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Phonon states on the (100), (110) and (111) aluminium surfaces
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ELSEVIER
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Surface Science331-333 (1995) 1414-1421
Phonon states on the (100), (110) and (111) aluminium surfaces E.V. Chulkov *, I.Yu. Sklyadneva Institute of Strength Physics and Materials Science of the Russian Academy of Sciences, pr. Academicheskii 2 / 1 , 634048 Tomsk, Russian Federation
Received 27 July 1994; acceptedfor publication 7 December 1994
Abstract
We present the calculation of vibrational states on the relaxed surfaces (100), (110) and (111) of aluminium. The surface phonon frequencies and polarizations are calculated with the use of the interatomic interaction potential obtained in the framework of the embedded-atom method. New surface modes have been found on the AI(100) surface along the FM direction and at the X and M symmetry points. On the AI(ll0) surface new surface phonon states have been obtained along the FX, FY directions and at the F and ~( points. For the AI(lll) surface the present calculation gives new surface modes along FK and FM. Two surface modes are observed within the energy gap at the I~ point. Keywords: Aluminum;Phonons;Semi-empirical models and model calculations
1. Introduction
Vibrational states on the low-index surfaces of aluminium were investigated both theoretically and experimentally [1-11]. Black et al. [1,2] calculated the surface phonons for AI(100) and A I ( l l l ) using the force-constant model with parameters fitted to experimental bulk phonon data. A comparison with the experimental measurements shows that the calculated frequencies tend to be significantly overestimated. Franchini et al. [3] analyzed the experimental surface phonon data obtained with helium-atom inelastic scattering (HAS) for A1 surfaces using a semiempirical model. Ditlevsen and N0rskov [4] calculated vibrational modes for A I ( l l l ) using the effective medium theory (EMT). The most accurate calculations were performed by Ho and Bohnen [5,6]
* Corresponding author.
and Eguiluz et al. [7-9]. Ho and Bohnen calculated surface phonons on AI(ll0) and AI(100) [5,6] at the symmetry points making use of the first-principles self-consistent pseudopotential method. Eguiluz et al. [7-9] performed accurate calculations of the screening response of the ion core by the conduction electrons. To obtain the total energy and phonon frequencies they used the second order of the pseudopotenfial perturbation theory, which appeared to give no way of adequate description of the behaviour of the resonance longitudinal mode propagating along the F X direction on AI(100) [3,8]. Experiments on vibrational states on the aluminium surfaces were performed by the HAS method for AI(100) [8], AI(ll0) [10], A I ( l l l ) [11] and by high-resolution electron-energy-loss spectroscopy (HREELS) for AI(100) [12]. In this paper we present systematic calculations performed by the embedded-atom method (EAM) [13,14] for the relaxed (100), (110) and (111) alu-
0039-6028/95/$09.50 © 1995 Elsevier ScienceB.V. All rights reserved SSDI 0039-6028(95)00197-2
E.V. Chulkov, L Yu. Sklyadneva / Surface Science 331-333 (1995) 1414-1421
minium surfaces. We have found a number of new surface phonon states for all surfaces of interest.
2. Calculation method To construct interatomic interaction potentials we use the embedded-atom method [13,14]. However contrary to Refs. [13,14] the free atom charge density for the embedding function is calculated using the local density approximation [15] for the valence configuration 3s 2"° 3p 1°. The parameters of the method were obtained by fitting to the experimental values of the equilibrium lattice constant, sublimation energy E, bulk modulus B, elastic constants Ca1 and C44 and vacancy-formation energy Eav. The lattice constant, sublimation energy, bulk modulus and vacancy-formation energy are exactly reproduced [ 1 6 18]. The difference between the experimental and calculated values of C n and C44 does not exceed 6%. This accuracy of fitting enables us to obtain a reasonable description of the bulk vibrations in AI and the surface vibrational modes too.
3. Calculation results and discussion
3.1. Bulk vibrations The calculated and measured bulk-phonon frequencies of aluminium along the symmetry directions (100), (110), (111) are shown in Fig. 1. As can
be seen from the figure the calculated dispersion curves agree with the experimental values. The best agreement is observed for the transverse modes. The largest differences ( ~ 10%) are found for the longitudinal modes near the symmetry points X and L. The disagreement is partly due to the fact that for the chosen valence electron configuration 3s 2"° 3p 1° it is impossible to reproduce C u determining the behaviour of the longitudinal modes at small q with an accuracy better than 6% and no fitting to the zoneboundary phonons has been done. It is interesting to note that in the case of Cu the EAM method also provides a good description of transverse modes but the frequencies of the longitudinal modes are ~ 6% higher than the experimental values [20].
3.2. Surface vibrations 3.2.1. The Al(lO0) surface The relaxation of the AI(100) surface has been found to be small. The changes in interlayer spacing are A12 = -- 2.7%, A23 = -- 0.1%. Our results agree with the values obtained with the effective medium theory by Ditlevsen and Ncrskov [4] A12 = --3%, while the pseudopotential calculation [6] shows a small expansion of the first two interlayer spacings A12 = 1.2%, A23 = 0.2%. Generally these results are in reasonable agreement with the experimental estimates of the LEED study [21] in which no relaxation was found for the clean (100) surface of A1 A12 = 0.
10 A'
,.._J
A• •
6 Z
)C
Z Z. "
A
cy 4 2
1415
T1
/
L
/
x o.5 X 0.75 F F L Fig. 1. Calculated and measured bulk-phononfrequencies of A1. The experimental points are taken from Ref. [19].
