Pressure-induced pseudoatom bonding collapse and isosymmetric phase transition in Zr2Cu: First-principles predictions Jinliang Ning, Xinyu Zhang, Suhong Zhang, Na Sun, Limin Wang, Mingzhen Ma, and Riping Liu Citation: The Journal of Chemical Physics 139, 234504 (2013); doi: 10.1063/1.4846995 View online: http://dx.doi.org/10.1063/1.4846995 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/139/23?ver=pdfcov Published by the AIP Publishing
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THE JOURNAL OF CHEMICAL PHYSICS 139, 234504 (2013)
Pressure-induced pseudoatom bonding collapse and isosymmetric phase transition in Zr2 Cu: First-principles predictions Jinliang Ning, Xinyu Zhang,a) Suhong Zhang, Na Sun, Limin Wang, Mingzhen Ma, and Riping Liua) State Key Laboratory of Metastable Materials Science and Technology, Yanshan University, Qinhuangdao 066004, People’s Republic of China
(Received 10 August 2013; accepted 27 November 2013; published online 19 December 2013) The structural evolution of tetragonal Zr2 Cu has been investigated under high pressures up to 70 GPa by means of density functional theory. Our calculations predict a pressure-induced isosymmetric transition where the tetragonal symmetry (I4/mmm) is retained during the entire compression as well as decompression process while its axial ratio (c/a) undergoes a transition from ∼3.5 to ∼4.2 at around 35 GPa with a hysteresis width of about 4 GPa accompanied by an obvious volume collapse of 1.8% and anomalous elastic properties such as weak mechanical stability, dramatically high elastic anisotropy, and low Young’s modulus. Crystallographically, the tetragonal axial ratio shift renders this transition analogous to a simple bcc-to-fcc structural transition, which implies it might be densification-driven. Electronically, the ambient Zr2 Cu is uncovered with an intriguing pseudo BaFe2 As2 -type structure, which upon the phase transition undergoes an electron density topological change and collapses to an atomic-sandwich-like structure. The pseudo BaFe2 As2 -type structure is demonstrated to be shaped by hybridized dxz + yz electronic states below Fermi level, while the high pressure straight Zr-Zr bonding is accommodated by electronic states near Fermi level with dx2 − y2 dominant features. © 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4846995] I. INTRODUCTION
Intermetallics based on the “early” and “late” transition metal elements (ET-LT) exhibit numerous intriguing phenomena such as martensitic transformation and shape memory effect (B2 phases NiTi1 and ZrCu2 ), quasicrystal formation,3 hydrogen absorption capability,4, 5 superconductivity (C16 phases like Zr2 Co6 ), good glass forming ability, charge density waves,7 and so on. Zr2 Cu as a typical ET-LT intermetallic compound has attracted considerable scientific and industrial interest. It has been considered as potential fuel element cladding materials for nuclear power reactors and its anomalous oxidation behavior has been reported with a decrease of the oxygen consumption rate in the temperature range 890– 975 ◦ C and a Zr-preferred selective oxidation.8 As a potential hydrogen storage material, the hydrogenation behavior of Zr2 Cu has raised a lot of research interest.4, 9, 10 Similarly, Zr2 Cu also plays a critical role during the formation and devitrification processes of ZrCu-based metallic glasses since it is one of the main competing crystalline phases. In spite of its technological interest, the knowledge of polymorphic transformation in Zr2 Cu is of fundamental importance and remains incomplete. The prevailing crystal structure for Zr2 Cu is the MoSi2 type, with unit cell parameters a = b = 3.2204, and c = 11.1832 Å, space group I4/mmm, first determined by Nevitt.11 A metastable NiTi2 -type “big cube” Zr2 Cu is reported to form by ball milling and annealing, but only in one source.12 The amorphous state in Zr2 Cu alloy can be easily obtained by rapid quenching, and occasionally a) Authors to whom correspondence should be addressed. Electronic ad-
dresses: [email protected] and [email protected]. 0021-9606/2013/139(23)/234504/9/$30.00
