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Surface phonon dispersion curves in GaAs(110) and Ge(111)2 × 1: a critical comparison
Surface Science 241 (1991) 346-352 North-Holland
346
Surface phonon dispersion curves in GaAs( 110) and Ge( 111) 2 a critical comparison
x
1:
P. Santini ‘, L. Miglio, G. Benedek Dipartimento di Fisica dell’ Universitci, via Celoria 16, I-20133 Milano, Iialy
and P. Ruggerone Max-Planck-Institut ftir StrGmungsforschung, Bunsenstrasse IO, D-3400 Giittingen, Germany
Received 1 June 1990; accepted for publication 6 September 1990
We compare the surface phonon dispersion curves of GaAs(ll0) and Ge(111)2 X 1 as calculated by the Bond Charge Model. The atomic masses, bulk force constants and crystallography of the topmost layer are nearly equal, still the substrate orientation and charge distributions are different. Similarities and discrepancies in the surface phonon dispersion curves are investigated as fingerprints of the structure configuration.
1. Introduction In a recent work [l] we reported the results of a Bond Charge Model (BCM) calculation of the Si(111)2 x 1 surface dynamics, which satisfactorily explain the existing experimental features. In particular, the 10 meV flat branch measured by He inelastic scattering [2] and the 56 meV EELS peak [3] were well described without any parameter fitting. The surface dispersion curves of an extensively reconstructed surface like Si(111)2 X 1 are extremely rich and the origin of the several branches can be attributed either to the purely geometric folding of the Brillouin zone or to the new structural characteristics produced by the reconstruction. A tool to analyze the entangled phonon spectrum of a reconstructed surface was indeed provided by the three following general criteria [l]: (i) the qualitative features of the dis’ Present address: lnstitut de Physique Theorique, Universite de Lausanne, CH-1015 Lausanne, Switzerland. 0039-6028/91/$03.50
persion curves largely reflect those of the ideal unreconstructed structure, after the folding of the larger surface Brillouin zone (SBZ) into the smaller one associated with the reconstructed phase (folding criterion); (ii) in the case of extensive reconstruction it is meaningful to speak of an interface between reconstructed and ideal regions, and of modes which may arise from this buried interface (interface criterion); (iii) the gaps opened along the folding lines arise from the new local arrangement of atoms in the reconstructed region and may be used to characterize the surface geometry (gap criterion). Such a connection between geometry and dynamics led us to compare the dispersion curves of Ge(111)2 x 1 and GaAs(llO), two systems with relevant structural similarities. Unfortunately neither EELS nor He scattering data are available at present for Ge(111)2 x 1, yet its reconstruction pattern is very similar to the one proposed by Pandey [4] for Si(ll1) [5]. This model suggests the formation of r-bonded atomic chains on the surface (see fig. l), with the upper chains
0 1991 - Elsevier Science Publishers B.V. (North-Holland)
P. Santini et al. / Surfnce phonon dispersion curves in GaAs(l IO) and Ge(l II)2 x I
characterized by a tilt, i.e. a difference in the vertical position between atoms of type 1 and type 2, as shown in fig. 1. The lower chains are affected by a tilt as well, but its value is smaller. The side view of the reconstructed surface in fig. 1 also shows the presence of alternating five-fold (4,3,5,8,6) and seven-fold (1,2,4,6,7,5,3) rings formed by the interface and the chain atoms. On the other hand, GaAs(ll0) is by far the most studied heteropolar semiconductor surface, and the qualitative arrangement of the equilibrium positions for the surface atoms is now fairly well established. The surface keeps the same periodicity of an ideal (110) plane, but it displays a relaxation of the topmost atoms, with As shifted above the ideal surface plane and Ga shifted towards of the topmost atoms, with As shifted above the ideal surface plane and Ga shifted towards the bulk. In detail, the relaxation involves a nearly bondlength-conserving rotation of the surface (ITO) chains by a tilt angle of about 30 O, while the subsurface chains are characterized by a much smaller tilt (see fig. 2) [6]. Discussions have been devoted to determine the exact values of the tilt angle and of the atomic displacements parallel to the surface, which accompany the tilt. The large amount of available He-scattering data along the boundaries of the irreducible part of the SBZ [7,9] permits a dynamical test of different crystallographic models for this surface. An extensive analysis of the low-energy part of the GaAs(ll0) spectrum with a discussion of the influence of the surface geometry on the dynamics can be found in ref. [9]. In the present paper we are interested in a somewhat different question. We investigate the dynamic effects of the structural analogies existing between GaAs(llO) and Ge(111)2 x 1: the masses of Ga, As and Ge, as well as the short-range force constants of GaAs and Ge, are nearly the same [lo]. Moreover, the chain structure produced in Ge(ll1) by the 2 X 1 reconstruction is very similar to the relaxed configuration of GaAs(ll0) [6] (see figs. 1 and 2): as we have already pointed out, in both cases we have inner and outer chains running parallel to the surface plane, the latter being particularly affected by an atomic buckling in order to minimize the energy. Yet, some relevant dif-
347
ferences exist, arising both from the matching of the chain geometry to the different (110) and (111) substrates and from the polar character of GaAs. In particular, the matching region displays, in the case of GaAs(llO), 6-fold rings (see fig. 2) instead of the 5- and 7-fold rings of Ge(111)2 X 1. Also, the electronic density maps for Ge(111)2 X 1 are very likely to be the same as for Si(111)2 X 1 [ll], where the charge in the surface bonds is located midway the chain atoms, since negligible charge transfer is seen to accompany the tilt. On the contrary, in the heteropolar case the filled state is located upon the upwards shifted As ions, since charge transfer does lower the surface energy [4]. The subsurface structure is responsible for many of the differences between the two spectra and therefore it is crucial to include in the calculations a suitable number of layers in order to reproduce properly the influence of the matching region between the bulk and surface structures. This task can be hardly achieved with first-principle calculations which are limited to very thin slab configurations. A satisfactory compromise is, in our opinion, to use a realistic dynamical model, which is able to account for the role of electrons in a
Fig. 1. Structureof Ge(111)2 X 1 (from ref. [5]).
