- Email: [email protected]
Lattice vibrations at the Be(101̄0) surface
Surface Science 377-379 (1997) 330-334
ELSEVIER
Lattice vibrations at the Be( lOlO) surface Ph. Hofmann a,*, E.W. Plummer b aDepartment of Physics and Astronomy,
University of Tennessee, Knoxville, b Oak Ridge National Laboratories, Oak Ridge, TN37831-6057,
TN37996, USA
USA
Received 1 August 1996; accepted for publication 15 October 1996
Abstract
We have measured the surface phonon dispersion of Be( lOi0) by electron energy loss spectroscopy (EELS) and found three surface resonances at the surface Brillouin zone centre as well as two surface phonons at the li and &I points. The experimental data do not agree with a calculation for the truncated bulk. In particular, the predicted loss associated with the Rayleigh wave at the zone boundaries is not observed. The character of the surface vibrations suggests that the bonding is more complicated than on Be(OOO1)where the qualitative features of the surface phonon dispersion can be described within a simple central force model. Keywords:
Alkaline earth metals; Electron energy loss spectroscopy; Low index single crystal surfaces
1. Introduction In the harmonic approximation the vibrational properties of a crystal are given by the forces on an atom when another atom is moved. For many simple and noble metals the phonon dispersions can be approximately described within the framework of a pairwise potential which depends only on the interatomic distances, i.e. a central force potential. Although beryllium (Be) is formally a simple (sp bonded) metal the bulk phonon dispersion shows pronounced qualitative effects of noncentral force contributions [ 11. This is caused by a substantial degree of directional bonding [2,3] which is also reflected in a very low electronic density-of-states (DOS) at the Fermi level. The bonding between Be atoms has to be formed from * Corresponding author. Fax: + 1423 576 8135; e-mail: [email protected]
a closed-shell configuration ls22s2 by promoting s electrons into p orbitals. The sp hybridisation is facilitated by the absence of a p core level due to the missing orthogonalization barrier and the resulting bonds may be viewed as partially filled sp and sp2 hybrid orbitals [2]. The energy for the sp promotion has to be compensated by the bonding. It is therefore hardly surprising that the bond strength, length and physical properties of a Be structure depend crucially on its dimensionality [4-61. Indeed, at the Be(OOO1) and Be(lOi0) surfaces the bond length between the first- and second-layer atoms are very different from the corresponding bulk values and electronic states which exist in wide gaps of the bulk projected band structure [7-lo] give rise to a high density-of-states at the Fermi level. In the case of Be(OOO1 ), Hannon, Mele and Plummer (HMP) have shown that the dispersion of the Rayleigh wave (RW) cannot be described in terms
0039-6028/97/$17.00 Copyright 0 1997 Elsevier Science B.V. All rights reserved PZZ SOO39-6028(96)01387-S
Ph. Hofmann, E. W. Plummer / Surface Science 377-379
of the bulk force constants but, at least qualitatively, is consistent with a decrease in the noncentral forces at the surface due to the more isotropic screening [ 111 which is caused by a freeelectron-like surface Fermi surface. In the present paper we report measurements of the surface phonon dispersion on Be(lOi0). The first layer of a hcp ( lOi0) surface consists of parallel, closed-packed atom rows, similar to a fee (110) surface. Two terminations of the truncated bulk are possible whose first interlayer spacings differ by a factor of two. In the case of Be(lOi0) only the termination with the shorter first interlayer spacing is present [8,12]. An oscillatory relaxation is found with a change in the first interlayer distance of as much as 25% with respect to the bulk [12]. Just as on Be(OOOl), the DOS at the Fermi level is high due to a surface state. However, the surface Fermi surface consists of two half-circles centred around the ;j; points of the surface Brillouin zone (SBZ). Hence, the screening on this surface will be highly anisotropic and the surface phonon dispersion can probably not be described within a central force model.
(1997) 330-334
331
taken, the quasi-elastic peak shows a substantially higher count rate than the loss peaks, which indicates the presence of surface roughness. A set of typical spectra taken at the high-symmetry points is given in Fig. 1.
