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Surface relaxation and surface dynamics of crystals of the rutile structure: MgF2 (110)
Journai of Electron Spectrosqy and Related Phenomena, W65 (1993) ‘769-775 OSSS2048/93/$06.00 @ 1993 - Elsevier science Publishers B.V. All rig& reserved
769
Surface relaxation and surface dynamics of crystals of the rutile structure: MgFz (110) G. Pelg”, U. Schrijdera, “Theoretische
Physik,
bMax-Planck-Institut ‘Department
W. Kressb, and F.W. de Wette’ Universit$t
Regensburg,
fiir Festkorperforschung,
of Physics,
University
93040 Regensburg, 70506 Stuttgart,
of Texas, Austin,
FRG FRG
Texas 78712, USA
We extend to crystals of the rutile structure our previous studies of the surface dynamics of ionic crystals within the framework of the shell model. These crystals have many interesting features due to their relatively open structure. Among the crystals of this type, MgFz has a closed-shell electronic configuration; we therefore expect that the surface properties can be calculated using potentials which are obtained from bulk dispersion We find that the bulk dynamics can be described very satisfactorily, if the static charge relations. distribution of the shell of the fluorine ion is shifted with respect to the core. The value of this shift is determined by the equilibrium condition. Using this model we calculate the surface relaxation pattern and dynamics of the (110) surface. Due to the open nature of the structure, we find relaxations both perpendicular to the surface and parallel to the surface. The calculated surface dynamics reveals a variety of surface modes which will be
discussed and compared with the results for other ionic crystals. 1. INTRODUCTION Rutile structure compounds occur in two groups: the fluorides, e.g. MgF2, MnFz, FeF2, etc., and the oxides, e.g. TiOs (rutile), &Ox, NbOz etc. As a result of their relatively open structure, these compounds exhibit many interesting features, such as phase transitions as a function of pressure and temperature. TiO2 is used as a catalyst, which points to the importance of its surface properties in particular, and stimulates interest in the surface properties of the rutile structure compounds in general. The fluorides provide a good starting point for studying the surface dynamics of these crystals; the oxides are more difficult to treat because of the peculiarities of the 02- ion. In this work we study the structural and dynamic properties of the (110) surface of MgF2 in the framework of a shell model. Since the anions in the rutile bulk structure display permanent induced dipole moments (in contrast to the alkali halides and perovskites), it is necessary to first develop a shell model for the bulk dynamics. This is done here in Sec. 2. In Sec. 3 we use the bulk shell model to evaluate the relaxation of the (110) surface. The dynamics of an eleven-lay-
er (llO)-oriented Sec. 4.
slab of MgF2 is discussed
2. BULK SHELL DYNAMICS
MODEL
AND
in
BULK
The bulk lattice dynamics of MgFz has been treated by Katiyar and Krishnan (1967) [l] within the rigid-ion model and by Almairac and Benoit (1974) [2] within the shell model. The present treatment distinguishes itself from that of Almairac and Benoit in two important ways: first, the short-range interactions are treated in a systematic way in terms of short-range potentials, and second, we take account of the fact that the fluorine-ions are at positions of non-zero electric field, which means that these ions have static dipole moments (but such that the total dipole moment of the unit cell is zero). The effect of these dipole moments on the dynamics of the related compound MnF2 has been treated by Cran and Sangster (1974) [3] in terms of a displacedshell model in which the core and the shell of the fluorine ions are allowed to have different equilibbrium positions. Here we adopt this model for
770
the treatment of MgF2. In the displaced-shell model it is assumed that in equilibrium the shells of the anions are centered at the crystalline lattice positions of these ions, while the core positions are shifted (because of symmetry) along the anion-anion axis by an amount which leads to the correct static dipole moment; the latter is obtained from a knowledge of the electric field at the fluorine equilibrium position and the free ion polarixability. The treatment of bulk dynamics in terms of the shell model is one of the standard procedures of lattice dynamics of insulators and needs not be discussed here. However, since the aim of this work is to use the shell model for the study of surface properties, we need to mention that, in addition to the long-range Coulomb potentials, the shortcrange overlap interaction of the ion pairs (i, j) also needs to be expressed in terms of pair potentials, which can then be used at the surface, where the interparticle distances may be different from those in the bulk. For these shortrange potentials we use the Born-Mayer form
xi(r)
=
Uij
eXp(-bijr).
