Solid State Communications, Voi.42, No.3, pp.179183, Printed in Great Britain.
1982.
00381098/82/15017905503.00/0 Pergamon Press Ltd.
ELASTIC ENERGY AND STAGING IN INTERCALATION COMPOUNDS J.R. Dahn, D.C. Dahn and R.R. Haering Department of Physics, University of British Columbia Vancouver, B.C., Canada V6T IW5 (Received 21 December 1981 by R. Barrie)
We present a simple model for the elastic energy associated with the intercalation of a layered host lattice. The model accounts for the staging phenomena observed in many intercalation systems and is compared with Safran's theory of staging. The model is used to discuss the thermodynamic behaviour of LixTiS 2 and is shown to give a good account of the experimental observatiuns.
Work on the effects of lattice strains in intercalation systems has shown that intercalantintercalant interactions can be mediated by the elastic strain field present in the intercalation host. Graphite intercalation compounds (GICs)I3$ hydrogen in metals 4 and lithium intercalated transition metal dichalcogenides, LixMX2 ' 5 have been discussed. Nagelberg and Worrell 6 have recently shown that a phenomenological inclusion of the elastic effects in LixMX 2 compounds can give rise to a two phase region at low intercalant content. In this note we present a simple model for the elastic energy in LixMX 2 compounds which explains the observed lattice parameter variation. We show how to incorporate the elastic energy in lattice gas models of intercalation systems. The BraggWilliams approximation is used to produce phase diagrams as a function of stage, temperature and intercalant content. We show that the inclusion of the elastic energy in such models breaks the symmetry of the phase diagrams about x = ½. Monte Carlo simulations are used to reproduce the observed electrochemical behaviour of Li/LixTiS 2 cells. We also briefly discuss the applicability of the model to other systems including graphite. Unlike the hydrogenmetal systems 7, lattice expansion in LixMX 2 systems is anisotropic and nonlinear as a function of x. Whittingham and Gamble 8 showed that the caxis usually exhibits a 10% expansion while the aaxis changes by approximately 1% upon the intercalation of one lithium atom per metal atom. We have made detailed measurements of the variation of c(x) and a(x) in LixTiS 2 9 (fig. l) which are in good agreement with those reported by Thompson and Symon 10. One observes that the caxis saturates as x nears one. The saturation of c near x = 1 is typical of most LixMX 2 systems. In the following model we consider only the contribution of the caxis expansion to the elastic energy of the system. An estimate of the error introduced by neglecting the aaxis expansion can be made in the case of LixNbSe 2 where the elastic constants of NbSe 2 are known. For
x = i, the lattice expands such that Aa/a = .013 = e I = e 2 and Ac/c = .080 = c 3. The measured elastic constants are Cll + c12 = 28.5 x 1011, c33 = 6.7 x i0 II and c13 = I.0 x 1011dynes/cm 2 Ii The elastic energy computed per unit volume is E = ½cijsie j 2.4 x 109 ergs/cm 3, of which 89% arises from the c33e ~ term. Thus the neglect of the aaxis expansion introduces an error of approximately 10% in the elastic energy of LixNbSe 2. The errors for other systems may be expected to be of a similar magnitude unless c33 is anomalously small. When a lithium atom is intercalated into an MX 2 structure it produces local distortions in the host lattice. The competition between the local distortions and the Van der Waals forces binding the layers together determines the resulting caxis behaviour. In cases where the intercalated lithium tends to expand the lattice, the MX 2 compound can be modeled by rigid plates connected together by springs of spring constant K and equilibrium length co, corresponding to the caxis of the MX 2 compound (fig. 2). These springs correspond to the Van der Waals bonds between the MX 2 "sandwiches". The elastic effects resulting from the intercalated lithium atoms are modeled by springs with spring constant k and equilibrium length c L > c 0. To calculate the caxis variation, we normalize the number of springs in the host lattice to one per metal atom; i.e. N lattice springs. We now insert n lithium atoms (x = n/N) and apply the condition for equilibrium: NK(c(x)  Co) = nk(c L  c(x))
(i)
which yields c(x)

c o
x
cL  c O
K
~ + x
•
a
=

k
(2)
The resulting elastic energy, per metal atom, is
179
=
Ec(x
=
x +
J
a
x
+
½kx
I cL (3)
180
ELASTIC ENERGY AND STAGING IN INTERCALATION COMPOUNDS
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Vol. 42, No. 3
3.46
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3.45
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,
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i
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Fig. 1
The measured lattice parameter variations for LixTiS 2 from reference (9) (points). The solid curve is a fit to the caxis expansion using the model described in the text.
