Crystal structure of incommensurate ηʺ-Cu_1.235Sn intermetallic

3.1 Development of the modulated structure model

The previously reported superstructures for Cu₆Sn₅ IMC listed Table 1 all have in common a translation vector b_m = −a_h + b_h, as indicted by the 2nd row of the matrix A according to Eq. (1). Hence, in accordance with Refs. [13], [14], [15], a projection along that direction can be employed to unambiguously depict the Cu(2) distribution (see right side of Figure 1). Such a projection is used in Figure 3 for the η′, η⁸, and η⁴⁺¹ superstructures, keeping the drawings in agreement with the A matrices from Table 1. The η⁶ structure is not considered further but only noted for completeness, since it cannot be described by the modulated structure model to be outlined in the following. The alloys with the applied heat treatments investigated in our own studies have given no indication for η⁶ type of order.

The η′, η⁸, and η⁴⁺¹ superstructures have, apart from the translation vector b_m = −a_h + b_h, further features in common, which partially have been also referred to in previous works [13], [14], [15], and which are visible from inspecting Figure 3. In particular, all Cu(2) atoms are assembled on planes parallel to (114‾)_h. The trace of such a plane is illustrated by a dashed green line in Figure 3. These planes are spanned by b_m and by a further translation vector c_m = 2a_h+2b_h + c_h (third row of the A matrices in Table 1). Assembling the Cu(2) on some of these planes and leaving others empty explains occurrence of superstructure reflections perpendicular to such planes (see analysis of diffraction patterns in Ref. [18]). Taking the positions of these superstructure reflections as modulation vectors, it succeeded to arrive at a general structure model featuring one-dimensional modulations, which is able

1. to describe the η′, η⁸, and η⁴⁺¹ superstructures as commensurately modulated structures, and

2. to act as a starting model for Rietveld refinement based on X-ray powder diffraction data from the ηʺ phase.

The modulated structures have at maximum a monoclinic symmetry due to the symmetry breaking associated with selecting one set of (114‾)_h planes on which the Cu(2) atoms are assembled. Hence, in line with the conventions to formulate one-dimensionally modulated structures an average structure is used in the following as a basis to describe the structures. This average structure has a unit cell spanned by the basis vectors a_av, b_av, c_av obtained using the transformation matrix A_av:

The unit cell of the average structure has close-to orthorhombic shape with a_av ≈ a_h, b_av ≈ 3^1/2a_h, c_av ≈ c_h, β_av ≈ 90°. With respect to this average structure, A from Eq. (1) can be reformulated as A = A′A_av (see Table 1) and (114‾)_h becomes (204‾)_av. The matrices A′ have been included in Table 1, too.

The modulation vector selected here to describe the modulated structure is of the type

with positive q₁ and q₃,^[1] and with * designating the basis vectors of the reciprocal lattice. The q vector leads to Bragg reflections at

Thereby, h, k, l are the conventional Laue indices and m is the order of the satellite reflection. The Cu(2) distribution of the η′, η⁸, and η⁴⁺¹ superstructures is reproduced correctly with the following structure features:

1. 3+1 dimensional superspace group symmetry is taken as C2/c(q₁0–q₃)00 [20]

2. the Cu(2) distributions is described by a Crenel type occupation modulation with a fractional length of the occupied region equal to the value of δ of the respective superstructure

3. q₁ and q₃ in Eq. (3) fulfill

and there is an appropriate t₀ value describing the position of the section of the three-dimensional structure through the 3+1 dimensional superspace structure. The t₀ determines the actual superstructure and its 3-dimensional space group symmetry.

Using the tools implemented into the Jana2006 software [19], atomic coordinates pertaining to the three η′, η⁸, and η⁴⁺¹ superstructures can be derived in the modulated structure description and as conventional superstructures. The basic parameters pertaining to these descriptions have been added to the lower part of Table 1. Thereby, a uniform type of a_m basis vector of

with integer valued r and s was adopted, as given in the first row of the A′ matrices on the right part of Table 1, where the original description of the η′ phase already adopts that setting. The accordingly generated atomic arrangement of the η⁴⁺¹ structure shows C2/c symmetry, whereas the space group pertaining to the original unit cell appears to be I2/c instead of P1 given in Ref. [15]. This higher symmetry is also confirmed when applying the findsym software tool [21] with very high tolerances to the fractional coordinates reported in Ref. [15].