The longitudinal and transverse force constants Aij and Bij which enter into the dynamical matrix (DM) are then given by
where e is the electric charge, V, the volume of the unit cell, and T$ the equilibrium distance between the ions i and j. For details about the use of short-range potentials for surface applications we refer to de Wette et al. (1985) [4], Kress et al. (1987) [5] and Reiger et al. (1989) [6]. In static equilibrium the total potential energy of the crystal should be a minimum. This can be
expressed by the conditions that the derivatives of the total potential energy 0 of the unit cell with respect to the lattice constants and the lattice parameters vanish in equilibrium. In the case of the normal shell model (i.e. undisplaced), this leads to one or more equilibrium conditions containing the Madelung constant and the short-range force constants {Bij). However, in the case of the displaced-shell model, there is an extra condition which expresses the fact that in equilibrium the core-shell displacements of the fluorine ions are kept in balance by the long-range interactions and the core-shell restoring force of the fluorine ions. The Coulomb interactions are evaluated with the Ewald method and are thus not cut off in distance. In deciding which short-range interactions to take into account, we followed the work of Almairac and Benoit (1974) [2]. The magnesium site is octahedral, with fluorine ions at each corner; however the distances ~4 - ~1 and ~4 - 1-2 (see Fig. 1) are slightly different. The fluorine surroundings are less symmetrical and involve 12 neighbors; besides the Mg-F interaction they involve two F-F interactions at different distances. Since the Mg-F and F-F short-range interactions are described by Born-Mayer potentials the effects of the variations in the distances are automatically taken into account. This set of interactions, together with the condition of charge neutrality (2~s = -2zF), leads to a shell model with eight parameters: ionic charge ZF; shell charge YF; core-shell coupling constant kF; Born-Mayer constants a, b of the Mg-F and F-F interactions; and the core-shell shifts Rn of the F ions (Mg is regarded to be unpolarizable) . The four equilibrium conditions reduce this set to four free parameters. Their values were determined by fitting the calculated phonon dispersion curves to the bulk experimental data [7] in the main symmetry directions of the Brillouin zone. The model parameters are given in Tab. 1.
aF_F[$A2] bF_F[_k-l]
u~~-F[EA~]bMvlg-FIA-l]
%+I
zF[el
kF[$l
RDIAl
1.1454 x lo6
1.5142 x lo6
-1.706
-0.822
547.8
0.00529
5.2929
Table 1. Parameters
5.1783
of the shell model for MgF2
771
3. RELAXATION FACE OF MgF2
OF THE
(110)
SUR-
The structure of the (110) surface is displayed in Fig. 1. Surface relaxation and reconstruction are the result of the imbalance of the forces acting on the ion cores and shells, when the ions at or near the surface are in their unrelaxed bulk positions. The symmetry of the (110) surface of the rutile structure is such that the surface ion displacements do not lead to a modified twodimensional surface unit cell (indicated by the heavy outline in Fig. l), compared to that for the case when the bulk is simply terminated at
was calculated by the same procedure as used earlier for alkali halide (001) surfaces (de Wette et al. 1985 [4]) and for the fluoridic perovskites (Reiger et al. 1989 [Sj). For details and the method we refer to these earlier publications. The calculations were performed for a twentyone layer (110) slab of MgFz, in which eight layers were allowed to relax. The results are shown in Fig. 2. 4. DYNAMICS SLAB
OF
THE
RELAXED
The bulk shell model discussed in Sec. 2 was used to calculate the dynamics of an eleven layer MgFz (110) slab in which four layers on each side were relaxed. The dynamical matrix was diagonalised for two-dimensional (2D) wave vectors q along the symmetry directions of the surface Brillouin zone (see Fig. 3).
x Figure
1. Structure
of the (110) surface,
Figure 3. Two-dimensional surface Brillouin Zone (SBZ) for the (110) surface of MgFs.