348
P. Santini et al. / Surface phonon dispersion curves in GaAs(ll0)
0
and Ge(l I I)2 x I
Ga
@ As l
bc
Fig. 2. Side and top views of GaAs(ll0).
The black dots indicate the positions of the bond charges.
phenomenological way and to include all the important modifications occurring at the surface.
2. Model Our calculations are based on the Bond Charge Model (BCM) [12]. This model provides bulk phonon dispersion curves which are in good agreement with experimental neutron scattering data, both for homopolar and heteropolar semiconductors. The number of adjustable parameters in the bulk is 4 for Si and Ge and 6 for GaAs. The BCM relies on the assumption that the valence charge
can be shared between a “core” charge located at the ions and a “bond charge” (BC), viewed as a massless pseudoparticle with its own degrees of freedom. The ratio of the BC to the ion charge is fixed in our model: in order to fulfill the neutrality condition, the ion charge (including core and valence states contributions) has to be assumed minus twice the bond charge. The equilibrium position of the bond charges lies midway between the bonds in Si and Ge, whereas in GaAs it is displaced towards the anion, dividing the bond in a ratio 3 : 5. Thus the polar character of III-V compounds is recovered.
P. Santiniet al. / Surfacephonon dispersioncurvesin GaAs(lIO)and Ge(l I I)2 x I
In addition to the long range Coulombic interactions among ions and bond charges, short range potentials are introduced in order to include the quantum mechanical repulsion in a phenomenological way: central interactions are set up between neighbouring cores and between cores and neighbouring bond charges. A bond-bending Keating-like potential between nearby bonds is finally considered to ensure the lattice stability against shear displacements. The bond charge coordinates are eliminated via the adiabatic condition only before the diagonalization of the dynamical matrix and therefore the new arrangement of the electronic distribution due to the surface is included in the calculation. Actually, for the surface problem we modify the core positions to reproduce the new geometry induced by the surface perturbation, as determined by total energy calculations and elastic scattering measurements [5,6]. Concerning the electronic degrees of freedom, we set the bond charges at the surface according to the relative maxima of the valence charge density contours. In Ge(111)2 X 1, for instance, the bond charges are always placed at the bond midpoints. Their charge is the same as in the bulk, except for the bond charges between chain atoms, which are 50% larger: in fact the latter have to include a supplementary n-bond electron coming from the saturated dangling bond [l]. In GaAs(ll0) the doubly occupied dangling bonds, belonging to the topmost As atoms, are described by bond charges located at the maximum of the charge distribution [13] (see fig. 2). The remaining bond charges keep the same positions as in the bulk. The calculated surface phonon dispersion appears to be remarkably sensitive to the position of the “dangling” bond charge, and considerable discrepancies with experiment appear if its position is not appropriately chosen. This latter fact provides sufficient confidence in the sensitivity of BCM as a dynamical tool for surface characterization. In both GaAs(l10) and Ge(111)2 X 1, static equilibrium conditions are imposed on all surface ions and bond charges: in this way we can obtain new values for the first derivatives of the central ion-ion and ion-BC potentials. The second de-
349
rivatives of these potentials do not appear under such conditions, and we set them equal to the bulk value. This choice is supported by the fact that interatomic distances at both surfaces remain almost the same as in the bulk. In Ge(l11)2 x 1 the only appreciable variation with respect to the bulk is found for the ion-BC interaction along the chain, due to the a-state incorporated in the bond charge. We find, indeed, a negative value of the first derivative coming from equilibrium conditions, while in the bulk such a parameter is zero. Since the first derivative affects the restoring force for the BC motion normal to the bond, we interpret this variation as being due to the superposition of B and core states induced by such a motion. In fact, the r-orbital possesses a nodal plane crossing the cores and its overlap with the core states along the bond direction is influenced