Atomic mean-square displacements for 2HMoS2

Solid State Communications, Vol. 37, Pp. 879—881. Pergamon Press Ltd. 1081. Printed in Great Britain.

0038—1098/1 10879—03$02.00/0

ATOMIC MEAN-SQUARE DISPLACEMENTS FOR 2H—MoS2 J.L. Feldman and L.L. Boyer Naval Research Laboratory, Washington, DC 20375, U.S.A. (Received 4 August 1980 by A.A. Maradudin) We have calculated the temperature dependence of the atomic meansquare displacements (MSD’s) for a reasonable harmonic lattice dynamical model considered to be appropriate to 2H—MoS2. Thermal expansion effects were estimated in a crude way using previous model calculations and thermal expansion data. Our results are shown to obey the Lindemann melting criterion. A PHENOMENOLOGICAL, essentially valence force field model, has been used to fit neutron-measured phonon dispersion curves in 2H—MoS2, a layered transition metal dichalcogenide [1]. Given this model it is of interest to obtain various phonon-averaged quantities such as the heat capacity, which has already been computed [1], and the mean-square displacements (MSD’s) or equivalently, the Debye—Waller factor. The MSD’s of 2H—MoS2 provide an independent check on the model because they do not depend solely on the frequency spectrum, but involve the polarization vectors as well. Thus calculations of the MSD’s wifi be interesting to compare with experimental information on the Debye— Wailer factor when that becomes available, In this work the MSD’s of 2H—MoS2 have been determined through a calculation in the harmonic approximation. The force model adopted is the valence force model including the parameter values used by Wakabayashi et al. [1] for the intralayer forces and a first and second neighbor bond stretching force model for the interlayer S—S interactions. The latter parameters were chosen to yield agreement with the neutronmeasured [1] “rigid layer” mode frequencies and are given by a 3 dynecm’ and a 1 = 1.98 x i0 2 = 2.58 x io~dyne cm’, where a1 and a2 are the 1st and 2nd neighbor force respectively. It should alsouse be mentioned thatconstants, these values were obtained with the of the structural parameters quoted by Wilson and Yoffe

Born—Huang relations and hence also does not give the correct velocity of sound associated with the ~3-symmetry transverse acoustic mode: As a quantitative measure of this, if the bond-bending force constant of the above mentioned interlayer axially symmetric model is assumed to arise from a central force alone then it yields a contribution to the uniaxial stress (along the c-axis) of about thirty kilobars [4]. Also, the discrepancy in the elastic constant c~,between determinations based on wave propagation parallel and perpendicularto the layers, is a factor of almost two [1] with the use of the model of [1]. It is also of interest to consider the model results for the full set of elastic constants here. They are: c11 = 17.4 (23.8), c11 —c12 = 14.7 (29.2), c33 = 5.0 (5.2), 2, c~= 2.2 c13 = 1.0 (2.3) 1011 dyne where the (1.9) valuesand in parenthesis wereintaken fromcm Feldman [5] and are based on X-ray-measured linear compressibiities [6] and on sound velocities extracted from the neutron measured dispersion curves. (We are not aware of a direct and independent determination of the elastic constants to which to compare our results). Furthermore, for the axially symmetric (interlayer) model we find essentially identical results to ours (within 2%) except for c13 which given by 0.32 x of 2. In considering theisabove comparison 10” dyne cm elastic constants it should be noted that the dispersion curve associated with c 11 c12 was not determined via the neutron measurements, and further that through use of expression 1 of [5] quite substantial uncertainties in c12 (and hence c11 c12) and c13 arise from the experimental uncertainties in c11, c33, and the linear cornpressibilities. These derived uncertainties which unfortunately were not considered explicitly in [5] are indeed large enough to yield approximate agreement between the model and available experimental data. Specifically, upon considering reasonable (-‘ 10%) experimental uncertainties in c11 and c33 as derived from —

[2] because of the fact that the force constants were apparently obtained by Wakabayashi eta!. with the use of those structural parameters. Probably only slight changes would be obtained if the somewhat different structural parameters quoted by Wyckoff [3] were used. Similar results for the MSD’s using the axially symmetric model (for the interlayer interactions) of Wakabayashi eta!. [1] were obtained. However, it is necessary, at least in principle, to consider our model because of the fact that the axially symmetric model violates the

879

—

880

ATOMIC MEAN-SQUARE DISPLACEMENTS FOR 2H—MoS2 I

I

I

I I

/

/ ~,

LO

/ ,//

/ // / ~

0 5

//

//

// / /

/

/

,

~

/// /

/// ~

// /

06

/ /

/ /~ /// 04

/

/

//~~/ / /~z

Y~ 02

~ /

~,//

~

I 200

I

I 400

600

T(K)

Fig. 1. Mean square atomic displacements for 2H—Mo52.

