Phonon spectra of alkali metals

316

Physica 114B (1982) 316-322 North-Holland Publishing Company

P H O N O N SPECTRA OF A L K A L I M E T A L S S. ZEKOVI(~, F. V U K A J L O V I C and V. VELJKOVI(~ Laboratory for Theoretical Physics, Boris Kidri6 Institute of Nuclear Sciences, P.O,B. 552, 11001 Beograd, Yugoslavia

Received 28 December 1981 Revised 5 April 1982 In this work we used a simple local model pseudopotential which includes screening for the phonon spectra calculations of alkali metals. The results obtained are in very good agreement with experimental data. In some branches of phonon spectra the differences between theoretical and experimental results are within 1-2%, while the maximum error is about 6%. The suggested form of the pseudopotential allows us to describe the phonon spectra of Na, K and Rb with only one, and, at the same time, a unique, parameter. In this case, the maximum disagreements from experiment are 9% for Na, 8% for K and 7% for Rb. 1. Introduction The pseudopotential concept was originally proposed for the band calculation. However, the perturbational treatment of the pseudopotential has opened possibilities for calculating various crystal properties directly without band calculations. The so-called second-order perturbation theory has been applied to simple metals and the higher-order perturbation theory has been applied to complex crystals (polyvalent metals, covalent crystals and semimetals) with many successful examples [1]. There are two types of pseudopotentials which have been used in the pseudopotential calculations: The first principle pseudopotential, starting from the free-ion data, and the phenomenological pseudopotential with parameters fitted from some properties of metals. It is well known that if we consistently take into account all factors in the framework of the first approach, for instance nonlocality, it leads to numerous calculation problems, even in second-order perturbation theory. Since the pioneering work on Toya [2], there have been many attempts to calculate phonon dispersion relations in alkali metals from first principles. Such work has been stimulated by experimental data from inelastic neutron scat-

tering [3-5] and also by the development of the pseudopotential theory [6]. The pseudopotential seen by the valence electron is essentially nonlocal, but most authors assume a local pseudopotential for simplicity in perturbational treatment, although several attempts have been made to include nonlocal corrections. The frequencies are generally within 10 to 20% of the observed ones for the first-principle pseudopotential, but the agreement is even better for those containing adjustable parameters. One of the most important problems in the pseudopotential theory of metals seems to be that of the accuracy and completeness of the description, i.e. the possibility to propose a model which gives a sufficiently accurate description of the maximally wide class of properties of the metal. Only such a model may be applied with some confidence to the more complicated p h e n o m e n a where direct experimental information is not available. For example, in order to describe the structure and dynamics of lattice defects it is desirable to know the accuracy of the theory not only for the elastic and phonon properties, but for the anharmonic effects, properties under pressure phase transformations and so on. From all of the properties of metals, phonon spectra are the most sensitive on the details of the pseudopotential. The local pseudopotentials

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S. Zekovid et al. / Phonon spectra of alkali metals have to work best for the alkali metals. Because of that, the elastic properties of these metals have been examined many times in the framework of the pseudopotential theory using various forms of pseudopotentials. In connection with this, we only mention one of the recent, but, in our opinion, the most successful attempt [7] and also the pseudopotential scheme suggested in [8]. In the present paper we proceed to study the simple-model one-parameter pseudopotential [9], obtained by fitting the A n i m a l u - H e i n e [10] form factors. We will show that it can be modified in such a way to give very good agreement with the experimental phonon spectra.

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Table I Metal /3

Na K Rb 0.53158 0.52620 0.53191

global agreement with the experiment as best as possible. T o clarify a later step, we mention that in the case of Na the parameter fl is obtained as the arithmetic mean value of fll and r2 fitted on the longitudinal branch at the points [0.3, 0, 0] and [0.7, 0, 0], respectively. The analogous procedure is valid for other alkali metals. The optimized r-values for alkali metals Na, K and Rb are given in table I. For these metals we find experimental results for phonon spectra.

2. Model potential Modification of the potential [9], in the form

(k + q l w l k ) = --~Edo (27r13 2--~F) ,

(1)

where EF and KF are Fermi energy and Fermi momentum, respectively, and j0 is the zero-order Bessel function, has been applied to calculate electrical properties of metals [11]. It was shown that the pseudopotential (1) leads to a quite satisfactory description of the electrical resistivity for liquid simple metals. The later modification of this pseudopotential in [12, 13] has given us very good band-structure results for some metals and semiconductors. On the other hand, the pseudopotential (1), with the parameter /3 obtained in [9] becomes unsatisfactory in phonon spectra calculations for alkali metals. So, we are faced with a logical question: Is it possible to determine the parameter fl in such a way that we obtain a reasonable phonon spectrum? The answer is positive and it can be done as follows: (a) We fit this parameter on the experimental phonon value in one of the points at a symmetrical direction of a crystal. (b) We optimize the r - v a l u e so as to have

3. The phonon spectra

The phonon frequency tOqs, q being the phonon wave vector and S the polarization index, is determined in the harmonic approximation by solving the dispersion equation

~_~ IMo)2qsS,. - D , . (q)le~s = O,

(2)

where M is the ionic mass and e~s is the component along t h e / x axis (/x = x, y, z) of the unit polarization vector eqs. T h e quantity DA~(q) is the Fourier component of the dynamical matrix which can be expressed as a sum of three contributions:

D = DC+DR+D

E,

(3)

where D c represents the direct Coulomb interaction between the ions, D R is a repulsive interaction arising from the exchange-overlap between the ions, which is insignificant for the alkali metals, and D E is due to the polarization of the conduction-electron gas by the vibrating ions. The Coulomb term D c can be evaluated by

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S. Zekovi6 et al. / Phonon spectra of alkali metals

W ( q ) = Wb(q)/e(q) is the screened potential, e(q) being the Hartree dielectric function with modifications for exchange through the factor 1 - f(q) [16]. In this paper we further use the following expression [17] for f(g)

standard methods [14]. However, for the principle crystallographic directions [100], [100], [110] and [111] in a cubic crystal it may be directly used from [15]. For D E, the second-order perturbation theory gives [16] D ~ ( q ) = Mog~ { ~ (q+

h),~(q+h)#~G(iq + hi)

Iq + hi 2

- ~ ~I#,1 G(Ihl)}

q2 f(q) = 2(q2+ K2v+ K~)'

2KF,

,

K~= - - ~ la.U.l •

(4)

where q is the phonon wave vector and the ion plasma frequency O)p is

For a cubic crystal in the principle crystallographic directions, phonon frequencies have been obtained by the relation

w~,= {4 ~'( Ze )Z/ M12 } 1/2,

¢o2(qs) =

12 is atomic volume and h is a reciprocal lattice vector. For a local pseudopotential, G(q) is given by

where

(47rZe2'~ -2 G(q)= ~-~q2 ]

o,~-

o92 ,

o92=M-1DC

and

(6)

og2=M-1De.

e(q)- 1 W b ( q ) ' l -'-'--/7-~"--J~t/)

(5)

4. Discussion and conclusion

Here, Wb(q) is the unscreened Fourier component of bare electron-ion pseudopotential,

Our results for the phonon spectra of the alkali metals are given in table II. Besides that, on fig.

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S. Zekovid et al. / Phonon spectra of alkali metals

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S. Zekovid et al. / Phonon spectra of alkali metals

1 one can see the results for K. For comparison we present the results of the paper [7] and the experimental data, as well. On the basis of the table II and fig. 1, we conclude that our theoretical values are in very good agreement with the experiment for tested alkali metals. In some directions this agreement is within 2% (as much as the experimental error), while the maximum disagreement is 5-6%. Such a good result is obtained after the optimization of parameter /3 which usually reduces the maximum disagreement with experiment by 2-3%. In conclusion, we have shown that the simple model potential (1) with the parameter 13 taken from table I is at least not worse than the other, more complicated, pseudopotentials in describing the phonon spectra. In addition, from table I it follows that the optimal parameter /3 is approximately constant for analysed alkali metals. This is in accordance with the results of [18], where it was shown for the first time that the parameters of the pseudopotential (1) have to show regular dependence on the atomic number. Bearing this in mind we take the average value of parameter /3 from table I and calculate the phonon spectra for Na, K and Rb. The results obtained are presented in table lII, together with the discrepancies from experimental data. On the basis of the unique parameter /3, it is possible to compute the phonon spectra of the mentioned alkali metals with a very good accuracy (maximal disagreement is 9% for Na, 8% for K and 7% for Rb). This is a very unexpected result. At first, a precise interpretation of a phonon spectrum needs knowledge of a pseudopotential in a whole region of wave vector q. This means that expression (1) with optimized/3 value, obtained in the present paper, gives a very good global behaviour of the pseudopotential for Na, K and Rb. On the other side, it is a well-known fact that the electronic band structure of metals and semiconductors depends on a few values of the

321

pseudopotential W(q), where q is a reciprocal lattice vector. But, these values must be known very exactly if one wants to get a correct band structure. In order to obtain such precise values it was necessary to modify a simple oneparameter potential by introducing some new parameters [12, 13]. However, at the moment the problem of accounting for the higher pseudopotential form factors in a band structure calculation has not yet been satisfactorily solved within the local approximation [20]. We think that the simple analytic form (1), which successfully describes phonon spectra may help us in answering the question about the real possibilities of local pseudopotentials in complete investigations of the band structure of metals. The form of the local pseudopotential presented in this paper can also be suggested for the investigations of anharmonicity of lattice vibrations, and of alloy problems.

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[12] D. Ma~ovi6 and S. Zekovi6, Phys. Stat. Sol. (b) 89 (1978) K57. [13] D. Magovi6 and S. Zekovi6, Phys. Stat. Sol. (b) 96 (1979) 469. [14] E.W. Kellerman, Phil. Trans. Roy. Soc. London A233 (1940) 513. [15] A.O.E. Animalu, Phys. Rev. 161 (1%7) 445.

[16] A.O.E. Animalu, Proc. Roy. Soc. Vol. 294, No. 1438 (1%6) 376. [17] L.J. Sham, Proc. Roy. Soc. A283 (1%5) 33. [18] V. Veljkovi6, Phys. Lett. 45A (1973) 41. [19] M. Davidovi6, F. Vukajlovi6, S. Zekovi6 and V. Veljovi6, Phil. Mag. 36 (1977) 1257. [20] J.P. Van Dyke, Phys. Rev. B7 (1973) 2358.