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Surface energy calculation – metals with 1 and 2 delocalized electrons per atom
Chemical Physics 278 (2002) 111–117 www.elsevier.com/locate/chemphys
Surface energy calculation – metals with 1 and 2 delocalized electrons per atom S. Halas a,b, T. Durakiewicz c,*, J.J. Joyce c a
c
Mass Spectrometry Laboratory, Institute of Physics, Maria Curie-Sklodowska University, 20-031 Lublin, Poland b Institute of Vacuum Technology, ul. Dluga 44/50, 00-241 Warsaw, Poland Los Alamos National Laboratory, Condensed Matter and Thermal Physics, MST-IV Group, Mailstop K 764, Los Alamos, NM 87545, USA Received 28 November 2001
Abstract In this paper we calculate surface energy (SE) of monovalent, divalent and some trivalent metals. For these metals for which SE can be solely expressed by dimensionless Wigner–Seitz density parameter, rs , of delocalized electrons: SE ¼ C1 rs5 þ C2 rs3:5 C3 rs4 ; where constants C1 , C2 and C3 have been calculated on the basis of Sommerfeld’s free electron and surface plasma models. Excellent agreement with experimental data was obtained. On the basis of our model SE values for Fr and Ra have been predicted as well. Ó 2002 Published by Elsevier Science B.V.
1. Introduction Surface energy (SE) is usually defined either as the difference between the free energy of the bulk and surface, or simply as energy needed to split the solid in two along a plane. Numerous theoretical attempts to calculate this important surface property were undertaken within the last decades. It seems, however, that none of the approaches so far resulted in satisfactory agreement with experimental values. In this regard one has to repeat the statement by Lang and Kohn in 1970 [1], namely
*
Corresponding author. Tel.: +1-5056674819; fax: +15056657652. E-mail address: [email protected] (T. Durakiewicz).
‘‘theories of metal surfaces have, relatively speaking, lagged far behind’’. Thirty years later one may still notice a discrepancy between the theories of bulk being usually capable of predicting numerous physical properties of solids with acceptable accuracy and surface theories the state of which regarding both the range of problems solved and accuracy is far from satisfactory. In their theory of metal surfaces, Lang and Kohn [1] employed the jellium model in which the inhomogeneous electron gas and exchange and correlation effects were taken into account. The uniform background model was suggested to be fully inadequate, as shown by unphysical decrease in calculated SE towards negative values for highelectron density metals (Mg, Zn and Al). This drawback was overcomed by utilizing the
0301-0104/02/$ - see front matter Ó 2002 Published by Elsevier Science B.V. PII: S 0 3 0 1 - 0 1 0 4 ( 0 2 ) 0 0 3 7 9 - 8
112
S. Halas et al. / Chemical Physics 278 (2002) 111–117
pseudopotential representation of ions, but overall agreement was still satisfactory only for lowdensity metals: Cs, Rb, K and Na, out of eight calculated. Interesting concept of utilizing the electron-ion plasma oscillations to calculate SE was introduced by Schmit and Lucas [2], who found SE to be proportional to rs5=2 , where rs is the electron density parameter. They investigated the modifications to the collective motions in the solid when surface is introduced and identified the surface correlation energy with zero-point energy of the plasma modes. Their calculated SE values for eight metals (Cs, Rb, K, Na, Li, Mg, Zn and Al) were systematically too large than those observed, except for Li, for which agreement was very good. This discrepancy was explained by authors as resulting from not including contributions, like shifts in kinetic, electrostatic, exchange and correlation energies that should add up to a negative value. Authors were unable to derive those terms. In this work we successfully utilize their concept of negative kinetic energy contribution to SE. The same authors [3] performed calculations for transition metals, for which, as in the case of simple metals, overall calculated values were too high. The problem of correlation energy was investigated by Vannimenus and Budd [4] who obtain SE from an approximate knowledge of electron density, in agreement with Lang and Kohn calculations [1], but also in poor agreement with experiments. Lang and Kohn’s calculations were recognized as a reference by Sahni and Ma [5] who calculated SE by applying variational principle to minimize the SE functional with respect to parametrized trial density. Authors obtained results equivalent to the self-consistent methods but in a much more simple manner. Stabilized jellium was found by Fiolhais and Perdew [6] to be ‘‘much more realistic’’ for highdensity metals than ordinary jellium in calculating surface properties by Kohn–Sham expansion of kinetic energy within the LDA; a conclusion similar to the one presented in the original paper by Lang and Kohn [1]. The comparison with experimental values for SE shows that agreement is ‘‘fixed’’ at Al for rs ¼ 2:07, and the calculated values are systematically too low by 10–20% for lower electron densities. The ‘‘ideal metal’’ model