1416
E.V. Chulkov, L Yu. Sklyadneva / Surface Science 331-333 (1995) 1414-1421
The phonon dispersion curves calculated for the relaxed 48-layer AI(100) film are shown in Fig. 2. Surface vibrational states are depicted by full circles. It is obvious from the figure that there are three surface modes propagating along the F X direction. The lowest state corresponds to shear horizontal movements of surface atoms. The second state is the Rayleigh mode. The third state is a group of surface resonances being predominantly of longitudinal character with a slight shear vertical component of surface atom displacements. The frequency difference does not exceed 0 . 1 - 0 . 2 THz. Four surface states are observed at the X point. The two lower modes are the continuations of the corresponding modes propagating along F X . The third state at 3.8 THz is characterized by shear horizontal displacements of second layer atoms. The fourth state is a typical gap mode with longitudinal movements of the top layer atoms. It follows from Fig. 2 that three surfaces states ~ e observed along the F M direction. The lowest one is the Rayleigh mode. The second state is characterized by shear horizontal displacements of atoms of the two upper layers. The third mode is a group of surface resonances with predominantly longitudinal polarization and a small portion of vertical displacemerits of the top layer atoms. A t the M point three surface states have been found. The upper state corresponds to shear vertical movements of second layer atoms. There are five surface states along XM. The two lower modes are the continuations of the
8.0
Table 1 Frequencies of the surface modes for the relaxed AI(100) surface Symmetry Mode Frequency(THz) point This work Expt. Ho, Bohnen
M
Unrel. Relax. [12]
[6]
r'l
z.3
2.7
-
3.0
Z1 112 X1 Z1 X1 = Y1 Z2
3.4 3.6 6.4 4.1 5.4 5.6
3.5 3.8 6.8 4.4 5.5 5.9
3.66___0.15 3.6 8.1 4.9 5.9 -
Atomic displacements X, Y, and Z correspond to [110], [-110] and [001] directions respectively. The subscripts 1 and 2 at the mode label denote the surface and second layers respectively. Rayleigh and transverse states propagating along F M. There are also two gap states. The calculated surface-phonon frequencies at and M are summarized in Table 1 together with results of the calculations given in Ref. [6] and the H R E E L S measurement [12]. For comparison we show here the surface-phonon frequencies calculated for the unrelaxed AI(100) surface. It is evident from the table that the experimental Rayleigh mode frequency at X and theoretical values are in good agreement. At the same time the number of surface states obtained by E A M is higher than that was found in the calculation [6]. A comparison of our results with those obtained in Ref. [6] shows that the frequencies calculated by E A M are 5 - 1 0 % lower
--
6.0
4.o N ~
2.0
0.0 Fig. 2. Calculated phonon dispersion curves for the relaxed 48-layer AI(100) film.
E. V. Chulkov, L Yu. Sklyadneva / Surface Science 331-333 (1995) 1414-1421
than those obtained in the first-principles approach [6]. This result is due to using charge density as a superposition of atomic densities rather than the self-consistent one. The greatest difference of 1.3 THz is observed for the longitudinal gap mode at X. This discrepancy is mainly explained by the effect of decreasing the longitudinal modes for bulk AI ~ 10% at X and L in this method. Our results are generally in fairly good agreement with theoretical [6] and experimental [12] values. Comparing our results obtained for the relaxed and unrelaxed geometries one can conclude that the relaxation has the effect of raising the frequencies of phonons and allows to obtain the better agreement with the experimental and theoretical values [6,12]. A comparison with the results of the microscopic self-consistent calculations [3,8,9], first-principles total-energy calculation [6] and HAS [3,8] and HREELS [12] measurements shows that we have found new longitudinally polarized surface mode and resonance propagating along FM. A new surface state characterized by shear vertical displacem__ents of the second layer atoms has been obtained at M, At the X point a new shear horizontal mode localized on the second layer is observed. The calculated surface resonance along F X is in agreement with the HAS measurements [3,8]. The dependence of the results on the number of film layers has been tested in the calculation of a 24-layer and 30-layer films. The resulting phonon dispersion curves for the relaxed 24-layer and 30-
1417
layer AI(100) films are practically the same. The changes of the frequency values do not exceed 0.1 THz. Therefore we will examine AI(110) and AI(111) surfaces on the basis of the calculations performed for 24-layer films. 3.2.2. The Al(110) surface
The results obtained for the (110) surface show a multilayer oscillatory character of relaxation. The changes in interlayer spacings are A12 = - - 7 . 4 % , A23 "~" q - 0 . 8 % , A34 = -- 0.6% and A45 = + 0.3%. Similar results were obtained from a LEED experiment [22,23]: their values are z112= - 8 . 5 _ 1.0%, A23 ~-- - - 5 . 0 "-[- 1.1%, A34 ~" -- 1.6 + 1.2%. The best agreement is observed for A12. As one goes into the bulk the change in interlayer spacings becomes rapidly smaller. Generally this trend agrees well with experiment, but the magnitude of the effect is overestimated. The first-principles pseudopotential calculation of Ho and Bohnen [5] gives Z~12 = - - 6 . 8 % , A23 ~- +3.5%, A34 = - 2 . 0 % . Eguiluz et al. [7] using pseudopotential perturbation theory found A~2 = --5.4%, A23 = +0.8%, A34 = --2.6%, while the values of Ditlevsen and NCrskov [4] obtained with effective medium theory are A12 = - - 6 % , Z123= + 1%. A qualitative agreement between the theoretical and experimental values [22,23] is rather good. The phonon dispersion curves calculated for the relaxed 24-layer AI(ll0) film are shown in Fig. 3. It is obvious from Fig. 3 there are three surface states at F. The lowest state with a frequency of 014 THz is
8.0
6.0
b 4-.0
2.0
0.0_ Fig. 3. Calculated phonon dispersion curves for the relaxed 24-layer AI(110) film.