There is, in fact, an infinite number of superstructures which can be derived based on the present ideal scheme with validity of Eq. (5), in the range 0 ≤ r/(r + s) ≤ 1. The η′-Cu_1.20Sn superstructure constitutes the endmember superstructure with r/(r + s) = 1, whereas r/(r + s) = 0 corresponds to a structure included as η^x-Cu_1.333Sn in Table 1 and in Figure 3.

Figure 3:

Distribution of the Cu(2) atoms in various ordered Cu_1+δSn superstructures listed in Table 1 (primed basis vector refer to Eq. (6)) by projecting the Cu(2) atoms on a plane parallel to (010)_av in accordance with Figure 1.

3.2 Rietveld refinement

According to Ref. [18] the heat treatment step at 438 K produced the ηʺ ordering from η at constant composition (i.e., partitionless) assessed as Cu_1.235Sn on the basis of the lattice parameters. Rietveld refinement on the basis of the PXRD data from this alloy (see Figure 4) succeeded based on an incommensurately modulated structure model, which is based on the structure description outlined in Section 3.1.

Figure 4:

Powder-X-ray diffraction data of ηʺ-Cu_1.235Sn with some Cu₃Sn impurity (Co-K_α1 radiation; black data points) together with calculated profile (red) and difference curve (black, bottom) as a result of Rietveld refinements. Reflection markers for the two phases are indicated with black reflection markers, which in the case of ηʺ pertain to fundamental reflections and green ones to satellite reflections up to order of two: (a) overview with square-root intensity scale, (b) zoomed low-angle region and (c) magnified region of the main fundamental reflections revealing the reflection splitting.

Upon starting to do full-pattern based refinements on the basis of the PXRD data, it first succeeded to get convincing fits to the fundamental reflections using the average structure. Thereby, it turned out that, against original expectation from the evaluation of PXRD patterns of the η′ phase [16], [17], visible reflection splitting occurred for some of the fundamental reflections (see Figure 4c). In terms of the monoclinic lattice parameters of the average structure, it turned out that the lattice angle β_av deviated from 90° by a significant amount of ±0.2°. Both directions correspond to equivalent structures and, hence, equivalent qualities of fit. Upon trying to establish q₁ and q₃ parameters for the modulation vector according to Eq. (3), performing a fit considering the satellite reflections, however, it turned out to be important to get the correct choice of the direction of the deviation from β_av = 90°, which turned out to be positive (90.2 … °) for a q vector according to Eq. (3). The q₁ and q₃ values calculated via Eq. (5) for the η⁴⁺¹ structure (see also lower part of Table 2) constituted a suitable starting point for a converging refinement of q₁ and q₃, but these values deviated significantly from the starting values. In any case, freely refined q₁ and q₃ values according to Eq. (3) appeared necessary, since the restriction according to Eq. (5), even with only imposing q₃/q₁ = 2 ends with remaining misfits between observed and calculated peak positions. Displacive modulation parameter were required for all sites to get reasonable satellite reflection intensities.

The refinement model further included a second, hexagonal phase used to describe the relevant reflections due to the Cu₃Sn phase and having a crystal structure according to Ref. [22], [23]. That phase was present due to the overall alloy composition corresponding to 43 at.% Sn and this phase was also visible in optical micrographs of the microstructure, see supplementary material of Ref. [18]. A simplified hexagonal close packed structure was adopted to keep the Rietveld refinement stable, which was well possible because the orthorhombic distortion due to the long-period superstructure is apparently negligible, and the superstructure reflections are very weak (one superstructure reflection is discernible at 2θ = 32.4°, see Figure 4b). Note that slight preferred orientation has been adopted for both phases, which might be related with anisotropic cleavage of the crystallites and an orientation relationship of the Cu₃Sn crystallites in the Cu₆Sn₅ matrix.

