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Dielectric relaxation of lithium-doped zinc fluoride
ARTICLE IN PRESS
Physica B 365 (2005) 147–154 www.elsevier.com/locate/physb
Dielectric relaxation of lithium-doped zinc fluoride Mark R. Levya,, K.J.W. Atkinsonb, Onyeka H. Odilia, Robin W. Grimesa a Department of Materials, Imperial College, London SW7 2BP, UK British Energy Generation Ltd., Barnett Way, Barnwood, Gloucester GL4 3RS, UK
b
Received 17 August 2004; accepted 7 May 2005
Abstract It has been claimed that Li-doped ZnF2 exhibits anisotropic dielectric relaxation. However, the mechanism for this behaviour has not been resolved. Here, we present a new analysis of the defect structures and reactions in this system based on results from atomistic simulations. In particular, the relative energies for the solution of LiF are predicted which lead to the structure of a split Liþ interstitial cluster. The implications of this to the dielectric relaxation processes are discussed. This requires an analysis of Liþ ion migration and cluster binding energies. r 2005 Elsevier B.V. All rights reserved. PACS: 61.66.f; 61.72.y; 61.72.Ji; 81.05.Je Keywords: ZnF2 ; Dielectric relaxation; Anisotropy; Atomistic simulation
1. Introduction Many materials, when subjected to an electric field, will polarise to oppose the field. One contribution to the polarisation involves the movement of some ions from their equilibrium lattice positions. When the field is removed, the ions will generally return to their equilibrium positions. If the field is imposed on a material at a high temperature, for some time, and the material is subsequently quenched in the field (typically to Corresponding author. Tel.: +44 20 7594 6802;
fax: +44 20 7594 6729. E-mail address: [email protected] (M.R. Levy).
liquid nitrogen temperature), the ionic contribution to the polarisation can, for some materials, be partly frozen-in. This implies that there is an energy barrier against the relaxation of the ions back to their equilibrium positions. If the material is slowly heated, this polarisation will be lost once the ions gain sufficient thermal energy to overcome the barrier. This process, the dielectric relaxation, can be monitored and quantified as an electric current [1,2]. It has been suggested by Roth [3,4] that zinc fluoride (ZnF2 ) doped with lithium exhibits a considerable anisotropy in its dielectric properties. In particular, it exhibits a strong relaxation in [0 0 1], while the effect in [1 0 0] and [1 1 0] is several
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orders of magnitude lower [3]. Roth initially proposed a relaxation mechanism involving the movement of fluorine vacancies, as the chargecompensating defects for Liþ substitutional ions [3]. He later suggested that the anisotropy is due to the motion of Liþ interstitial ions, as an alternative charge compensating species [4]. The rationale behind the change in mechanism was an analysis of the experimental data which resulted in a low energy (0.32 eV) for defect cluster reorientation. Roth considered it unlikely that a vacancy model would be compatible with such a low energy but that an interstitial ion model would, because similarly low Liþ interstitial migration energies in materials with the rutile structure were already known [5]. In this regard, he suggested that the orientation of the dipole, formed by a substitutional Liþ ion–interstitial Liþ ion cluster, is initially randomised throughout the lattice with the substitutional lithium defect immobile. Intriguingly the anisotropy in the dielectric relaxation was only observed when the doped material was subjected to a prior heat treatment. The reason for this treatment differed between the two Roth publications: in the first, it ensures a sufficiently high concentration of fluorine vacancies [3] and in the second it is to dissolve defect clusters and increase the intensity of the dielectric relaxation [4]. The discrepancies between the two Roth publications were commented on by Nowick [6]. In particular, he stated that the same dielectric response in the [1 1 0] and [1 0 0] directions is not a test of the model, but a requirement of the theory. In addition, the experimental data was reanalysed to yield a dipole reorientation energy of about 0.7 eV and as such, the necessity for an interstitial lithium mechanism was removed. Finally, Nowick suggested that the dielectric anisotropy could be due merely to differences in the heat treatment of the samples. Our aim here is to use atomic scale computer simulation to predict the structures and energies of defects associated with lithium solution in ZnF2 . In this way, we will provide a mechanistic basis for the defect processes that underpin such dielectric relaxation process in this material.
