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Surface energy of metals: theory and experiment
Surface Science 437 (1999) 285–288 www.elsevier.nl/locate/susc
Surface energy of metals: theory and experiment K.F. Wojciechowski * Institute of Experimental Physics, University of Wroclaw, Pl. Maxa Borna 9, 50-204 Wroclaw, Poland Received 12 January 1999; accepted for publication 10 June 1999
Abstract The correlation between the known theoretical calculated surface energies and the ’free’ mobile electron density parameter, r , is studied. It is shown that the experimental data, both for simple and transition metals, are better m fitted by the function r−4 than by r−3 , which follows from the Miedema model of cohesion. © 1999 Elsevier Science m m B.V. All rights reserved. Keywords: Simple metals; Surface energy; Transition metals
The surface energy (SE), s, has been the subject of many theoretical studies [1,2] and semi-empirical investigations [3,4]. The local density approximation (LDA) gives [1,5,6 ] s as a function of the electron density parameter, r , which has a maxis mum near r ~1.8 a.u. amounting to about s 1.2 J/m2. The jellium like free-electron models for electrons in a finite square potential [6,7] suggest that the SE is approximately proportional to the fourth power of the Fermi momentum. The collective mode treatment [9] and phenomenological approach proposed by Kohn and Yaniv [10] (see also Ref. [11]) and the optimized variational theory based on LDA (the Fermi hypernettedchain theory) [12] for simple metals have given the relation [12] s~r-2.5. The ab-initio calculations s of Skriver and Rosengaard [13] yield SE values which, for simple metals, are proportional to ~r−3. Such a linear relationship between the s experimental data for SEs and the interatomic electron density, n (proportional to r−3 for s-p s * Fax: +48-71-328-7365. E-mail address: [email protected] ( K.F. Wojciechowski)
metals), is known as the Miedema model of cohesion [3]. Some years ago, Perrot and Rasolt [14] (P–R) reported a list of the ‘effective’ r ¬r values for s m which the experimental data for SEs, s (M ), of exp the individual metal (M ), can be represented by an unexplicit function F(r )1 of r : m m s (M )=F(r ). (1) exp m The above statement that s (M ) correlates with exp r in any metal M, allows the empirical discrete m function s (rM) to be used as the ‘reference exp m function’ for the comparison of the theoretical continuous functions, s (r), of the electron density th parameter, r, obtained from different theories in the whole metallic range of r (1≤r≤6). A comparison of the functions s (rM) and s (r=rM) allows exp m th m a theory to be qualified that predicts this s (r) th function. In the present paper, we give such a qualification 1 In Ref. [14], this function is denoted by C(r ); since, howm ever, such a denotation may be misunderstood as the Euler integral, we simply use: F(r ). m
0039-6028/99/$ – see front matter © 1999 Elsevier Science B.V. All rights reserved. PII: S0 0 39 - 6 0 28 ( 99 ) 0 07 4 1 -4
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K.F. Wojciechowski / Surface Science 437 (1999) 285–288
Fig. 1. Experimental surface energies of simple metals as functions of the conventional electron density parameter, r (circles), s and of the effective (mobile) electron density parameter, r m (crosses). Theoretical values of the surface energy are displayed as a dashed line ( jellium model ), a dotted-dashed line (stabilized jellium) and a solid line ( T–O model [8]).
Fig. 2. Same as in Fig. 2 with the theoretical results displayed for an ’ideal metal’ by a dotted line and for the A–K theory by a solid line [12].