The final parameters of the Rietveld refinement are listed Table 2. The final choice of the parameters to be refined, in particular the selected orders of the modulation functions, was a compromise between improving the residuals and keeping stability of the refinement as determined by the correlations between the refined modulation parameters.

4.1 Properties of the ideal superstructures

The ordered superstructures which are basis for the general modulated structure model derived in Section 3.1 (see Figure 3) feature very specific close distances between Cu(2) atoms. In the context of this work these distances can be described in terms of different types of connectivities between Cu(2)Sn₅ trigonal bipyramids via common Sn atoms. Possible close connectivities are depicted in Figure 5. It is directly evident from an inspection of the superstructures depicted in Figure 3 that only ac type edge-sharing of Cu(2)Sn₅ trigonal pyramids occurs, which are shown in Figure 5c.

Figure 5:

Type of close Cu(2)–Cu(2) distances depicted (a) schematically as in in Figure 3 with solid circles as the Cu(2) atoms and (b, c, d) as illustrated in terms of connected Cu(2)Sn₅ trigonal bipyramids. Only the ac type of edge-sharing according to (c) occurs in the superstructures depicted in Figure 3.

Quite naturally the degree of connectivity between Cu(2)Sn₅ trigonal bipyramids should increase with increasing Cu(2) content, i.e., with increasing δ. In case of the η′ superstructure at δ = 0.2 the Cu(2)Sn₅ trigonal bipyramids are isolated in the sense that there are no Sn atoms in common to two or more of such trigonal bipyramids. This has to change for δ > 0.2. Until, δ = 0.25, this edge sharing occurs in the form of isolated ac-type edge-sharing pairs of Cu(2)Sn₅ trigonal bipyramids (isolated Cu(2)₂Sn₈ units). All Cu(2) atoms in the η⁸ superstructure belong to such pairs. The η⁴⁺¹ superstructure consists of both the isolated Cu(2)Sn₅ trigonal bipyramids (composing the η′ superstructure) and the Cu(2)₂Sn₈ units. Strikingly the compositional range of the Cu₆Sn₅ IMC, especially when considering the disordered η high-temperature phase (see Figure 2), is approximately confined to the range 0.2 ≤ δ ≤ 0.25. Limiting the homogeneity range to δ ≤ 0.25 allows avoiding likely unfavorable higher connectivities between Cu(2)Sn₅ trigonal bipyramids as, e.g., expected for the hypothetical η^x-Cu_1.33Sn superstructure (see Figure 3). Note, however, that disordered η phase with δ < 0.2 appears to exist at relatively low temperatures implying existence of Sn having no direct Cu(2) neighbors atoms at some low temperatures [17].

4.2 Refined atomic structure of the ηʺ phase

The q₁ and q₃ parameters resulting from Rietveld refinement (Table 2) determining the modulation vector q according to Eq. (3) do not fulfill the relation q₃/q₁ = 2 implied by Eq. (5), which is fulfilled for all the ideal superstructures. Moreover, neither q₁ nor q₃ exactly fulfills the relation with the parameter δ, neither if the latter value is taken as assessed from the lattice parameters as described in Ref. [18] nor taking the refined value of δ, which is by 3 standard deviations smaller than the assessed parameter (0.229 vs. 0.235). Hence, the Cu(2) ordering must be defective as compared to the ideal structure model derived in Section 3.1 if Eq. (5) would prevail. An approximant structure has been derived based on the refined structure parameters using the tools by Jana2006 and is depicted schematically in Figure 6, highlighting the types of defects occurring in the ordering in terms of dislocation dipoles in the projected modulated structure (double “T”s). The occurrences of such disturbances in the modulated Cu(2) distribution originate from the deviations from q₃/q₁ = 2. These dipoles, on a larger spatial scale can be conceived to form boundaries between more well-ordered regions, and thereby resemble (albeit being geometrically different) domain boundaries showing up in approximant structures of incommensurate Ni₂In/NiAs type Cu_1+δIn phase with δ > 0.5 [25]. It has to be emphasized that the diffraction data available here and the refinement unlikely contain significant information about the exact way in which the deviations from the ideal structural model are realized on a local level, in particular at the dislocations dipoles in Figure 6.