where J = ½K(c L  c0) 2. In a typical case, J is of the order of 0. i ev, and a is small (= < i) so that the elastic energy increases most rapidly for small x (x ~ a). It is of interest to note the similarity of equations (2) and (3). A fit of equation (2) to the measured caxis variation of LixTiS 2 is shown in figure 1. The parameters used were co = 5.695 ~, c L = 6.305 ~ and ~ = 0.2. The agreement is reasonably good and we shall therefore consider the effect of the elastic energy on the thermodynamics of LixMX 2 systems. Lattice gas models of intercalation systems have been widely investigated I'12'13 The models used when discussing GICs have been three dimensional, while in the past, LixMX 2 systems have been modeled as a two dimensional triangular lattice gas. Attempts to explain the observed V(x) and ~x/~V behaviour of Li/LixTiS 2 cells 12,14 on the basis of two dimensional models have failed. Recent Xray diffraction results 9 regarding the variation of (00£) Bragg peak widths as a function of x in LixTi~ 2 suggest that a three dimensional model is needed. Experiments revealing staged structures in LixNbSe215 , Ag x TaS 2 and AgxTiS216 have also been performed. With this in mind, we introduce the following three dimensional Hamiltonian in the spirit of reference 13. H = NE0x +
~N klE~U,XkXk+l + U
+ Z Z U XikXjk k i,j
"x
Xk kXk+2 + J   ~ X k ~ (4)
In this expression, x is the average intercalant concentration of the lattice, ~k is the average concentration in layer k, xi~ is the occupancy of site i in layer k (Xik = ~ or I), N is the number of available sites and L is the number of layers. We have introduced a nearest neighbour interaction between intercalant atoms in the same plane (U) and interactions between atoms intercalated in Ist and 2nd neighbouring layers (U' and U") which we have taken to depend only on the average occupancy of the involved layers. The elastic energy, from equation (3), applied to
each layer and a site energy, E0, are also included. We first set U = 0 and examine equation (4) within the framework of the BraggWilliams approximation. By neglecting U, we reduce the problem to a one dimensional one with a continuous variable, x k. We expect ground states exhibiting staged structures at x = ~ 3 and x = ½ . We therefore divide the system into 6 sublattices, commensurate with the expected ground state structure. We then obtain self consistent equations for each of the x i ( 1 ~ i ~ 6) as described in reference 13. We find xi =
[I + e~[Xi~/kTJ I~
i = 16
(5)
e~xi~ = E0 ~ + U'(xi+ 1 + xi_ I) + U"(xi+ 2 xi_ 2) +
J
(~ + xi52
These equations can be solved iteratively 17 When the procedure converges one obtains x i as a function of the chemical potential, ~ , and T. When the set of equations has more than one solution, we take only the solution which has the lowest free energy. The phase diagram for the case U' = J, U" = .IU' and e = 0.2 is shown in figure 3a. The width of the stage 2 and stage 3 phases are controlled by the magnitude of U' and U" respectively. T h e p h a s e s 6' and 6" have 6 layer unit cells and are described at T = 0 as having 4 and 5 of the 6 sublattices filled and the others empty. This phase diagram should be compared with the one produced by Safran i and shown in figure 3b. The overall features are quite similar; the presence of stages I, 2 and 3, the first order phase transitions between these phases at low temperature and a region at high temperature where the stage 1  2 transition becomes continuous. In our case it is the inclusion of the elastic energy which makes the transitions first order at low temperature. Safran introduces an attractive interaction be
Vol. 42, No. 3
ELASTIC ENERGY AND STAGING IN INTERCALATION COMPOUNDS
K
Fig. 2
18
T Co ±
The springs which represent the Van der Waals bonds between the host layers (K, c O ) and those which represent the local distortions near the intercalant (k, CL). The host layers are modeled as rigid plates.