Figure 2. Relaxation of the (110) surface of MgF2 (bold: relaxed, light: unrelaxed). Side view of the two outer “layers”, each consisting of 3 planes of ions. The relaxation shifts are enhanced by a factor of 1.3. the surface. We call this restructuring surface relaxation, although it involves shifts of the fluorine ions parallel to the surface plane. The relaxation
The slab dispersion curves of the modes with their main vibrational amplitude in the first layer are plotted in Fig. 4. The shaded areas represent the bulk modes of the slab. Since the bulk unit cell contains six particles (eighteen degrees of freedom), there are eighteen bulk dispersion curves, three of which are acoustic and fifteen optic. Each of these bands can give rise to surface localized modes or resonances; in Fig. 4 these are indicated by bold and dashed lines, respectively. The type of surface modes and resonances that occur for each individual compound is determined by the structure of gaps between the bulk bands and by the way the bands overlap. Because the mass ratio of Mg and F is small, there is significant over-
772
lap of the bulk bands, with the result that there are relatively few well-pronounced surface modes and relatively many surface resonances. In Fig. 4
a more detailed discussion of the character and classification of these surface modes we refer to Kress et al. (1987) [5] and Reiger et al. (1989) [6].
20
15
10
5
0
0
0
r Figure 4a. Dispersion curves for an eleven-layer (110) slab of MgFz. The solid and dashed lines are SPI surface modes and resonances, respectively.
0
r
3
PT
xs
Figure 4b. SPll surface modes and resonances. the surface modes and resonances have been presented according to their predominant vibrational character: SP stands for sagittal plane (plane through the surface-normal and the wave vector); SPl and SPll are vibrations in the SP, with displacements predominantly perpendicular and parallel to the surface; SH (shear horizontal) indicates vibrations perpendicular to the SP. For
3
Yr
x’s
Figure 4c. SH surface modes and resonances, The identification of surface modes and surface resonances is in principle done by examination of the vibrational amplitudes as a function of the distance from the surface. However a convenient way to obtain the most important information of this sort is to display, in a three-dimensional plot, the participation of the surface particles in the slab modes contained in two-dimensional (2D) intervals AqAw of the dispersion curve diagrams. Let &(l, IC;qp) be the o-component of the polarization vector associated with particle h: in layer 1 in the mode characterized by the 2D wave vector q and polarization index p (= 1. . .3n; n is the number of particles in the slab unit cell). Since we want to display the predominant vibrational character of the surface modes we choose a representation in which (Y represents the mutually orthogonal SPl, SPll and SH directions. Then the appropriate quantity to be plotted is the normalized surface
squared amplitude:
A,(1 = 1; 9, w) =
Here I = 1 singles out the surface layer and g(q, w) is the density of slab modes in (?j,w)-space, (i.e. g(a, w)&Aw is the number of slab modes whose
773
@, w-values lie in AqAw). The division by g(@,U) assures that the accumulation of a large number of bulk contributions in a given 2D interval Aqhw, does not result in the appearance of a
spurious surface mode or resonance in that interval. The function A, is displayed in Fig. 5 for o = SPl, SPll and SH.
20
20
15
10
5
0 Figure 5a. Three-dimensional plot of the normalized corresponding to the dispersion
surface squared amplitude A,(1 = 1; Q, U) of MgFz curves of Fig. 4 for CI = SPl.
20
15
10
5
0 i-
3 Figure 5b. &(I
7
f
x
= 1; Q,w) of MgFs for (Y= SPll.