Figure 6:

Approximant structure of ηʺ-Cu_1.235Sn calculated using the refined parameters resulting from Rietveld refinement as depicted by the arrangement of Cu(2) atoms in the projection described in Figure 1 and as used in Figure 3 and Figure 5 above. Due to deviation of the refined values of q₁ and q₃ from q₃/q₁ = 2, disturbances in the modulated structure occur, which are highlighted as double-“T”s, indicating that in the projection these disturbances can be regarded as edge dislocation dipoles in the ordered arrangement.

Deviations from ideal occupational ordering as observed here and also previously in related intermetallics (e.g., [26], [27]) might originate from peculiarities of the electronic structure. They will, however, in the present case more likely originate from a dependence of the energy of the modulated structure from the degree of ordering. Onset of spatially modulated occupational order can generally be described in terms of static concentration waves [28], which modulate the occupancies of the ordering sites. While the optimum, composition(δ)-adapted ordered superstructures according Section 3.1 and Eq. (5) might indeed be of lowest-energy if perfectly established, incomplete ordering, e.g., during the onset of ordering might deviate from the restrictions of Eq. (5). That argument has already been used in Refs. [26], [27], where the order was definitely incomplete, as also demonstrated by the observed temperature dependence of the degree of order [26]. But also in the present case, where the Crenel function appears suitable to describe the ordering, the early stage of ordering or the short-range order in the short-range ordered η phase may cause onset of ordering with q₁ and q₂ deviating from Eq. (5). During further improving the state of ordering, approaching order in accordance with Eq. (5), would require longer-range redistribution of Cu(2) atoms in order to accomplish the required changes of q₁ and q₃. Hence, it is expected that the degree of ordering will first improve with “somewhat incorrect” q₁ and q₃ values, while only a very long-time annealing might bring about changes in q₁ and q₃ toward values in accordance with Eq. (5). Annealing experiments at different temperatures and different durations and analysis of changes in q₁ and q₃ with the annealing program, may yield information about these ideas. Note, however, that according to the current ideas about the phase diagram [18] (see Figure 2), the ηʺ phase is in any case metastable with respect to unmixing, which might obstruct an analysis of structure changes of the ordering of the ηʺ phase by means of long-term annealing experiments.

In any case, disregarding the disturbances of the order due to the dislocation dipoles, the Cu(2) ordering in Figure 6 strongly resembles the ordering pattern of the ideal η⁴⁺¹ superstructure, which contains both pairs of ac edge-sharing Cu(2)Sn₅ trigonal bipyramids (Cu(2)₂Sn₈) and isolated Cu(2)Sn₅ trigonal bipyramids.

The refined parameters from Table 2 can be used to calculate interatomic distances, which are most conveniently depicted for an incommensurate structure as a function of the internal coordinate t given by t=x4−q⋅(T+r0), while T is a translation vector of the average structure and r₀ the position vector in the average structure. Figure 7 focusses on the distances from the Cu(2) atoms to other atoms. Especially, when comparing these distances with the “unmodulated” distances shown as dashed lines, the following can be concluded from Figure 7:

1. The Cu(2)–Sn distances get expanded, in particular, for the equatorial triangles surrounding the Cu(2) atoms in the a_av–b_av plane. This agrees with the trends described previously in similar Ni₂In/NiAs structures [13], [26], [27], [29], [30]. Minor t ranges of short Cu(2)–Sn distances are likely insignificant in view of the limited orders of satellite reflections visible for evaluation.

2. The Cu(2)–Cu(1) distances get somewhat contracted, which is also in agreement with previous observations [13], [26], [29].

3. Cu(2)–Cu(2) get strongly expanded. This is the distance between the two Cu(2) atoms in the centers of two ac edge-sharing Cu(2)Sn₅ trigonal bipyramids (Cu(2)₂Sn₈) as shown in Figure 5c. This distance increase is a consequence of the increase of the lengths of the Cu(2)–Sn distances for the Sn atoms bridging the two trigonal bipyramids.

