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On the anisotropy of surface tension in transition metals
Surface Science 50 (1975) 257-261 0 North-Holland Publishing Company
ON THE ANISOTROPY IN TRANSITION
OF SURFACE TENSION
METALS
Received 6 February 1975; revised manuscript received 12 March 1975
Surface tension measurements exist mainly on liquid or polycrystalline cleaved transition metals and they show a parabolic behaviour with the d band filling [ 11. Very few experiments have been done on single-crystal faces leading to determinations of the anisotropy of surface tension, and their results are somewhat spread. From a theoretical point of view, the determinatio~l of the surface tension of transition metals has been approached, in the tight-binding scheme, using crude models for the density of states. Nevertheless, the parabolic behaviour of the variation of the surface tension in terms of the band filling was shown to be due to the d character of the electron states [2,3] _ We have recently performed [4] a systematic study of surface densities of states of cleaved transition metals including d band degeneracy and fully taking into aecount the charge rearrangement on the surface. It seems, therefore, interesting to use our results to study the variation of the surface tension and its an~sotropy with the d band filling for the three crystalline structures: fee, hcp, bee of transition metals. The densities of states are calculated through a moment’s expansion and built up from their first moments using the continuous fraction technique [S] . Relaxation and reconstruction effects are neglected but the possible influence of relaxation will be discussed at the end of this letter. The variation of energy per surface atom when cleaving the crystal is given by the formula [3J
Es=l~t’Ph~an,(E,uo)dL;-~,u,-t(Zs-~ (1) s i=l
where u. is the surface perturbing potential due to the charge rearrangement at the surface; ani(E, ~0) is the variation of the local density of states on the ith plane, and Z, and Zs are respectively the number ofd electrons per atom in the bulk and on the surface. The last two terms of (I) are added to prevent the double counting of Coulomb interactions occurring in the integral. We have used the local densities of states and perturbing surface potential u. previously determined for various transition metals [4]. As we already explained, these quantities depend on the values of the overlap integrals between first and
ipif.C?.Desjonqueres, F. Cyrot-t~ck?~anPl~itrlisotropy of surface tension
258
second nearest neighbours, as defined by Slater and Koster (61. There are no direct calculations of these overlap integrals but one can reasonably assume [7] that they have a constant ratio for various band fillings, their absolute value being fitted to the cohesive energy. As in the case of narrow s band [S] , calculations have shown that the sum in the integral in (I) can be limited, with very good accuracy, to the surface plane and only a few layers below because as one proceeds into the crystal, the local density of states tends rapidly to the bulk density of states. The surface tension ys, equal to the variation of energy per surface unit, is also a function of the lattice parameter a which varies with the band filling. Results for ysa2, in band width units, are shown in fig. I for (1 I I) and (100) cleaved fee crystals and in fig. 2 for (1 lo), (100) and (111) cleaved bee crystals. For fee crystals we have also calculated ys for a (110) cleaved crystal and a band filling which is of physical interest for this structure (zM = 0.9, i.e. the case of Ni, Pd, Pt). We thus obtained (110)>yf100)>y(3W,
7s
s
with #’ lu)/y(‘ou) = 1.07, S
S
$1 ll)//y(~uo) = 0.73, s
s
(3)
The order is the same as that given by a broken bond model [9 ] . From an experimental point of view, there have been a few nleasurements on Ni [lo] which have
0.4
03
0.2
0.1
Fig. 1. The variation of y,a2, in bandwidth units (w), in terms of the band filling (2~ = Z~/l0) for (111) and (100) cleaved fee crystals (ys is the surface tension per surface unit and a is the lattice parameter).
M.C. Desjonqueres,
F. Cyrot-LuckmannlAnisotropy
of surface tension
259
Fig. 2. The variation of y,a2, in bandwidth units (w), in terms of the band filling (2~ = Z~/10) for (1 lo), (100) and (111) cleaved bee crystals (ys is the surface tension per surface unit and a is the lattice parameter).
led to the same order as in our calculations with a very good agreement for yillo)/ greater than ours. y(loo) but a value of y,(l 11) relative to $“) S For bee crystals, we observed a peculiar behaviour pattern. Indeed, owing to the sharp peak of surface states present in the middle of the band in the surface density of states of a (100) cleaved bee crystal [4], the ~1~“) curve is somewhat peaked in that region and it crosses the -yg” ‘) curve. Consequently, the anisotropy depends on the band filling. For most bee transition metals, the band is nearly half-filled and we have: $00,
> yjlll)
> $lO),
(4)
whereas for zM N 0.2 and 0.7 (i.e. the case of Fe) #ill)
>+OO)
>,1”0).