E . K Chulkov, LYu. Sklyadneva / S u r f a c e Science 3 3 1 - 3 3 3 (1995) 1 4 1 4 - 1 4 2 1
1418
m
a mode of longitudinal movements of surface layer atoms. The second state describes shear horizontal displacements of surface atoms. The squared amplitude of atom vibration of these states decreases slowly upon movement inside the film. The upper state is characterized by shear vertical movements of atoms of two upper layers. Along the F X symmetry direction five surface states are observed. The lowest state is the Rayleigh mode with shear vertical displacements of surface atoms. It becomes appreciable at kx > 0.5 I FXI. The second state corresponds to shear horizontal movements of surface atoms. The third phonon state is longitudinal in the first layer at small k x. A t F this state is the lowest one. On increasing k x the longitudinal portion of displacements of atoms decreases and shear vertical portion of displacements generated by the second layer atoms increases. Simultaneously we have a change of this state weight. About 60% of this state is localized in the surface layer and 24% corresponds to the second layer at k x = 0.5 I F X 1. At the X point about 10% of this state weight corresponds to the surface layer and 50% of this state is localized in the second la_yer. The fourth state at 5.6 THz at F extends along F X up to k~ = 0.35 I FXI. On increasing k x the longitudinal portion of displacements of atoms increases. The
_
_
fifth surface state begins at k x = 0.7 ] F X [ . Near the X point this state is a typical gap first layer dominated longitudinal mode. As follows from Fig. 3 there are six surface states along the F Y symmetry direction. The frequency values of the two lower states are very close and their difference does not exceed 0.1 THz at a n y ky. The lowest surface state has longitudinal polarization in the outermost layer and shear vertical polarization in the second layer. The other state of this couple is characterized by shear horizontal displacements of surface layer atoms. The third state is predominantly longitudinal in the surface layer a t k y ~ 0 . 5 [ F Y ] and contains a small portion of vertical atom displacements. Upon increase of kv the z-polarized contribution increases and at the Y point this state is a distinct Rayleigh mode, characterized by shear vertical displacements of the outermost layer atoms. Below the energy gap at Y there is a fourth surface state localized in the second layer whose atoms move along the [ - 1 1 0 ] direction. The fifth surface state is located at the bottom of the projection of the bulk-phonon structure. It is longitudinal in the second and the fourth layers and polarized perpendicular to the surface in the first and third layers. The sixth state is a typical gap mode longitudinal in the surface layer. As one
Table 2 Frequencies of the surface modes for the relaxed AI(ll0) Symmetry point
Mode
surface
Frequency (THz) This work
~(
E x p t . [10]
H o , B o h n e n [5]
E g u i l u z et al. [7]
Unrel.
Relax.
Z1, Z e
5.3
5.6
-
-
-
Z1
3.3
3.43
3.53 ± 0.05
4.2
4.6
X1Z e
3.6
3.6
-
4.1
-
Y1
3.8
4.4
-
5.5
-
ZI X 2
-
6.0
-
6.0
-
Y2
5.5
6.0
-
6.3
-
X1
6.2
6.3
-
7.7
-
Ya
2.73
2.76
2.15 _+ 0 . 0 7
1.9
3.1
Xx Z1
2.8 3.3
2.84 3.5
3 . 2 6 + 0.05
3.4
3.3
Xz
4.0
4.1
-
3.6
-
Z1Y2
-
](1
4.9
4.7 5.5
-
6.0
-
Y1
3.0
3.3
-
-
-
Z1 )(1
3.4 5.9
3.5 6.6
-
-
-
A t o m i c d i s p l a c e m e n t s X , Y, a n d Z c o r r e s p o n d to [ - 110], [001] a n d [110] d i r e c t i o n s , r e s p e c t i v e l y .