2. Methodology The calculations presented here are based upon the Born description of an ionic lattice [7,8]. The forces acting between ions are resolved into longrange coulombic and a short-range pair terms. The long-range terms were summed using the Ewald method [9], while the short-range components were modelled with parameterised Buckingham pair potentials [10]. Initially, the perfect lattice is defined by tessellating the unit cell throughout space using periodic boundary conditions which are defined by the crystallographic lattice vectors. The perfect lattice energy (E L ) is given by " ! !# X X qi qj rij C ij EL ¼ þ Aij exp , 4p0 rij rij r6ij j4i i (1) where Aij , rij and C ij are adjustable short-range parameters, 0 is the permittivity of free space, and rij is the interionic separation between ions i and j of charges qi and qj , respectively. The shortrange potential parameters used in this work are reported in Table 1. The electronic polarisability of the fluorine ions was accounted for through the shell model [11]. Here, a shell of charge Y ¼ 1:3776jej is coupled to a core of charge X ¼ þ0:3776jej via an isotropic, harmonic spring of force constant, ( 2 . k ¼ 24:36 eV A In order to model the effect of a charged defect, a multi-region approach in which the lattice is partitioned into concentric, spherical regions centered about the defect is adopted. The region immediately surrounding the defect is termed region I, in which all ions are explicitly relaxed, subject to the forces described by the inter-atomic potentials. Surrounding this is region IIa in which Table 1 Short-range potential parameters Interaction
A (eV)
r (A˚)
C (eV A˚6 )
F -F Liþ -F Zn2þ -F
1317.50 574.80 918.41
0.27530 0.25530 0.28481
13.8 0.0 0.0
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the forces between the ions are determined via a dielectric continuum approximation [12] and all the ions can be relaxed in a single step. The interaction energies between the ions in interfacial region IIa and region I are, however, calculated explicitly. Beyond region IIa is the outer region IIb, which is a point charge array and provides the Madelung field of the remaining lattice; the response energy for this region is determined using the Mott–Littleton approximation [13]. Migration energies are calculated assuming a thermally activated process is in operation. Thus an ion is moved between sites through a saddle point which defines the activation energy. The saddle point is identified by rastering the migrating ion through possible positions on the migration pathway. The migrating ion is fixed and the surrounding lattice is relaxed, thus allowing the location and evaluation of the migration activation energy. All calculations presented here were carried out using the CASCADE code [14] which has been used successfully in many previous related studies of defects in fluorides, e.g. Refs. [15–17]. 3. Results for undoped ZnF2 ZnF2 adopts the tetragonal rutile crystal structure with space group 136 (P4=mnm). This structure can be considered as a body centered tetragonal lattice of cations (zinc) which are then octahedrally coordinated by the anions (fluorine) (see Fig. 1a). The comparison between experimental and calculated lattice parameters, presented in Table 2, shows excellent agreement. Elastic constants for ZnF2 have also been reported experimentally [18–20] and a comparison with values predicted here are presented in Table 3. This shows good agreement meaning that displacements of ions around their equilibrium positions are well reproduced. Since we are concerned with a lithium-doped material, the lithium–fluorine potential was derived so that the predicted lattice parameter of ¯ LiF (space group 225 (Fm3m) [21]) reproduces the experimental data, i.e. a lattice parameter of 4.027 A˚.
Fig. 1. The ZnF2 tetragonal unit cell showing the three mechanisms discussed in this publication. (a) The ZnF2 tetragonal unit cell. The large, dark spheres represent fluorine ions and the small, paler spheres represent zinc ions. (b) Represents a schematic of Roth’s first proposed mechanism [3]. (c) A schematic of Roth’s second mechanism [4]. (d) The mechanism proposed here. The transparent cube in (b) represents a fluorine vacancy, the transparent cube in (d) is a zinc vacancy and the white spheres represent lithium substitutional ions in (b) and (c), but interstitial ions in (d).
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The energies for intrinsic defect processes in ZnF2 were also calculated (see Appendix for details). The results, shown in Table 4, suggest that the anion Frenkel reaction is slightly preferred over the Schottky reaction while the cation Frenkel energy is substantially higher. The anion Frenkel defects (i.e. the fluorine vacancy and fluorine interstitial) are therefore the majority defects in this material. However, the Schottky reaction is only slightly higher in energy than the Table 2 Comparison of calculated and experimental lattice parameters for ZnF2 a (A˚)
c (A˚)
c a
Volume (A˚3 )
Calc. 4.6526 3.2046 0.6888 69.37 Expt. [24] 4.7034 3.1335 0.6662 69.32 Percentage difference 1.11 2.26 3.41 0.002
Table 3 Calculated and experimental elastic constants ð 1011 Nm2 ) for ZnF2 C 11 Calc. Expt. [19,20] Expt. [18]
C 12
131.8 126 130
91.4 93 97
C 13 60.8 84 89
C 33 214.2 192 199
C 44 50.1 39.2 39.5
anion Frenkel. This reaction will therefore provide an important concentration of intrinsic defects. Finally, it was found that interstitial F adopts a split-interstitial configuration similar to that observed in other fluorite structures [15,16]. In this, the interstitial fluorine ion displaces a lattice F ion into a neighbouring interstitial site. Thus, the two F ions effectively share the vacant fluorine lattice site that is created. The splitinterstitial configuration remains in the ab lattice plane.
4. Solution mechanisms for LiF-doped ZnF2 The incorporation of monovalent Liþ ions onto divalent zinc sites requires charge compensation. This can be facilitated in one of three ways: via a Zn2þ interstitial ion (Eq. (2)), a Liþ interstitial ion (Eq. (3)) or a F vacancy (Eq. (4)). In Kro¨ger– Vink notation [22], these are: 0 2LiF þ 2ZnX Zn Ð 2LiZn þ Zni þ ZnF2 ,
(2)
0 2LiF þ ZnX Zn Ð LiZn þ Lii þ ZnF2 ,
(3)
0 X LiF þ ZnX Zn þ FF Ð LiZn þ VF þ ZnF2 .
(4)
C 66 92.4 80.7 81.4
Table 4 Intrinsic defect process energies (eV) for ZnF2 Schottky
Anion Frenkel
Cation Frenkel
2.59
2.17
4.31
The normalised energies for these processes are reported in Table 5 (see Appendix for details). It is clear that the energies of these extrinsic defect processes are considerably smaller than for intrinsic processes. As such, at equilibrium, extrinsic defects will dominate intrinsic defects. If we assume the defects remain spatially isolated, the lowest energy charge compensating defect is a Liþ interstitial (Eq. (3)). However, Eq. (4) which describes charge compensation via
Table 5 Solution and cluster binding energies (eV) for Liþ accommodation in ZnF2 Solution mechanism
Reaction 2 Reaction 3 Reaction 4 a
Charge compensating defect
Zni Lii VF
Solution energy
Cluster binding energy
Isolateda
Clustered
2.22 1.27 1.46
1.82 0.73 1.52
Note: these energies have been normalised as described in the Appendix.