by Shore and Rose [7], in which the jellium is improved by introducing mechanical equilibrium of electron gas, was used by Ainsworth and Krotscheck [8] to calculate SE by optimized variational theory, in which some limitations of LDA are overcome by application of the JastrowFeenberg wave function. As a result they obtain SE as a function of rs where good agreement is obtained for low-electron density metals and results are too large for high-electron densities. Authors conclude that the ‘‘high-density regime is conceptually problematic’’ and were ‘‘not concerned’’ with the inaccuracy of their approach for metals with rs 6 2. Full potential linear-muffin-tinorbital (LMTO) approach was used by Methfessel et al. [9] to calculate work functions and SE for the seven-layer slabs of 4d transition metals. Acceptable agreement with experimental values deduced either from liquid metal or enthalpies of atomization was shown, but the calculated SE values were significantly too high when compared with direct measurement of liquid metal SE’s. Jellium-like model was exercised again by Takahashi and Ozawa [10] who approximated the effective oneelectron potential by the step function shifted to satisfy the charge neutrality. Calculated SE values are too low for low- and medium-electron density metals and significantly too high for high-density metals. Probably the first thorough study of SE was done by Vitos et al. [11] who calculated the database of SE values for low index surfaces of 60 metals by use of full-charge-density linear-muffintin-orbital approximation. Obtained agreement with experimental values was satisfactory. Search for the jellium-based functional dependence of ES on electron density was summarized by Wojciechowski [12] shortly before he passed away. From this interesting comparison one can learn that the Takahashi and Onzawa [10] and Ainsworth and Krotscheck [8] theories (mentioned above) which are not self-consistent, give a much better agreement with experiment for simple metals than the self-consistent LDA approaches. This was one of the reasons for the authors of this paper to go back to the ‘‘pure’’ free electron picture. Stabilized jellium is once again invoked by Brajczewska et al. [13] as ‘‘the simplest model
S. Halas et al. / Chemical Physics 278 (2002) 111–117
which yields realistic results for the physical properties of simple metals’’ who fits the powerlaw SEðrs ) functions to calculated values. Most recently a comparison of various self-consistent schemes was given by Pitarke and Eguiluz [14] where the authors went beyond LDA by evaluating the density response function of a bounded free-electron gas within the random-phase-approximation. Unfortunately enough, results were not compared with experimental values. In our model we utilize some of the ideas discussed above, like negative kinetic SE contribution, but generally we simplify the problem drastically by coming back directly to the free electron formalism.
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due to bulk electron pressure and (2) their vibrations in the electrostatic field of the nearest ions. Assuming that kinetic energy of free electrons in the surface layer for x and y degrees of freedom is identical to that in the bulk, we will consider below SE as difference of energies accounted for z degree of freedom only. It is convenient to calculate SE in atomic units Ry=pa20 , where 1 Ry ¼ e2 =ð8pe0 a0 Þ ¼ 13:6 eV, e0 is the electric constant and a0 is the Bohr radius. The following conversion formula was used in the expression of final results: Ry J ¼ 247:79 2 : 2 pa0 m
ð1Þ
3. Electrostatic energy of dipole layer 2. Model According to SE definition, we have to consider the energy difference of delocalized or free electrons in two identical monoatomic layers: one on the surface and another in the bulk. These two layers are schematically shown in Fig. 1. Free electrons in the bulk layer have only kinetic energy, the distribution of which can be calculated by the Sommerfeld model, whereas in the surface layer the electrons have similar kinetic energy for two degrees of freedom (in x, y plane according to Fig. 1) but not for the degree along z-axis perpendicular to the surface. Additionally, electrons in the surface layer have a certain potential energy originating from: (1) an overall shift versus ions
The average shift of free electrons, ds , in the surface layer due to action of bulk electron pressure, Pe , may be calculated from the static equilibrium condition: 2
Pe pðrs a0 Þ ¼ Fc ;
ð2Þ
where rs is dimensionless Wigner–Seitz electron density parameter and Fc is the resultant Coulomb force of the nearest unscreened ions. For small shifts Fc is directly proportional to ds : Fc ¼ kds ;
ð3Þ
where k is the force constant in vibrational motion of surface electron. The vibrational frequency of surface electron may be expressed in terms of electron density, n [15] as follows: x2 ¼
k ne2 ¼ ; m 2e0 m
ð4Þ
where m is the mass of electron. Hence Eq. (1) may be rewritten in the following form: 2
Pe pðrs a0 Þ ¼
ne2 ds ; 2e0
ð5Þ
from which ds is calculated. Having ds , we may calculate the contribution of surface dipole layer to SE: Fig. 1. Schematic representation of free electrons shifted versus ions in the surface layer.