-5
T
s
P
T
w
3
Figure 5c. A,(1 = 1; Q,w) of MgFz for cy = SH. The relatively strong surface relaxation gives rise to three modes which, in the I’SYl’ and XS directions, lie above all bulk subbands; two of these modes have their main vibrational amplitude in the top layer; the third mode has its main contribution in the second layer and is not shown in the figures. The highest frequency mode is a small vibration of the F’-ion in i/-direction and a larger vibration of the Mg4-ion in y’-direction. The increase of frequency of this mode is due to the strong relaxation of the Mg4-ion in z’-direction, which reduces the distance between the Mg4- and F’-ions and therefore leads to a stiffening of the F’ - Mg4-bond. In the l%-direction this mode is predominantly SH and in the SY-direction its character is changing from SH to SPlb In the Yl?direction this mode is SPll. In the I’X-direction this mode shows a strong hybridization with the neighboring bulk subband, so that it is only possible to identify it by the increased eigenvector components in the top layer, as can be seen in the three-dimensional plot of the surface’squared amplitude. The mode with the second highest frequency has SPll character and is mainly a vibration of the F2- and F5-ions in the x’-direction
and of the Mg3-ion in the ?/-direction. The frequency shift of this mode is due to the relaxation of the F2- and F5-ions in the x/-direction, which increases the interaction between F2 and Mg3 and between the F5- and Mg3-ions. The lowest frequency mode in Fig. 4a is the macroscopic Rayleigh mode, which has at S a frequency of 3.46 THz. The amplitude of this mode decreases exponentially with distance from the surface, and its penetration depth is proportional to the wavelength. This is why it cannot be found for small values of the wavevector p. However for large values of f the vibration is well localized at the surface. The Rayleigh mode has SPl character and is found to be mainly a vibration of the F2- and F5-ions in the z’-direction. Another mode has been found which, in the i?$YP and XS directions, is well below all bulk subbands (Fig. 4b). This is a microscopic surface mode which has at the zone center a frequency of 1.9 THa. In this mode the F’-ions are vibrating in the x/-direction. Because of the large distance between the F’ ions in the x’-direction, the interaction between the vibrating particles is small; this explains the low frequency of this mode. In the i?ii direction this
775
mode hybridizes with the surrounding bulk bands, In the TX direction all modes can be classified as SP and SH modes. In the SH modes all particle movements are in the y’-direction, whereas the wavevector q is pointing in the x’direction; hence these modes have an almost constant frequency, as can be seen in Fig. 4b. Except for the Rayleigh mode, the modes just described are all microscopic surface modes, which means that their amplitudes diminish very rapidly away from the surface, for all wavelengths. The Rayleigh mode is a macroscopic surface mode, in that its penetration into the crystal increases with the wavelength (see above). The other well-known example of macroscopic modes are the optically active Fuchs-Kliewer modes (FK), which have SP polarization and appear as the loci of hybridizing branches in the longitudinal optical (Lq) and transverse optical (TO) bulk b an d s near I’ (for a detailed description, see Chen et al. 1977 [8]). At r‘ the upper branch of the FK modes starts at the top of the upper LO band, and the lower branch starts at the bottom of the upper TO band. As g increases the two branches merge in the upper LO band. AS the slab thickness is increased the merging occurs closer and closer to I?, so that in a macroscopic crystal, the macroscopic FK mode has, at I’, the frequency of the merged branches. In principle FK mode pairs can occur in each of the five LOTO band pairs that occur in MgFz (110). We have found evidence of at least three FK mode pairs with frequencies of 9.1 THz, 16.5 THz and 16.6 THz near l?. In summary, we predict that the (110) surface of MgFz exhibits a rich spectrum of surface modes, making this an interesting system for experimental investigations, either with high resolution electron energy spectroscopy (HREELS) or inelastic helium atom scattering (HAS).
ACKNOWLEDGEMENTS The authors thank Dr. A. Jockisch for making available his very general computer program with which these calculations were carried out. One of us (F.W. de W.) acknowledges support by the Robert A. Welch Foundation and by the donors of the Petroleum Research Fund, administered by the American Chemical Society. A NATO travel grant is gratefully acknowledged.
REFERENCES 1 R.S. Katiyar and R.S. Krishnan, 45 (1967) 3079.
Can. J. Phys.