4.84 1.81 1.40
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a F vacancy is only 0.19 eV higher, and may, on this basis alone, yield a further minor concentration of defects. The coulombic interaction between the isolated defects leads to the formation of defect clusters. The lowest energy solution process, based on defect clusters, still involves compensation via a Liþ interstitial defect (reaction 3). This mechanism is now substantially favoured over the other two reactions. The reason for this is that the Liþ interstitial/substitution pair cluster undergoes considerable lattice relaxation, into a split interstitial configuration, which lowers the overall solution energy. (This is isostructural to the split F interstitial ion reported in the intrinsic defect analysis.) The structure of this cluster, as shown in Fig. 1d, consists of two equivalent Liþ interstitials either side of a vacant zinc site (i.e. fLii : V00Zn : Lii gX rather than fLi0Zn : Lii gX ). Eq. (5) describes the equilibrium between Liþ ions when isolated and when part of a defect cluster (indicated by braces). Li0Zn þ Lii Ð fLii : V00Zn : Lii gX .
(5)
The corresponding mass action equation is [22] ½fLii : V00Zn : Lii gX DH b ¼ exp , (6) ½Li0Zn ½Lii kT where DH b is the cluster binding enthalpy, k is the Boltzman constant, T is the absolute temperature and square brackets indicate a concentration. Using the electro-neutrality condition between the charged defects (Eq. (7)), Eq. (6) can be simplified to yield an expression for the ratio of lithium in clusters to that which is isolated (see Eq. (8)). ½Li0Zn ¼ ½Lii ,
(7)
clustered ½fLii : V00Zn : Lii gX ¼ . isolated 2½Li0Zn
(8)
If the assumption is made that the total lithium concentration is 100 ppm, as stated by Roth in the experimental studies [3], at the heat treatment temperature of 473 K and the calculated energy 1:81 eV, this ratio has the value 3 107 ; ergo, at equilibrium, Liþ ions will be in the form of clusters. On the basis of this analysis, the reason
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for the heat treatment given by Roth in his second paper [4], i.e., to dissolve the clusters, is not valid. It is also possible to used the calculated defect energies to predict an equilibrium solution limit and compare this against the value given by Roth [3]. In this case, we assume defect clusters dominate and rewrite Eq. (3) in the form X 00 2LiF þ ZnX Zn Ð fLii : VZn : Lii g þ ZnF2 .
(9)
The corresponding reduced mass action equation is DH sol X 00 ½fLii : VZn : Lii g ¼ exp , (10) kT where DH sol ¼ 0:73 eV, which is the cluster solution energy given in Table 5. Using Eq. (10) the equilibrium solution limit at 473 K is 0.01 ppm. This is substantially below the 100 ppm Liþ ion concentration value suggested to be present [3]. An assumption of the above analysis is that the Liþ ion content can reach equilibrium. It is therefore necessary to confirm that the kinetics of the Liþ interstitial defect are sufficiently fast as to allow access to equilibrium or even for defect clustering to occur in a non-equilibrium material. The migration activation energy for an isolated Liþ interstitial ion to migrate through the ZnF2 lattice was therefore calculated. The migration energy in [0 0 1] direction was determined to be 0.3 eV, whilst in the basal (ab) plane, it was 1.6 eV. Since interstitial Liþ ion migration along [0 0 1] is a much lower energy process, it is probable that the isolated Liþ defects migrate along the [0 0 1] until they are either trapped in vacant zinc sites or become part of clusters. The situation where the Liþ ion is initially trapped at a zinc site was therefore, considered. In this case, the calculated activation energy is that necessary to move the Liþ from the substitutional site (leaving behind a vacancy) to a nearest stable interstitial site. If that interstitial site remains in the ab plane, the activation energy is 5.0 eV. If the interstitial site has a component in the c direction, the activation energy is 3.1 eV. In either case, clearly, once the Liþ defect becomes incorporated at a vacant zinc site, it is strongly trapped.
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5. Discussion and comment on the dielectric relaxation Solution energy calculations clearly support the second of Roth’s models, i.e. that involving lithium interstitial ions. However, the cluster configuration predicted here is quite distinct from the models envisaged by Roth (as shown in Fig. 1b and 1c). The implications of this to the dielectric relaxation process must now be considered. First, we must investigate how this defect cluster couples to an applied field. This split lithium cluster is a trimer defect (i.e. +:2-:+) oriented in the [1 5 0] direction. As such, if a field is applied parallel to the ab plane, this defect will not couple without reorientation towards the dipole configuration envisaged by Roth (compare Figs. 1c and 1d). To facilitate this, one Liþ defect is restricted to remain at the substitutional site and the Liþ interstitial is allowed to relax. The calculated energy difference between the two configurations is 0.66 eV which is the same as that suggested as the dipole reorientation energy as reanalysed by Nowick [6]. The resulting calculated dipole of the Roth cluster is 7:0 1029 C m. The problem with this model is, however, when no field is present the defect relaxes back to the split orientation with no energy barrier. When a field is applied in the [0 0 1] direction, the split interstitial defect cluster will couple with the field and form a dipole most readily by the relaxation of both Lii defects in a [0 0 1] direction. If the field imparts the same energy (i.e. 0.62 eV) to displace the Lii ions, a similar dipole of 2:9 1029 C m is formed. Once again, when the field is removed, the Liþ ions will relax back to the fLii : V00Zn : Lii g tripole defect with no net polarisation of the lattice. Thus, reorientation of the equilibrium defect cluster cannot result in a dielectric relaxation of the type observed by Roth [3,4]. In order to consider the possible role of the equilibrium lithium cluster in the dielectric relaxation response, the energy of the cluster was considered as a function of defect separation. This was facilitated by placing one Liþ at a substitutional site and the second Liþ ion at first, then second and finally third neighbour interstitial sites.