1419
E. K Chulkov, L Yu. Sklyadneva / Surface Science 331-333 (1995) 1414-1421
can see i n F i g . 3__all surface states in symmetry directions XS and YS are the continuations of corresponding states extending along F X and F Y symmetry lines. The calculated values of surface-phonon frequencies at the symmetry points F, X, Y and S are summarized in Table 2. One can see from the table the calculation gives higher frequencies for the relaxed geometry than for the unrelaxed one. It agrees to the tendency for the AI(100) surface and to the calculations [5,7] performed for AI(ll0). For comparison the results of first-principles and microscopic calculations [5,7] and experimental values [10] are shown. As follows from the table the surface mode at F characterized by shear vertical displacements of atoms of the first and second layers is obtained solely in the present calculations. It is not clear why this state has not been found in the calculations of Refs. [5,7] and in experiment [10]. A similar state was observed on the Ag(ll0), Cu(ll0) and Pd(ll0) surfaces [24,25]. A comparison of the theoretical results with the experimental value of the Rayleigh mode frequency at X shows that the best agreement is obtained in the present calculation. The calculations [5,7] tend to be overestimated. Comparing our results with those obtained by Ho and Bohnen [5] one can see that EAM gives lower frequency values at X than the first-principles method. The greatest difference is obtained for the gap longitudinal mode. A different situation is observed at the Y point. Here
all our frequencies are higher than those calculated by Ho and Bohnen [5] except for the high frequency longitudinal mode. It is interesting to note that all three calculations give a value of the Rayleighmode in good agreement with experiment [10] at Y. The best agreement was achieved by Eguiluz et al. [7]. The present calculation generally gives the surface states in agreement with other calculations [5,7] and the experiment [10]. New results obtained are the surface mode at F, two surface modes at Y and three surface modes at the S point. 3.2.3. The Al(111) surface
The embedded-atom calculation of the (111) surface relaxation gives the values for A12 = --1.8%, A23 = + 0.1%. These results are in reasonable agreement with the values obtained by Ditlevsen and Norskov [4], A12 = - 1%, but differ from the experimental estimates [23], A12 = 0.9%. The phonon dispersion curves obtained for the relaxed 24-layer A1(111) film are shown in Fig. 4. As one can see in the figure there are four surface states along the F K direction. The :lowest one is the Rayleigh mode. The second state is characterized by displacements of the top layer atoms in the plane perpendicular to the wave vector and by shear vertical movements of the second layer atoms. The third state is a gap mode which corresponds to movements of the atoms in the surface plane. The fourth state is a group of resonance states characterized by longitu-
8.0
"~ 6.0
c~
4.0
2.0
0.0
n f
2
~
Fig. 4. Calculatedphonon dispersioncurvesfor the relaxed 24-layerA1(111)film.
f
1420
E.V. Chulkov, LYu. Sklyadneva / Surface Science 331-333 (1995) 1414-1421
Table 3 Frequencies of the surface modes for the relaxed A I ( l l l ) surface Symmetry point
Mode
Frequency (THz) This work Unrel.
Relax.
Z1 (XY)IZ 2
3.8
(XY) 1 (XY)I Z1 (XY) 1 (XY) 1
5.9 6.2 3.5 3.7 7.1
4.0 4.7 6.0 6.4 3.6 3.7 7.3
4.6
Expt. [11]
Black et al. [2]
Ditlevsen, NCrskov [4]
4.4 4.0 -
5.8 7.0 8.2 5.3 5.7 10.0
4.1 6.0 3.8 7.2
Atomic displacements X, Y, and Z correspond to [1 - 10], [11 - 2] and [11t] directions, respectively.
dinal displacements of the atoms of the first and second layers. The frequency value of these resonance states are very close and their difference does not exceed 0.1 THz. We have found four surface modes at the K point. Two of them characterized by horizontal movements of surface layer atoms are in the energy gap. The second lower state also corresponds to the horizontal displacements of surface atoms. There are four surface states along the F M direction. The difference between the frequencies of the two lower states is ~ 0.1 THz. The lowest state is the Rayleigh mode. The next state corresponds to the horizontal movements of the surface layer atoms. The third mode is a group of horizontally polarized resonances. The frequency difference does not exceed 0.10-0.15 THz. The fourth state is a gap mode. All surface states in the KM direction are the continuation of the corresponding states obtained in the F K and F M directions. The resulting frequencies at the symmetry points K and M are given in Table 3 together with those calculated in Refs. [2,4] and measured by HAS method in Ref. [11]. As follows from the table the values obtained with EAM and EMT are very close to each other, but they are lower than the experimental value. At the same time the parametrized forceconstant model tends to overestimate the frequencies. The new results obtained for A I ( l l l ) are as follows: (1) two surface states in the energy gap at the K point, (2) two surface resonance states in the F K direction, (3) two surface resonance modes along the FM symmetry line.
4. Conclusion The calculations of the surface phonon frequencies for the low-index aluminium surfaces have been performed using the embedded-atom method. New surface modes have been found on all surfaces of interest.
References [1] J.E. Black, D.A. Campbell and R.F. Wallis, Surf. Sci. 116 (1982) 161. [2] J.E. Black, F.C. Shanes and R.F. Wallis, Surf. Sci. 133 (1983) 199. [3] A. Franchini, V. Bortolani, G. Santoro, J.F. Gaspar and A.G. Eguiluz, Surf. Sci. 269/270 (1992) 146. [4] P.D. Ditlevsen and J.K. NCrskov, Surf. Sci. 254 (1991) 261. [5] K.M. Ho and K.P. Bohnen, Phys. Rev. Lett. 56 (1986) 934. [6] K.P. Bohneu and K.M. Ho, Surf. Sci. 207 (1988) 105. [7] A.G. Eguiluz, A.A. Maradudin and R.F. Wallis, Phys. Rev. Lett. 60 (1988) 309. [8] J.A. Gaspar, A.G. Eguiluz, M. Gester, A. Lock and J.P. Toennies, Phys. Rev. Lett. 66 (1991) 337. [9] J.A. Gaspar and A.G. Eguiluz, Phys. Rev. B 40 (1989) 11976. [10] J.P. Toennies and Ch. W/511, Phys. Rev. B 36 (1987) 4475. [11] A. Lock, J.F. Toennies, Ch. Woll, V.B. Bortolani, A. Franchini and G. Santoro, Phys. Rev. B 37 (1988) 7087. [12] M.H. Mohamed and L.L. Kesmodel, Phys. Rev. B 37 (1988) 6519. [13] M.S. Daw and M.I. Baskes, Phys. Rev. B 29 (1984) 6443. [14] S.M. Foiles, M.I. Baskes and M.S. Daw, Phys. Rev. B 33 (1986) 7983. [15] W. Kohn and L.J. Sham, Phys. Rev. A 140 (1965) 1133. [16] G. Simmons and H. Wang, Single Crystal Elastic Constants and Calculated Aggregate Properties: A Handbook (MIT Press, Cambridge, 1971).