also by ball milling13 and electron irradiation.14 A metastable fcc solid solution of Zr2 Cu can form during the crystallization of amorphous Zr2 Cu subject to high energy ball milling13 or electron irradiation.14 In very few sources,15 a low temperature structure different from the prevailing C11b structure is proposed. It is described to be a superstructure of the high temperature C11b type, but no detailed crystallographic data are given. The application of high pressure is another effective way to modify both electronic structures and atomic arrangements of materials. High pressure research on Zr2 Cu and other ET-LT intermetallics is fundamental both itself and to the understanding of high pressure behavior of ET-LT-based metallic glasses, typified by Zr41 Ti14 Cu12.5 Ni10 Be22.5 .16 It has been reported that high pressure has a complex effect on ZrTiCuNiBe metallic glasses and results in numerous phenomena such as pressure-tuned glass forming ability,17 primary precipitate phase, and precipitate sequence.18 As both the glass formation and devitrification are processes dominated by phase competing, detailed information about the pressure-driven phase transition of related competing crystalline phases is necessary. However, such investigations are still dormant. In this paper, we focus on the possibility of a special pressure-induced phase transition in Zr2 Cu when pondering deeply over the nature of its MoSi2 -type structure. This structure can be described as a superstructure of the bcc structure with three subcells stacked along [001] direction. On the basis of their axial ratios, MoSi2 -type phases can be classified into subgroups such as large groups (with rich members) with c/a ratios of either ∼2.45 or ∼3.45 and smaller ones (with poor members) with c/a ratios at 3.0 and 4.24.19 In general,
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both electronic structure and atomic packing play a role in determining the structure (mainly c/a) of these phases. At ambient conditions, Zr2 Cu takes an axial ratio of 3.473, corresponding to a distorted bcc superstructure, while another two phases with similar compositions, Ti2 Ag with c/a = 4.014 and CdTi2 with c/a = 4.684,20 are more like an fcc lattice. The intriguing subgrouping of MoSi2 -type phases and drastic c/a oscillation of the three above-mentioned C11b phases with similar compositions might imply the feasibility of drastic axial ratio shifts between different subgroups under certain conditions. Given the isostructural feature of such structural changes, they should be tagged as the “isosymmetric phase transitions (IPTs).” Though temperature21 or magnetic field22 is shown to be driving forces to induce the transition, most reported IPTs are induced by pressure, which in addition is more ready to theoretical calculations. By virtue of the plane wave pseudopotential method, we managed to examine our hypothesis and identify the pressure-induced isosymmetric phase transition in Zr2 Cu with an axial ratio shift from a subgroup (∼3.45) to another (∼4.24). IPTs are phenomena relatively exotic and rare due to their symmetry-retaining character. Such transitions are usually accompanied by discontinuities in pressure dependences of macroscopic properties, such as volume collapses, elastic anomalies, and magnetic transitions. For the case of IPT in Zr2 Cu, our subsequent calculations probing into the pressure dependences of structural, elastic, and electronic properties reveal that it is pressure-induced densification-driven, analogous to a bcc-to-fcc transition and results from the collapse of an intriguing pseudoatom bonding. II. COMPUTATION METHOD
The present calculations are implemented in the CASTEP code23 based on density functional theory (DFT) framework.24 The Vanderbilt ultrasoft pseudopotential25 with the cutoff energy 450 eV is employed to treat the ionelectron interaction and the original valence configurations are 3d10 4s1 and 4s2 4p6 4d2 5s2 for Cu and Zr, respectively. For the calculations of structural properties without external pressure, both the local density approximation (LDA)26 and the Perdew–Burke–Ernzerh form27 of the generalized gradient approximation (GGA) are adopted to describe the exchange and correlation potentials for electron–electron interaction. And the rest of the calculations are performed within GGA. The sampling over the Brillouin zone is treated by Monkhorst–Pack mesh28 with the k-points separation set as 0.025/Å. During the geometry optimization, the structure was relaxed within the Broyden–Fletcher–Goldfarb–Shanno minimization methods,29 until the average force on each atom is less than 0.01 eV/Å and the energy change is less than 5.0 × 10−6 eV/atom. The elastic coefficients are derived by applying two necessary strain patterns and calculating the resultant stress when optimizing the internal degrees of freedoms. Three negative and three positive amplitudes are applied for each strain pattern with the maximum strain value of 0.003. The convergence tolerance when optimizing the atomic internal freedoms is set as follows: maximum ionic displacement within 1 × 10−4 Å, ionic Hellmann–Feynman forces