by the shear but not by the BC radial vibration. Actually, no additional contribution to the second derivative was assumed to come from the “r-part” of the BC. In GaAs a noticeable variation concerns only the first derivatives of the ion-BC potentials, whereas the the ion-ion parameters are essentially unaffected. The modification of the bond angles occurring at the surfaces is included in the geometric part of the Keating force constant matrices, which are calculated in the actual bond configuration. The intensities of the Keating parameters were taken to be the same as in the bulk. Actually, values different from the bulk cannot be ruled out a priori, but Raman data for diamond under stress [14] indicate that corrections due to angle variations are quite small. For Ge no experimental support for this choice can be given, yet we rely on the good agreement that we achieved in the Si(111)2 x 1 case [l], where the same parametrization was previously used. Moreover, the general criterion of our approach is to exploit the intrinsic features of the surface dynamics, that is, to use the bulk parameters as far as possible and avoid the introduction of adjustable parameters. Our calculations are performed for a slab with 24 atomic planes in Ge and 23 in GaAs. To exploit the symmetry properties, which are very helpful in the analysis of the eigenvectors, the thickness of the slab must be chosen carefully: in
P. Santini et al. / Surface
3.50
phonon
dispersion curves in GaAs(ll0) and Ge(iI1)2 x I
Fig. 3. Phonon spectnun of Ge(111)2 x 1. The shaded areas represent the surface projection of the bulk bands, the solid lines and the broken lines indicate surface modes and weak resonances, respectively.
1
4ol-
-q
t
2
5+-
>
“-
r
x
i-4
i 0
-6
X
Fig. 4. Phonon spectrum of GaAs(ll0).
i=
P. Santiniet al. / Swjace phonon dispersioncurves in GaAs(il0) and Ge(ilI)2 x I
fact, with the present sizes of our slabs the systems have, in the medial plane, a center of inversion for the (111) slab and a mirror plane for the (110) case.
3. Results In figs. 3 and 4 we display the surface phonon dispersion curves (solid lines) and the projected bulk bands (shaded areas) for Ge(111)2 X 1 and for GaAs(ll0) along the borders of their SBZ’s (we remark that the two SBZ’s have the same geometry). Due to the nearly equal masses of Ga, As and Ge, and to the similarity between the two sets of force constants, the same energy range is spanned by the two bulk compounds. However, the position and shape of the gaps are determined by the different orientations of the two surfaces. As expected, some features are common to both spectra: one is the flat branch of transverse acoustical (TA) origin, sparming m and I%’ in the energy range between 9 and 11 meV for GaAs, and between 5 and 7.5 meV for Ge. In both cases, the surface phonon turns to a broad resonance as it approaches the zone center, due to its large penetration into the bulk. The displacement pattern of this branch is analogous for both surfaces: it is mostly normal to the surface, starting at l? as an in-phase vibration of the two atoms in the surface chain, and switching to the single motion of the topmost ion (in fig. 1 it is marked by 1, while in fig. 2 it corresponds to the As ion) near x. Consequently, the flat branch at the zone border takes a character ~mplementa~ to the Rayleigh wave (RW), which is a vertical vibration of the lower ion in the surface chain (2 in fig. 1, Ga in fig. 2). The behaviour enlightens the origin of the flat branch as a folded RW. Thus agrees qualitatively with the isolated chain dynamics on top of a rigid lattice, as calculated by Wang and Duke for GaAs(ll0) in the framework of the sp3s* tightbinding model [15]: at the zone center such vibration is found to be a bondlength conserving libration of the topmost chain, in analogy with the behaviour of the 10 meV flat branch in Si(111)2 X 1 [l]. Actually, as we have already pointed out in Si [l], the transition from the 1 x 1
351