(5)
S
The anisotropy
is, here again, rather strong:
y(llu)/y(luo) S
= 0.68,
S
r;%j
loo) = 0.89,
(6)
when ZM = 0.5 (Cr, MO, W). Very little is known with certainty about the experimental anisotropy of bee transition metals. Some measurements have been performed on MO and W but the results are rather various. While Miiller and Drechsler [ 111 with a field emission microscope have found 7:’ l”)/$oo’ 2: 1 and ye1 “I/ plo) v 1.03 for W, Nelson et al. [ 121 give a ratio y(’ l”)/-y(lOO) 2: 0.88 for MO and more recently Cordwell and Hull [ 131 have achievedSyd’ l”)&~loo) z 0.72 for W. This last value is in better agreement with our results but the absolute values they give are much greater than those usually obtained. It seems then impossible to conclude with certainty about the anisotropy of surface tension in bee crystals. We have also calculated the surface tension for (0001) and (IOiO) cleaved hcp crystals. For zlll = 0.7 (i.e. the case of Ru and OS) we have obtained
y(ioio)pool) = 1.5. S
S
(7)
260
M.C. Desjonqueres,
F. Cyrot-LackmannfAnisotropy
of surface tension
Unfortunately no experimental data are available about the anisotropy of surface tension in hcp transition metals. Let us add that the absolute values of surface tension that can be obtained from our calculations are in rather good agreement with experiment when the overlap integrals are chosen to fit the cohesive energy. Measured mean surface tensions must be minimum for an equilibrium shape and are to be compared with our results for dense planes which are always the smallest. For Nb, MO, Ta and W Hodkin et al. [ 141 have measured 2550, 2050,2680,2830 erg/cm2 respectively while our results are in the same order 2380, 2220,254O and 2800 erg/cm2. Furthermore, Bettler and Barnes [ 1.51 have obtained 2800 f 10% erg/cm2 for W (110). The agreement is then quite good. For (100) and (110) Ni we obtained 2150 and 2300 erg/cm2, respectively, while Maiya and Blakely [lo] have measured 1821 f 182 and 1911 + 190erg/ cm2. However, we must notice that in this case the d band width is much too large. Indeed, for fee transition metals, such as Ni, the band is almost completely filled and the main contribution to the cohesive energy is due to s-d hybridization and crystal field integrals, which we have neglected. Crystal field integrals would be quite easy to include in our calculations if their values were known but the introduction of s-d mixing is much more difficult and is still to study. As a conclusion, our calculations give a larger anisotropy but with the same order of planes as experimental results for fee crystals. For bee crystals the calculated anisotropy is also rather large, it depends on the band fuling but experimental data are too spread to conclude anything. The absolute values of surface tension are in good agreement with the experimental ones. The effect of a possible relaxation can be included in principle very easily in our calculations. One introduces as in ref. [16] the Born-Mayer repulsive energy term already used to calculate elastic constants of the bulk metal [ 171, but the computations show that the relaxation energy is so small a fraction of the surface tension that the results depend heavily on the model chosen for the density of states. The only conclusion we can draw is that the relaxation of the lattice parameter seems to be an oscillating function of the filling of the d-band, but its exact shape changes according to the various densities of states one chooses. No definitive conclusion can then be given for its magnitude or its sign. However, the relaxation energy being a very small fraction of the surface tension, one can expect that it will not affect very much our results. Institut Laue-Langevin, B.P. 156, 38042 Grenoble Ct?dex, France, and Groupe des Transitions de Phases, CNRS, B.P. 166, 38042 Grenoble Cidex, France
M.C. DESJONQUEKES
and F. CYROT-LACKMANN