J. Chem. Phys. 139, 234504 (2013)
within 0.002 eV/Å and difference on the total energy within 1 × 10−6 eV/atom. The Fermi surface calculations are carried out by means of the plane wave pseudopotential method, as implemented in the Quantum espresso package.30 The electron–electron interactions are also treated with the PBE form of GGA in ultrasoft pseudopotentials. The electronic wave function was expanded with a plane-wave basis set with a kinetic energy cutoff of 60 Ry. The augmentation of charges was expanded up to 480 Ry. A Gaussian smearing of 0.02 Ry has been applied. The lattice-dynamics are calculated using the density functional perturbation theory (DFPT) through the Quantum-ESPRESSO package.30
III. RESULTS AND DISCUSSIONS A. Structural evolution under pressure
We first examine the ground state structure of Zr2 Cu. The MoSi2 -type structure, which has been described as a superstructure of the bcc or CsCl structure with three subcells stacked along [001] direction, is illustrated by the inset of Fig. 1(b). The detailed geometry optimizing results, including Wyckoff positions of atoms and lattice parameters, are listed in Table I, showing good agreement with available experiments11 and previous calculations.31 Meanwhile, compared with LDA, GGA gives slightly larger lattice parameters, yet closer to the experimental results. Thus, we choose GGA for subsequent calculations. We then investigate the variation of the crystal geometry during both compression and decompression processes. The applied external hydrostatic pressure is increased from 0 to 70 GPa in discrete upward steps to simulate the compression process, and then reverts back to zero in discrete downward steps to model the decompression process. At every step, the crystal geometry obtained from the last step is fully optimized again and is used as the initial structure for the next compression (decompression) step. For the sake of studying phase transitions, symmetry constraints are not imposed. No noticeable symmetry deviation from the original I4/mmm space group occurs at any concerned pressure. So, at hydrostatic pressures (0–70 GPa) and low temperatures Zr2 Cu’s tetragonal symmetry can survive. The pressure evolution of the reduced lattice parameters (a/a0 , c/c0 ), axial ratio (c/a), and the unit-cell volume of tetragonal Zr2 Cu are illustrated in Fig. 1. A hysteresis loop about 4 GPa is pronounced in both Figs. 1(a) and 1(b). Though both lowand high-pressure phases are tetragonal, there is an evident volume change (1.8%) upon the phase transition. A volume discontinuity and hysteresis loop is a necessary feature for reversible thermodynamic first-order transitions. Thus, a firstorder isosymmetric phase transition in Zr2 Cu can now be determined. This IPT share several features of that in sillimanite reported by Oganov.32 It is reversible, first-order and exhibits negative linear compressibility along the c-axis. It also exhibits the “preparation” behavior that the lattice parameters of both low- and high-pressure phases tend to merge towards the transition. So our example is another case standing for the view that in cases where the symmetries of the phases are
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FIG. 2. Pressure dependence of elastic constants.
between phases with a distorted bcc lattice and a distorted fcc lattice. B. Pressure dependence of elastic properties
As an IPT is often accompanied or indicated by discontinuities or anomalies in the pressure dependence of elastic properties, the elastic parameters of Zr2 Cu as a function of pressure are calculated and discussed. The full set of elastic constants calculated in stress-strain method33 are shown in Fig. 2. Generally, it can be seen that all elastic constants increase with pressure, but the anomalous tendency is remarkable in the hysteresis region, in contrast to the relatively simple trends at low and high pressure regions. Additionally, according to the mechanical stability criteria for a tetragonal crystal,34, 35 FIG. 1. Pressure evolution of (a) the reduced lattice parameters and (b) axial ratio and (c) the unit-cell volume of tetragonal Zr2 Cu. (b) Inset is the plot of unit cell of Zr2 Cu.