to the 2 x 1 structure in Ge leads to a folding of the surface branches into the new smaller SBZ and the inequivalent positions of atoms 1 and 2 provide an interpretation for the gaps opened at the folding points (x, S, %?). The (110) surface of GaAs is unreconstructed (but just relaxed) with no change in the SBZ. However, the surface region of GaAs(ll0) (with 4 ions located in two layers) is geometrically equivalent to the surface region of Ge(l11)2 x 1 (with 4 ions located in one layer) and the mass difference is very small. Therefore, in the case of the heteropolar surface a folding ~gument can be invoked for the flat branch as well. The gap at the 52 point between the RW and the folded branch is thus proportional to the inequivalence of the two ions in the chain. For Ge such an inequivalence is produced only by the chain tilt (apart from fourth nearest-neighbour effects). Therefore, if a He-scattering measurement of the gap would be available, a dynamical estimation of the tilt angle would be possible, as in the case of Si(111)2 x 1 [l]. In GaAs the two ions in the surface chain are also inequivalent due to the slightly different masses, but mainly because of the different effective charges. However even in this case the gap between the RW and the flat branch at 10 meV can be used for the geometrical characterization of the surface. In fact, we performed several calculations with different values of the tilt angle and found, as expected, a decreasing gap for decreasing tilt (see ref. [9]). One final remark should be added for the energy position at x of the flat branch and the RW: they have remarkably different energies in GaAs and Ge. The subsurface ~nfigurations are different and this seems to be the reason for such a dissimilarity. Particularly, the displacement pattern of these modes in Ge involves deformations of the sevenfold rings, which are less stiff than the ideal six-fold rings present in GaAs(ll0). An important feature, which appears in our calculations, is a mode characterizing both GaAs and Ge systems and already described in the case of Si(111)2 X 1 fl]. This mode is located in the high part of the energy spectrum (at r its energy is 25 meV in GaAs and 31.7 meV in Ge, whereas in Si it appears at 52.5 mev). The mode is indi-
352
P. Santini et al. / Surface phonon dispersion curves in GaAs(IlOJ and G-(1 I I )2 x I
cated by an arrow in the GaAs and Ge spectra (figs. 3 and 4). The dispersion and the displacement pattern associated with it are very similar in the three cases: for this optical mode the atoms of the upper chain vibrate parallel to the chain direction. This feature turns out to be a real fingerprint of the chain structure at the surface, which is common to the three materials. Another branch displays a shear horizontal displacement pattern very localized at the surface chains, i.e. the nearly flat mode at 10 meV, along the symmetry direction m’ for GaAs and m for Ge: actually the dispersion relations in both cases are very similar. Besides the common chain modes described above, the high energy part of the spectra for the two systems shows important features which are strictly related to the different nature of Ge and GaAs. The Fuchs and Kliewer (FK) mode is an example: its origin arises from the polar character of the crystal, thus it appears only in the GaAs spectrum with a nearly flat dispersion at about 34 meV just below the upper bulk band edge. The calculated frequency of this macroscopic mode at i? (- 35 meV) is in good agreement with the EELS data [16]. Even Ge(ll1) exhibits peculiar modes having no counterparts in GaAs(ll0). We refer to the modes involving deformations of the stiff five-fold rings sparming the interface. Two of these modes have shear horizontal character and they are located just above the bulk projected band, whereas the two with sagittal character have at r energies of 32.3 and 33.1 meV respectively. This latter vibration would be responsible for the EELS peak, showing the same behaviour as the EELS active mode at 56 meV in Si [l]. The absence of the matching structures in the case of the GaAs surface explains the lack of these vibrations in the GaAs spectrum. In conclusion, we have shown that the two surfaces display several dynamical analogies, when the vibrations are localized at the very surface. In this case the dispersions and the displacement patterns of the modes are alike and this criterion enables a dynamical ~h~acte~zation of the surface. When the modes penetrate deeply into the crystals, however, the analogies disappear because of the different geometries of the substrates.
Acknowledgments The present calculation, performed on a Cray XMP computer, was financially supported by the Consiglio Nazionale delle Ricerche of Italy. The authors thank A. Lock and D. Kleinhesselink (Gottingen) for a critical reading of the manuscript. One of us (P.R.) thanks the Alexander-vonHumboldt Foundation for a fellowship.