Table 6 shows the formation energy of the cluster constrained (unrelaxed) and after lattice relaxation. It is immediately clear that when the two Liþ ions are in either 1st or 2nd neighbour positions (see Fig. 2) but unconstrained, subsequent energy minimisation results in a relaxation to the ‘‘splitinterstitial’’ orientation. Only, in the 3rd neighbour position, do they remain apart, i.e. there is an energy barrier against relaxation to the split interstitial configuration. The resulting 3rd neighbour configuration as shown in Fig. 2 is oriented in
Table 6 fLii : V00Zn : Lii g cluster energies (eV) Cluster neighbour positions
Relaxation
No relaxation
1st 2nd 3rd
0.73 0.73 1.81
1.39 1.64 1.85
Fig. 2. The ZnF2 tetragonal unit cell showing the three different unrelaxed neighbour positions of the Liþ interstitial defect (1st, 2nd, 3rd) which together with the Li0Zn defect will generate the defect cluster fLi0Zn : Lii gX .
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the [2 0 3] direction and as such has very similar dipole components both in the ab plane and in the [0 0 1] direction. However, the existence of symmetry-related equivalent orientations of the 3rd neighbour cluster means that in a purely [0 0 1] field a distribution of such defects can together have a net dipole only in the [0 0 1] direction (e.g. [2 0 3] + [2¯ 0 3] and [0 2 3] + [0 2¯ 3]). Unfortunately, the equivalent holds true for a ¯ field applied in the ab plane (e.g. [2 0 3] + [0 2 3], etc.), so it is possible to construct a dipole response that is restricted to the ab plane. On the basis of the above analysis, the split Liþ interstitial cluster cannot be responsible for the anisotropic dielectric relaxation reported by Roth [3]. However, the solution energy analysis suggests that Roth’s material was far from equilibrium. The supersaturated solid solution consisted of essentially immobile substitutional Liþ ions and charge compensating Liþ interstitial ions. These later defects are confined to move preferentially in [0 0 1]. Once they encounter a Liþ substitutional ion they become trapped and form the split Liþ interstitial cluster. Any untrapped Liþ interstitial ions will move in [0 0 1] subject to an applied field. It is the displacement of these nonequilibrium, residual defects that may conceivably result in a polarization of the crystal. Certainly such a polarization will occur much more strongly in [0 0 1] than in the ab plane. Furthermore, the polarization will decay once the temperature is raised to overcome the activation energy barrier. Nevertheless, such defects should eventually become trapped and as such the anisotropic relaxation effects may be subject to a form of thermal cycling fatigue. An experiment based on this premise may provide one way of further investigating this material. Alternatively, samples could be subjected to higher temperature heat treatments so that they are closer to obtaining an equilibrium defect configuration.
Appendix Throughout, the mass action approach and Kro¨ger–Vink notation [22] were employed in order to describe and analyse defect processes. For
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example, the anion Frenkel reaction is 0 FX F Ð Fi þ VF
(11)
and the corresponding mass action equation [23] is ½F0i ½VF DH F ¼ exp , (12) kT ½FX F where DH F is the total calculated energy for Eq. (11), k is the Boltzmann’s constant, T is the temperature and square brackets indicate defect concentration. If this is the dominant reaction, then electroneutrality would dictate that ½F0i ¼ ½VF and Eq. (12) can be rewritten to yield DH F ½F0i ¼ exp . 2kT
(13)
(14)
The energy that is of significance in order to determine the F ion interstitial concentration (or fluorine vacancy concentration) is therefore DH F =2. The value 2 is the normalisation factor for this reaction. Only normalised defect process energies can be compared to each other. Following the same approach, but for the Schottky reaction yields a normalisation factor of 3. An equivalent analysis for extrinsic reactions can also be carried out. For example, assuming Eq. (9), Liþ interstitial compensation, were dominant, electroneutrality would dictate that ½Lii ¼ ½Li0Zn and the mass action equation becomes DH sol 0 ½LiZn ¼ exp . 2kT
(15)
(16)
Overall, this type of analysis yields normalisation factors for reactions (2), (3) and (4) to be 3, 2 and 2, respectively. The energies for the isolated defect reactions reported in Table 5 have been normalised. When defect clusters are formed, the component defect positions are spatially correlated. This limits the configurational contribution and alters the mass action equations so that the normalisation factor becomes unity (in the case where all defects are bound into a cluster as for example in Eq. (5)).
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References [1] C. Bucci, R. Fieschi, Phys. Rev. Lett. 12 (1964) 16. [2] C. Bucci, R. Fieschi, G. Guidi. Phys. Rev. 148 (1966) 816. [3] T. Roth, J. Appl. Phys. 42 (1970) 246. [4] T. Roth, J. Appl. Phys. 44 (1973) 1056. [5] O.W. Johnson, Phys. Rev. 136 (1964) 284. [6] A.S. Nowick, J. Appl. Phys. 44 (1973) 5169. [7] M. Born, J.E. Mayer, Z. Physik. 75 (1932) 1. [8] M. Born, J.E. Mayer, J. Chem. Phys. 1 (1933) 270. [9] P.P. Ewald, Ann. Physik. 64 (1921) 253. [10] R.A. Buckingham, Philos. Trans. R. Soc. London, Ser. A 168 (1938) 264. [11] B.G. Dick, A.W. Overhauser, Phys. Rev. 112 (1958) 90. [12] N.F. Mott, R. W Gurney, Trans. Faraday Soc. (1938) 506. [13] N.F. Mott, M. Littleton, Trans. Faraday Soc. 34 (1938) 485. [14] M. Leslie, DL/SCI/TM31T, Technical Report, SERC Daresbury Laboratory (1982).
