a distance of 1.49 A ˚ length is 2.38 A. Characteristic modes of the monohydride complex are the stretching and bending vibrations of the adsorbed hydrogen atoms as well as hindered translations and rotations of the monohydride unit, which is bonded to the second substrate layer. By means of DFPT, the following vibrational frequencies were obtained at the C9 point of the surface Brillouin zone (SBZ ): the individual stretching oscillations of the two hydrogen atoms are coupled to an antisymmetric vibration at 269.7 meV and a symmetric stretching mode at 270.7 meV. The computed frequencies are in good agreement with 259.1 and 260.6 meV, as measured by infra-red spectroscopy [12]. The splitting of 1.5 meV is slightly underestimated by theory. For the bending oscillations in H:Si(001) (2×1), DFPT yields four states with C9 point energies of about 71.4, 72.6, 73.1, and 74.9 meV. The modes at 73.1 and 74.9 meV involve atomic motions of the adsorbed hydrogen atoms parallel to the dimer rows. The two other eigenstates are polarized in the plane perpendicular to the dimer rows. In contrast with the stretching modes, the lower vibrational energy is associated here with the symmetric bending mode. With the help of electron-energy loss spectroscopy, a bondbending oscillation was resolved at about 78.5 meV [13]. In the Si(001) (2×1) monohydride geometry, tilted dimers are no longer favored over symmetric dimers. As a result of the derelaxation, the frequency of the dimer-rocking mode is increased. In the dispersion along the dimer rows, it appears at about 26–27 meV. The respective frequencies computed for the clean Si(001) surface lie in the range 20–23 meV [8]. A detailed discussion of all vibrational features in H:Si(001) (2×1) is given in Ref. [14]. The deposition of group-III elements on Si(111) leads to the formation of (앀3×앀3)R30° superlattices. The particular order of the reconstruction is related to the fact that the adsorbed group-III atoms can saturate three dangling bonds of the free surface. With the help of refined experiments [15–17] and total energy calculations [17,18], the T position was identified as the adsorption site. 4 In the (앀3×앀3)R30° adsorption geometry, the
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J. Fritsch et al. / Surface Science 427–428 (1999) 58–63
atoms relax under the constraint of preserving the C symmetry. Hence, the atoms in the second and 3v third substrate layer are allowed to relax only perpendicular to the surface, while relaxations in the first substrate layer are allowed along and radially to the symmetry axis. Here, we discuss our results obtained for the chemisorption of gallium [19]. The first substrate layer atoms are ˚ and relax towards the shifted upwards by 0.03 A ˚ adatom by 0.14 A. The atoms in the second substrate layer that are directly below the adatoms ˚ , while all other are shifted downwards by 0.33 A ˚ . Fig. 3 atoms in this layer relax upwards by 0.17 A illustrates the phonon dispersion computed by DFPT for the Ga:Si(111) (앀3×앀3) R30° adsorption configuration [10,19] and compares the results with the data from HREELS [5] and HAS [6 ]. Most of the features recorded by means of inelastically scattered electrons and helium atoms can be attributed to localized surface states identified in the density-functional calculation. For the explanation of the flat branch at about 40 meV and the other vibrational features seen by HREELS, the projected density of states
vector, j, was chosen to lie in the sagittal plane at an angle of 30° with respect to the surface normal. As can be seen from the right-hand side of Fig. 3, all of the branches resolved in the HREELS measurements follow maxima of the projected density of states. A prominent feature in the measured and calculated phonon dispersion of Ga:Si(111) ( 앀3×앀3) R30° is a flat branch above the continuum of the bulk states, with a computed C9 point frequency of about 67.7 meV, which is in good agreement with the experimental value of about 68.0 meV. The corresponding eigenvector is dominated by a bond-stretching oscillation of the atoms, which are in the second and third substrate layers and directly below the adatoms. The gallium atoms and the atoms in the first substrate layer are essentially at rest. A detailed analysis of all vibrational modes obtained for the adsorption of gallium and other group-III elements on Si(111) is given in [19].
∑ |j · Vs (q: )|2 (1) Z(v, q: ) dv=∑ ∑ a s vse(v,v+dv) a was calculated. Here, Vs (q: ) is the displacement of a the ath atom given by the normalized eigenvector of the sth phonon branch at the wavevector q: . The summation over a is restricted to the outermost four atomic layers. To account for atomic displacements normal and parallel to the surface, the
We thank S. Baroni and P. Giannozzi for providing numerical support. Our calculations have been accomplished using the Cray-YMP supercomputers of the HLRZ of the KFA in Ju¨lich under Contract No. K2710000 and the Leibniz Rechenzentrum in Mu¨nchen. This work has been supported by a grant of the Deutsche Forschungsgemeinschaft through the Graduiertenkolleg ‘Komplexita¨t in Festko¨rpern: Phononen, Elektronen und Strukturen’.
Acknowledgements
References
Fig. 3. Phonon dispersion of the Ga:Si(111) (앀3×앀3) R30° surface computed by means of DFPT [10,19] in comparison with the results from high-resolution electron-energy loss spectroscopy (circles) [5] and helium-atom scattering (triangles) [6 ]. The gray scale on the right-hand side of the figure illustrates the projected density of states as defined in Eq. (1).
[1] P. Hohenberg, W. Kohn, Phys. Rev. B 136 (1964) 864. [2] W. Kohn, L.J. Sham, Phys. Rev. A 140 (1965) 1133. [3] S. Baroni, P. Giannozzi, A. Testa, Phys. Rev. Lett. 58 (1987) 1861. [4] P. Giannozzi, S. de Gironcoli, P. Pavone, S. Baroni, Phys. Rev. B 43 (1991) 7231. [5] J. Schmid, H. Ibach, J.E. Mu¨ller, Phys. Rev. B 51 (1995) 5233. [6 ] R.B. Doak, J. Vac. Sci. Technol. B 7 (1989) 1252. [7] J.P. Perdew, A. Zunger, Phys. Rev. B 23 (1981) 5048. [8] J. Fritsch, P. Pavone, Surf. Sci. 344 (1995) 159.
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