Figure 7:

{two columns} Plot of the interatomic distances Cu(2)–Sn, Cu(2)–Cu(1) and Cu(2)–Cu(2) as full lines versus the internal coordinate t as results from the Rietveld refinement on the basis of PXRD data from ηʺ-Cu_1.235Sn. The distances only occur in a certain range of t, in which the Cu(2) sites are occupied. Since for Cu(2)–Cu(2) pairs both sites have to be occupied, the t range for which such pairs occur, is smaller than the ranges for Cu(2)–Sn and Cu(2)–Cu(1). Dashed lines are obtained from setting all displacement modulation parameters u (see Table 2) equal to zero.

4.3 Metrical distortion of the ηʺ phase

The Cu(2)–Cu(2) pairs from the pairs ac edge-sharing trigonal bipyramids Cu(2)₂Sn₈, which do not occur in the η′ superstructure, may be made responsible for the metrical distortion of the ηʺ-Cu_1.235Sn structure which has been mentioned in Section 3.2 and which has highlighted in Figure 4c. The type of distortion will now be worked out in terms of special types of strain tensors. In Ref. [16], it was shown how the symmetry breaking distortions of a superstructure can be expressed in terms of a symmetric strain tensor following the idea of the Aizu strain [31]. Such a strain tensor fulfills the symmetry restrictions of the crystal class of the analyzed crystal structure. Apart from a zero trace, which is generally required, further restrictions are imposed depending on the crystal class of the parent structure, with respect to which the distortion is considered. Upon describing a 2nd rank tensor like the Aizu strain tensor, it has to refer to a Cartesian coordinate system defined in a specific way with respect to the crystallographic coordinate system of the high-symmetry structure. In case of the monoclinic η′ superstructure derived from the hexagonal η, the Cartesian basis vectors were e₁ parallel to a_h + b_h, e₂ parallel to − a_h + b_h and e₃ parallel to c_h. For that the strain tensor has been determined on the basis of the lattice parameters determined for the η′ phase [16]:

Thereby, the chosen Cartesian basis vectors e₁–e₃ are also parallel with the basis vectors of the undistorted average unit cell used for the modulated structure description of the η′ and also compatible with the q vector listed in the lower part of Table 1.

Applying the same procedure to the refined cell parameters of the average structure of the ηʺ (see Table 2), one obtains (technically following the somewhat more compact but equivalent procedure described in Ref. [32]):

As already emphasized in Ref. [16], the components of εη→η′ are very small, which can explain the not directly detectable peak splitting in PXRD patterns of η′-Cu_1.20Sn, at least not in view of the resolution of the diffraction pattern, which largely corresponds to that in the current case. In comparison to this the 7 fold larger εm13 component of εη→η″ (which is simply ½cosβ_av) brings about a visible splitting of some reflections (see Figure 4c). Figure 8 compares the tensor surfaces of εη→η′ and εη→η″. The green lobes point into the direction of expansion approximately in the (+e₁−e₃) direction, which is about parallel to the Cu(2)−Cu(2) pairs in the modulated structure. Evidently these pairs and the ac edge-sharing trigonal bipyramids Cu(2)₂Sn₈ act as elastic dipoles [33] with very anisotropic space requirement. The density of such dipoles determines the overall extent of the symmetry breaking strain. In fact, checking the lattice parameters of the alloys showing diffraction patterns from Cu₆Sn₅ IMC with ηʺ phase ordering imply that the deviation from β_av decreases with increasing Sn content/decreasing δ.

Figure 8:

{single column} Tensor surfaces (green lobes: positive, red lobes: negative) depicting the direction dependence of symmetry-breaking strain for ηʺ-Cu_1.235Sn (Eq. (8), this work) and for η′-Cu_1.20Sn (Eq. (7), [16]), revealing, in view of the common scale (bottom), a much more pronounced distortion for the case of ηʺ-Cu_1.235Sn. That distortion is also visible by splitting of fundamental reflections in the powder diffraction pattern (see Figure 4c).