Groupe des Transitions de Phases, CNRS, B.P. 166, 38042 Grenoble Cidex, France
MC. Desjongueres,
F. Cyrot-ImkmannfAnisotropy
of surface tension
References [l] B.C. Allen, Trans. AIME 227 (1963) 1175. [2] F. Cyrot-Lackmann, Surface Sci. 15 (1969) 535; F. Cyrot-Lackmann, J. Phys. (Paris) 31 (1970) Cl 67. [3] Cl. Allan, Ann. Phys. (Paris) 5 (1970) 169. [4] M.C. Desjonqueres and F. Cyrot-Lackmann, J. Phys. (Paris) Lettres 36 (1975) L45. [5] J.P. Gaspard and F. Cyrot-Lackmann, J. Phys. C (Solid State Phys.) 6 (1973) 3077. [6] J.C. Slater and C.F. Koster, Phys. Rev. 94 (1954) 1498. [7] F. Ducastelle and F. Cyrot-Lackmann, J. Phys. Chem. Solids 31 (1970) 1295. (81 F. Cyrot-Lackmann, M.C. Desjonqu&es and J.P. Gaspard, J. Phys. C (Solid State Phys.) 7 (1974) 925. [9] J.J. Burton and G. Jura, J. Phys. Chem. 71 (1967) 1937. [lo] P.S. Maiya and J.M. Blakely, J. Appl. Phys. 38 (1967) 698; B.E. Sundquist, Acta Met. 12 (1964) 67; U. Jeschkowski and E. Menzel, Surface Sci. 15 (1969) 333. [ll] A. Miller and M. Drechsler, Surface Sci. 13 (1969) 471. (121 R.S. Nelson, D.J. Mazey and R.S. Barnes, Phil. Mag. 11 (1965) 91. [13] J.E. Cordwell and D. Hull, Phil. Mag. 26 (1972) 215. [14] E.N. Hodkin, M.G. Nicholas and D.M. Poole, J. Less-Common Metals 20 (1970) 93. [15] P.C. Bettler and G. Barnes, Surface Sci. 10 (1968) 165. 1161 G. Allan and M. Lannoo, Surface Sci. 40 (1973) 375. [17] F. Ducastelle, J. Phys. (Paris) 31 (1970) 1055.
261
ON THE ANISOTROPY IN TRANSITION
OF SURFACE TENSION
METALS
Received 6 February 1975; revised manuscript received 12 March 1975
Surface tension measurements exist mainly on liquid or polycrystalline cleaved transition metals and they show a parabolic behaviour with the d band filling [ 11. Very few experiments have been done on single-crystal faces leading to determinations of the anisotropy of surface tension, and their results are somewhat spread. From a theoretical point of view, the determinatio~l of the surface tension of transition metals has been approached, in the tight-binding scheme, using crude models for the density of states. Nevertheless, the parabolic behaviour of the variation of the surface tension in terms of the band filling was shown to be due to the d character of the electron states [2,3] _ We have recently performed [4] a systematic study of surface densities of states of cleaved transition metals including d band degeneracy and fully taking into aecount the charge rearrangement on the surface. It seems, therefore, interesting to use our results to study the variation of the surface tension and its an~sotropy with the d band filling for the three crystalline structures: fee, hcp, bee of transition metals. The densities of states are calculated through a moment’s expansion and built up from their first moments using the continuous fraction technique [S] . Relaxation and reconstruction effects are neglected but the possible influence of relaxation will be discussed at the end of this letter. The variation of energy per surface atom when cleaving the crystal is given by the formula [3J
Es=l~t’Ph~an,(E,uo)dL;-~,u,-t(Zs-~ (1) s i=l
where u. is the surface perturbing potential due to the charge rearrangement at the surface; ani(E, ~0) is the variation of the local density of states on the ith plane, and Z, and Zs are respectively the number ofd electrons per atom in the bulk and on the surface. The last two terms of (I) are added to prevent the double counting of Coulomb interactions occurring in the integral. We have used the local densities of states and perturbing surface potential u. previously determined for various transition metals [4]. As we already explained, these quantities depend on the values of the overlap integrals between first and
ipif.C?.Desjonqueres, F. Cyrot-t~ck?~anPl~itrlisotropy of surface tension
258