[15] K.J.W. Atkinson, R.W. Grimes, S. Owens, Solid State Ionics 150 (2002) 443. [16] A.M. Cormack, C.R.A. Catlow, S. Ling, Phys. Rev. B 40 (1989) 3278. [17] M.S. Islam, R.C. Baetzold, J. Phys. Chem. Solids 53 (1992) 1105. [18] S. Hart, S.A. Afr, J. Phys. 1 (1978) 65. [19] J.C. Jamieson, D.S. Rimai, Bull. Am. Phys. Soc. 22 (1977) 353. [20] D.S. Rimai, Phys. Rev. B 16 (1977) 2200. [21] K. Recker, F. Wallrafen, K. Dupre, Naturwissenschaften 75 (1988) 156. [22] F.A. Kroger, The Chemistry of Imperfect Crystals, second ed., vol. 2, North-Holland Publishing Company, Ltd., Amsterdam, 1974, p. 82. [23] F.A. Kroger, H.J. Vink, Relationships between the concentrations of imperfections in crystalline solids, in: F. Seitz, D. Turnbull (Eds.), Solid State Physics: Advances in Research and Applications, vol. 3, Academic Press, New York, 1956, p. 312. [24] J.K. Vassiliou, J. Appl. Phys. 59 (1985) 1125.
Physica B 365 (2005) 147–154 www.elsevier.com/locate/physb
Dielectric relaxation of lithium-doped zinc fluoride Mark R. Levya,, K.J.W. Atkinsonb, Onyeka H. Odilia, Robin W. Grimesa a Department of Materials, Imperial College, London SW7 2BP, UK British Energy Generation Ltd., Barnett Way, Barnwood, Gloucester GL4 3RS, UK
b
Received 17 August 2004; accepted 7 May 2005
Abstract It has been claimed that Li-doped ZnF2 exhibits anisotropic dielectric relaxation. However, the mechanism for this behaviour has not been resolved. Here, we present a new analysis of the defect structures and reactions in this system based on results from atomistic simulations. In particular, the relative energies for the solution of LiF are predicted which lead to the structure of a split Liþ interstitial cluster. The implications of this to the dielectric relaxation processes are discussed. This requires an analysis of Liþ ion migration and cluster binding energies. r 2005 Elsevier B.V. All rights reserved. PACS: 61.66.f; 61.72.y; 61.72.Ji; 81.05.Je Keywords: ZnF2 ; Dielectric relaxation; Anisotropy; Atomistic simulation
1. Introduction Many materials, when subjected to an electric field, will polarise to oppose the field. One contribution to the polarisation involves the movement of some ions from their equilibrium lattice positions. When the field is removed, the ions will generally return to their equilibrium positions. If the field is imposed on a material at a high temperature, for some time, and the material is subsequently quenched in the field (typically to Corresponding author. Tel.: +44 20 7594 6802;
fax: +44 20 7594 6729. E-mail address: [email protected] (M.R. Levy).
liquid nitrogen temperature), the ionic contribution to the polarisation can, for some materials, be partly frozen-in. This implies that there is an energy barrier against the relaxation of the ions back to their equilibrium positions. If the material is slowly heated, this polarisation will be lost once the ions gain sufficient thermal energy to overcome the barrier. This process, the dielectric relaxation, can be monitored and quantified as an electric current [1,2]. It has been suggested by Roth [3,4] that zinc fluoride (ZnF2 ) doped with lithium exhibits a considerable anisotropy in its dielectric properties. In particular, it exhibits a strong relaxation in [0 0 1], while the effect in [1 0 0] and [1 1 0] is several
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orders of magnitude lower [3]. Roth initially proposed a relaxation mechanism involving the movement of fluorine vacancies, as the chargecompensating defects for Liþ substitutional ions [3]. He later suggested that the anisotropy is due to the motion of Liþ interstitial ions, as an alternative charge compensating species [4]. The rationale behind the change in mechanism was an analysis of the experimental data which resulted in a low energy (0.32 eV) for defect cluster reorientation. Roth considered it unlikely that a vacancy model would be compatible with such a low energy but that an interstitial ion model would, because similarly low Liþ interstitial migration energies in materials with the rutile structure were already known [5]. In this regard, he suggested that the orientation of the dipole, formed by a substitutional Liþ ion–interstitial Liþ ion cluster, is initially randomised throughout the lattice with the substitutional lithium defect immobile. Intriguingly the anisotropy in the dielectric relaxation was only observed when the doped material was subjected to a prior heat treatment. The reason for this treatment differed between the two Roth publications: in the first, it ensures a sufficiently high concentration of fluorine vacancies [3] and in the second it is to dissolve defect clusters and increase the intensity of the dielectric relaxation [4]. The discrepancies between the two Roth publications were commented on by Nowick [6]. In particular, he stated that the same dielectric response in the [1 1 0] and [1 0 0] directions is not a test of the model, but a requirement of the theory. In addition, the experimental data was reanalysed to yield a dipole reorientation energy of about 0.7 eV and as such, the necessity for an interstitial lithium mechanism was removed. Finally, Nowick suggested that the dielectric anisotropy could be due merely to differences in the heat treatment of the samples. Our aim here is to use atomic scale computer simulation to predict the structures and energies of defects associated with lithium solution in ZnF2 . In this way, we will provide a mechanistic basis for the defect processes that underpin such dielectric relaxation process in this material.