~

U~(SP;~—~ (U.~(S)>, X

—

—~

(U~(Mo));

2

Table 1. Anisotropy parameter, A = (U~)/(U~),for 2H—MoS2 _____________________________________________________

T (K)

AM,,,

A~

10

1.11 1.40 1.60 1.62

0.89 1.04 1.15 1.16

600

neutron data and the recent measurements [6] of the linear compressibilities with their (perhaps too small [7]) estimated uncertainties [6] one can achieve agreement with our model for c13 and one can approach our model result for c11 —cr2 to within about 10%. We note too that the discrepancy in c11 between theory and experiment iseta!. indicative 12—13% of the Wakabayashi modeloftothe data for thefit appropriate dispersion curve. These uncertainties in the experimental results for the elastic constants should be borne in mind when assessing the accuracy of the predictions of the model. The atomic site-symmetries yield two non-zero independent atomic displacement correlation functions (per atomic specie) which determine the Debye—Waller

Vol. 37, No. 11

factor of 2H—MoS2. These are MSD’s along the c- and a-axes, (UZ~) and (U~),respectively. The standard normal mode expressions [8] in terms of lattice frequencies and complex polarization vectors were used to determine the MSD’s. Two computationally important features of the dispersion curves are (1) a strong anisotropy in q-space arising from the relatively weak interlayer forces and (2) approximate isotropy of the acoustic modes with respect to q11 where q11 is the component of the wave vector parallel to the hexagonal faces of the Brillouin zone; in the long wavelength limit it is well known that this isotropy,is exact. Corresponding to the first feature, the dispersion curves are two dimension-like for q11 > ~ where ~ 0.4 of the distance (along a hexagonal face) to the Brillouin zone boundary. Therefore, in performing normal mode sums, a relatively crude grid of points along q,~is sufficient especially for q11 > q,1.1~. Integrations in q space were performed by sampling both over octahedral cellular grids, consisting of on the order of 100 points in the irreducible element of the Brillouin zone, and over grids appropriate to cylindrical coordinates. In the latter case, due to the second feature of the dispersion curves, the variation in the value of the integrand of the angular integral (where the integral over q11 is performed first) is only a few percent; this was determined by calculations of the angular integrand for angles 0°, 150 and 30°,which spans the irreducible elementofof the Brillouin zone. From a comparison of different calculations our results which are given in .

.

Fig. 1, are reliable with respect to q space integration to within a few percent. The reader is reminded that high temperature MSD’s at reflect low T only the the masses atomic alsocoupling affect the constants results. Possibly [9], whereas the relatively large S atom values of ( UX~)and (Ui) at low T are a mass effect. In view of large anisotropies found in many physical properties of layered transition metal dichalcogenides [2], it is also noteworthy that the anisotropy in the MSD’s, represented by the quantity A = (U~) /( U~),can be seen to be less than two; values of A for the Mo and S atoms at several temperatures are given in Table 1. An estimate of thermal expansion effects in the quantity2(s)>+MMO( U2(Mo)) (1) 2M~U can be obtained from an analysis of thermal data [10]. The necessary data are not available but a rough estimate can be achieved from results of available thermal expansion data [11] and of 73(T) based on our model calculations [4]. The latter calculations suggest that 73(T) may be quite large at low temperatures; it is noted that low temperature values of 73(T) play an important