2 R. Almairac and C. Benoit, J. Phys. State Phys. 7 (1974) 2614.
C: Solid
3 G.C. Cran and M.J.L. Sangster, J. Phys. Solid State Phys. 7 (1974) 1937.
C:
4 F.W. de Wette, W. Kress, and IT. Schrijder, Phys. Rev. B 32 (1985) 4143. 5 W. Kress, F.W. de Wette, A.D. Kulkarni, and U. Schrijder, Phys. Rev. B 35 (1987) 5783. 6 R. Reiger, J. Prade, U. Schriider, F.W. de Wette, A.D. Kulkarni, and W. Kress, Phys. Rev. B 39 (1989) 7938. 7 H.G. Smith and N. Wakabayashi, Topics in Current Physics Vol. 3, Eds. S.W. Lovesey and T. Springer, Springer Verlag, Heidelberg/Berlin, 1977. 8 T.S. Chen, F.W. de Wette, and G.P. Alldredge, Phys. Rev. B 15 (1977) 1167.
769
Surface relaxation and surface dynamics of crystals of the rutile structure: MgFz (110) G. Pelg”, U. Schrijdera, “Theoretische
Physik,
bMax-Planck-Institut ‘Department
W. Kressb, and F.W. de Wette’ Universit$t
Regensburg,
fiir Festkorperforschung,
of Physics,
University
93040 Regensburg, 70506 Stuttgart,
of Texas, Austin,
FRG FRG
Texas 78712, USA
We extend to crystals of the rutile structure our previous studies of the surface dynamics of ionic crystals within the framework of the shell model. These crystals have many interesting features due to their relatively open structure. Among the crystals of this type, MgFz has a closed-shell electronic configuration; we therefore expect that the surface properties can be calculated using potentials which are obtained from bulk dispersion We find that the bulk dynamics can be described very satisfactorily, if the static charge relations. distribution of the shell of the fluorine ion is shifted with respect to the core. The value of this shift is determined by the equilibrium condition. Using this model we calculate the surface relaxation pattern and dynamics of the (110) surface. Due to the open nature of the structure, we find relaxations both perpendicular to the surface and parallel to the surface. The calculated surface dynamics reveals a variety of surface modes which will be
discussed and compared with the results for other ionic crystals. 1. INTRODUCTION Rutile structure compounds occur in two groups: the fluorides, e.g. MgF2, MnFz, FeF2, etc., and the oxides, e.g. TiOs (rutile), &Ox, NbOz etc. As a result of their relatively open structure, these compounds exhibit many interesting features, such as phase transitions as a function of pressure and temperature. TiO2 is used as a catalyst, which points to the importance of its surface properties in particular, and stimulates interest in the surface properties of the rutile structure compounds in general. The fluorides provide a good starting point for studying the surface dynamics of these crystals; the oxides are more difficult to treat because of the peculiarities of the 02- ion. In this work we study the structural and dynamic properties of the (110) surface of MgF2 in the framework of a shell model. Since the anions in the rutile bulk structure display permanent induced dipole moments (in contrast to the alkali halides and perovskites), it is necessary to first develop a shell model for the bulk dynamics. This is done here in Sec. 2. In Sec. 3 we use the bulk shell model to evaluate the relaxation of the (110) surface. The dynamics of an eleven-lay-
er (llO)-oriented Sec. 4.
slab of MgF2 is discussed
2. BULK SHELL DYNAMICS
MODEL
AND
in
BULK
The bulk lattice dynamics of MgFz has been treated by Katiyar and Krishnan (1967) [l] within the rigid-ion model and by Almairac and Benoit (1974) [2] within the shell model. The present treatment distinguishes itself from that of Almairac and Benoit in two important ways: first, the short-range interactions are treated in a systematic way in terms of short-range potentials, and second, we take account of the fact that the fluorine-ions are at positions of non-zero electric field, which means that these ions have static dipole moments (but such that the total dipole moment of the unit cell is zero). The effect of these dipole moments on the dynamics of the related compound MnF2 has been treated by Cran and Sangster (1974) [3] in terms of a displacedshell model in which the core and the shell of the fluorine ions are allowed to have different equilibbrium positions. Here we adopt this model for
770
the treatment of MgF2. In the displaced-shell model it is assumed that in equilibrium the shells of the anions are centered at the crystalline lattice positions of these ions, while the core positions are shifted (because of symmetry) along the anion-anion axis by an amount which leads to the correct static dipole moment; the latter is obtained from a knowledge of the electric field at the fluorine equilibrium position and the free ion polarixability. The treatment of bulk dynamics in terms of the shell model is one of the standard procedures of lattice dynamics of insulators and needs not be discussed here. However, since the aim of this work is to use the shell model for the study of surface properties, we need to mention that, in addition to the long-range Coulomb potentials, the shortcrange overlap interaction of the ion pairs (i, j) also needs to be expressed in terms of pair potentials, which can then be used at the surface, where the interparticle distances may be different from those in the bulk. For these shortrange potentials we use the Born-Mayer form
xi(r)
=
Uij
eXp(-bijr).