C44 > 0, C66 > 0, C11 > |C12 | , and
identical, both structures show a tendency to approach each other in the vicinity of the transition.32 Furthermore, we highlight the remarkable structural character of this transition. It is a transition between phases with low c/a ratio (∼3.45) and high c/a ratio (∼4.24) at their respective pressure ranges. Considering the bcc superstructure nature of the MoSi2 -type structure, it is natural to compare this isosymmetric phase transition to a simple bcc-to-fcc structural phase transition. It can be looked on as a transition
F = C11 + C12 −
2 2C13 > 0. C33
(1)
Zr2 Cu is stable in the concerned pressure range (0–70 GPa), which is confirmed by the calculated phonon dispersions in Fig. 3 where no imaginary frequencies appear in any branch of the dispersions. However, note that the F index is extremely low in the hysteresis region (see Fig. 5), which indicates its weak stability against tetragonal distortion and is consistent with the abrupt change of axial ratio. Then some derivative elastic parameters under compression are obtained, including the single crystal Young’s modulus E (reflecting the resistance of materials against uniaxial
TABLE I. Crystallographic data of Zr2 Cu. Unit cell internal parameters Type Present (GGA) Present (LDA) Previous calc.a Expt.b a b
Zr(4e) 0 0 0 0
0 0 0 0
0.346 0.347 0.348 0.34
Lattice parameters (Å) Cu(2a) 0 0 0 0
0 0 0 0
0 0 0 0
a
c
3.229 3.153 3.2224 3.2204
11.193 10.930 11.094 11.1832
Reference 31. Reference 11.
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FIG. 3. Phonon dispersion for Zr2 Cu at 20 (a), 35 (b), 50 (d) GPa during compression and 35 GPa (c) during decompression. The notation of the highsymmetry points is as follows: Z (0.5, 0.5, −0.5), (0, 0, 0), X (0, 0, 0.5), P (0.25, 0.25, 0.25), N (0, 0.5, 0). The phonon frequencies are in cm−1 .
tensions), the polycrystalline bulk modulus B (a measure of the resistance of a material against volume change under hydrostatic pressure), and the shear modulus G (describing the resistance to shape change caused by a shearing force) on the basis of the Voigt-Reuss-Hill approximation method.36 Also, the universal elastic anisotropy index AU 37 is obtained by AU = 5
GV BV + − 6. GR BR
(2)
As shown in the upper panel of Fig. 4, inside the hysteresis region, we have observed large discrepancy between the Voigt and Reuss estimates of shear modulus and bulk modulus, where both BR and GR are much lower than BV and GV , re-
FIG. 4. Upper panel: Pressure dependence of the Bulk modulus B and shear modulus G in respective Voigt and Reuss forms. Lower panel: Pressure dependence of the universal elastic anisotropy index AU derived from polycrystalline elastic moduli.
FIG. 5. The pressure dependence of Young’s moduli Ex = Ey , Ez and their ratio Ez /Ex together with the elastic stability factor F.
spectively. This discrepancy results from the high single crystal elastic anisotropy which can be measured by the index AU , as shown in the lower panel of Fig. 4. The calculated Young’s moduli along a-axis (Ex ) and c-axis (Ez ) are plotted together with the F index in Fig. 5. It is notable that, like F, they are also anomalously low in the hysteresis region. This interesting synchronization phenomenon is not a coincidence and will be further discussed later. In addition, their ratio Ez /Ex , much lower than 1 in the hysteresis region, also confirms the high elastic anisotropy discussed above by the index AU . To get further understanding of the phase transition, we extend the application of volume-conserving Bain path to Zr2 Cu. Though typically applied to simple elemental substances38, 39 along tetragonal distortion path connecting bcc and fcc structures, the Bain path model will still be powerful applied on tetragonal complexes or bcc superstructures, as in this work and other reported cases.40 A plot of energy variation with volume-conserving axial ratio, as shown in Fig. 6, may provide a clear picture of the peculiar structural and elastic response of Zr2 Cu to compression.