Vol. 37, No. 11

ATOMIC MEAN-SQUARE DISPLACEMENTS FOR 2H—MoS2

role in determining the effective Gruneisen parameter for the high temperature thermal expansion effect in expression (1). Expression (1) is proportional to a product of the temperature and the minus second moment of the frequency spectrum at high temperatures. Somewhat arbitrarily choosing the appropriate moment y’s [10] to be 4 and 1 corresponding to c-axis and a-axis strains, respectively, leads to about a 4% increase in the minus second moment from room temperature to 600 K, and a 14% increase in that moment between room temperature and 1123K, the maximum temperatureconsideredin [11]. Debye temperatures associated with the minus first and minus second moments of the frequency spectrum relate to mass weighted sums over the MSD’s (expression 1) at T = OK and in the classical (high temperature) limit, respectively. Using our values for MSD’s we find = 492 K and 02 = 414K which are within the range of specific heat Debye temperatures [11 for this model as might have been expected. Only a cursory comparison with available experimental LEED intensity data [12] is possible at present since, for example, no multiple scatteringor TDS corrections have been made to those data. We also note that the effective Debye temperature quoted in that work is incorrect it is a factor of almost two too large [13]. Further, when converted to a MSD the corrected Debye temperature yields values of (U~(Mo))[12, 13] which are substantially (about a factor of two) higher than our values, but a meaningful comparison between theory and experiment must await further analysis of the data. These results are also of interest from the standpoint of melting theory [14]. According to2/3)”2, the is of Lindemann theinteratomic root-MSD,distance a (U at melting. the order ofcriterion 10% of an Ignoring anharmonicity we obtain UMO = 0.208 A and US = 0.236 A at 3000 K, the apparent melting temperature of 2H—MoS 2 [15]. These values represent fractions, 0.086 and 0.098 of the (room temperature) smallest interatomic distance, i.e. the Mo—S distance, so it appears that the Lindemann criterion may be satisfied. We stress, however, the importance of including anharmonic effects in considerations such as these.

REFERENCES 1.

2. 3. 4. 5.

6. 7.

—

Acknowledgement This paper has greatly benefited from discussions with B.J. Mrstik.

881

8.

io.9. 11. 12.

13. 14.

—

15.

N. Wakabayashi, H.G.(1975). Smith & Phys. Rev. B12, 659 SeeR.M. also Nicklow, N. Wakabayashi & R.M. Nicklow, Electrons and Phonons in Layered Crystal Structures (Edited by T.J. Wieting & M. Schiuter). Reidel, Dordrecht (1979). J.A. Wilson & A.D. Yoffe,Adv. Phys. 18, 193 (1969). R.W.G. Wyckoff, Crystal Structures, Vol. 1. Interscience, New York (1960). J.L. Feldman & L.L. Boyer, Physica 99B, 347 (1980). J.L. Feldman, J. Phys. Chem. Solids 37, 1141 (1976). The values in parenthesis for c33 and c~ also correspond to results of similar (interlayer) model calculations which are presented in this reference. The 15% difference in c~between the two calculations is due to the fact that the Ramanmeasured [J.L. Verble & T.J. Wieting, Phys. Rev. Lett. shear rigid layer was fit25, in362 this (1970)] reference, whereas in the frequency present work neutron measurements were solely relied upon. A.W. Webb, J.L. Feldman, E.F. Skelton, L.C. Towle, C.Y. Uu & I.L. Spain, J. Phys. Chem. Solids 37, 329 (1976). See J.L. Feldman, C.L. Vold, E.F. Skelton, S.C. Yu & I.L. Spain, Phys~Rev. B18, 5820 (1978) for a somewhat different treatment of X-ray data (on 2H—TaSe2) as a function of pressure than that of [6]. B.T.M. Willis & A.W. Pryor, Thermal Vibrations in Crystallography, Cambridge University Press, London (1975). See also A.A. Maradudin, E.W. Montroil, G.H. Weiss & I.P. Ipatova, Theory of Lattice Dynamics in the Harmonic Approximation, p. 62. Academic Press, New York (1971). M. Blackman, Cryst. 9, 735 T.H.K. BarronActa & R.W. Munn, Phil.(1956). Mag. 15,85 (1967). S.H. El-Mahalaway & B.L. Evans, J. AppL Cryst. 9, 403 B.J. (1976). Mrstik, R. Kaplan, T.L. Reinecke, M. Van Hove & S.Y. Tong, Phys. Rev. B15, 897 (1977); B.J. Mrstik, S.Y. Tong, R. Kaplan & A.K. Ganguly, Solid State Commun. 17, 755 (1975). B.J. Mrstik, (private communication). It was previously found (see [4]) that a high ternperature instability in the quasi-harmonic equation of state based on a Lennard—Jones S—S potential happens to occur near the melting temperature. That this type of instability could be related to melting was first proposed by K.F. Herzfeld and M. Goeppert-Mayer [Phys. Rev. 46, 995 (1934)] and further developed by L.L. Boyer [Phys. Rev. Lett. 42, 584 (1979) and ibid. 45, 1858 (1980).]. P. Cannon, Nature 183, 1612 (1959).