2. Methodology The calculations presented here are based upon the Born description of an ionic lattice [7,8]. The forces acting between ions are resolved into longrange coulombic and a short-range pair terms. The long-range terms were summed using the Ewald method [9], while the short-range components were modelled with parameterised Buckingham pair potentials [10]. Initially, the perfect lattice is defined by tessellating the unit cell throughout space using periodic boundary conditions which are defined by the crystallographic lattice vectors. The perfect lattice energy (E L ) is given by " ! !# X X qi qj rij C ij EL ¼ þ Aij exp , 4p0 rij rij r6ij j4i i (1) where Aij , rij and C ij are adjustable short-range parameters, 0 is the permittivity of free space, and rij is the interionic separation between ions i and j of charges qi and qj , respectively. The shortrange potential parameters used in this work are reported in Table 1. The electronic polarisability of the fluorine ions was accounted for through the shell model [11]. Here, a shell of charge Y ¼ 1:3776jej is coupled to a core of charge X ¼ þ0:3776jej via an isotropic, harmonic spring of force constant, ( 2 . k ¼ 24:36 eV A In order to model the effect of a charged defect, a multi-region approach in which the lattice is partitioned into concentric, spherical regions centered about the defect is adopted. The region immediately surrounding the defect is termed region I, in which all ions are explicitly relaxed, subject to the forces described by the inter-atomic potentials. Surrounding this is region IIa in which Table 1 Short-range potential parameters Interaction
A (eV)
r (A˚)
C (eV A˚6 )
F -F Liþ -F Zn2þ -F
1317.50 574.80 918.41
0.27530 0.25530 0.28481
13.8 0.0 0.0
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the forces between the ions are determined via a dielectric continuum approximation [12] and all the ions can be relaxed in a single step. The interaction energies between the ions in interfacial region IIa and region I are, however, calculated explicitly. Beyond region IIa is the outer region IIb, which is a point charge array and provides the Madelung field of the remaining lattice; the response energy for this region is determined using the Mott–Littleton approximation [13]. Migration energies are calculated assuming a thermally activated process is in operation. Thus an ion is moved between sites through a saddle point which defines the activation energy. The saddle point is identified by rastering the migrating ion through possible positions on the migration pathway. The migrating ion is fixed and the surrounding lattice is relaxed, thus allowing the location and evaluation of the migration activation energy. All calculations presented here were carried out using the CASCADE code [14] which has been used successfully in many previous related studies of defects in fluorides, e.g. Refs. [15–17]. 3. Results for undoped ZnF2 ZnF2 adopts the tetragonal rutile crystal structure with space group 136 (P4=mnm). This structure can be considered as a body centered tetragonal lattice of cations (zinc) which are then octahedrally coordinated by the anions (fluorine) (see Fig. 1a). The comparison between experimental and calculated lattice parameters, presented in Table 2, shows excellent agreement. Elastic constants for ZnF2 have also been reported experimentally [18–20] and a comparison with values predicted here are presented in Table 3. This shows good agreement meaning that displacements of ions around their equilibrium positions are well reproduced. Since we are concerned with a lithium-doped material, the lithium–fluorine potential was derived so that the predicted lattice parameter of ¯ LiF (space group 225 (Fm3m) [21]) reproduces the experimental data, i.e. a lattice parameter of 4.027 A˚.
Fig. 1. The ZnF2 tetragonal unit cell showing the three mechanisms discussed in this publication. (a) The ZnF2 tetragonal unit cell. The large, dark spheres represent fluorine ions and the small, paler spheres represent zinc ions. (b) Represents a schematic of Roth’s first proposed mechanism [3]. (c) A schematic of Roth’s second mechanism [4]. (d) The mechanism proposed here. The transparent cube in (b) represents a fluorine vacancy, the transparent cube in (d) is a zinc vacancy and the white spheres represent lithium substitutional ions in (b) and (c), but interstitial ions in (d).
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The energies for intrinsic defect processes in ZnF2 were also calculated (see Appendix for details). The results, shown in Table 4, suggest that the anion Frenkel reaction is slightly preferred over the Schottky reaction while the cation Frenkel energy is substantially higher. The anion Frenkel defects (i.e. the fluorine vacancy and fluorine interstitial) are therefore the majority defects in this material. However, the Schottky reaction is only slightly higher in energy than the Table 2 Comparison of calculated and experimental lattice parameters for ZnF2 a (A˚)
c (A˚)
c a
Volume (A˚3 )
Calc. 4.6526 3.2046 0.6888 69.37 Expt. [24] 4.7034 3.1335 0.6662 69.32 Percentage difference 1.11 2.26 3.41 0.002
Table 3 Calculated and experimental elastic constants ð 1011 Nm2 ) for ZnF2 C 11 Calc. Expt. [19,20] Expt. [18]
C 12
131.8 126 130
91.4 93 97
C 13 60.8 84 89
C 33 214.2 192 199
C 44 50.1 39.2 39.5
anion Frenkel. This reaction will therefore provide an important concentration of intrinsic defects. Finally, it was found that interstitial F adopts a split-interstitial configuration similar to that observed in other fluorite structures [15,16]. In this, the interstitial fluorine ion displaces a lattice F ion into a neighbouring interstitial site. Thus, the two F ions effectively share the vacant fluorine lattice site that is created. The splitinterstitial configuration remains in the ab lattice plane.