1 2 e0 SEdipole ¼ Pe ds ¼ Pe2 pðrs a0 Þ 2 : 2 ne
ð6Þ
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S. Halas et al. / Chemical Physics 278 (2002) 111–117
The factor 12 in the above equation results from considering the average force acting along the shift. The electron gas pressure may be generally expressed in terms of electron density and its kinetic energy, hEkin i, which in the case of simple metals may be replaced by 35 of Fermi energy, EF . Hence 2 2 Pe ¼ nhEkin i ¼ nEF : 3 5
ð7Þ
2
merical coefficient at pðrs a0 Þ in Eq. (11) is not far from unity. In all the calculations below we assume that the numerical coefficient in Eq. (11) is exactly equal to 1. In other words we have as2 sumed one free electron per area of pðrs a0 Þ for all the considered metals: alkali metals, alkaline earth metals and coinage metals. The contribution of average vibrational energy to SE may be calculated as 1 2 2 SEvib ¼ kA ð12Þ pðrs a0 Þ ; 2
Substituting the above expression for Pe into formula (6) one obtains: 2 1 4pe0 a0 EF r s a 0 n SEdipole ¼ 5 e2 1 ¼ ðEF rs Þ2 a0 n=Ry: ð8Þ 50
where k is the force constant and A is the amplitude of vibration. Replacing the classical formula by the quantum-mechanical one, we have to introduce the average number hN i of oscillators per pðrs a0 Þ2 :
Finally we express n by rs a0 and substitute to this formula for EF the value from Sommerfeld model:
1 2 SEvib ¼ hN i hix=pðrs a0 Þ ; 2
EF ¼
50:1 eV 50:1 Ry ¼ : rs2 13:6rs2
ð9Þ
Hence SEdipole
2 50:1 a0 13:6rs ð4=3Þpðrs a0 Þ3 2 Ry 3 50:1 ¼ 2 rs5 pa0 200 13:6 J ¼ 50:44 2 rs5 : m Ry ¼ 50
where h is Planck constant and x is given by Eq. (4). Assuming linear drop of A2 with r, one obtains hA2 i ¼ A20 =3, where A0 is the maximum amplitude. Based on Bohr’s correspondence principle we may asses that 1 hN i ¼ : 3
ð14Þ 2
ð10Þ
4. Vibrational energy In order to calculate electron vibrational and kinetic energy contributions to SE we have to realize that in simple metals the average area occupied by simple electron surface layer is very close to pðrs a0 Þ2 . Strictly speaking we have one electron per nh area, where n is the electron density and h is the average thickness of the layer, which is close to 1.5 of atom radius R. Hence the surface electron density is 1 ð4=3Þpðrs a0 Þ3 4 r s a0 2 ¼ pðrs a0 Þ : nh 4:5 R 1:5R
ð13Þ
ð11Þ
It can be demonstrated that the ratio ðrs a0 Þ=R for alkali metals is close to 1.14, therefore the nu-
In other words we have one oscillator per 3pðrs a0 Þ area of surface layer. We shall see below that this assessment of hN i is crucial in SE calculation. From Eqs. (13), (14) and (4), one obtains sffiffiffiffiffiffiffiffiffiffi, h ne2 2 SEvib ¼ pðrs a0 Þ 6 2e0 m sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 e2 ¼ 2 pa0 ð4=3Þprs3 a30 2e0 m pffiffiffi rffiffiffiffiffiffiffiffi J 3 hc Ry 3:5 ¼ 101:1 2 rs3:5 : ð15Þ ¼ r m 6p a30 mc2 s 5. Kinetic energy The kinetic energy of the surface electrons (along z-axis) is distributed in similar manner as the vibrational energy, but its maximum falls onto the position of ion (in the x, y plane), whereas the
S. Halas et al. / Chemical Physics 278 (2002) 111–117
kinetic energy totally disappears in the mid-position between ions. The contribution of the kinetic energy to SE may be calculated as follows: 1 2 2 mvz SEkin ¼ pðrs a0 Þ 2 1 2 ¼ mv20z =pðrs a0 Þ ; ð16Þ 6 where v20z is the square of electron velocity along z-axis in the position of surface ions. Assuming that this v20z is exactly equal to the mean hv2z i in bulk, we may rewrite Eq. (16) in the following form: SEkin
1 2 ¼ hEz i=pðrs a0 Þ ; 3
ð17Þ
where hEz i is the average kinetic energy of electron per one degree of freedom. For simple metals this energy may be also expressed by rs : 1 13 1 50:1 eV EF ¼ hEz i ¼ hEkin i ¼ : 3 35 5 rs2
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where C1 ¼ 50:44
J ; m2
C2 ¼ 101:1
J : m2
The third term in the above formula is negative: 2 2 1 50:1 Ry 4 hEz i=pðrs a0 Þ2 ¼ r 3 3 5 13:6 pa20 s J ¼ 121:7 2 rs4 : m
Therefore the final formula for total SE of simple metals may be expressed solely in terms of dimensionless electron density parameter: SE ¼ C1 rs5 þ C2 rs3:5 C3 rs4 ;
ð21Þ
where C1 and C2 are given above, and C3 ¼ 121:7
J : m2
7. Comparison with experiment ð18Þ
6. Total SE Total SE is calculated by addition of all the above contributions to SE and by subsequent subtracting the total energy (kinetic only) in bulk 2 layer per pðrs a0 Þ . Hence from formulae (10), (15), (16), (17) and (18) one obtains 1 hEz i hEz i SE ¼ C1 rs5 þ C2 rs3:5 þ pðrs a0 Þ2 ; 3 ð19Þ
Plotting SE versus rs according to Eq. (21) we noticed that calculated SE are somewhat higher than experimental values for small rs . Therefore we tried to adjust the force constant k, which may be lower than calculated from Eq. (4) for metals with low rs values. For these metals we have both: (i) large electron shift and (ii) high vibrational energy, see Table 1. If we assume that vibrational frequency is reduced only by 7% (which is hardly measured with greater accuracy) then the agreement between calculated SE and experimental data is excellent, see Table 2 and Fig. 2. Note that if we reduce C2 by a factor 1/1.07, we have to reduce C1 2 by factor 1=ð1:07Þ because this constant is directly
Table 1 The contribution of SE dipole for various rs rs
1.5 2.0 2.5 3.0 4.0 5.0 6.0
ð20Þ
C1 ¼ 44:06, C2 ¼ 94:49, C3 ¼ 121:7
C1 ¼ 50:44, C2 ¼ 101:1, C3 ¼ 121:7
SEdipole
SEtotal
SEdipole
SEtotal
5.80 1.38 0.45 0.18 0.04 0.01 0.006
4.62 2.12 1.60 0.70 0.31 0.16 0.09
6.64 1.58 0.52 0.21 0.05 0.016 0.006
7.06 2.91 1.49 0.87 0.36 0.18 0.10
The first set was calculated for adjusted C1 and C2 constants as described in text.