E. V. Chulkov, L Yu. Sklyadneva / Surface Science 331-333 (1995) 1414-1421 [17] R.W. Balluffi, J. Nucl. Mater. 69 70 (1978) 240. [18] C Kittel, Introduction to Solid State Physics (Wiley, New York, 1976). [19] R. Stedman and G. Nilsson, Phys. Rev. 145 (1966) 492. [20] A.V. Bertch, E.V. Chnlkov, S.V. Eremeev A.G. Lipnitskii, G.G. Rusina and I.Yu. Sldyadneva, Vacuum, in press. [21] D.W. Jepsen, P.M. Marcus and F. Joua, Phys. Rev. B 5 (1972) 3933.
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[22] J.N. Andersen, H.B. Nielsen, L. Peterson and D.L. Adams, J. Phys. C 17 (1984) 173. [23] J.R. Noonan and H.L. Davies, Phys. Rev. B 29 (1984) 4349. [24] G. Bracco, R. Tatarek, F. Tommasini, V. Linke and M. Persson, Phys. Rev. B 36 (1987) 2928. [25] A.M. Lance, J,P. Toennies and Ch. Woll, Surf. Sci. 191 (1987) 529.
ELSEVIER
~ :
~ ,~, ~i~ ~: ~::~i~ i
Surface Science331-333 (1995) 1414-1421
Phonon states on the (100), (110) and (111) aluminium surfaces E.V. Chulkov *, I.Yu. Sklyadneva Institute of Strength Physics and Materials Science of the Russian Academy of Sciences, pr. Academicheskii 2 / 1 , 634048 Tomsk, Russian Federation
Received 27 July 1994; acceptedfor publication 7 December 1994
Abstract
We present the calculation of vibrational states on the relaxed surfaces (100), (110) and (111) of aluminium. The surface phonon frequencies and polarizations are calculated with the use of the interatomic interaction potential obtained in the framework of the embedded-atom method. New surface modes have been found on the AI(100) surface along the FM direction and at the X and M symmetry points. On the AI(ll0) surface new surface phonon states have been obtained along the FX, FY directions and at the F and ~( points. For the AI(lll) surface the present calculation gives new surface modes along FK and FM. Two surface modes are observed within the energy gap at the I~ point. Keywords: Aluminum;Phonons;Semi-empirical models and model calculations
1. Introduction
Vibrational states on the low-index surfaces of aluminium were investigated both theoretically and experimentally [1-11]. Black et al. [1,2] calculated the surface phonons for AI(100) and A I ( l l l ) using the force-constant model with parameters fitted to experimental bulk phonon data. A comparison with the experimental measurements shows that the calculated frequencies tend to be significantly overestimated. Franchini et al. [3] analyzed the experimental surface phonon data obtained with helium-atom inelastic scattering (HAS) for A1 surfaces using a semiempirical model. Ditlevsen and N0rskov [4] calculated vibrational modes for A I ( l l l ) using the effective medium theory (EMT). The most accurate calculations were performed by Ho and Bohnen [5,6]
* Corresponding author.
and Eguiluz et al. [7-9]. Ho and Bohnen calculated surface phonons on AI(ll0) and AI(100) [5,6] at the symmetry points making use of the first-principles self-consistent pseudopotential method. Eguiluz et al. [7-9] performed accurate calculations of the screening response of the ion core by the conduction electrons. To obtain the total energy and phonon frequencies they used the second order of the pseudopotenfial perturbation theory, which appeared to give no way of adequate description of the behaviour of the resonance longitudinal mode propagating along the F X direction on AI(100) [3,8]. Experiments on vibrational states on the aluminium surfaces were performed by the HAS method for AI(100) [8], AI(ll0) [10], A I ( l l l ) [11] and by high-resolution electron-energy-loss spectroscopy (HREELS) for AI(100) [12]. In this paper we present systematic calculations performed by the embedded-atom method (EAM) [13,14] for the relaxed (100), (110) and (111) alu-
0039-6028/95/$09.50 © 1995 Elsevier ScienceB.V. All rights reserved SSDI 0039-6028(95)00197-2
E.V. Chulkov, L Yu. Sklyadneva / Surface Science 331-333 (1995) 1414-1421
minium surfaces. We have found a number of new surface phonon states for all surfaces of interest.
2. Calculation method To construct interatomic interaction potentials we use the embedded-atom method [13,14]. However contrary to Refs. [13,14] the free atom charge density for the embedding function is calculated using the local density approximation [15] for the valence configuration 3s 2"° 3p 1°. The parameters of the method were obtained by fitting to the experimental values of the equilibrium lattice constant, sublimation energy E, bulk modulus B, elastic constants Ca1 and C44 and vacancy-formation energy Eav. The lattice constant, sublimation energy, bulk modulus and vacancy-formation energy are exactly reproduced [ 1 6 18]. The difference between the experimental and calculated values of C n and C44 does not exceed 6%. This accuracy of fitting enables us to obtain a reasonable description of the bulk vibrations in AI and the surface vibrational modes too.