The longitudinal and transverse force constants Aij and Bij which enter into the dynamical matrix (DM) are then given by
where e is the electric charge, V, the volume of the unit cell, and T$ the equilibrium distance between the ions i and j. For details about the use of short-range potentials for surface applications we refer to de Wette et al. (1985) [4], Kress et al. (1987) [5] and Reiger et al. (1989) [6]. In static equilibrium the total potential energy of the crystal should be a minimum. This can be
expressed by the conditions that the derivatives of the total potential energy 0 of the unit cell with respect to the lattice constants and the lattice parameters vanish in equilibrium. In the case of the normal shell model (i.e. undisplaced), this leads to one or more equilibrium conditions containing the Madelung constant and the short-range force constants {Bij). However, in the case of the displaced-shell model, there is an extra condition which expresses the fact that in equilibrium the core-shell displacements of the fluorine ions are kept in balance by the long-range interactions and the core-shell restoring force of the fluorine ions. The Coulomb interactions are evaluated with the Ewald method and are thus not cut off in distance. In deciding which short-range interactions to take into account, we followed the work of Almairac and Benoit (1974) [2]. The magnesium site is octahedral, with fluorine ions at each corner; however the distances ~4 - ~1 and ~4 - 1-2 (see Fig. 1) are slightly different. The fluorine surroundings are less symmetrical and involve 12 neighbors; besides the Mg-F interaction they involve two F-F interactions at different distances. Since the Mg-F and F-F short-range interactions are described by Born-Mayer potentials the effects of the variations in the distances are automatically taken into account. This set of interactions, together with the condition of charge neutrality (2~s = -2zF), leads to a shell model with eight parameters: ionic charge ZF; shell charge YF; core-shell coupling constant kF; Born-Mayer constants a, b of the Mg-F and F-F interactions; and the core-shell shifts Rn of the F ions (Mg is regarded to be unpolarizable) . The four equilibrium conditions reduce this set to four free parameters. Their values were determined by fitting the calculated phonon dispersion curves to the bulk experimental data [7] in the main symmetry directions of the Brillouin zone. The model parameters are given in Tab. 1.
aF_F[$A2] bF_F[_k-l]
u~~-F[EA~]bMvlg-FIA-l]
%+I
zF[el
kF[$l
RDIAl
1.1454 x lo6
1.5142 x lo6
-1.706
-0.822
547.8
0.00529
5.2929
Table 1. Parameters
5.1783
of the shell model for MgF2
771
3. RELAXATION FACE OF MgF2
OF THE
(110)
SUR-
The structure of the (110) surface is displayed in Fig. 1. Surface relaxation and reconstruction are the result of the imbalance of the forces acting on the ion cores and shells, when the ions at or near the surface are in their unrelaxed bulk positions. The symmetry of the (110) surface of the rutile structure is such that the surface ion displacements do not lead to a modified twodimensional surface unit cell (indicated by the heavy outline in Fig. l), compared to that for the case when the bulk is simply terminated at
was calculated by the same procedure as used earlier for alkali halide (001) surfaces (de Wette et al. 1985 [4]) and for the fluoridic perovskites (Reiger et al. 1989 [Sj). For details and the method we refer to these earlier publications. The calculations were performed for a twentyone layer (110) slab of MgFz, in which eight layers were allowed to relax. The results are shown in Fig. 2. 4. DYNAMICS SLAB
OF
THE
RELAXED
The bulk shell model discussed in Sec. 2 was used to calculate the dynamics of an eleven layer MgFz (110) slab in which four layers on each side were relaxed. The dynamical matrix was diagonalised for two-dimensional (2D) wave vectors q along the symmetry directions of the surface Brillouin zone (see Fig. 3).
x Figure
1. Structure
of the (110) surface,
Figure 3. Two-dimensional surface Brillouin Zone (SBZ) for the (110) surface of MgFs.