of the theories of Ainsworth and Krotscheck [12] (A–K ), Takahashi and Onzawa [8] ( T–O) and of the ab-initio calculations performed by Skriver and Rosengaard (S–R) [13]. In Figs. 1 and 2 the discrete function, s (r), for simple metals, is exp
represented by open circles for conventional r=r s and by crosses for r=r . The experimental data m are taken from [3]. The theoretical functions, s (r), are represented in Fig. 1 by a dashed line th ( jellium model [1]), a dotted-dashed line (stabilized jellium [5]) and a solid line ( T–O model [8]). Fig. 2 shows s (r ) and s (r ), together with exp s exp m the s (r) functions calculated in the LDA for an th ‘ideal metal’ [6 ] (dotted line) and by A–K [12] (solid line). From Fig. 1, it can be seen that in accordance with the P–R statement [14], the theoretical curves sTO(r)~r−4 (eq. (16) in [8]) are in th a better agreement with s (r ) than with exp m s (r ). From Fig. 2 it can be seen, however, that exp s the theoretical curve sAK(r)~r−2.5, taken after th [12], is rather in better agreement with s (r ) exp s than with s (r ). exp m The fit of the function F(r)=A(l)r−l,
(2)
to the s (r ) discrete function for 17 metals exp m (alkali-, alkaline earth-metals, and Be, Al, Zn, Mg, Pb, Ga, In, Cd and Hg) shows that A=21.69 and l=3.37 (the units of SE are J/m2). Comparing the above-mentioned theoretical curves, sTO(r) and th sAK(r) to this result, one can conclude that the th Miedema rule [3] (s~r−3) for simple metals is fulfilled only approximately. Eq. (2), fitted to the ab-initio caculations [13], gives l=2.82 and A=11.42 with a smaller sum of square deviations and mean square error than the fit of the function Ar−3, which additionally confirms the above conclusion. Fig. 3 displays (circles) the ab-initio calculations [13] of S–R, sTO (r) and sAK (r) curves (dashed th th and dotted line, respectively) together with the experimental data [3] (crosses). A comparison of Figs. 1–3 shows that the T–O and A–K theories, which are not self-consistent, give a much better agreement with the experiment for simple metals than the theories based on the LDA [1,5–7]. This fact is due probably to the avoidance in T–O and A–K models the limitations of the LDA. In transition metals, in addition, the d-electrons contribute to the average density of the ‘free’ mobile part of the electron gas, and an examination of the correlation between SE and the electron density parameter r is much more complicated
K.F. Wojciechowski / Surface Science 437 (1999) 285–288
Fig. 3. Theoretical results for the surface energy s(r) of simple metals: S–R [13] (circles); T–O [8] (dashed line); A–K [12] (dotted line). The experimental values of s are marked by crosses.
than in the case of simple metals [15]. Therefore, we restrict ourselves only to a correlation of the experimental data and the ab-initio calculations with the effective r through the expression for SE m of the form given by Eq. (2). The fit of this function with l=2.5, 3 and 4 to the s (r ), for exp m the 3d, 4d, and 5d series, shows that the best fit is obtained with l=4 giving A(4)=49.57, 72.61 and 83.00 for the mentioned series, respectively. The fit of Ar−l to the s (r ) gives l>4, and since exp m the value of l=4 has the theoretical motivation [7,8], in Fig. 4, we compare SEs calculated from eq. 16 of [8] and the classical cleavage energy (calculated according to eq. 8.24 of [2] with r=r ) (full circles), with the experimental data [3] m
287
Fig. 4. Calculated surface energy of transition metal: S–R [13], (open circles); T–O ([8]+cleavage energy) (full circles); Ar−4 , m [fitted to s (r )] (dotted circles). The experimental values exp m s are marked by crosses. exp
(crosses) and with the ab-initio calculations [13] (open circles). From this figure, it can be seen that the simple T–O theory [8] gives a fairly good agreement with the ab-initio calculations [13] and gives the same trends as the experimental ones [3]2 (with the exclusion of 3d magnetic metals). For the 5d series, the agreement is only qualitative, but a reasonable agreement with the experiment for all three series is given by the functions A(nd)r−4 (dotted circles in Fig. 4) with A(3d)= 49.57, A(4d )=72.61, and A(5d )=82.88, which 2 The jellium model for a square potential based on LDA (see Ref. [7]) yields values that are smaller (by approximately a factor of 0.7) for the calculated SEs than the T–O model [8].
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may be treated as the F(r) functions suggested by P–R [14]. The results of the present paper can be summarized as follows. For simple metals, the theoretical predictions of the SE dependence on the electron density parameter correlate better with the effective r values determined by Perrot and Resolt m [14] than with the conventional r values [16 ], s even for the ab-initio calculations given in [13]. For these metals, the theory agrees well with the experiment, using the simple approach for the electronic structure of semi-infinite jellium as proposed by Takahashi and Onzawa [8], in which exchange, correlation and other contributions are obscurely enclosed in Fermi energy and in work function, sum of which is assumed as the barrier height. In the case of simple metals, the ‘unexplicit’ function in Eq. (1), is therefore the explicit function F(r )=A(4)r−4. For 4d and 5d transition m metals, the function F(r ) can be approximately m represented by the explicit functions of the form Ar−l; however, for the 3d metals, a better description of the experimental data is given by an unexplicit function of r of the shape illustrated m in fig. 6 of [14]. Finally, since for both simple and transition metals, the best agreement with the experimental data is achieved for s ~r−4 but not th for s ~r−3, we conclude that the Miedema model th of cohesion [3] may be treated only as a very approximated model. Generally, one can conclude that for s-p metals and for the 4d and 5d series of transition metals, the SE data correlates with r m through the function Ar−4 , which results from m simple quantum mechanics.