3. Results and discussion The position of the projected bulk bands and the surface phonon dispersion for Be(lOi0) was calculated using the method of Allen et al. as implemented by Hannon [ 11,13,14]. In this approach the dynamical matrix for a slab with a tiite thickness and two-dimensional periodicity is evaluated from the atomic force constant matrices by summing over nearest-neighbour shells up to a certain cut-off. We have used force constant matrices for up to seventh nearest neighbour interactions as obtained by HMP from a fit to the
2. Experimental The sample was cleaned by cycles of Ne ion bombardment and annealing, both at 450°C [ 121. This resulted in a sharp (1 x 1) LEED pattern and a clean surface as judged by EELS. The EELS data were taken in two mutually perpendicular directions of the SBZ, along i?-A and along f-&I. The impact energy was varied between 60 and 80 eV. The energy dependence of the loss intensities was not investigated in detail but no indications of substantial variations were found. For energies outside this range the elastic intensity in the specular beam was so small that it was not possible to tune the voltage settings of the instiument for resolution optimisation. The scattering geometry was chosen such that the electron analyser was kept fixed at an angle of 45” off the surface normal and the electron monochromator was moved to angles 245” off-normal to achieve the desired momentum transfer. The typical energy resolution was at 3-4meV. For all the spectra
I 0
I
I
I
20 40 60 energy loss (meV)
I
80
Fig. 1. A set of typical loss spectra taken at the high-symmetry points of the surface Brillouin zone.
Ph. Hofmann,
332
E. W Plummer / Surface Science 377-379
experimentally determined bulk phonon dispersion of Stedman et al. [ 151. Fig. 2 shows the measured phonon dispersion together with the calculation for a 104 layer slab with a truncated bulk termination. Clear losses have been marked as filled circles, very weak losses as open circles. At the A point a surface mode is found which disperses upwards as q is lowered and is still observed as it enters the bulk band continuum. Three modes are found at r which all lie within the projected bulk bands: the lowest exists far into both directions, the second is only visible along i; -A and the highest mainly along r - ti. At ti one mode which shows little dispersion is located below the bulk band edge. Judged from the experimental data alone the possibility cannot be excluded that it is a continuation of the lowest mode at r. In the calculation, modes which have more than 50% of the total displacement localised in the first two layers have been highlighted. We adopt the convention of calling the calculated modes at the
80
60 2 5 B $ 5
40 1
20
0
-I_
I
-0.5 A-i’
I
0.0 (A-‘)
I
I
1 .o o’5 p__M
Fig. 2. Measured surface phonon dispersion for Be( lOTO) together with a calculation for a bulk-terminated 104 layer slab. The filled circles correspond to clear losses, the open circles to very weak losses. In the calculation modes with a total displacement of more than 50% in the first two layers have been highlighted. The inset shows the irreducible quarter of the surface Brillouin zone.
(1997) 330-334
high-symmetry points by their ordering, starting with the lowest energy, i.e., A(l), A(2) and so on. The agreement between the measured and calculated surface modes is rather poor, indicating that the force constants at the surface are very different from the bulk. However, we may use the displacement patterns of the calculated modes for a tentative assignment of the observed losses. Note that these displacements are mainly dominated by the symmetry of the surface rather than by the precise magnitude of the force constants. First, we try to assign the loss at A to one of the three modes found in the calculation: the highest mode, A(3), has a shear-horizontal (SH) character consisting only of displacements perpendicular to q. Since in this scattering geometry, and also in the other azimuth, the sagittal plane coincides with a mirror plane of the crystal, such vibrations cannot be excited [ 161. The two possible modes are the RW A( 1) and the mode A(2) which are both polarised in the sagittal plane and consist mainly of a motion perpendicular to the surface in the second and first layer, respectively. Given the upward dispersion with decreasing q it is more likely that the loss must be assigned to A(2). The situation is very similar at ti: the lower loss clearly falls into a region with no bulk bands while the upper loss in the vicinity of $I might correspond to an onset of loss channels at the bulk band edge a(3) and M(4) at 50.2 and 50.36 meV are SH and cannot be observed due to symmetry. Again, the RW ti( 1) and ti(2) are both polarised in the sagittal plane and, given the dispersion character of the modes, it is more likely that the experimental losses are due to M(2). In contrast to A(2) it consists mostly of a vibration perpendicular to the surface located in the second layer, whereas the main displacements of a(l) are in the first layer. From this localisation, however, we would expect to observe stronger losses due to ti( 1) than due to ti(2). At r three surface modes are found in the calculation just as in the experiment. We assign r( 1) to the lowest loss observed in the experiment. The displacement is perpendicular to the surface at r and gains more and more SH character as q is increased. Hence, the mode should be observed in both directions but, eventually, must disappear.