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S. Halas et al. / Chemical Physics 278 (2002) 111–117
Table 2 Calculated surface energies for metals with well known rs values (taken from [17]) Element
Li Na K Rb Cs Fr Be Mg Ca Sr Ba Ra Sc Y La Cu Ag Au a
Valence
1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 2 2 2.5
rs (bohr)
3.26 3.99 4.95 5.31 5.75 6.12 1.87 2.65 3.26 3.55 3.73 3.92 2.38 2.61 2.64 2.12 2.39 2.22
SE (J=m2 )
dSE=SE
Calculated
Experimental
0.55 0.31 0.16 0.13 0.10 0.08 2.54 0.99 0.55 0.43 0.37 0.32 0.92 0.71 1.00 1.81 1.31 1.60
0.525 0.261 0.145 0.117 0.095 ) 2.7 0.785 0.502 0.419 0.38 ) 1.275 1.125 1.02 1.825 1.246 1.51
a
0.05 0.19 0.10 0.11 0.05 )0.06 0.26 0.09 0.03 )0.03 0.04 )0.08 )0.02 )0.01 0.05 0.06
Data compiled from [18,19].
proportional to the force constant, k, whilst C2 is proportional to the square root of k. It seems that anharmonicity effect is responsible for lower force constant and lower vibrational frequency.
8. Discussion We have demonstrated that (like in the case of work function calculation [16,17]) the SE can be calculated semiclassically with a high degree of accuracy. We totally neglect in our considerations the exchange and correlation energies, which are major contributions in any self-consistent calculations. As it was already found in the case of work function calculations [16,17] and for SE [12], we observe that the self-consistent calculations (except for the work by Vitos et al. [11]) in spite of their enormous level of complication, are unable to give very good agreement with experimental values for a broad range of electron densities.
9. Conclusions
Fig. 2. Surface energies of simple metals. Dots are experimental data, while squares are predicted values.
(1) The model presented in this paper is based on limited number of reliable assumptions, which are: (i) Identity of lateral kinetic energies of electrons in surface and bulk layers.
S. Halas et al. / Chemical Physics 278 (2002) 111–117 2
(ii) One electron occupies pðrs a0 Þ of surface layer. (iii) Maximum kinetic energy per degree of freedom perpendicular to the layer is identical as average kinetic energy in bulk. (iv) Vibrational and kinetic energies of surface layer electrons drop linearly with r from their maximum values to zero at r ¼ rs . The last assumption is less obvious than the remaining ones, but it leads to perfect agreement with experiment. One can also imagine gaussian distribution which lead to identical result. (2) Surface energy of alkali metals, alkaline earth scandides and coinage metals, can be expressed as a function rs only. The function given by Eq. (21) is relatively simple. For alkali metals with large rs the contribution of dipole layer may be neglected, but for metals with small rs it is the most important term. (3) Reliable SE values for radioactive elements Fr and Ra have been predicted by this model.
Acknowledgements Thanks are due to Dr. Janina Szaran for paper processing. This study was supported by State Committee for Scientific Research, Warsaw, in the framework of 8T19C 01418 grant and by the US Department of Energy, Office of Basic Energy Sciences, Division of Materials Science.