Acknowledgement This work was supported by University, Grant No. 2016/W/99.
Wroclaw
References [1] N.D. Lang, in: S. Lundquist, N.H. March ( Eds.), Theory of the Inhomogeneous Electron Gas, Plenum, New York, 1983. [2] A. Kiejna, K.F. Wojciechowski, Metal Surface Electron Physics, Pergamon, Oxford, 1996. [3] F.R. de Boer, R. Boom, W.C.M. Mattens, A.R. Miedema, A.K. Niessen, Cohesion in Metals, North-Holland, Amsterdam, 1988. [4] A.M. Rodriguez, G. Bozzolo, J. Ferrante, Surf. Sci. 289 (1993) 100. [5] J.P. Perdew, H.Q. Tran, E.D. Smith, Phys. Rev. B 42 (1990) 11627. [6 ] H.B. Shore, J.H. Rose, Phys. Rev. Lett. 66 (1991) 2519. [7] G. Paasch, H. Wonn, Phys. Stat. Sol. (b) 70 (1975) 555. [8] K. Takahashi, T. Onzawa, Phys. Rev. B 48 (1993) 5689. [9] J. Schmit, A.A. Lucas, Solid State Commun. 11 (1972) 415. [10] W. Kohn, A. Yaniv, Phys. Rev. B 20 (1979) 4948. [11] N.H. March, B.V. Paranjape, Phys. Rev. B 30 (1984) 3131. [12] T.L. Ainsworth, E. Krotscheck, Phys. Rev. B 45 (1992) 8779. [13] H.L. Skriver, N.M. Rosengaard, Phys. Rev. B 46 (1992) 7157. [14] F. Perrot, M. Rasolt, J. Phys.: Condens. Matter 6 (1994) 1473. [15] K.F. Wojciechowski, Mod. Phys. Lett. B 12 (1998) 658. [16 ] Ch. Kittel, Introduction to Solid State Physics, Wiley, New York, 1986.
Surface energy of metals: theory and experiment K.F. Wojciechowski * Institute of Experimental Physics, University of Wroclaw, Pl. Maxa Borna 9, 50-204 Wroclaw, Poland Received 12 January 1999; accepted for publication 10 June 1999
Abstract The correlation between the known theoretical calculated surface energies and the ’free’ mobile electron density parameter, r , is studied. It is shown that the experimental data, both for simple and transition metals, are better m fitted by the function r−4 than by r−3 , which follows from the Miedema model of cohesion. © 1999 Elsevier Science m m B.V. All rights reserved. Keywords: Simple metals; Surface energy; Transition metals
The surface energy (SE), s, has been the subject of many theoretical studies [1,2] and semi-empirical investigations [3,4]. The local density approximation (LDA) gives [1,5,6 ] s as a function of the electron density parameter, r , which has a maxis mum near r ~1.8 a.u. amounting to about s 1.2 J/m2. The jellium like free-electron models for electrons in a finite square potential [6,7] suggest that the SE is approximately proportional to the fourth power of the Fermi momentum. The collective mode treatment [9] and phenomenological approach proposed by Kohn and Yaniv [10] (see also Ref. [11]) and the optimized variational theory based on LDA (the Fermi hypernettedchain theory) [12] for simple metals have given the relation [12] s~r-2.5. The ab-initio calculations s of Skriver and Rosengaard [13] yield SE values which, for simple metals, are proportional to ~r−3. Such a linear relationship between the s experimental data for SEs and the interatomic electron density, n (proportional to r−3 for s-p s * Fax: +48-71-328-7365. E-mail address: [email protected] ( K.F. Wojciechowski)
metals), is known as the Miedema model of cohesion [3]. Some years ago, Perrot and Rasolt [14] (P–R) reported a list of the ‘effective’ r ¬r values for s m which the experimental data for SEs, s (M ), of exp the individual metal (M ), can be represented by an unexplicit function F(r )1 of r : m m s (M )=F(r ). (1) exp m The above statement that s (M ) correlates with exp r in any metal M, allows the empirical discrete m function s (rM) to be used as the ‘reference exp m function’ for the comparison of the theoretical continuous functions, s (r), of the electron density th parameter, r, obtained from different theories in the whole metallic range of r (1≤r≤6). A comparison of the functions s (rM) and s (r=rM) allows exp m th m a theory to be qualified that predicts this s (r) th function. In the present paper, we give such a qualification 1 In Ref. [14], this function is denoted by C(r ); since, howm ever, such a denotation may be misunderstood as the Euler integral, we simply use: F(r ). m
0039-6028/99/$ – see front matter © 1999 Elsevier Science B.V. All rights reserved. PII: S0 0 39 - 6 0 28 ( 99 ) 0 07 4 1 -4
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K.F. Wojciechowski / Surface Science 437 (1999) 285–288
Fig. 1. Experimental surface energies of simple metals as functions of the conventional electron density parameter, r (circles), s and of the effective (mobile) electron density parameter, r m (crosses). Theoretical values of the surface energy are displayed as a dashed line ( jellium model ), a dotted-dashed line (stabilized jellium) and a solid line ( T–O model [8]).