4. Solution mechanisms for LiF-doped ZnF2 The incorporation of monovalent Liþ ions onto divalent zinc sites requires charge compensation. This can be facilitated in one of three ways: via a Zn2þ interstitial ion (Eq. (2)), a Liþ interstitial ion (Eq. (3)) or a F vacancy (Eq. (4)). In Kro¨ger– Vink notation [22], these are: 0 2LiF þ 2ZnX Zn Ð 2LiZn þ Zni þ ZnF2 ,
(2)
0 2LiF þ ZnX Zn Ð LiZn þ Lii þ ZnF2 ,
(3)
0 X LiF þ ZnX Zn þ FF Ð LiZn þ VF þ ZnF2 .
(4)
C 66 92.4 80.7 81.4
Table 4 Intrinsic defect process energies (eV) for ZnF2 Schottky
Anion Frenkel
Cation Frenkel
2.59
2.17
4.31
The normalised energies for these processes are reported in Table 5 (see Appendix for details). It is clear that the energies of these extrinsic defect processes are considerably smaller than for intrinsic processes. As such, at equilibrium, extrinsic defects will dominate intrinsic defects. If we assume the defects remain spatially isolated, the lowest energy charge compensating defect is a Liþ interstitial (Eq. (3)). However, Eq. (4) which describes charge compensation via
Table 5 Solution and cluster binding energies (eV) for Liþ accommodation in ZnF2 Solution mechanism
Reaction 2 Reaction 3 Reaction 4 a
Charge compensating defect
Zni Lii VF
Solution energy
Cluster binding energy
Isolateda
Clustered
2.22 1.27 1.46
1.82 0.73 1.52
Note: these energies have been normalised as described in the Appendix.
4.84 1.81 1.40
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a F vacancy is only 0.19 eV higher, and may, on this basis alone, yield a further minor concentration of defects. The coulombic interaction between the isolated defects leads to the formation of defect clusters. The lowest energy solution process, based on defect clusters, still involves compensation via a Liþ interstitial defect (reaction 3). This mechanism is now substantially favoured over the other two reactions. The reason for this is that the Liþ interstitial/substitution pair cluster undergoes considerable lattice relaxation, into a split interstitial configuration, which lowers the overall solution energy. (This is isostructural to the split F interstitial ion reported in the intrinsic defect analysis.) The structure of this cluster, as shown in Fig. 1d, consists of two equivalent Liþ interstitials either side of a vacant zinc site (i.e. fLii : V00Zn : Lii gX rather than fLi0Zn : Lii gX ). Eq. (5) describes the equilibrium between Liþ ions when isolated and when part of a defect cluster (indicated by braces). Li0Zn þ Lii Ð fLii : V00Zn : Lii gX .
(5)
The corresponding mass action equation is [22] ½fLii : V00Zn : Lii gX DH b ¼ exp , (6) ½Li0Zn ½Lii kT where DH b is the cluster binding enthalpy, k is the Boltzman constant, T is the absolute temperature and square brackets indicate a concentration. Using the electro-neutrality condition between the charged defects (Eq. (7)), Eq. (6) can be simplified to yield an expression for the ratio of lithium in clusters to that which is isolated (see Eq. (8)). ½Li0Zn ¼ ½Lii ,
(7)
clustered ½fLii : V00Zn : Lii gX ¼ . isolated 2½Li0Zn
(8)
If the assumption is made that the total lithium concentration is 100 ppm, as stated by Roth in the experimental studies [3], at the heat treatment temperature of 473 K and the calculated energy 1:81 eV, this ratio has the value 3 107 ; ergo, at equilibrium, Liþ ions will be in the form of clusters. On the basis of this analysis, the reason
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for the heat treatment given by Roth in his second paper [4], i.e., to dissolve the clusters, is not valid. It is also possible to used the calculated defect energies to predict an equilibrium solution limit and compare this against the value given by Roth [3]. In this case, we assume defect clusters dominate and rewrite Eq. (3) in the form X 00 2LiF þ ZnX Zn Ð fLii : VZn : Lii g þ ZnF2 .
(9)
The corresponding reduced mass action equation is DH sol X 00 ½fLii : VZn : Lii g ¼ exp , (10) kT where DH sol ¼ 0:73 eV, which is the cluster solution energy given in Table 5. Using Eq. (10) the equilibrium solution limit at 473 K is 0.01 ppm. This is substantially below the 100 ppm Liþ ion concentration value suggested to be present [3]. An assumption of the above analysis is that the Liþ ion content can reach equilibrium. It is therefore necessary to confirm that the kinetics of the Liþ interstitial defect are sufficiently fast as to allow access to equilibrium or even for defect clustering to occur in a non-equilibrium material. The migration activation energy for an isolated Liþ interstitial ion to migrate through the ZnF2 lattice was therefore calculated. The migration energy in [0 0 1] direction was determined to be 0.3 eV, whilst in the basal (ab) plane, it was 1.6 eV. Since interstitial Liþ ion migration along [0 0 1] is a much lower energy process, it is probable that the isolated Liþ defects migrate along the [0 0 1] until they are either trapped in vacant zinc sites or become part of clusters. The situation where the Liþ ion is initially trapped at a zinc site was therefore, considered. In this case, the calculated activation energy is that necessary to move the Liþ from the substitutional site (leaving behind a vacancy) to a nearest stable interstitial site. If that interstitial site remains in the ab plane, the activation energy is 5.0 eV. If the interstitial site has a component in the c direction, the activation energy is 3.1 eV. In either case, clearly, once the Liþ defect becomes incorporated at a vacant zinc site, it is strongly trapped.