tween intercalant atoms in one plane which drives the first order transitions and a repulsive interaction between intercalant atoms in different layers which decays as a power law as the layer separation increases. The major difference between the theories is the breaking of the symmetry about x = ½ exhibited by Safran's phase diagram. It is the inclusion of the elastic energy, which varies most rapidly at small x, that breaks the symmetry. It has recently been discovered 15 that LixNbSe 2 exhibits a phase diagram much like that in figure 3a. As lithium is intercalatedinto NbSe 2 , several regions of coexisting phases are observed between x = 0 and x = .3. LixNbSe 2 with x = . 1 5 is a stage 2 compound, and with x = .3 it is a stage 1 compound. For compositions .15 ~ x ~ .3, a mixture of stage 2 and stage 1 structures is observed. An imperfect stage 3 structure is observed near Li.07NbSe2 and coexisting stage 2 and stage 3 regions are observed for .07 ~ x ~ .15. The existence of higher stages at smaller x cannot be ruled out by these experiments. In order to extend this treatment to include the Uterms in equation 4 and in order to include effects due to short range order ( which are not included in the BraggWilliams approximation), we have made Monte Carlo simulations of the Hamiltonian (equation 4) on a 6 x 6 x 6 lattice, triangular in two dimensions with periodic boundary conditions. This was done to explain the V(x) and ~x/~V behaviour of Li/LixTiS 2 electrochemical cells. The data 9 and Monte Carlo results are plotted in figure 4. The parameters used were U" = 0.0, U' = 3.5kT, U = 2.0kT, J = 1.4kT and E 0 = 2.4547 ev with kT = 25.7 mev. It is the inclusion of the elastic energy which produces the flattening of the voltage curve at small x. This is in accord with the phenomenological treatment of Nagelberg and Worrell 5 The dip in the 6x/6V curve exhibited by the Monte Carlo results arises from a tendency of the LixTiS 2 system to form a shortranged staged structure near x = .16. This is consistent with
the Xray data 9 which shows the (00£) Bragg peak widths narrowing at x = .16 as the order in the cdirection becomes greater. It should be pointed out that the value of a used in fitting the lattice parameter variations (figure I) is the same as the one used in the Monte Carlo simulations of LixTiS 2. One notes that U' > U in this fit to the data. These interactions represent the combinations of electronic and elastic effects not taken into explicit account in our model. The strain mediated intercalant  intercalant interaction is expected to be attractive between atoms in the same plane and repulsive between atoms located in different layers 5 The screened Coulomb interaction is expected to be repulsive in both cases. The sum of these effects could lead to the values needed to fit the data. LixTiS 2 is a system where longrange staging does not occur at room temperature but should occur as the temperature is lowered in samples near Li.16TiS 2. Experiments searching for staged structures at low temperatures are in progress. We believe that the treatment of elastic effects associated with intercalation which we have presented should be applicable to most of the layered intercalation compounds including those of graphite. GICs produce fully staged structures at room temperature with well defined inplane concentrations (in most cases) and lattice parameters. This is in accord with the low temperature portion of the phase diagram of figure 3. It would be of interest to examine the high temperature behaviour of GICs to determine how well our model can explain the lattice parameter variation and the departures from stoichiometry of GICs. Acknowledgement  The authors would like to thank A. J. Berlinsky and W.G. Unruh for stimulating discussions. The support of the Natural Sciences and Engineering Research Council of Canada and of the B.C. Science Council is gratefully acknowledged.
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ELASTIC ENERGY AND STAGING IN INTERCALATION COMPOUNDS
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Phase diagram for U' = IOU" = J, U = 0 and ~ = 0.2 plotted as a function of ~ T / J and x. The integers are the single phase pure stages. The shaded areas denote twophase regions. The transition from stage i to stage 2 becomes continuous over the dashed portion of the diagram. The dotted portions of the single phase stage 3, 6' and 6" phases have not been determined precisely. The 6' and 6" phases are discussed in the text.
(b)
Phase diagram after Safran (i). Although the phase diagram exhibits symmetry about x = ½ due to the binary interactions between intercalant atoms, Safran has ommitted from the diagram the phases related by symmetry to 3, 4 and 5.
Vol. 42, No. 3
Vol. 42, No. 3
ELASTIC ENERGY AND STAGING IN INTERCALATION COMPOUNDS
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REFERENCES i. 2. 3. 4. 5. 6. 7. 8. 9. 10. II. 12. 13. 14. 15. 16. 17.
S.A. Safran, Phys. Rev. Lett. 44, 937 (1980) S.A. Safran and D.R. Hamann, Phys. Rev. B. 22, 606 (1980) S. Ohnishi and S. Sugano, Solid State Comm. 36, 823 (1980) H. Wagner in Hydrogen in Metals I edited by G. Alefeld and J. Volkl, SpringerVerlag (1978) W.R. McKinnon and R.R. Haering, Solid State lonics I, 111 (1980) A.S. Nagelberg and W. Worell, J. Solid State Chem. 38, 321 (1981) H. Piesl in Hydrogen in Metals i edited by G. Alefeld and J. Volkl, SpringerVerlag (1978) M.S. Whittingham and F.R. Gamble, Mat. Res. Bull. I0, 363 (1975) J.R. Dahn and R.R. Haering, Solid State Comm. 40, 245 (1981) A.H. Thompson and C.R. Symon, Solid State lonics, 3, 175 (1981) O. Sezerman, A.M. Simpson and M.H. Jericho, Solid State Comm. 40, 245 (1980) A.J. Berllnsky, W.G. Unruh, W.R. McKinnon and R.R. Haering, Solid State Comm. 31, 135 (1979) CR. Lee, H. Aoki and H. Kamimura, J. Phys. Soc. Japan 49, 870 (1980) R. Osorio and L.M. Falicov, J. Phys. Chem. Solids (in press) D.C. Dahn and R.R. Haering (to be published) G.A. Scholz and R.F. Frindt, Mat. Res. Bull. 15, 1703 (1980) A.B. Harris and A.J. Berlinsky, Can. J. Phys. 57, 1859 (1979)
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