References [l] L. Miglio, P. Santini, P. Ruggerone and G. Benedek, Phys. Rev. Lett. 62 (1989) 3070. [2] U. Harten, J.P. Toenmes and Ch. Wiill, Phys. Rev. Lett. 57 (1986) 2947. [3] H. Ibach, Phys. Rev. Lett. 27 (1971) 253. [4] K.C. Pandey, Phys. Rev. Lett. 47 (1981) 1913; 49 (1982) 223. [5] J.E. Northrup and M.L. Cohen, Phys. Rev. B 27 (1983) 6553. [6] G. Qian, R.M. Martin and D.J. Chadi, Phys. Rev. B 37 (1988) 1303; C.B. Duke, S.L. Richardson, A. Paton and A. Kahn, Surf. Sci. 127 (1983) L135; S.Y. Tong, W.N. Mei and G. Xu, J. Vat. Sci. Technol. B 2 (1984) 393. [7] U. Harten and J.P. Toennies, Europhys. Lett. 4 (1987) 833. [8] R.B. Doak and D.B. Nguyen, J. Electron Spectrosc. Relat. Phenom. 44 (1987) 205: _ [91 P. Santini, L. Migho, G. Benedek, U. Harten, P. Ruggerone and J.P. Toennies, to be published. UOI L. Migfio and L. Coiombo, Phys. Ser. 40 (1989) 238. illI J.E. Northrup and M.L. Cohen, J. Vat. Sci. TectUnol. 21 (1982) 333. WI W. Weber, Phys. Rev. B 15 (1976) 4789; K.C. Rustagi and W. Weber, Solid State Commun. 18 (1979) 673. P31 J.R. Chelikowsky and M.L. Cohen, Phys. Rev. B 20 (1979) 4150. 1141 M.H. Grimsditch, E. Anastassakis and M. Cardona, Phys. Rev. B 18 (1978) 901. WI Y.R. Wang and C.B. Duke, Surf. Sci. 205 (1988) L755: C.B. Duke and Y.R. Wang, J. Vat. Sci. Technol. B 7 (1989) 1027. WI U. del Pennino, M.G. Betti and C. Mariani, Surf. Sci. 211/212 (1989) 557.
346
Surface phonon dispersion curves in GaAs( 110) and Ge( 111) 2 a critical comparison
x
1:
P. Santini ‘, L. Miglio, G. Benedek Dipartimento di Fisica dell’ Universitci, via Celoria 16, I-20133 Milano, Iialy
and P. Ruggerone Max-Planck-Institut ftir StrGmungsforschung, Bunsenstrasse IO, D-3400 Giittingen, Germany
Received 1 June 1990; accepted for publication 6 September 1990
We compare the surface phonon dispersion curves of GaAs(ll0) and Ge(111)2 X 1 as calculated by the Bond Charge Model. The atomic masses, bulk force constants and crystallography of the topmost layer are nearly equal, still the substrate orientation and charge distributions are different. Similarities and discrepancies in the surface phonon dispersion curves are investigated as fingerprints of the structure configuration.
1. Introduction In a recent work [l] we reported the results of a Bond Charge Model (BCM) calculation of the Si(111)2 x 1 surface dynamics, which satisfactorily explain the existing experimental features. In particular, the 10 meV flat branch measured by He inelastic scattering [2] and the 56 meV EELS peak [3] were well described without any parameter fitting. The surface dispersion curves of an extensively reconstructed surface like Si(111)2 X 1 are extremely rich and the origin of the several branches can be attributed either to the purely geometric folding of the Brillouin zone or to the new structural characteristics produced by the reconstruction. A tool to analyze the entangled phonon spectrum of a reconstructed surface was indeed provided by the three following general criteria [l]: (i) the qualitative features of the dis’ Present address: lnstitut de Physique Theorique, Universite de Lausanne, CH-1015 Lausanne, Switzerland. 0039-6028/91/$03.50
persion curves largely reflect those of the ideal unreconstructed structure, after the folding of the larger surface Brillouin zone (SBZ) into the smaller one associated with the reconstructed phase (folding criterion); (ii) in the case of extensive reconstruction it is meaningful to speak of an interface between reconstructed and ideal regions, and of modes which may arise from this buried interface (interface criterion); (iii) the gaps opened along the folding lines arise from the new local arrangement of atoms in the reconstructed region and may be used to characterize the surface geometry (gap criterion). Such a connection between geometry and dynamics led us to compare the dispersion curves of Ge(111)2 x 1 and GaAs(llO), two systems with relevant structural similarities. Unfortunately neither EELS nor He scattering data are available at present for Ge(111)2 x 1, yet its reconstruction pattern is very similar to the one proposed by Pandey [4] for Si(ll1) [5]. This model suggests the formation of r-bonded atomic chains on the surface (see fig. l), with the upper chains
0 1991 - Elsevier Science Publishers B.V. (North-Holland)
P. Santini et al. / Surfnce phonon dispersion curves in GaAs(l IO) and Ge(l II)2 x I