Ph. Hofmann,
E. W. Plumlner 1 Surface Science 377-379
This supports the view that it is not a continuation of the lowest mode at M. It is tempting to assign the two remaining losses just by their ordering to r(2) and r(3). This, however, would contradict the selection rule imposed by symmetry: r(3) for instance, shows only a high surface localisation along r--M but at the same time it has a SH polarisation along this direction and should not be observed in the experiment. We also have to keep in mind that the peaks in the EELS spectra do not necessarily have to correspond to surface modes when they fall within the projected bulk bands. According to HMP the dispersion of the Rayleigh wave on Be( 000 1) can be described qualitatively by a purely central force model. In the present case the situation is different: even a qualitative agreement between the calculated modes and our data cannot be achieved within a model where the interactions in the first two layers are described by central forces. By changing the force constants at the surface it is possible to lower M(2) and A(2) such that their energies match the observed losses. However, this also results in a much softer RW, which is not observed. It does not seem to be possible to change the ordering of the RW and the second mode in our simple model. This may not be surprising if we keep in mind the physical origin of A(2) and M(2): A hcp metal has two atoms per unit cell. The bulk dispersion along T-A shows two acoustic and two optical branches, the latter being just the back-folded acoustic modes. Probably A( 1) and A(2) are just the surface modes derived from these bulk phonons. This would also explain the quasi-degeneracy of A( 1) and A(2) which seems to be independent of the force constants used. At M the situation is slightly different because this point of the SBZ does not correspond to a high-symmetry point in the BZ; it is the point three-quarters of the way in the T-K direction. This position itself also causes a back-folding of the bands so that M(2) is probably the back-folded RW. Note that an understanding of the surface phonons at M will be of particular interest since it is the K-point of the BZ which shows the pronounced effects of non-central forces in the bulk [ 11. Also, we have to keep in mind that the electronic
(1997) 330-334
333
structure of this surface is very different from Be(OOO1). As mentioned above, the surface Fermi surface will lead to very anisotropic screening here, whereas the free-electron-like surface state on Be(OOO1) causes the screening to be isotropic and leads to a surface phonon dispersion which can be described within a central force model. In addition, oscillatory relaxations of the layer distances as found for Be( 1010) are likely to change the force constants and thus the phonon dispersion [ 171.
4. Conclusion In conclusion, we have presented a measurement of the surface phonon dispersion on Be(lOi0) showing three surface resonances at the r point as well as two surface phonons at the A and M points of the SBZ. The measured phonon dispersion does not agree with a slab calculation for the truncated bulk mainly because the RW predicted in the calculations is not found in the experiment. It is unlikely that these disagreements can be resolved within the framework of a purely central force model as on Be(OOO1) because of the nonfree-electron-like surface Fermi surface, which will lead to highly anisotropic screening. A better understanding of the microscopic origin of the forces determining both the interlayer relaxations and the surface phonon dispersion of Be(lOi0) could be achieved by a first-principles calculation.
Acknowledgements
The authors gratefully acknowledge stimulating discussions with J.B. Hannon, K. Pohl and T.S. Rahman. We also thank K. Pohl for experimental assistance and J.B. Hannon for help with the slab calculations. This work was supported by the National Science Foundation under Grant No. NSF-9510132. Ph. Hofmann thanks the Alexander v. Humboldt-Stiftung for a Fellowship. Part of this study was conducted at ORNL and supported by the US Department of Energy under contract DE-AC05-960R22464 with Lockheed Martin Energy Research Corp.