FIG. 6. The energy variation of Zr2 Cu as a function of axial ratio c/a for volumes: V0, V30, V35, V40, V60. Vx denotes the volume of the unit cell geometry optimized by CASTEP at a hydrostatic pressure x GPa. The energy of equilibrium Zr2 Cu is set as the energy reference.
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The reliability of the geometry optimizing calculations under pressure implemented in the CASTEP code is validated to some extent by that for each of the five energy-c/a curves, the energy minimum falls at the very axial ratio which belongs to the CASTEP-optimized geometry with whose unit cell volume the corresponding Bain path is developed. The c/a ratio at the energy minimum increases with pressure, as demonstrated by the red dotted curve linking each energy minimum marked by red stars, reproducing the trend in Fig. 1(b). It manifests itself that low compression (0–30 GPa) favors a relatively bcc-like structure with c/a ratio of ∼3.5 and high compression (40–70 GPa) prefers a relatively fcc-like structure with c/a ratio of ∼4.2. Frequently, simple metals and some transition metals have a pronounced double-well feature (the “well” here refers to the shape of the energy-c/a curve around an energy minimum) in their Bain path.38 In contrast, the Bain path for Zr2 Cu takes a single-well feature for low and high compression states, and the “well” is deeper former than latter. The V0 Bain path of Zr2 Cu is similar to that of FeAl, CoAl, and NiAl and different from that of TiAl, VAl, and CrAl.40 More notable is the becoming not pronounced of energy minimum for intermediately compressed Zr2 Cu. Instead, a rather wide and plain basin-like bottom is in sight. This anomalous feature relates to the low mechanical stability and low Young’s modulus of Zr2 Cu within the hysteresis region, since intuitively there is almost no energy barrier for tetragonal deformation in terms of the basin-like feature. Now we discuss in detail the correlation between the basin-like feature of Bain path and the low F and low Young’s modulus. Marcus et al. states that F is given by the curvature at the minima of epitaxial Bain path,41 1 d 2 E EBP (a) F = , (3) c0 da 2 a0 and thus is always positive dictated by the mathematical nature of the minima in a curve. As to this work, fairly good correspondence is obtained between the curvatures at minima of the Bain path and F values. It can be easily seen that, for the case of both low and high compressed states, the pronounced well-like feature gives a high curvature around minima of the Bain path and thus leads to a high value of F; for the case of intermediate pressure range, the basin-like feature of Bain path means low curvature around the minima and hence a low F value. Note that the Bain path used here is a volumeconserving one for simplicity. So, we have explicitly revealed the origin of the low F values at the hysteresis region. The same argument goes for the low Young’s modulus since they are calculated by 2 /C33 (C11 − C12 ) C11 + C12 − 2C13 (4a) Ex = Ey = 2 C11 − C13 /C33 and
2 /C33 C11 + C12 − 2C13 , Ez = (C11 + C12 ) /C33
(4b)
and thus are dominated by F, as shown in Fig. 5. Therefore, at the hysteresis region, Zr2 Cu shows an anomalous softening in stark contrast to low (0–30 GPa) and high (40–70 GPa) pressure regions, including weak mechan-
ical stability and low Young’s modulus (both are illuminated by the basin-like feature of Bain path), and dramatically high elastic anisotropy. These findings consolidate our original view that Zr2 Cu undergoes a pressure-induced bcc-to-fcc-like structural phase transition through a series intermediate transition states. The only difference compared with a real bccto-fcc structural phase transition, which can be accomplished through intermediate tetragonal states as demonstrated by the Bain path model, is that the parent phase in this case takes a distorted bcc lattice but not a strict one and the product phase a distorted fcc structure. Both the parent and product phases are tetragonal, the same as the intermediate states. These facts degrade the group-subgroup relations that exist in an exact bcctetragonal-fcc structural phase transition, and thus leaving the phase transition isosymmetric. IPTs can originate from structural changes, but the ones with such close connections to a simple structural phase transition have never been reported. Although similar pressure-induced tetragonal axial ratio fluctuations occur in In and its alloys,42 they are anti-isostructural phase transitions between structures with opposite signs of tetragonal distortion from an fcc structure, different from the case in Zr2 Cu. We might term such IPTs like in Zr2 Cu as quasistructural phase transitions (QPTs). QPTs may occur in systems for which the parent phase and product phase share the symmetry of intermediate states but can be connected to different high-symmetry lattices, i.e., associated with degraded group-subgroup relations between the isosymmetric parent (and product) phase and intermediate phase.