117
References [1] N.D. Lang, W. Kohn, Phys. Rev. B 1 (1970) 4555. [2] J. Schmit, A.A. Lucas, Solid State Commun. 11 (1972) 415. [3] J. Schmit, A.A. Lucas, Solid State Commun. 11 (1972) 419. [4] J. Vannimenus, H.F. Budd, Solid State Commun. 15 (1974) 1739. [5] V. Sahni, C.Q. Ma, Phys. Rev. B 20 (1979) 3511. [6] C. Fiolhais, J.P. Perdew, Phys. Rev. B 45 (1992) 6207. [7] H.B. Shore, J.H. Rose, Phys. Rev. Lett. 66 (1991) 2519. [8] T.L. Ainsworth, E. Krotscheck, Phys. Rev. B 45 (1992) 8779. [9] M. Methfessel, D. Henning, M. Sheffler, Phys. Rev. B 46 (1992) 4816. [10] K. Takahashi, T. Onzawa, Phys. Rev. B 48 (1993) 5689. [11] L. Vitos, A.V. Ruban, H.L. Skriver, J. Kollar, Surf. Sci. 411 (1998) 186. [12] K.F. Wojciechowski, Surf. Sci. 437 (1999) 285. [13] M. Brajczewska, C. Henriques, C. Fiolhais, Vacuum 63 (2001) 135. [14] J.M. Pitarke, A.G. Eguiluz, Phys. Rev. B 63 (2001), 045116-1/11. [15] N.W. Ashcroft, N.D. Mermin, Solid State Physics, Holt, Rinehart and Winston, New York; Saunders College, Philadelphia, PA, 1976. [16] S. Halas, T. Durakiewicz, J. Phys. Cond. Matt. 10 (1998) 10815. [17] T. Durakiewicz, S. Halas, A. Arko, J.J. Joyce, D.P. Moore, Phys. Rev. B 64 (2001), 045101/1-8. [18] W.R. Tyson, W.A. Miller, Surf. Sci. 62 (1977) 267. [19] F.R. de Boer, R. Boom, W.C.M. Mattens, A.R. Miedema, A.K. Niessen, Cohesion in Metals, North-Holland, Amsterdam, 1988.
Surface energy calculation – metals with 1 and 2 delocalized electrons per atom S. Halas a,b, T. Durakiewicz c,*, J.J. Joyce c a
c
Mass Spectrometry Laboratory, Institute of Physics, Maria Curie-Sklodowska University, 20-031 Lublin, Poland b Institute of Vacuum Technology, ul. Dluga 44/50, 00-241 Warsaw, Poland Los Alamos National Laboratory, Condensed Matter and Thermal Physics, MST-IV Group, Mailstop K 764, Los Alamos, NM 87545, USA Received 28 November 2001
Abstract In this paper we calculate surface energy (SE) of monovalent, divalent and some trivalent metals. For these metals for which SE can be solely expressed by dimensionless Wigner–Seitz density parameter, rs , of delocalized electrons: SE ¼ C1 rs5 þ C2 rs3:5 C3 rs4 ; where constants C1 , C2 and C3 have been calculated on the basis of Sommerfeld’s free electron and surface plasma models. Excellent agreement with experimental data was obtained. On the basis of our model SE values for Fr and Ra have been predicted as well. Ó 2002 Published by Elsevier Science B.V.
1. Introduction Surface energy (SE) is usually defined either as the difference between the free energy of the bulk and surface, or simply as energy needed to split the solid in two along a plane. Numerous theoretical attempts to calculate this important surface property were undertaken within the last decades. It seems, however, that none of the approaches so far resulted in satisfactory agreement with experimental values. In this regard one has to repeat the statement by Lang and Kohn in 1970 [1], namely
*
Corresponding author. Tel.: +1-5056674819; fax: +15056657652. E-mail address: [email protected] (T. Durakiewicz).
‘‘theories of metal surfaces have, relatively speaking, lagged far behind’’. Thirty years later one may still notice a discrepancy between the theories of bulk being usually capable of predicting numerous physical properties of solids with acceptable accuracy and surface theories the state of which regarding both the range of problems solved and accuracy is far from satisfactory. In their theory of metal surfaces, Lang and Kohn [1] employed the jellium model in which the inhomogeneous electron gas and exchange and correlation effects were taken into account. The uniform background model was suggested to be fully inadequate, as shown by unphysical decrease in calculated SE towards negative values for highelectron density metals (Mg, Zn and Al). This drawback was overcomed by utilizing the
0301-0104/02/$ - see front matter Ó 2002 Published by Elsevier Science B.V. PII: S 0 3 0 1 - 0 1 0 4 ( 0 2 ) 0 0 3 7 9 - 8
112
S. Halas et al. / Chemical Physics 278 (2002) 111–117
pseudopotential representation of ions, but overall agreement was still satisfactory only for lowdensity metals: Cs, Rb, K and Na, out of eight calculated. Interesting concept of utilizing the electron-ion plasma oscillations to calculate SE was introduced by Schmit and Lucas [2], who found SE to be proportional to rs5=2 , where rs is the electron density parameter. They investigated the modifications to the collective motions in the solid when surface is introduced and identified the surface correlation energy with zero-point energy of the plasma modes. Their calculated SE values for eight metals (Cs, Rb, K, Na, Li, Mg, Zn and Al) were systematically too large than those observed, except for Li, for which agreement was very good. This discrepancy was explained by authors as resulting from not including contributions, like shifts in kinetic, electrostatic, exchange and correlation energies that should add up to a negative value. Authors were unable to derive those terms. In this work we successfully utilize their concept of negative kinetic energy contribution to SE. The same authors [3] performed calculations for transition metals, for which, as in the case of simple metals, overall calculated values were too high. The problem of correlation energy was investigated by Vannimenus and Budd [4] who obtain SE from an approximate knowledge of electron density, in agreement with Lang and Kohn calculations [1], but also in poor agreement with experiments. Lang and Kohn’s calculations were recognized as a reference by Sahni and Ma [5] who calculated SE by applying variational principle to minimize the SE functional with respect to parametrized trial density. Authors obtained results equivalent to the self-consistent methods but in a much more simple manner. Stabilized jellium was found by Fiolhais and Perdew [6] to be ‘‘much more realistic’’ for highdensity metals than ordinary jellium in calculating surface properties by Kohn–Sham expansion of kinetic energy within the LDA; a conclusion similar to the one presented in the original paper by Lang and Kohn [1]. The comparison with experimental values for SE shows that agreement is ‘‘fixed’’ at Al for rs ¼ 2:07, and the calculated values are systematically too low by 10–20% for lower electron densities. The ‘‘ideal metal’’ model