334
Ph. Hofmann, E. W. Plummer / Surface Science 377-379 (1997) 330-334
References [l] A.P. Roy, B.A. Dasannachaya, C.L. Thape and P.K. Iyengar, Phys. Rev. Lett. 30 (1973) 906. [2] Y.W. Yang and P. Coppens, Acta Cryst. A 34 (1978) 61. [3] N.A.W. Holzwarth and Y. Zeng, Phys. Rev. B 51 (1995) 13653. [4] V.E. Bondybey and J.H. English, J. Chem. Phys. 80 (1984) 568. [5] E. Wimmer, J. Phys. F 14 (1984) 681. [6] J.C. Boettger and S.B. Trickey, Phys. Rev. B 32 (1985) 1356. [7] R.A. Bartynski, E. Jensen, T. Gustafsson and E.W. Phrmmer, Phys. Rev. B 32 (1985) 1921. [S] Ph. Hofinann, R. Stumpf, V.M. Silkin, E.V. Chulkov and E.W. Plummer, Surf. Sci. 355 (1996) L278. [9] E.V. Chulkov, V.M. Silkin and E.N. Shirykalov, Surf. Sci. 188 (1987) 287.
[lo] V.M. Silkin and E.V. Chulkov, Phys. Solid State 37 (1995) 1540. [ 1l] J.B. Hannon, E.J. Mele and E.W. Phunmer, Phys. Rev. B 53 (1996) 2090. [12] Ph. Hofmann, K. Pohl, R. Stumpf and E.W. Plummer, Phys. Rev. B 53 (1996) 13751. [13] R.E. Allen, G.P. Alldredge and F.W. de Wette, Phys. Rev. B 4 (1971) 1648. [ 141 J.B. Hannon, Dissertation, University of Pennsylvania, 1994. [15] R. Stedman, Z. Amilius, R. Pauli and 0. Sundin, J. Phys. F 6 (1976) 157. [16] H. Ibach and D.L. Mills, Electron Energy Loss Spectroscopy and Surface Vibrations (Academic Press, New York, 1982). 1171A.G. Eguiluz, Third International Conference on Phonon Physics (World Scientmc, Singapore, 1990) p. 851.
ELSEVIER
Lattice vibrations at the Be( lOlO) surface Ph. Hofmann a,*, E.W. Plummer b aDepartment of Physics and Astronomy,
University of Tennessee, Knoxville, b Oak Ridge National Laboratories, Oak Ridge, TN37831-6057,
TN37996, USA
USA
Received 1 August 1996; accepted for publication 15 October 1996
Abstract
We have measured the surface phonon dispersion of Be( lOi0) by electron energy loss spectroscopy (EELS) and found three surface resonances at the surface Brillouin zone centre as well as two surface phonons at the li and &I points. The experimental data do not agree with a calculation for the truncated bulk. In particular, the predicted loss associated with the Rayleigh wave at the zone boundaries is not observed. The character of the surface vibrations suggests that the bonding is more complicated than on Be(OOO1)where the qualitative features of the surface phonon dispersion can be described within a simple central force model. Keywords:
Alkaline earth metals; Electron energy loss spectroscopy; Low index single crystal surfaces
1. Introduction In the harmonic approximation the vibrational properties of a crystal are given by the forces on an atom when another atom is moved. For many simple and noble metals the phonon dispersions can be approximately described within the framework of a pairwise potential which depends only on the interatomic distances, i.e. a central force potential. Although beryllium (Be) is formally a simple (sp bonded) metal the bulk phonon dispersion shows pronounced qualitative effects of noncentral force contributions [ 11. This is caused by a substantial degree of directional bonding [2,3] which is also reflected in a very low electronic density-of-states (DOS) at the Fermi level. The bonding between Be atoms has to be formed from * Corresponding author. Fax: + 1423 576 8135; e-mail: [email protected]
a closed-shell configuration ls22s2 by promoting s electrons into p orbitals. The sp hybridisation is facilitated by the absence of a p core level due to the missing orthogonalization barrier and the resulting bonds may be viewed as partially filled sp and sp2 hybrid orbitals [2]. The energy for the sp promotion has to be compensated by the bonding. It is therefore hardly surprising that the bond strength, length and physical properties of a Be structure depend crucially on its dimensionality [4-61. Indeed, at the Be(OOO1) and Be(lOi0) surfaces the bond length between the first- and second-layer atoms are very different from the corresponding bulk values and electronic states which exist in wide gaps of the bulk projected band structure [7-lo] give rise to a high density-of-states at the Fermi level. In the case of Be(OOO1 ), Hannon, Mele and Plummer (HMP) have shown that the dispersion of the Rayleigh wave (RW) cannot be described in terms
0039-6028/97/$17.00 Copyright 0 1997 Elsevier Science B.V. All rights reserved PZZ SOO39-6028(96)01387-S
Ph. Hofmann, E. W. Plummer / Surface Science 377-379