3. Calculation results and discussion
3.1. Bulk vibrations The calculated and measured bulk-phonon frequencies of aluminium along the symmetry directions (100), (110), (111) are shown in Fig. 1. As can
be seen from the figure the calculated dispersion curves agree with the experimental values. The best agreement is observed for the transverse modes. The largest differences ( ~ 10%) are found for the longitudinal modes near the symmetry points X and L. The disagreement is partly due to the fact that for the chosen valence electron configuration 3s 2"° 3p 1° it is impossible to reproduce C u determining the behaviour of the longitudinal modes at small q with an accuracy better than 6% and no fitting to the zoneboundary phonons has been done. It is interesting to note that in the case of Cu the EAM method also provides a good description of transverse modes but the frequencies of the longitudinal modes are ~ 6% higher than the experimental values [20].
3.2. Surface vibrations 3.2.1. The Al(lO0) surface The relaxation of the AI(100) surface has been found to be small. The changes in interlayer spacing are A12 = -- 2.7%, A23 = -- 0.1%. Our results agree with the values obtained with the effective medium theory by Ditlevsen and Ncrskov [4] A12 = --3%, while the pseudopotential calculation [6] shows a small expansion of the first two interlayer spacings A12 = 1.2%, A23 = 0.2%. Generally these results are in reasonable agreement with the experimental estimates of the LEED study [21] in which no relaxation was found for the clean (100) surface of A1 A12 = 0.
10 A'
,.._J
A• •
6 Z
)C
Z Z. "
A
cy 4 2
1415
T1
/
L
/
x o.5 X 0.75 F F L Fig. 1. Calculated and measured bulk-phononfrequencies of A1. The experimental points are taken from Ref. [19].
1416
E.V. Chulkov, L Yu. Sklyadneva / Surface Science 331-333 (1995) 1414-1421
The phonon dispersion curves calculated for the relaxed 48-layer AI(100) film are shown in Fig. 2. Surface vibrational states are depicted by full circles. It is obvious from the figure that there are three surface modes propagating along the F X direction. The lowest state corresponds to shear horizontal movements of surface atoms. The second state is the Rayleigh mode. The third state is a group of surface resonances being predominantly of longitudinal character with a slight shear vertical component of surface atom displacements. The frequency difference does not exceed 0 . 1 - 0 . 2 THz. Four surface states are observed at the X point. The two lower modes are the continuations of the corresponding modes propagating along F X . The third state at 3.8 THz is characterized by shear horizontal displacements of second layer atoms. The fourth state is a typical gap mode with longitudinal movements of the top layer atoms. It follows from Fig. 2 that three surfaces states ~ e observed along the F M direction. The lowest one is the Rayleigh mode. The second state is characterized by shear horizontal displacements of atoms of the two upper layers. The third mode is a group of surface resonances with predominantly longitudinal polarization and a small portion of vertical displacemerits of the top layer atoms. A t the M point three surface states have been found. The upper state corresponds to shear vertical movements of second layer atoms. There are five surface states along XM. The two lower modes are the continuations of the
8.0
Table 1 Frequencies of the surface modes for the relaxed AI(100) surface Symmetry Mode Frequency(THz) point This work Expt. Ho, Bohnen
M
Unrel. Relax. [12]
[6]
r'l
z.3
2.7
-
3.0
Z1 112 X1 Z1 X1 = Y1 Z2
3.4 3.6 6.4 4.1 5.4 5.6
3.5 3.8 6.8 4.4 5.5 5.9
3.66___0.15 3.6 8.1 4.9 5.9 -
Atomic displacements X, Y, and Z correspond to [110], [-110] and [001] directions respectively. The subscripts 1 and 2 at the mode label denote the surface and second layers respectively. Rayleigh and transverse states propagating along F M. There are also two gap states. The calculated surface-phonon frequencies at and M are summarized in Table 1 together with results of the calculations given in Ref. [6] and the H R E E L S measurement [12]. For comparison we show here the surface-phonon frequencies calculated for the unrelaxed AI(100) surface. It is evident from the table that the experimental Rayleigh mode frequency at X and theoretical values are in good agreement. At the same time the number of surface states obtained by E A M is higher than that was found in the calculation [6]. A comparison of our results with those obtained in Ref. [6] shows that the frequencies calculated by E A M are 5 - 1 0 % lower