by Shore and Rose [7], in which the jellium is improved by introducing mechanical equilibrium of electron gas, was used by Ainsworth and Krotscheck [8] to calculate SE by optimized variational theory, in which some limitations of LDA are overcome by application of the JastrowFeenberg wave function. As a result they obtain SE as a function of rs where good agreement is obtained for low-electron density metals and results are too large for high-electron densities. Authors conclude that the ‘‘high-density regime is conceptually problematic’’ and were ‘‘not concerned’’ with the inaccuracy of their approach for metals with rs 6 2. Full potential linear-muffin-tinorbital (LMTO) approach was used by Methfessel et al. [9] to calculate work functions and SE for the seven-layer slabs of 4d transition metals. Acceptable agreement with experimental values deduced either from liquid metal or enthalpies of atomization was shown, but the calculated SE values were significantly too high when compared with direct measurement of liquid metal SE’s. Jellium-like model was exercised again by Takahashi and Ozawa [10] who approximated the effective oneelectron potential by the step function shifted to satisfy the charge neutrality. Calculated SE values are too low for low- and medium-electron density metals and significantly too high for high-density metals. Probably the first thorough study of SE was done by Vitos et al. [11] who calculated the database of SE values for low index surfaces of 60 metals by use of full-charge-density linear-muffintin-orbital approximation. Obtained agreement with experimental values was satisfactory. Search for the jellium-based functional dependence of ES on electron density was summarized by Wojciechowski [12] shortly before he passed away. From this interesting comparison one can learn that the Takahashi and Onzawa [10] and Ainsworth and Krotscheck [8] theories (mentioned above) which are not self-consistent, give a much better agreement with experiment for simple metals than the self-consistent LDA approaches. This was one of the reasons for the authors of this paper to go back to the ‘‘pure’’ free electron picture. Stabilized jellium is once again invoked by Brajczewska et al. [13] as ‘‘the simplest model
S. Halas et al. / Chemical Physics 278 (2002) 111–117
which yields realistic results for the physical properties of simple metals’’ who fits the powerlaw SEðrs ) functions to calculated values. Most recently a comparison of various self-consistent schemes was given by Pitarke and Eguiluz [14] where the authors went beyond LDA by evaluating the density response function of a bounded free-electron gas within the random-phase-approximation. Unfortunately enough, results were not compared with experimental values. In our model we utilize some of the ideas discussed above, like negative kinetic SE contribution, but generally we simplify the problem drastically by coming back directly to the free electron formalism.
113
due to bulk electron pressure and (2) their vibrations in the electrostatic field of the nearest ions. Assuming that kinetic energy of free electrons in the surface layer for x and y degrees of freedom is identical to that in the bulk, we will consider below SE as difference of energies accounted for z degree of freedom only. It is convenient to calculate SE in atomic units Ry=pa20 , where 1 Ry ¼ e2 =ð8pe0 a0 Þ ¼ 13:6 eV, e0 is the electric constant and a0 is the Bohr radius. The following conversion formula was used in the expression of final results: Ry J ¼ 247:79 2 : 2 pa0 m
ð1Þ
3. Electrostatic energy of dipole layer 2. Model According to SE definition, we have to consider the energy difference of delocalized or free electrons in two identical monoatomic layers: one on the surface and another in the bulk. These two layers are schematically shown in Fig. 1. Free electrons in the bulk layer have only kinetic energy, the distribution of which can be calculated by the Sommerfeld model, whereas in the surface layer the electrons have similar kinetic energy for two degrees of freedom (in x, y plane according to Fig. 1) but not for the degree along z-axis perpendicular to the surface. Additionally, electrons in the surface layer have a certain potential energy originating from: (1) an overall shift versus ions
The average shift of free electrons, ds , in the surface layer due to action of bulk electron pressure, Pe , may be calculated from the static equilibrium condition: 2
Pe pðrs a0 Þ ¼ Fc ;
ð2Þ
where rs is dimensionless Wigner–Seitz electron density parameter and Fc is the resultant Coulomb force of the nearest unscreened ions. For small shifts Fc is directly proportional to ds : Fc ¼ kds ;
ð3Þ
where k is the force constant in vibrational motion of surface electron. The vibrational frequency of surface electron may be expressed in terms of electron density, n [15] as follows: x2 ¼
k ne2 ¼ ; m 2e0 m
ð4Þ
where m is the mass of electron. Hence Eq. (1) may be rewritten in the following form: 2
Pe pðrs a0 Þ ¼
ne2 ds ; 2e0
ð5Þ
from which ds is calculated. Having ds , we may calculate the contribution of surface dipole layer to SE: Fig. 1. Schematic representation of free electrons shifted versus ions in the surface layer.