Fig. 2. Same as in Fig. 2 with the theoretical results displayed for an ’ideal metal’ by a dotted line and for the A–K theory by a solid line [12].
of the theories of Ainsworth and Krotscheck [12] (A–K ), Takahashi and Onzawa [8] ( T–O) and of the ab-initio calculations performed by Skriver and Rosengaard (S–R) [13]. In Figs. 1 and 2 the discrete function, s (r), for simple metals, is exp
represented by open circles for conventional r=r s and by crosses for r=r . The experimental data m are taken from [3]. The theoretical functions, s (r), are represented in Fig. 1 by a dashed line th ( jellium model [1]), a dotted-dashed line (stabilized jellium [5]) and a solid line ( T–O model [8]). Fig. 2 shows s (r ) and s (r ), together with exp s exp m the s (r) functions calculated in the LDA for an th ‘ideal metal’ [6 ] (dotted line) and by A–K [12] (solid line). From Fig. 1, it can be seen that in accordance with the P–R statement [14], the theoretical curves sTO(r)~r−4 (eq. (16) in [8]) are in th a better agreement with s (r ) than with exp m s (r ). From Fig. 2 it can be seen, however, that exp s the theoretical curve sAK(r)~r−2.5, taken after th [12], is rather in better agreement with s (r ) exp s than with s (r ). exp m The fit of the function F(r)=A(l)r−l,
(2)
to the s (r ) discrete function for 17 metals exp m (alkali-, alkaline earth-metals, and Be, Al, Zn, Mg, Pb, Ga, In, Cd and Hg) shows that A=21.69 and l=3.37 (the units of SE are J/m2). Comparing the above-mentioned theoretical curves, sTO(r) and th sAK(r) to this result, one can conclude that the th Miedema rule [3] (s~r−3) for simple metals is fulfilled only approximately. Eq. (2), fitted to the ab-initio caculations [13], gives l=2.82 and A=11.42 with a smaller sum of square deviations and mean square error than the fit of the function Ar−3, which additionally confirms the above conclusion. Fig. 3 displays (circles) the ab-initio calculations [13] of S–R, sTO (r) and sAK (r) curves (dashed th th and dotted line, respectively) together with the experimental data [3] (crosses). A comparison of Figs. 1–3 shows that the T–O and A–K theories, which are not self-consistent, give a much better agreement with the experiment for simple metals than the theories based on the LDA [1,5–7]. This fact is due probably to the avoidance in T–O and A–K models the limitations of the LDA. In transition metals, in addition, the d-electrons contribute to the average density of the ‘free’ mobile part of the electron gas, and an examination of the correlation between SE and the electron density parameter r is much more complicated
K.F. Wojciechowski / Surface Science 437 (1999) 285–288
Fig. 3. Theoretical results for the surface energy s(r) of simple metals: S–R [13] (circles); T–O [8] (dashed line); A–K [12] (dotted line). The experimental values of s are marked by crosses.