characterized by a tilt, i.e. a difference in the vertical position between atoms of type 1 and type 2, as shown in fig. 1. The lower chains are affected by a tilt as well, but its value is smaller. The side view of the reconstructed surface in fig. 1 also shows the presence of alternating five-fold (4,3,5,8,6) and seven-fold (1,2,4,6,7,5,3) rings formed by the interface and the chain atoms. On the other hand, GaAs(ll0) is by far the most studied heteropolar semiconductor surface, and the qualitative arrangement of the equilibrium positions for the surface atoms is now fairly well established. The surface keeps the same periodicity of an ideal (110) plane, but it displays a relaxation of the topmost atoms, with As shifted above the ideal surface plane and Ga shifted towards of the topmost atoms, with As shifted above the ideal surface plane and Ga shifted towards the bulk. In detail, the relaxation involves a nearly bondlength-conserving rotation of the surface (ITO) chains by a tilt angle of about 30 O, while the subsurface chains are characterized by a much smaller tilt (see fig. 2) [6]. Discussions have been devoted to determine the exact values of the tilt angle and of the atomic displacements parallel to the surface, which accompany the tilt. The large amount of available He-scattering data along the boundaries of the irreducible part of the SBZ [7,9] permits a dynamical test of different crystallographic models for this surface. An extensive analysis of the low-energy part of the GaAs(ll0) spectrum with a discussion of the influence of the surface geometry on the dynamics can be found in ref. [9]. In the present paper we are interested in a somewhat different question. We investigate the dynamic effects of the structural analogies existing between GaAs(llO) and Ge(111)2 x 1: the masses of Ga, As and Ge, as well as the short-range force constants of GaAs and Ge, are nearly the same [lo]. Moreover, the chain structure produced in Ge(ll1) by the 2 X 1 reconstruction is very similar to the relaxed configuration of GaAs(ll0) [6] (see figs. 1 and 2): as we have already pointed out, in both cases we have inner and outer chains running parallel to the surface plane, the latter being particularly affected by an atomic buckling in order to minimize the energy. Yet, some relevant dif-
347
ferences exist, arising both from the matching of the chain geometry to the different (110) and (111) substrates and from the polar character of GaAs. In particular, the matching region displays, in the case of GaAs(llO), 6-fold rings (see fig. 2) instead of the 5- and 7-fold rings of Ge(111)2 X 1. Also, the electronic density maps for Ge(111)2 X 1 are very likely to be the same as for Si(111)2 X 1 [ll], where the charge in the surface bonds is located midway the chain atoms, since negligible charge transfer is seen to accompany the tilt. On the contrary, in the heteropolar case the filled state is located upon the upwards shifted As ions, since charge transfer does lower the surface energy [4]. The subsurface structure is responsible for many of the differences between the two spectra and therefore it is crucial to include in the calculations a suitable number of layers in order to reproduce properly the influence of the matching region between the bulk and surface structures. This task can be hardly achieved with first-principle calculations which are limited to very thin slab configurations. A satisfactory compromise is, in our opinion, to use a realistic dynamical model, which is able to account for the role of electrons in a
Fig. 1. Structureof Ge(111)2 X 1 (from ref. [5]).
348
P. Santini et al. / Surface phonon dispersion curves in GaAs(ll0)
0
and Ge(l I I)2 x I
Ga
@ As l
bc
Fig. 2. Side and top views of GaAs(ll0).
The black dots indicate the positions of the bond charges.
phenomenological way and to include all the important modifications occurring at the surface.
2. Model Our calculations are based on the Bond Charge Model (BCM) [12]. This model provides bulk phonon dispersion curves which are in good agreement with experimental neutron scattering data, both for homopolar and heteropolar semiconductors. The number of adjustable parameters in the bulk is 4 for Si and Ge and 6 for GaAs. The BCM relies on the assumption that the valence charge
can be shared between a “core” charge located at the ions and a “bond charge” (BC), viewed as a massless pseudoparticle with its own degrees of freedom. The ratio of the BC to the ion charge is fixed in our model: in order to fulfill the neutrality condition, the ion charge (including core and valence states contributions) has to be assumed minus twice the bond charge. The equilibrium position of the bond charges lies midway between the bonds in Si and Ge, whereas in GaAs it is displaced towards the anion, dividing the bond in a ratio 3 : 5. Thus the polar character of III-V compounds is recovered.