of the bulk force constants but, at least qualitatively, is consistent with a decrease in the noncentral forces at the surface due to the more isotropic screening [ 111 which is caused by a freeelectron-like surface Fermi surface. In the present paper we report measurements of the surface phonon dispersion on Be(lOi0). The first layer of a hcp ( lOi0) surface consists of parallel, closed-packed atom rows, similar to a fee (110) surface. Two terminations of the truncated bulk are possible whose first interlayer spacings differ by a factor of two. In the case of Be(lOi0) only the termination with the shorter first interlayer spacing is present [8,12]. An oscillatory relaxation is found with a change in the first interlayer distance of as much as 25% with respect to the bulk [12]. Just as on Be(OOOl), the DOS at the Fermi level is high due to a surface state. However, the surface Fermi surface consists of two half-circles centred around the ;j; points of the surface Brillouin zone (SBZ). Hence, the screening on this surface will be highly anisotropic and the surface phonon dispersion can probably not be described within a central force model.
(1997) 330-334
331
taken, the quasi-elastic peak shows a substantially higher count rate than the loss peaks, which indicates the presence of surface roughness. A set of typical spectra taken at the high-symmetry points is given in Fig. 1.
3. Results and discussion The position of the projected bulk bands and the surface phonon dispersion for Be(lOi0) was calculated using the method of Allen et al. as implemented by Hannon [ 11,13,14]. In this approach the dynamical matrix for a slab with a tiite thickness and two-dimensional periodicity is evaluated from the atomic force constant matrices by summing over nearest-neighbour shells up to a certain cut-off. We have used force constant matrices for up to seventh nearest neighbour interactions as obtained by HMP from a fit to the
2. Experimental The sample was cleaned by cycles of Ne ion bombardment and annealing, both at 450°C [ 121. This resulted in a sharp (1 x 1) LEED pattern and a clean surface as judged by EELS. The EELS data were taken in two mutually perpendicular directions of the SBZ, along i?-A and along f-&I. The impact energy was varied between 60 and 80 eV. The energy dependence of the loss intensities was not investigated in detail but no indications of substantial variations were found. For energies outside this range the elastic intensity in the specular beam was so small that it was not possible to tune the voltage settings of the instiument for resolution optimisation. The scattering geometry was chosen such that the electron analyser was kept fixed at an angle of 45” off the surface normal and the electron monochromator was moved to angles 245” off-normal to achieve the desired momentum transfer. The typical energy resolution was at 3-4meV. For all the spectra
I 0
I
I
I
20 40 60 energy loss (meV)
I
80
Fig. 1. A set of typical loss spectra taken at the high-symmetry points of the surface Brillouin zone.
Ph. Hofmann,
332
E. W Plummer / Surface Science 377-379
experimentally determined bulk phonon dispersion of Stedman et al. [ 151. Fig. 2 shows the measured phonon dispersion together with the calculation for a 104 layer slab with a truncated bulk termination. Clear losses have been marked as filled circles, very weak losses as open circles. At the A point a surface mode is found which disperses upwards as q is lowered and is still observed as it enters the bulk band continuum. Three modes are found at r which all lie within the projected bulk bands: the lowest exists far into both directions, the second is only visible along i; -A and the highest mainly along r - ti. At ti one mode which shows little dispersion is located below the bulk band edge. Judged from the experimental data alone the possibility cannot be excluded that it is a continuation of the lowest mode at r. In the calculation, modes which have more than 50% of the total displacement localised in the first two layers have been highlighted. We adopt the convention of calling the calculated modes at the
80
60 2 5 B $ 5
40 1
20
0
-I_
I
-0.5 A-i’
I
0.0 (A-‘)
I
I
1 .o o’5 p__M
Fig. 2. Measured surface phonon dispersion for Be( lOTO) together with a calculation for a bulk-terminated 104 layer slab. The filled circles correspond to clear losses, the open circles to very weak losses. In the calculation modes with a total displacement of more than 50% in the first two layers have been highlighted. The inset shows the irreducible quarter of the surface Brillouin zone.