Figure 2. Relaxation of the (110) surface of MgF2 (bold: relaxed, light: unrelaxed). Side view of the two outer “layers”, each consisting of 3 planes of ions. The relaxation shifts are enhanced by a factor of 1.3. the surface. We call this restructuring surface relaxation, although it involves shifts of the fluorine ions parallel to the surface plane. The relaxation
The slab dispersion curves of the modes with their main vibrational amplitude in the first layer are plotted in Fig. 4. The shaded areas represent the bulk modes of the slab. Since the bulk unit cell contains six particles (eighteen degrees of freedom), there are eighteen bulk dispersion curves, three of which are acoustic and fifteen optic. Each of these bands can give rise to surface localized modes or resonances; in Fig. 4 these are indicated by bold and dashed lines, respectively. The type of surface modes and resonances that occur for each individual compound is determined by the structure of gaps between the bulk bands and by the way the bands overlap. Because the mass ratio of Mg and F is small, there is significant over-
772
lap of the bulk bands, with the result that there are relatively few well-pronounced surface modes and relatively many surface resonances. In Fig. 4
a more detailed discussion of the character and classification of these surface modes we refer to Kress et al. (1987) [5] and Reiger et al. (1989) [6].
20
15
10
5
0
0
0
r Figure 4a. Dispersion curves for an eleven-layer (110) slab of MgFz. The solid and dashed lines are SPI surface modes and resonances, respectively.
0
r
3
PT
xs
Figure 4b. SPll surface modes and resonances. the surface modes and resonances have been presented according to their predominant vibrational character: SP stands for sagittal plane (plane through the surface-normal and the wave vector); SPl and SPll are vibrations in the SP, with displacements predominantly perpendicular and parallel to the surface; SH (shear horizontal) indicates vibrations perpendicular to the SP. For
3
Yr
x’s
Figure 4c. SH surface modes and resonances, The identification of surface modes and surface resonances is in principle done by examination of the vibrational amplitudes as a function of the distance from the surface. However a convenient way to obtain the most important information of this sort is to display, in a three-dimensional plot, the participation of the surface particles in the slab modes contained in two-dimensional (2D) intervals AqAw of the dispersion curve diagrams. Let &(l, IC;qp) be the o-component of the polarization vector associated with particle h: in layer 1 in the mode characterized by the 2D wave vector q and polarization index p (= 1. . .3n; n is the number of particles in the slab unit cell). Since we want to display the predominant vibrational character of the surface modes we choose a representation in which (Y represents the mutually orthogonal SPl, SPll and SH directions. Then the appropriate quantity to be plotted is the normalized surface
squared amplitude:
A,(1 = 1; 9, w) =
Here I = 1 singles out the surface layer and g(q, w) is the density of slab modes in (?j,w)-space, (i.e. g(a, w)&Aw is the number of slab modes whose
773
@, w-values lie in AqAw). The division by g(@,U) assures that the accumulation of a large number of bulk contributions in a given 2D interval Aqhw, does not result in the appearance of a
spurious surface mode or resonance in that interval. The function A, is displayed in Fig. 5 for o = SPl, SPll and SH.
20
20
15
10
5
0 Figure 5a. Three-dimensional plot of the normalized corresponding to the dispersion
surface squared amplitude A,(1 = 1; Q, U) of MgFz curves of Fig. 4 for CI = SPl.
20
15
10
5
0 i-
3 Figure 5b. &(I
7
f
x
= 1; Q,w) of MgFs for (Y= SPll.