P. Santiniet al. / Surfacephonon dispersioncurvesin GaAs(lIO)and Ge(l I I)2 x I
In addition to the long range Coulombic interactions among ions and bond charges, short range potentials are introduced in order to include the quantum mechanical repulsion in a phenomenological way: central interactions are set up between neighbouring cores and between cores and neighbouring bond charges. A bond-bending Keating-like potential between nearby bonds is finally considered to ensure the lattice stability against shear displacements. The bond charge coordinates are eliminated via the adiabatic condition only before the diagonalization of the dynamical matrix and therefore the new arrangement of the electronic distribution due to the surface is included in the calculation. Actually, for the surface problem we modify the core positions to reproduce the new geometry induced by the surface perturbation, as determined by total energy calculations and elastic scattering measurements [5,6]. Concerning the electronic degrees of freedom, we set the bond charges at the surface according to the relative maxima of the valence charge density contours. In Ge(111)2 X 1, for instance, the bond charges are always placed at the bond midpoints. Their charge is the same as in the bulk, except for the bond charges between chain atoms, which are 50% larger: in fact the latter have to include a supplementary n-bond electron coming from the saturated dangling bond [l]. In GaAs(ll0) the doubly occupied dangling bonds, belonging to the topmost As atoms, are described by bond charges located at the maximum of the charge distribution [13] (see fig. 2). The remaining bond charges keep the same positions as in the bulk. The calculated surface phonon dispersion appears to be remarkably sensitive to the position of the “dangling” bond charge, and considerable discrepancies with experiment appear if its position is not appropriately chosen. This latter fact provides sufficient confidence in the sensitivity of BCM as a dynamical tool for surface characterization. In both GaAs(l10) and Ge(111)2 X 1, static equilibrium conditions are imposed on all surface ions and bond charges: in this way we can obtain new values for the first derivatives of the central ion-ion and ion-BC potentials. The second de-
349
rivatives of these potentials do not appear under such conditions, and we set them equal to the bulk value. This choice is supported by the fact that interatomic distances at both surfaces remain almost the same as in the bulk. In Ge(l11)2 x 1 the only appreciable variation with respect to the bulk is found for the ion-BC interaction along the chain, due to the a-state incorporated in the bond charge. We find, indeed, a negative value of the first derivative coming from equilibrium conditions, while in the bulk such a parameter is zero. Since the first derivative affects the restoring force for the BC motion normal to the bond, we interpret this variation as being due to the superposition of B and core states induced by such a motion. In fact, the r-orbital possesses a nodal plane crossing the cores and its overlap with the core states along the bond direction is influenced by the shear but not by the BC radial vibration. Actually, no additional contribution to the second derivative was assumed to come from the “r-part” of the BC. In GaAs a noticeable variation concerns only the first derivatives of the ion-BC potentials, whereas the the ion-ion parameters are essentially unaffected. The modification of the bond angles occurring at the surfaces is included in the geometric part of the Keating force constant matrices, which are calculated in the actual bond configuration. The intensities of the Keating parameters were taken to be the same as in the bulk. Actually, values different from the bulk cannot be ruled out a priori, but Raman data for diamond under stress [14] indicate that corrections due to angle variations are quite small. For Ge no experimental support for this choice can be given, yet we rely on the good agreement that we achieved in the Si(111)2 x 1 case [l], where the same parametrization was previously used. Moreover, the general criterion of our approach is to exploit the intrinsic features of the surface dynamics, that is, to use the bulk parameters as far as possible and avoid the introduction of adjustable parameters. Our calculations are performed for a slab with 24 atomic planes in Ge and 23 in GaAs. To exploit the symmetry properties, which are very helpful in the analysis of the eigenvectors, the thickness of the slab must be chosen carefully: in
P. Santini et al. / Surface
3.50
phonon
dispersion curves in GaAs(ll0) and Ge(iI1)2 x I
Fig. 3. Phonon spectnun of Ge(111)2 x 1. The shaded areas represent the surface projection of the bulk bands, the solid lines and the broken lines indicate surface modes and weak resonances, respectively.
1
4ol-
-q
t
2
5+-
>
“-
r
x
i-4
i 0
-6
X
Fig. 4. Phonon spectrum of GaAs(ll0).
i=
P. Santiniet al. / Swjace phonon dispersioncurves in GaAs(il0) and Ge(ilI)2 x I
fact, with the present sizes of our slabs the systems have, in the medial plane, a center of inversion for the (111) slab and a mirror plane for the (110) case.