second nearest neighbours, as defined by Slater and Koster (61. There are no direct calculations of these overlap integrals but one can reasonably assume [7] that they have a constant ratio for various band fillings, their absolute value being fitted to the cohesive energy. As in the case of narrow s band [S] , calculations have shown that the sum in the integral in (I) can be limited, with very good accuracy, to the surface plane and only a few layers below because as one proceeds into the crystal, the local density of states tends rapidly to the bulk density of states. The surface tension ys, equal to the variation of energy per surface unit, is also a function of the lattice parameter a which varies with the band filling. Results for ysa2, in band width units, are shown in fig. I for (1 I I) and (100) cleaved fee crystals and in fig. 2 for (1 lo), (100) and (111) cleaved bee crystals. For fee crystals we have also calculated ys for a (110) cleaved crystal and a band filling which is of physical interest for this structure (zM = 0.9, i.e. the case of Ni, Pd, Pt). We thus obtained (110)>yf100)>y(3W,
7s
s
with #’ lu)/y(‘ou) = 1.07, S
S
$1 ll)//y(~uo) = 0.73, s
s
(3)
The order is the same as that given by a broken bond model [9 ] . From an experimental point of view, there have been a few nleasurements on Ni [lo] which have
0.4
03
0.2
0.1
Fig. 1. The variation of y,a2, in bandwidth units (w), in terms of the band filling (2~ = Z~/l0) for (111) and (100) cleaved fee crystals (ys is the surface tension per surface unit and a is the lattice parameter).
M.C. Desjonqueres,
F. Cyrot-LuckmannlAnisotropy
of surface tension
259
Fig. 2. The variation of y,a2, in bandwidth units (w), in terms of the band filling (2~ = Z~/10) for (1 lo), (100) and (111) cleaved bee crystals (ys is the surface tension per surface unit and a is the lattice parameter).
led to the same order as in our calculations with a very good agreement for yillo)/ greater than ours. y(loo) but a value of y,(l 11) relative to $“) S For bee crystals, we observed a peculiar behaviour pattern. Indeed, owing to the sharp peak of surface states present in the middle of the band in the surface density of states of a (100) cleaved bee crystal [4], the ~1~“) curve is somewhat peaked in that region and it crosses the -yg” ‘) curve. Consequently, the anisotropy depends on the band filling. For most bee transition metals, the band is nearly half-filled and we have: $00,
> yjlll)
> $lO),
(4)
whereas for zM N 0.2 and 0.7 (i.e. the case of Fe) #ill)
>+OO)
>,1”0).
(5)
S
The anisotropy
is, here again, rather strong:
y(llu)/y(luo) S
= 0.68,
S
r;%j
loo) = 0.89,
(6)
when ZM = 0.5 (Cr, MO, W). Very little is known with certainty about the experimental anisotropy of bee transition metals. Some measurements have been performed on MO and W but the results are rather various. While Miiller and Drechsler [ 111 with a field emission microscope have found 7:’ l”)/$oo’ 2: 1 and ye1 “I/ plo) v 1.03 for W, Nelson et al. [ 121 give a ratio y(’ l”)/-y(lOO) 2: 0.88 for MO and more recently Cordwell and Hull [ 131 have achievedSyd’ l”)&~loo) z 0.72 for W. This last value is in better agreement with our results but the absolute values they give are much greater than those usually obtained. It seems then impossible to conclude with certainty about the anisotropy of surface tension in bee crystals. We have also calculated the surface tension for (0001) and (IOiO) cleaved hcp crystals. For zlll = 0.7 (i.e. the case of Ru and OS) we have obtained
y(ioio)pool) = 1.5. S
S
(7)
260
M.C. Desjonqueres,
F. Cyrot-LackmannfAnisotropy
of surface tension