--
6.0
4.o N ~
2.0
0.0 Fig. 2. Calculated phonon dispersion curves for the relaxed 48-layer AI(100) film.
E. V. Chulkov, L Yu. Sklyadneva / Surface Science 331-333 (1995) 1414-1421
than those obtained in the first-principles approach [6]. This result is due to using charge density as a superposition of atomic densities rather than the self-consistent one. The greatest difference of 1.3 THz is observed for the longitudinal gap mode at X. This discrepancy is mainly explained by the effect of decreasing the longitudinal modes for bulk AI ~ 10% at X and L in this method. Our results are generally in fairly good agreement with theoretical [6] and experimental [12] values. Comparing our results obtained for the relaxed and unrelaxed geometries one can conclude that the relaxation has the effect of raising the frequencies of phonons and allows to obtain the better agreement with the experimental and theoretical values [6,12]. A comparison with the results of the microscopic self-consistent calculations [3,8,9], first-principles total-energy calculation [6] and HAS [3,8] and HREELS [12] measurements shows that we have found new longitudinally polarized surface mode and resonance propagating along FM. A new surface state characterized by shear vertical displacem__ents of the second layer atoms has been obtained at M, At the X point a new shear horizontal mode localized on the second layer is observed. The calculated surface resonance along F X is in agreement with the HAS measurements [3,8]. The dependence of the results on the number of film layers has been tested in the calculation of a 24-layer and 30-layer films. The resulting phonon dispersion curves for the relaxed 24-layer and 30-
1417
layer AI(100) films are practically the same. The changes of the frequency values do not exceed 0.1 THz. Therefore we will examine AI(110) and AI(111) surfaces on the basis of the calculations performed for 24-layer films. 3.2.2. The Al(110) surface
The results obtained for the (110) surface show a multilayer oscillatory character of relaxation. The changes in interlayer spacings are A12 = - - 7 . 4 % , A23 "~" q - 0 . 8 % , A34 = -- 0.6% and A45 = + 0.3%. Similar results were obtained from a LEED experiment [22,23]: their values are z112= - 8 . 5 _ 1.0%, A23 ~-- - - 5 . 0 "-[- 1.1%, A34 ~" -- 1.6 + 1.2%. The best agreement is observed for A12. As one goes into the bulk the change in interlayer spacings becomes rapidly smaller. Generally this trend agrees well with experiment, but the magnitude of the effect is overestimated. The first-principles pseudopotential calculation of Ho and Bohnen [5] gives Z~12 = - - 6 . 8 % , A23 ~- +3.5%, A34 = - 2 . 0 % . Eguiluz et al. [7] using pseudopotential perturbation theory found A~2 = --5.4%, A23 = +0.8%, A34 = --2.6%, while the values of Ditlevsen and NCrskov [4] obtained with effective medium theory are A12 = - - 6 % , Z123= + 1%. A qualitative agreement between the theoretical and experimental values [22,23] is rather good. The phonon dispersion curves calculated for the relaxed 24-layer AI(ll0) film are shown in Fig. 3. It is obvious from Fig. 3 there are three surface states at F. The lowest state with a frequency of 014 THz is
8.0
6.0
b 4-.0
2.0
0.0_ Fig. 3. Calculated phonon dispersion curves for the relaxed 24-layer AI(110) film.
E . K Chulkov, LYu. Sklyadneva / S u r f a c e Science 3 3 1 - 3 3 3 (1995) 1 4 1 4 - 1 4 2 1
1418
m
a mode of longitudinal movements of surface layer atoms. The second state describes shear horizontal displacements of surface atoms. The squared amplitude of atom vibration of these states decreases slowly upon movement inside the film. The upper state is characterized by shear vertical movements of atoms of two upper layers. Along the F X symmetry direction five surface states are observed. The lowest state is the Rayleigh mode with shear vertical displacements of surface atoms. It becomes appreciable at kx > 0.5 I FXI. The second state corresponds to shear horizontal movements of surface atoms. The third phonon state is longitudinal in the first layer at small k x. A t F this state is the lowest one. On increasing k x the longitudinal portion of displacements of atoms decreases and shear vertical portion of displacements generated by the second layer atoms increases. Simultaneously we have a change of this state weight. About 60% of this state is localized in the surface layer and 24% corresponds to the second layer at k x = 0.5 I F X 1. At the X point about 10% of this state weight corresponds to the surface layer and 50% of this state is localized in the second la_yer. The fourth state at 5.6 THz at F extends along F X up to k~ = 0.35 I FXI. On increasing k x the longitudinal portion of displacements of atoms increases. The
_
_
fifth surface state begins at k x = 0.7 ] F X [ . Near the X point this state is a typical gap first layer dominated longitudinal mode. As follows from Fig. 3 there are six surface states along the F Y symmetry direction. The frequency values of the two lower states are very close and their difference does not exceed 0.1 THz at a n y ky. The lowest surface state has longitudinal polarization in the outermost layer and shear vertical polarization in the second layer. The other state of this couple is characterized by shear horizontal displacements of surface layer atoms. The third state is predominantly longitudinal in the surface layer a t k y ~ 0 . 5 [ F Y ] and contains a small portion of vertical atom displacements. Upon increase of kv the z-polarized contribution increases and at the Y point this state is a distinct Rayleigh mode, characterized by shear vertical displacements of the outermost layer atoms. Below the energy gap at Y there is a fourth surface state localized in the second layer whose atoms move along the [ - 1 1 0 ] direction. The fifth surface state is located at the bottom of the projection of the bulk-phonon structure. It is longitudinal in the second and the fourth layers and polarized perpendicular to the surface in the first and third layers. The sixth state is a typical gap mode longitudinal in the surface layer. As one
Table 2 Frequencies of the surface modes for the relaxed AI(ll0) Symmetry point
Mode
surface
Frequency (THz) This work
~(
E x p t . [10]
H o , B o h n e n [5]
E g u i l u z et al. [7]
Unrel.
Relax.
Z1, Z e
5.3
5.6
-
-
-
Z1
3.3
3.43
3.53 ± 0.05
4.2
4.6
X1Z e
3.6
3.6
-
4.1
-
Y1
3.8
4.4
-
5.5
-
ZI X 2
-
6.0
-
6.0
-
Y2
5.5
6.0
-
6.3
-
X1
6.2
6.3
-
7.7
-
Ya
2.73
2.76
2.15 _+ 0 . 0 7
1.9
3.1
Xx Z1
2.8 3.3
2.84 3.5
3 . 2 6 + 0.05
3.4
3.3
Xz
4.0
4.1
-
3.6
-
Z1Y2
-
](1
4.9
4.7 5.5
-
6.0
-
Y1
3.0
3.3
-
-
-
Z1 )(1
3.4 5.9
3.5 6.6
-
-
-
A t o m i c d i s p l a c e m e n t s X , Y, a n d Z c o r r e s p o n d to [ - 110], [001] a n d [110] d i r e c t i o n s , r e s p e c t i v e l y .