1 2 e0 SEdipole ¼ Pe ds ¼ Pe2 pðrs a0 Þ 2 : 2 ne
ð6Þ
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S. Halas et al. / Chemical Physics 278 (2002) 111–117
The factor 12 in the above equation results from considering the average force acting along the shift. The electron gas pressure may be generally expressed in terms of electron density and its kinetic energy, hEkin i, which in the case of simple metals may be replaced by 35 of Fermi energy, EF . Hence 2 2 Pe ¼ nhEkin i ¼ nEF : 3 5
ð7Þ
2
merical coefficient at pðrs a0 Þ in Eq. (11) is not far from unity. In all the calculations below we assume that the numerical coefficient in Eq. (11) is exactly equal to 1. In other words we have as2 sumed one free electron per area of pðrs a0 Þ for all the considered metals: alkali metals, alkaline earth metals and coinage metals. The contribution of average vibrational energy to SE may be calculated as 1 2 2 SEvib ¼ kA ð12Þ pðrs a0 Þ ; 2
Substituting the above expression for Pe into formula (6) one obtains: 2 1 4pe0 a0 EF r s a 0 n SEdipole ¼ 5 e2 1 ¼ ðEF rs Þ2 a0 n=Ry: ð8Þ 50
where k is the force constant and A is the amplitude of vibration. Replacing the classical formula by the quantum-mechanical one, we have to introduce the average number hN i of oscillators per pðrs a0 Þ2 :
Finally we express n by rs a0 and substitute to this formula for EF the value from Sommerfeld model:
1 2 SEvib ¼ hN i hix=pðrs a0 Þ ; 2
EF ¼
50:1 eV 50:1 Ry ¼ : rs2 13:6rs2
ð9Þ
Hence SEdipole
2 50:1 a0 13:6rs ð4=3Þpðrs a0 Þ3 2 Ry 3 50:1 ¼ 2 rs5 pa0 200 13:6 J ¼ 50:44 2 rs5 : m Ry ¼ 50
where h is Planck constant and x is given by Eq. (4). Assuming linear drop of A2 with r, one obtains hA2 i ¼ A20 =3, where A0 is the maximum amplitude. Based on Bohr’s correspondence principle we may asses that 1 hN i ¼ : 3
ð14Þ 2
ð10Þ
4. Vibrational energy In order to calculate electron vibrational and kinetic energy contributions to SE we have to realize that in simple metals the average area occupied by simple electron surface layer is very close to pðrs a0 Þ2 . Strictly speaking we have one electron per nh area, where n is the electron density and h is the average thickness of the layer, which is close to 1.5 of atom radius R. Hence the surface electron density is 1 ð4=3Þpðrs a0 Þ3 4 r s a0 2 ¼ pðrs a0 Þ : nh 4:5 R 1:5R
ð13Þ
ð11Þ
It can be demonstrated that the ratio ðrs a0 Þ=R for alkali metals is close to 1.14, therefore the nu-
In other words we have one oscillator per 3pðrs a0 Þ area of surface layer. We shall see below that this assessment of hN i is crucial in SE calculation. From Eqs. (13), (14) and (4), one obtains sffiffiffiffiffiffiffiffiffiffi, h ne2 2 SEvib ¼ pðrs a0 Þ 6 2e0 m sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 e2 ¼ 2 pa0 ð4=3Þprs3 a30 2e0 m pffiffiffi rffiffiffiffiffiffiffiffi J 3 hc Ry 3:5 ¼ 101:1 2 rs3:5 : ð15Þ ¼ r m 6p a30 mc2 s 5. Kinetic energy The kinetic energy of the surface electrons (along z-axis) is distributed in similar manner as the vibrational energy, but its maximum falls onto the position of ion (in the x, y plane), whereas the
S. Halas et al. / Chemical Physics 278 (2002) 111–117
kinetic energy totally disappears in the mid-position between ions. The contribution of the kinetic energy to SE may be calculated as follows: 1 2 2 mvz SEkin ¼ pðrs a0 Þ 2 1 2 ¼ mv20z =pðrs a0 Þ ; ð16Þ 6 where v20z is the square of electron velocity along z-axis in the position of surface ions. Assuming that this v20z is exactly equal to the mean hv2z i in bulk, we may rewrite Eq. (16) in the following form: SEkin
1 2 ¼ hEz i=pðrs a0 Þ ; 3
ð17Þ
where hEz i is the average kinetic energy of electron per one degree of freedom. For simple metals this energy may be also expressed by rs : 1 13 1 50:1 eV EF ¼ hEz i ¼ hEkin i ¼ : 3 35 5 rs2
115
where C1 ¼ 50:44
J ; m2
C2 ¼ 101:1
J : m2
The third term in the above formula is negative: 2 2 1 50:1 Ry 4 hEz i=pðrs a0 Þ2 ¼ r 3 3 5 13:6 pa20 s J ¼ 121:7 2 rs4 : m
Therefore the final formula for total SE of simple metals may be expressed solely in terms of dimensionless electron density parameter: SE ¼ C1 rs5 þ C2 rs3:5 C3 rs4 ;
ð21Þ
where C1 and C2 are given above, and C3 ¼ 121:7
J : m2
7. Comparison with experiment ð18Þ
6. Total SE Total SE is calculated by addition of all the above contributions to SE and by subsequent subtracting the total energy (kinetic only) in bulk 2 layer per pðrs a0 Þ . Hence from formulae (10), (15), (16), (17) and (18) one obtains 1 hEz i hEz i SE ¼ C1 rs5 þ C2 rs3:5 þ pðrs a0 Þ2 ; 3 ð19Þ