C. Pressure evolution of electronic properties
As the macroscopic structural and mechanical properties reflect the microscopic bonding structure of materials, to obtain further insight into the relationship between them in Zr2 Cu and shed light on the underling mechanism of the IPT, the electronic structures and their pressure dependences are discussed as follows. Figure 7 shows the calculated total charge density distribution of the bcc-like phase (at 0 GPa) and fcc-like phase (at 60 GPa) during compression, by using contour plots in (100) and (110) planes and a schematic picture of the bonding structure with sticks representing the bond paths. The HohenbergKohn theorem24 states that a system’s ground-state properties are a consequence of its charge density, which can be denoted as a scalar field ρ(r). From this theorem, the relationship between the topology of the charge density and many properties of materials can be rationalized. The topology of this scalar field is given in terms of its critical points (CPs), which are the zeros of the gradient of this field. In Fig. 7, each bond path is given by following the gradient of the charge density from a bond critical point (BCP) toward the bound atoms. A BCP is a (3, −1) CP that has two negative and one positive curvatures in three orthogonal directions and thus usually guarantees a ridge of maximum charge density connecting two nuclei (see details in Ref. 43). For the bcc-like phase, the most interesting character we revealed here is the pseudo BaFe2 As2 -type structure with the same symmetry I4/mmm. There is a considerable charge accumulation in the interstitial of each Zr tetrahedron,
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FIG. 7. The contour plots of total charge density in (100) and (110) planes and schematic picture of the bonding structure of Zr2 Cu at 0 GPa (a) and at 60 GPa (b). The small dark blue balls in the schematic bonding picture of (a) denote the pseudoatoms.
indicated by the local maximum of charge density in the (100) plane contour plot. This charge accumulation serves just like a pseudoatom which is bound to four nearest Zr atoms, similar to the situation in hcp Zr43 and compressed fcc Ca.44 In addition, there is no straight Zr-Zr bonding CPs, but (3, +1) CPs at the boundaries of nearest Zr atoms. This bonding picture renders ambient Zr2 Cu an interesting pseudo BaFe2 As2 -type structure where the tetrahedral interstitial charge density maxima serve as the Fe-position pseudoatoms. The BaFe2 As2 type structure is famous in iron-based superconductors and is also adopted by A2 BH2 hydrides (A = Zr, Ti, Hf; B = Cu, Pd) which are considered as potential hydrogen storage materials. With external hydrostatic pressure increasing towards the IPT pressure, each pseudoatom separates into two which gradually fade out when moving up and down along c-axis toward their respective adjacent Zr (001) layers. In the meantime, the two BCPs in the illustrated (100) plane between the pseudoatom and nearest Zr atoms will also move toward the midpoint of the two Zr atoms until they meet each other near that point. After the transition, each Zr atom now is bound to four nearest Zr atoms in the same (001) plane and four nearest Cu atoms in the adjacent, through approximately straight bonds, as shown in Fig. 7(b). And note that the interaction between interlayer Zr atoms becomes even weaker. Therefore, the exotic pseudo BaFe2 As2 -type structure of ambient Zr2 Cu collapses upon the IPT to a layered structure formed by the periodic stacking of Zr-Cu-Zr atomic sandwiches along caxis. An IPT accompanied by similar topological changes in the charge density was reported in fcc Ca44 but in a somehow opposite way. For the low-pressure phase, the nearest neighboring Ca atoms are connected through bifurcated bond paths with deep minima in the octahedral holes, while for the highpressure one, the atoms bond through non-nuclear maxima in the octahedral holes. In addition, the above electron density topological analysis agrees, in principle, but not in detail, with our previous viewpoint that the IPT in Zr2 Cu is analogous to a simple bccto-fcc structural phase transition. By the Bain path model, a tetragonal axial ratio of about 3 makes the low pressure phase a bcc superstructure, while 4.24 the high pressure phase an