1419
E. K Chulkov, L Yu. Sklyadneva / Surface Science 331-333 (1995) 1414-1421
can see i n F i g . 3__all surface states in symmetry directions XS and YS are the continuations of corresponding states extending along F X and F Y symmetry lines. The calculated values of surface-phonon frequencies at the symmetry points F, X, Y and S are summarized in Table 2. One can see from the table the calculation gives higher frequencies for the relaxed geometry than for the unrelaxed one. It agrees to the tendency for the AI(100) surface and to the calculations [5,7] performed for AI(ll0). For comparison the results of first-principles and microscopic calculations [5,7] and experimental values [10] are shown. As follows from the table the surface mode at F characterized by shear vertical displacements of atoms of the first and second layers is obtained solely in the present calculations. It is not clear why this state has not been found in the calculations of Refs. [5,7] and in experiment [10]. A similar state was observed on the Ag(ll0), Cu(ll0) and Pd(ll0) surfaces [24,25]. A comparison of the theoretical results with the experimental value of the Rayleigh mode frequency at X shows that the best agreement is obtained in the present calculation. The calculations [5,7] tend to be overestimated. Comparing our results with those obtained by Ho and Bohnen [5] one can see that EAM gives lower frequency values at X than the first-principles method. The greatest difference is obtained for the gap longitudinal mode. A different situation is observed at the Y point. Here
all our frequencies are higher than those calculated by Ho and Bohnen [5] except for the high frequency longitudinal mode. It is interesting to note that all three calculations give a value of the Rayleighmode in good agreement with experiment [10] at Y. The best agreement was achieved by Eguiluz et al. [7]. The present calculation generally gives the surface states in agreement with other calculations [5,7] and the experiment [10]. New results obtained are the surface mode at F, two surface modes at Y and three surface modes at the S point. 3.2.3. The Al(111) surface
The embedded-atom calculation of the (111) surface relaxation gives the values for A12 = --1.8%, A23 = + 0.1%. These results are in reasonable agreement with the values obtained by Ditlevsen and Norskov [4], A12 = - 1%, but differ from the experimental estimates [23], A12 = 0.9%. The phonon dispersion curves obtained for the relaxed 24-layer A1(111) film are shown in Fig. 4. As one can see in the figure there are four surface states along the F K direction. The :lowest one is the Rayleigh mode. The second state is characterized by displacements of the top layer atoms in the plane perpendicular to the wave vector and by shear vertical movements of the second layer atoms. The third state is a gap mode which corresponds to movements of the atoms in the surface plane. The fourth state is a group of resonance states characterized by longitu-
8.0
"~ 6.0
c~
4.0
2.0
0.0
n f
2
~
Fig. 4. Calculatedphonon dispersioncurvesfor the relaxed 24-layerA1(111)film.
f
1420
E.V. Chulkov, LYu. Sklyadneva / Surface Science 331-333 (1995) 1414-1421
Table 3 Frequencies of the surface modes for the relaxed A I ( l l l ) surface Symmetry point
Mode
Frequency (THz) This work Unrel.
Relax.
Z1 (XY)IZ 2
3.8
(XY) 1 (XY)I Z1 (XY) 1 (XY) 1
5.9 6.2 3.5 3.7 7.1
4.0 4.7 6.0 6.4 3.6 3.7 7.3
4.6
Expt. [11]
Black et al. [2]
Ditlevsen, NCrskov [4]
4.4 4.0 -
5.8 7.0 8.2 5.3 5.7 10.0
4.1 6.0 3.8 7.2
Atomic displacements X, Y, and Z correspond to [1 - 10], [11 - 2] and [11t] directions, respectively.
dinal displacements of the atoms of the first and second layers. The frequency value of these resonance states are very close and their difference does not exceed 0.1 THz. We have found four surface modes at the K point. Two of them characterized by horizontal movements of surface layer atoms are in the energy gap. The second lower state also corresponds to the horizontal displacements of surface atoms. There are four surface states along the F M direction. The difference between the frequencies of the two lower states is ~ 0.1 THz. The lowest state is the Rayleigh mode. The next state corresponds to the horizontal movements of the surface layer atoms. The third mode is a group of horizontally polarized resonances. The frequency difference does not exceed 0.10-0.15 THz. The fourth state is a gap mode. All surface states in the KM direction are the continuation of the corresponding states obtained in the F K and F M directions. The resulting frequencies at the symmetry points K and M are given in Table 3 together with those calculated in Refs. [2,4] and measured by HAS method in Ref. [11]. As follows from the table the values obtained with EAM and EMT are very close to each other, but they are lower than the experimental value. At the same time the parametrized forceconstant model tends to overestimate the frequencies. The new results obtained for A I ( l l l ) are as follows: (1) two surface states in the energy gap at the K point, (2) two surface resonance states in the F K direction, (3) two surface resonance modes along the FM symmetry line.
4. Conclusion The calculations of the surface phonon frequencies for the low-index aluminium surfaces have been performed using the embedded-atom method. New surface modes have been found on all surfaces of interest.
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[22] J.N. Andersen, H.B. Nielsen, L. Peterson and D.L. Adams, J. Phys. C 17 (1984) 173. [23] J.R. Noonan and H.L. Davies, Phys. Rev. B 29 (1984) 4349. [24] G. Bracco, R. Tatarek, F. Tommasini, V. Linke and M. Persson, Phys. Rev. B 36 (1987) 2928. [25] A.M. Lance, J,P. Toennies and Ch. Woll, Surf. Sci. 191 (1987) 529.