Plotting SE versus rs according to Eq. (21) we noticed that calculated SE are somewhat higher than experimental values for small rs . Therefore we tried to adjust the force constant k, which may be lower than calculated from Eq. (4) for metals with low rs values. For these metals we have both: (i) large electron shift and (ii) high vibrational energy, see Table 1. If we assume that vibrational frequency is reduced only by 7% (which is hardly measured with greater accuracy) then the agreement between calculated SE and experimental data is excellent, see Table 2 and Fig. 2. Note that if we reduce C2 by a factor 1/1.07, we have to reduce C1 2 by factor 1=ð1:07Þ because this constant is directly
Table 1 The contribution of SE dipole for various rs rs
1.5 2.0 2.5 3.0 4.0 5.0 6.0
ð20Þ
C1 ¼ 44:06, C2 ¼ 94:49, C3 ¼ 121:7
C1 ¼ 50:44, C2 ¼ 101:1, C3 ¼ 121:7
SEdipole
SEtotal
SEdipole
SEtotal
5.80 1.38 0.45 0.18 0.04 0.01 0.006
4.62 2.12 1.60 0.70 0.31 0.16 0.09
6.64 1.58 0.52 0.21 0.05 0.016 0.006
7.06 2.91 1.49 0.87 0.36 0.18 0.10
The first set was calculated for adjusted C1 and C2 constants as described in text.
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S. Halas et al. / Chemical Physics 278 (2002) 111–117
Table 2 Calculated surface energies for metals with well known rs values (taken from [17]) Element
Li Na K Rb Cs Fr Be Mg Ca Sr Ba Ra Sc Y La Cu Ag Au a
Valence
1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 2 2 2.5
rs (bohr)
3.26 3.99 4.95 5.31 5.75 6.12 1.87 2.65 3.26 3.55 3.73 3.92 2.38 2.61 2.64 2.12 2.39 2.22
SE (J=m2 )
dSE=SE
Calculated
Experimental
0.55 0.31 0.16 0.13 0.10 0.08 2.54 0.99 0.55 0.43 0.37 0.32 0.92 0.71 1.00 1.81 1.31 1.60
0.525 0.261 0.145 0.117 0.095 ) 2.7 0.785 0.502 0.419 0.38 ) 1.275 1.125 1.02 1.825 1.246 1.51
a
0.05 0.19 0.10 0.11 0.05 )0.06 0.26 0.09 0.03 )0.03 0.04 )0.08 )0.02 )0.01 0.05 0.06
Data compiled from [18,19].
proportional to the force constant, k, whilst C2 is proportional to the square root of k. It seems that anharmonicity effect is responsible for lower force constant and lower vibrational frequency.
8. Discussion We have demonstrated that (like in the case of work function calculation [16,17]) the SE can be calculated semiclassically with a high degree of accuracy. We totally neglect in our considerations the exchange and correlation energies, which are major contributions in any self-consistent calculations. As it was already found in the case of work function calculations [16,17] and for SE [12], we observe that the self-consistent calculations (except for the work by Vitos et al. [11]) in spite of their enormous level of complication, are unable to give very good agreement with experimental values for a broad range of electron densities.
9. Conclusions
Fig. 2. Surface energies of simple metals. Dots are experimental data, while squares are predicted values.
(1) The model presented in this paper is based on limited number of reliable assumptions, which are: (i) Identity of lateral kinetic energies of electrons in surface and bulk layers.
S. Halas et al. / Chemical Physics 278 (2002) 111–117 2
(ii) One electron occupies pðrs a0 Þ of surface layer. (iii) Maximum kinetic energy per degree of freedom perpendicular to the layer is identical as average kinetic energy in bulk. (iv) Vibrational and kinetic energies of surface layer electrons drop linearly with r from their maximum values to zero at r ¼ rs . The last assumption is less obvious than the remaining ones, but it leads to perfect agreement with experiment. One can also imagine gaussian distribution which lead to identical result. (2) Surface energy of alkali metals, alkaline earth scandides and coinage metals, can be expressed as a function rs only. The function given by Eq. (21) is relatively simple. For alkali metals with large rs the contribution of dipole layer may be neglected, but for metals with small rs it is the most important term. (3) Reliable SE values for radioactive elements Fr and Ra have been predicted by this model.
Acknowledgements Thanks are due to Dr. Janina Szaran for paper processing. This study was supported by State Committee for Scientific Research, Warsaw, in the framework of 8T19C 01418 grant and by the US Department of Energy, Office of Basic Energy Sciences, Division of Materials Science.
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