than in the case of simple metals [15]. Therefore, we restrict ourselves only to a correlation of the experimental data and the ab-initio calculations with the effective r through the expression for SE m of the form given by Eq. (2). The fit of this function with l=2.5, 3 and 4 to the s (r ), for exp m the 3d, 4d, and 5d series, shows that the best fit is obtained with l=4 giving A(4)=49.57, 72.61 and 83.00 for the mentioned series, respectively. The fit of Ar−l to the s (r ) gives l>4, and since exp m the value of l=4 has the theoretical motivation [7,8], in Fig. 4, we compare SEs calculated from eq. 16 of [8] and the classical cleavage energy (calculated according to eq. 8.24 of [2] with r=r ) (full circles), with the experimental data [3] m
287
Fig. 4. Calculated surface energy of transition metal: S–R [13], (open circles); T–O ([8]+cleavage energy) (full circles); Ar−4 , m [fitted to s (r )] (dotted circles). The experimental values exp m s are marked by crosses. exp
(crosses) and with the ab-initio calculations [13] (open circles). From this figure, it can be seen that the simple T–O theory [8] gives a fairly good agreement with the ab-initio calculations [13] and gives the same trends as the experimental ones [3]2 (with the exclusion of 3d magnetic metals). For the 5d series, the agreement is only qualitative, but a reasonable agreement with the experiment for all three series is given by the functions A(nd)r−4 (dotted circles in Fig. 4) with A(3d)= 49.57, A(4d )=72.61, and A(5d )=82.88, which 2 The jellium model for a square potential based on LDA (see Ref. [7]) yields values that are smaller (by approximately a factor of 0.7) for the calculated SEs than the T–O model [8].
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K.F. Wojciechowski / Surface Science 437 (1999) 285–288
may be treated as the F(r) functions suggested by P–R [14]. The results of the present paper can be summarized as follows. For simple metals, the theoretical predictions of the SE dependence on the electron density parameter correlate better with the effective r values determined by Perrot and Resolt m [14] than with the conventional r values [16 ], s even for the ab-initio calculations given in [13]. For these metals, the theory agrees well with the experiment, using the simple approach for the electronic structure of semi-infinite jellium as proposed by Takahashi and Onzawa [8], in which exchange, correlation and other contributions are obscurely enclosed in Fermi energy and in work function, sum of which is assumed as the barrier height. In the case of simple metals, the ‘unexplicit’ function in Eq. (1), is therefore the explicit function F(r )=A(4)r−4. For 4d and 5d transition m metals, the function F(r ) can be approximately m represented by the explicit functions of the form Ar−l; however, for the 3d metals, a better description of the experimental data is given by an unexplicit function of r of the shape illustrated m in fig. 6 of [14]. Finally, since for both simple and transition metals, the best agreement with the experimental data is achieved for s ~r−4 but not th for s ~r−3, we conclude that the Miedema model th of cohesion [3] may be treated only as a very approximated model. Generally, one can conclude that for s-p metals and for the 4d and 5d series of transition metals, the SE data correlates with r m through the function Ar−4 , which results from m simple quantum mechanics.
Acknowledgement This work was supported by University, Grant No. 2016/W/99.
Wroclaw
References [1] N.D. Lang, in: S. Lundquist, N.H. March ( Eds.), Theory of the Inhomogeneous Electron Gas, Plenum, New York, 1983. [2] A. Kiejna, K.F. Wojciechowski, Metal Surface Electron Physics, Pergamon, Oxford, 1996. [3] F.R. de Boer, R. Boom, W.C.M. Mattens, A.R. Miedema, A.K. Niessen, Cohesion in Metals, North-Holland, Amsterdam, 1988. [4] A.M. Rodriguez, G. Bozzolo, J. Ferrante, Surf. Sci. 289 (1993) 100. [5] J.P. Perdew, H.Q. Tran, E.D. Smith, Phys. Rev. B 42 (1990) 11627. [6 ] H.B. Shore, J.H. Rose, Phys. Rev. Lett. 66 (1991) 2519. [7] G. Paasch, H. Wonn, Phys. Stat. Sol. (b) 70 (1975) 555. [8] K. Takahashi, T. Onzawa, Phys. Rev. B 48 (1993) 5689. [9] J. Schmit, A.A. Lucas, Solid State Commun. 11 (1972) 415. [10] W. Kohn, A. Yaniv, Phys. Rev. B 20 (1979) 4948. [11] N.H. March, B.V. Paranjape, Phys. Rev. B 30 (1984) 3131. [12] T.L. Ainsworth, E. Krotscheck, Phys. Rev. B 45 (1992) 8779. [13] H.L. Skriver, N.M. Rosengaard, Phys. Rev. B 46 (1992) 7157. [14] F. Perrot, M. Rasolt, J. Phys.: Condens. Matter 6 (1994) 1473. [15] K.F. Wojciechowski, Mod. Phys. Lett. B 12 (1998) 658. [16 ] Ch. Kittel, Introduction to Solid State Physics, Wiley, New York, 1986.