3. Results In figs. 3 and 4 we display the surface phonon dispersion curves (solid lines) and the projected bulk bands (shaded areas) for Ge(111)2 X 1 and for GaAs(ll0) along the borders of their SBZ’s (we remark that the two SBZ’s have the same geometry). Due to the nearly equal masses of Ga, As and Ge, and to the similarity between the two sets of force constants, the same energy range is spanned by the two bulk compounds. However, the position and shape of the gaps are determined by the different orientations of the two surfaces. As expected, some features are common to both spectra: one is the flat branch of transverse acoustical (TA) origin, sparming m and I%’ in the energy range between 9 and 11 meV for GaAs, and between 5 and 7.5 meV for Ge. In both cases, the surface phonon turns to a broad resonance as it approaches the zone center, due to its large penetration into the bulk. The displacement pattern of this branch is analogous for both surfaces: it is mostly normal to the surface, starting at l? as an in-phase vibration of the two atoms in the surface chain, and switching to the single motion of the topmost ion (in fig. 1 it is marked by 1, while in fig. 2 it corresponds to the As ion) near x. Consequently, the flat branch at the zone border takes a character ~mplementa~ to the Rayleigh wave (RW), which is a vertical vibration of the lower ion in the surface chain (2 in fig. 1, Ga in fig. 2). The behaviour enlightens the origin of the flat branch as a folded RW. Thus agrees qualitatively with the isolated chain dynamics on top of a rigid lattice, as calculated by Wang and Duke for GaAs(ll0) in the framework of the sp3s* tightbinding model [15]: at the zone center such vibration is found to be a bondlength conserving libration of the topmost chain, in analogy with the behaviour of the 10 meV flat branch in Si(111)2 X 1 [l]. Actually, as we have already pointed out in Si [l], the transition from the 1 x 1
351
to the 2 x 1 structure in Ge leads to a folding of the surface branches into the new smaller SBZ and the inequivalent positions of atoms 1 and 2 provide an interpretation for the gaps opened at the folding points (x, S, %?). The (110) surface of GaAs is unreconstructed (but just relaxed) with no change in the SBZ. However, the surface region of GaAs(ll0) (with 4 ions located in two layers) is geometrically equivalent to the surface region of Ge(l11)2 x 1 (with 4 ions located in one layer) and the mass difference is very small. Therefore, in the case of the heteropolar surface a folding ~gument can be invoked for the flat branch as well. The gap at the 52 point between the RW and the folded branch is thus proportional to the inequivalence of the two ions in the chain. For Ge such an inequivalence is produced only by the chain tilt (apart from fourth nearest-neighbour effects). Therefore, if a He-scattering measurement of the gap would be available, a dynamical estimation of the tilt angle would be possible, as in the case of Si(111)2 x 1 [l]. In GaAs the two ions in the surface chain are also inequivalent due to the slightly different masses, but mainly because of the different effective charges. However even in this case the gap between the RW and the flat branch at 10 meV can be used for the geometrical characterization of the surface. In fact, we performed several calculations with different values of the tilt angle and found, as expected, a decreasing gap for decreasing tilt (see ref. [9]). One final remark should be added for the energy position at x of the flat branch and the RW: they have remarkably different energies in GaAs and Ge. The subsurface ~nfigurations are different and this seems to be the reason for such a dissimilarity. Particularly, the displacement pattern of these modes in Ge involves deformations of the sevenfold rings, which are less stiff than the ideal six-fold rings present in GaAs(ll0). An important feature, which appears in our calculations, is a mode characterizing both GaAs and Ge systems and already described in the case of Si(111)2 X 1 fl]. This mode is located in the high part of the energy spectrum (at r its energy is 25 meV in GaAs and 31.7 meV in Ge, whereas in Si it appears at 52.5 mev). The mode is indi-
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P. Santini et al. / Surface phonon dispersion curves in GaAs(IlOJ and G-(1 I I )2 x I
cated by an arrow in the GaAs and Ge spectra (figs. 3 and 4). The dispersion and the displacement pattern associated with it are very similar in the three cases: for this optical mode the atoms of the upper chain vibrate parallel to the chain direction. This feature turns out to be a real fingerprint of the chain structure at the surface, which is common to the three materials. Another branch displays a shear horizontal displacement pattern very localized at the surface chains, i.e. the nearly flat mode at 10 meV, along the symmetry direction m’ for GaAs and m for Ge: actually the dispersion relations in both cases are very similar. Besides the common chain modes described above, the high energy part of the spectra for the two systems shows important features which are strictly related to the different nature of Ge and GaAs. The Fuchs and Kliewer (FK) mode is an example: its origin arises from the polar character of the crystal, thus it appears only in the GaAs spectrum with a nearly flat dispersion at about 34 meV just below the upper bulk band edge. The calculated frequency of this macroscopic mode at i? (- 35 meV) is in good agreement with the EELS data [16]. Even Ge(ll1) exhibits peculiar modes having no counterparts in GaAs(ll0). We refer to the modes involving deformations of the stiff five-fold rings sparming the interface. Two of these modes have shear horizontal character and they are located just above the bulk projected band, whereas the two with sagittal character have at r energies of 32.3 and 33.1 meV respectively. This latter vibration would be responsible for the EELS peak, showing the same behaviour as the EELS active mode at 56 meV in Si [l]. The absence of the matching structures in the case of the GaAs surface explains the lack of these vibrations in the GaAs spectrum. In conclusion, we have shown that the two surfaces display several dynamical analogies, when the vibrations are localized at the very surface. In this case the dispersions and the displacement patterns of the modes are alike and this criterion enables a dynamical ~h~acte~zation of the surface. When the modes penetrate deeply into the crystals, however, the analogies disappear because of the different geometries of the substrates.
Acknowledgments The present calculation, performed on a Cray XMP computer, was financially supported by the Consiglio Nazionale delle Ricerche of Italy. The authors thank A. Lock and D. Kleinhesselink (Gottingen) for a critical reading of the manuscript. One of us (P.R.) thanks the Alexander-vonHumboldt Foundation for a fellowship.
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