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5. Discussion and comment on the dielectric relaxation Solution energy calculations clearly support the second of Roth’s models, i.e. that involving lithium interstitial ions. However, the cluster configuration predicted here is quite distinct from the models envisaged by Roth (as shown in Fig. 1b and 1c). The implications of this to the dielectric relaxation process must now be considered. First, we must investigate how this defect cluster couples to an applied field. This split lithium cluster is a trimer defect (i.e. +:2-:+) oriented in the [1 5 0] direction. As such, if a field is applied parallel to the ab plane, this defect will not couple without reorientation towards the dipole configuration envisaged by Roth (compare Figs. 1c and 1d). To facilitate this, one Liþ defect is restricted to remain at the substitutional site and the Liþ interstitial is allowed to relax. The calculated energy difference between the two configurations is 0.66 eV which is the same as that suggested as the dipole reorientation energy as reanalysed by Nowick [6]. The resulting calculated dipole of the Roth cluster is 7:0 1029 C m. The problem with this model is, however, when no field is present the defect relaxes back to the split orientation with no energy barrier. When a field is applied in the [0 0 1] direction, the split interstitial defect cluster will couple with the field and form a dipole most readily by the relaxation of both Lii defects in a [0 0 1] direction. If the field imparts the same energy (i.e. 0.62 eV) to displace the Lii ions, a similar dipole of 2:9 1029 C m is formed. Once again, when the field is removed, the Liþ ions will relax back to the fLii : V00Zn : Lii g tripole defect with no net polarisation of the lattice. Thus, reorientation of the equilibrium defect cluster cannot result in a dielectric relaxation of the type observed by Roth [3,4]. In order to consider the possible role of the equilibrium lithium cluster in the dielectric relaxation response, the energy of the cluster was considered as a function of defect separation. This was facilitated by placing one Liþ at a substitutional site and the second Liþ ion at first, then second and finally third neighbour interstitial sites.
Table 6 shows the formation energy of the cluster constrained (unrelaxed) and after lattice relaxation. It is immediately clear that when the two Liþ ions are in either 1st or 2nd neighbour positions (see Fig. 2) but unconstrained, subsequent energy minimisation results in a relaxation to the ‘‘splitinterstitial’’ orientation. Only, in the 3rd neighbour position, do they remain apart, i.e. there is an energy barrier against relaxation to the split interstitial configuration. The resulting 3rd neighbour configuration as shown in Fig. 2 is oriented in
Table 6 fLii : V00Zn : Lii g cluster energies (eV) Cluster neighbour positions
Relaxation
No relaxation
1st 2nd 3rd
0.73 0.73 1.81
1.39 1.64 1.85
Fig. 2. The ZnF2 tetragonal unit cell showing the three different unrelaxed neighbour positions of the Liþ interstitial defect (1st, 2nd, 3rd) which together with the Li0Zn defect will generate the defect cluster fLi0Zn : Lii gX .
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the [2 0 3] direction and as such has very similar dipole components both in the ab plane and in the [0 0 1] direction. However, the existence of symmetry-related equivalent orientations of the 3rd neighbour cluster means that in a purely [0 0 1] field a distribution of such defects can together have a net dipole only in the [0 0 1] direction (e.g. [2 0 3] + [2¯ 0 3] and [0 2 3] + [0 2¯ 3]). Unfortunately, the equivalent holds true for a ¯ field applied in the ab plane (e.g. [2 0 3] + [0 2 3], etc.), so it is possible to construct a dipole response that is restricted to the ab plane. On the basis of the above analysis, the split Liþ interstitial cluster cannot be responsible for the anisotropic dielectric relaxation reported by Roth [3]. However, the solution energy analysis suggests that Roth’s material was far from equilibrium. The supersaturated solid solution consisted of essentially immobile substitutional Liþ ions and charge compensating Liþ interstitial ions. These later defects are confined to move preferentially in [0 0 1]. Once they encounter a Liþ substitutional ion they become trapped and form the split Liþ interstitial cluster. Any untrapped Liþ interstitial ions will move in [0 0 1] subject to an applied field. It is the displacement of these nonequilibrium, residual defects that may conceivably result in a polarization of the crystal. Certainly such a polarization will occur much more strongly in [0 0 1] than in the ab plane. Furthermore, the polarization will decay once the temperature is raised to overcome the activation energy barrier. Nevertheless, such defects should eventually become trapped and as such the anisotropic relaxation effects may be subject to a form of thermal cycling fatigue. An experiment based on this premise may provide one way of further investigating this material. Alternatively, samples could be subjected to higher temperature heat treatments so that they are closer to obtaining an equilibrium defect configuration.
Appendix Throughout, the mass action approach and Kro¨ger–Vink notation [22] were employed in order to describe and analyse defect processes. For
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example, the anion Frenkel reaction is 0 FX F Ð Fi þ VF
(11)
and the corresponding mass action equation [23] is ½F0i ½VF DH F ¼ exp , (12) kT ½FX F where DH F is the total calculated energy for Eq. (11), k is the Boltzmann’s constant, T is the temperature and square brackets indicate defect concentration. If this is the dominant reaction, then electroneutrality would dictate that ½F0i ¼ ½VF and Eq. (12) can be rewritten to yield DH F ½F0i ¼ exp . 2kT
(13)
(14)
The energy that is of significance in order to determine the F ion interstitial concentration (or fluorine vacancy concentration) is therefore DH F =2. The value 2 is the normalisation factor for this reaction. Only normalised defect process energies can be compared to each other. Following the same approach, but for the Schottky reaction yields a normalisation factor of 3. An equivalent analysis for extrinsic reactions can also be carried out. For example, assuming Eq. (9), Liþ interstitial compensation, were dominant, electroneutrality would dictate that ½Lii ¼ ½Li0Zn and the mass action equation becomes DH sol 0 ½LiZn ¼ exp . 2kT
(15)
(16)
Overall, this type of analysis yields normalisation factors for reactions (2), (3) and (4) to be 3, 2 and 2, respectively. The energies for the isolated defect reactions reported in Table 5 have been normalised. When defect clusters are formed, the component defect positions are spatially correlated. This limits the configurational contribution and alters the mass action equations so that the normalisation factor becomes unity (in the case where all defects are bound into a cluster as for example in Eq. (5)).
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