(1997) 330-334
high-symmetry points by their ordering, starting with the lowest energy, i.e., A(l), A(2) and so on. The agreement between the measured and calculated surface modes is rather poor, indicating that the force constants at the surface are very different from the bulk. However, we may use the displacement patterns of the calculated modes for a tentative assignment of the observed losses. Note that these displacements are mainly dominated by the symmetry of the surface rather than by the precise magnitude of the force constants. First, we try to assign the loss at A to one of the three modes found in the calculation: the highest mode, A(3), has a shear-horizontal (SH) character consisting only of displacements perpendicular to q. Since in this scattering geometry, and also in the other azimuth, the sagittal plane coincides with a mirror plane of the crystal, such vibrations cannot be excited [ 161. The two possible modes are the RW A( 1) and the mode A(2) which are both polarised in the sagittal plane and consist mainly of a motion perpendicular to the surface in the second and first layer, respectively. Given the upward dispersion with decreasing q it is more likely that the loss must be assigned to A(2). The situation is very similar at ti: the lower loss clearly falls into a region with no bulk bands while the upper loss in the vicinity of $I might correspond to an onset of loss channels at the bulk band edge a(3) and M(4) at 50.2 and 50.36 meV are SH and cannot be observed due to symmetry. Again, the RW ti( 1) and ti(2) are both polarised in the sagittal plane and, given the dispersion character of the modes, it is more likely that the experimental losses are due to M(2). In contrast to A(2) it consists mostly of a vibration perpendicular to the surface located in the second layer, whereas the main displacements of a(l) are in the first layer. From this localisation, however, we would expect to observe stronger losses due to ti( 1) than due to ti(2). At r three surface modes are found in the calculation just as in the experiment. We assign r( 1) to the lowest loss observed in the experiment. The displacement is perpendicular to the surface at r and gains more and more SH character as q is increased. Hence, the mode should be observed in both directions but, eventually, must disappear.
Ph. Hofmann,
E. W. Plumlner 1 Surface Science 377-379
This supports the view that it is not a continuation of the lowest mode at M. It is tempting to assign the two remaining losses just by their ordering to r(2) and r(3). This, however, would contradict the selection rule imposed by symmetry: r(3) for instance, shows only a high surface localisation along r--M but at the same time it has a SH polarisation along this direction and should not be observed in the experiment. We also have to keep in mind that the peaks in the EELS spectra do not necessarily have to correspond to surface modes when they fall within the projected bulk bands. According to HMP the dispersion of the Rayleigh wave on Be( 000 1) can be described qualitatively by a purely central force model. In the present case the situation is different: even a qualitative agreement between the calculated modes and our data cannot be achieved within a model where the interactions in the first two layers are described by central forces. By changing the force constants at the surface it is possible to lower M(2) and A(2) such that their energies match the observed losses. However, this also results in a much softer RW, which is not observed. It does not seem to be possible to change the ordering of the RW and the second mode in our simple model. This may not be surprising if we keep in mind the physical origin of A(2) and M(2): A hcp metal has two atoms per unit cell. The bulk dispersion along T-A shows two acoustic and two optical branches, the latter being just the back-folded acoustic modes. Probably A( 1) and A(2) are just the surface modes derived from these bulk phonons. This would also explain the quasi-degeneracy of A( 1) and A(2) which seems to be independent of the force constants used. At M the situation is slightly different because this point of the SBZ does not correspond to a high-symmetry point in the BZ; it is the point three-quarters of the way in the T-K direction. This position itself also causes a back-folding of the bands so that M(2) is probably the back-folded RW. Note that an understanding of the surface phonons at M will be of particular interest since it is the K-point of the BZ which shows the pronounced effects of non-central forces in the bulk [ 11. Also, we have to keep in mind that the electronic
(1997) 330-334
333
structure of this surface is very different from Be(OOO1). As mentioned above, the surface Fermi surface will lead to very anisotropic screening here, whereas the free-electron-like surface state on Be(OOO1) causes the screening to be isotropic and leads to a surface phonon dispersion which can be described within a central force model. In addition, oscillatory relaxations of the layer distances as found for Be( 1010) are likely to change the force constants and thus the phonon dispersion [ 171.