Unfortunately no experimental data are available about the anisotropy of surface tension in hcp transition metals. Let us add that the absolute values of surface tension that can be obtained from our calculations are in rather good agreement with experiment when the overlap integrals are chosen to fit the cohesive energy. Measured mean surface tensions must be minimum for an equilibrium shape and are to be compared with our results for dense planes which are always the smallest. For Nb, MO, Ta and W Hodkin et al. [ 141 have measured 2550, 2050,2680,2830 erg/cm2 respectively while our results are in the same order 2380, 2220,254O and 2800 erg/cm2. Furthermore, Bettler and Barnes [ 1.51 have obtained 2800 f 10% erg/cm2 for W (110). The agreement is then quite good. For (100) and (110) Ni we obtained 2150 and 2300 erg/cm2, respectively, while Maiya and Blakely [lo] have measured 1821 f 182 and 1911 + 190erg/ cm2. However, we must notice that in this case the d band width is much too large. Indeed, for fee transition metals, such as Ni, the band is almost completely filled and the main contribution to the cohesive energy is due to s-d hybridization and crystal field integrals, which we have neglected. Crystal field integrals would be quite easy to include in our calculations if their values were known but the introduction of s-d mixing is much more difficult and is still to study. As a conclusion, our calculations give a larger anisotropy but with the same order of planes as experimental results for fee crystals. For bee crystals the calculated anisotropy is also rather large, it depends on the band fuling but experimental data are too spread to conclude anything. The absolute values of surface tension are in good agreement with the experimental ones. The effect of a possible relaxation can be included in principle very easily in our calculations. One introduces as in ref. [16] the Born-Mayer repulsive energy term already used to calculate elastic constants of the bulk metal [ 171, but the computations show that the relaxation energy is so small a fraction of the surface tension that the results depend heavily on the model chosen for the density of states. The only conclusion we can draw is that the relaxation of the lattice parameter seems to be an oscillating function of the filling of the d-band, but its exact shape changes according to the various densities of states one chooses. No definitive conclusion can then be given for its magnitude or its sign. However, the relaxation energy being a very small fraction of the surface tension, one can expect that it will not affect very much our results. Institut Laue-Langevin, B.P. 156, 38042 Grenoble Ct?dex, France, and Groupe des Transitions de Phases, CNRS, B.P. 166, 38042 Grenoble Cidex, France
M.C. DESJONQUEKES
and F. CYROT-LACKMANN
Groupe des Transitions de Phases, CNRS, B.P. 166, 38042 Grenoble Cidex, France
MC. Desjongueres,
F. Cyrot-ImkmannfAnisotropy
of surface tension
References [l] B.C. Allen, Trans. AIME 227 (1963) 1175. [2] F. Cyrot-Lackmann, Surface Sci. 15 (1969) 535; F. Cyrot-Lackmann, J. Phys. (Paris) 31 (1970) Cl 67. [3] Cl. Allan, Ann. Phys. (Paris) 5 (1970) 169. [4] M.C. Desjonqueres and F. Cyrot-Lackmann, J. Phys. (Paris) Lettres 36 (1975) L45. [5] J.P. Gaspard and F. Cyrot-Lackmann, J. Phys. C (Solid State Phys.) 6 (1973) 3077. [6] J.C. Slater and C.F. Koster, Phys. Rev. 94 (1954) 1498. [7] F. Ducastelle and F. Cyrot-Lackmann, J. Phys. Chem. Solids 31 (1970) 1295. (81 F. Cyrot-Lackmann, M.C. Desjonqu&es and J.P. Gaspard, J. Phys. C (Solid State Phys.) 7 (1974) 925. [9] J.J. Burton and G. Jura, J. Phys. Chem. 71 (1967) 1937. [lo] P.S. Maiya and J.M. Blakely, J. Appl. Phys. 38 (1967) 698; B.E. Sundquist, Acta Met. 12 (1964) 67; U. Jeschkowski and E. Menzel, Surface Sci. 15 (1969) 333. [ll] A. Miller and M. Drechsler, Surface Sci. 13 (1969) 471. (121 R.S. Nelson, D.J. Mazey and R.S. Barnes, Phil. Mag. 11 (1965) 91. [13] J.E. Cordwell and D. Hull, Phil. Mag. 26 (1972) 215. [14] E.N. Hodkin, M.G. Nicholas and D.M. Poole, J. Less-Common Metals 20 (1970) 93. [15] P.C. Bettler and G. Barnes, Surface Sci. 10 (1968) 165. 1161 G. Allan and M. Lannoo, Surface Sci. 40 (1973) 375. [17] F. Ducastelle, J. Phys. (Paris) 31 (1970) 1055.
261