-5
T
s
P
T
w
3
Figure 5c. A,(1 = 1; Q,w) of MgFz for cy = SH. The relatively strong surface relaxation gives rise to three modes which, in the I’SYl’ and XS directions, lie above all bulk subbands; two of these modes have their main vibrational amplitude in the top layer; the third mode has its main contribution in the second layer and is not shown in the figures. The highest frequency mode is a small vibration of the F’-ion in i/-direction and a larger vibration of the Mg4-ion in y’-direction. The increase of frequency of this mode is due to the strong relaxation of the Mg4-ion in z’-direction, which reduces the distance between the Mg4- and F’-ions and therefore leads to a stiffening of the F’ - Mg4-bond. In the l%-direction this mode is predominantly SH and in the SY-direction its character is changing from SH to SPlb In the Yl?direction this mode is SPll. In the I’X-direction this mode shows a strong hybridization with the neighboring bulk subband, so that it is only possible to identify it by the increased eigenvector components in the top layer, as can be seen in the three-dimensional plot of the surface’squared amplitude. The mode with the second highest frequency has SPll character and is mainly a vibration of the F2- and F5-ions in the x’-direction
and of the Mg3-ion in the ?/-direction. The frequency shift of this mode is due to the relaxation of the F2- and F5-ions in the x/-direction, which increases the interaction between F2 and Mg3 and between the F5- and Mg3-ions. The lowest frequency mode in Fig. 4a is the macroscopic Rayleigh mode, which has at S a frequency of 3.46 THz. The amplitude of this mode decreases exponentially with distance from the surface, and its penetration depth is proportional to the wavelength. This is why it cannot be found for small values of the wavevector p. However for large values of f the vibration is well localized at the surface. The Rayleigh mode has SPl character and is found to be mainly a vibration of the F2- and F5-ions in the z’-direction. Another mode has been found which, in the i?$YP and XS directions, is well below all bulk subbands (Fig. 4b). This is a microscopic surface mode which has at the zone center a frequency of 1.9 THa. In this mode the F’-ions are vibrating in the x/-direction. Because of the large distance between the F’ ions in the x’-direction, the interaction between the vibrating particles is small; this explains the low frequency of this mode. In the i?ii direction this
775
mode hybridizes with the surrounding bulk bands, In the TX direction all modes can be classified as SP and SH modes. In the SH modes all particle movements are in the y’-direction, whereas the wavevector q is pointing in the x’direction; hence these modes have an almost constant frequency, as can be seen in Fig. 4b. Except for the Rayleigh mode, the modes just described are all microscopic surface modes, which means that their amplitudes diminish very rapidly away from the surface, for all wavelengths. The Rayleigh mode is a macroscopic surface mode, in that its penetration into the crystal increases with the wavelength (see above). The other well-known example of macroscopic modes are the optically active Fuchs-Kliewer modes (FK), which have SP polarization and appear as the loci of hybridizing branches in the longitudinal optical (Lq) and transverse optical (TO) bulk b an d s near I’ (for a detailed description, see Chen et al. 1977 [8]). At r‘ the upper branch of the FK modes starts at the top of the upper LO band, and the lower branch starts at the bottom of the upper TO band. As g increases the two branches merge in the upper LO band. AS the slab thickness is increased the merging occurs closer and closer to I?, so that in a macroscopic crystal, the macroscopic FK mode has, at I’, the frequency of the merged branches. In principle FK mode pairs can occur in each of the five LOTO band pairs that occur in MgFz (110). We have found evidence of at least three FK mode pairs with frequencies of 9.1 THz, 16.5 THz and 16.6 THz near l?. In summary, we predict that the (110) surface of MgFz exhibits a rich spectrum of surface modes, making this an interesting system for experimental investigations, either with high resolution electron energy spectroscopy (HREELS) or inelastic helium atom scattering (HAS).
ACKNOWLEDGEMENTS The authors thank Dr. A. Jockisch for making available his very general computer program with which these calculations were carried out. One of us (F.W. de W.) acknowledges support by the Robert A. Welch Foundation and by the donors of the Petroleum Research Fund, administered by the American Chemical Society. A NATO travel grant is gratefully acknowledged.
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