4. Conclusion In conclusion, we have presented a measurement of the surface phonon dispersion on Be(lOi0) showing three surface resonances at the r point as well as two surface phonons at the A and M points of the SBZ. The measured phonon dispersion does not agree with a slab calculation for the truncated bulk mainly because the RW predicted in the calculations is not found in the experiment. It is unlikely that these disagreements can be resolved within the framework of a purely central force model as on Be(OOO1) because of the nonfree-electron-like surface Fermi surface, which will lead to highly anisotropic screening. A better understanding of the microscopic origin of the forces determining both the interlayer relaxations and the surface phonon dispersion of Be(lOi0) could be achieved by a first-principles calculation.
Acknowledgements
The authors gratefully acknowledge stimulating discussions with J.B. Hannon, K. Pohl and T.S. Rahman. We also thank K. Pohl for experimental assistance and J.B. Hannon for help with the slab calculations. This work was supported by the National Science Foundation under Grant No. NSF-9510132. Ph. Hofmann thanks the Alexander v. Humboldt-Stiftung for a Fellowship. Part of this study was conducted at ORNL and supported by the US Department of Energy under contract DE-AC05-960R22464 with Lockheed Martin Energy Research Corp.
334
Ph. Hofmann, E. W. Plummer / Surface Science 377-379 (1997) 330-334
References [l] A.P. Roy, B.A. Dasannachaya, C.L. Thape and P.K. Iyengar, Phys. Rev. Lett. 30 (1973) 906. [2] Y.W. Yang and P. Coppens, Acta Cryst. A 34 (1978) 61. [3] N.A.W. Holzwarth and Y. Zeng, Phys. Rev. B 51 (1995) 13653. [4] V.E. Bondybey and J.H. English, J. Chem. Phys. 80 (1984) 568. [5] E. Wimmer, J. Phys. F 14 (1984) 681. [6] J.C. Boettger and S.B. Trickey, Phys. Rev. B 32 (1985) 1356. [7] R.A. Bartynski, E. Jensen, T. Gustafsson and E.W. Phrmmer, Phys. Rev. B 32 (1985) 1921. [S] Ph. Hofinann, R. Stumpf, V.M. Silkin, E.V. Chulkov and E.W. Plummer, Surf. Sci. 355 (1996) L278. [9] E.V. Chulkov, V.M. Silkin and E.N. Shirykalov, Surf. Sci. 188 (1987) 287.
[lo] V.M. Silkin and E.V. Chulkov, Phys. Solid State 37 (1995) 1540. [ 1l] J.B. Hannon, E.J. Mele and E.W. Phunmer, Phys. Rev. B 53 (1996) 2090. [12] Ph. Hofmann, K. Pohl, R. Stumpf and E.W. Plummer, Phys. Rev. B 53 (1996) 13751. [13] R.E. Allen, G.P. Alldredge and F.W. de Wette, Phys. Rev. B 4 (1971) 1648. [ 141 J.B. Hannon, Dissertation, University of Pennsylvania, 1994. [15] R. Stedman, Z. Amilius, R. Pauli and 0. Sundin, J. Phys. F 6 (1976) 157. [16] H. Ibach and D.L. Mills, Electron Energy Loss Spectroscopy and Surface Vibrations (Academic Press, New York, 1982). 1171A.G. Eguiluz, Third International Conference on Phonon Physics (World Scientmc, Singapore, 1990) p. 851.