fcc. From the perspective of bonding structure, in the low pressure phase, each atom has eight coordinates (Cu has 8 nearest Zr neighbors and Zr is bound to 4 Cu atoms and 4 pseudoatoms) and is relatively open compared with the high pressure phase in terms of the notable volume collapse upon the IPT and the fact that in the high pressure phase each Cu gets additional 4 Cu coordinates. The special open structure of ambient Zr2 Cu is accommodated by its unique bonding structure. The incipient change of the bonding structure of Zr2 Cu should be the origin of the high anisotropy of compressibility and the negative linear compressibility along the c-axis in the hysteresis region. The formation of the Zr-Zr and Cu-Cu straight bonding in (001) plane and the collapse of the pseudoatom bonding soften C11 and reinforce C33 , as shown in Fig. 2, and thus result in the collapse of the a-axis and the elongation of the c-axis. Similar negative linear compressibility and underlying origin in bonding changes are also reported in AuTe2 45 and Al2 SiO5 .32 To further probe into the bonding evolution of Zr2 Cu in real space, especially the collapse of the pseudoatom bonding, we investigate the evolution of the electronic structure in k space. We calculated the band-specific contribution to the charge density distribution, i.e., the orbitals, to find which band(s) contributes to the pseudoatom bonding character and its collapse. In CASTEP, the calculated “orbital” is the square of the absolute value of the wavefunction for a given electronic band, summed over all k-points. A band is defined by the position of its eigenvalue in the ordered list of electronic energies at each k-point. For example, orbital (or band) number 3 is obtained by summation over all k-points of the wavefunctions with the third lowest energy. The sum of the “orbital densities” obtained in this way gives the total electron density of the crystal. We find that mainly 6 bands, band 31 (B31) to B36 highlighted in Fig. 8, are related to the phase transition. The calculated orbitals (Fig. 9) indicate that the pseudoatom of ambient Zr2 Cu is mainly contributed from B31 and B32, of which the charge density distributions have the same pseudoatom bonding character. And the straight bonding character of the high-pressure phase is mainly contributed from B35 and B36. Upon the IPT, both B31 and B32 lose their
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FIG. 8. Band structure of Zr2 Cu at 0 GPa (a) and 60 GPa (b). The six bold and colored bands are bands 31 to 36 from lower to upper. Horizontal dashed line indicates the Fermi level. The notation of the high-symmetry points is as follows: Z (0, 0, 0.5), A (0.5, 0.5, 0.5), M (0.5, 0.5, 0), G (0, 0, 0), R (0, 0.5, 0.5), X (0, 0.5, 0). FIG. 10. Partial DOS of Zr in Zr2 Cu at zero pressure (a) and 60 GPa (b). Vertical dashed line indicates the Fermi level.
pseudoatom characters, at least partially. For B31, there still exists a tetrahedral interstitial accumulation region of charge density, but it becomes spatially extended especially along caxis, which makes it not so typical as a pseudoatom like in the ambient phase. While B32 loses its original character completely and puts on a straight bonding character. So, it seems that the ambient pseudoatom bonding is shaped by lower level bands below the Fermi level (B31, −2.024–−1.002 eV; B32, −1.757–−0.845 eV), while the straight Zr-Zr bonding character is accommodated by upper level bands crossing the Fermi level (B35, −1.078–0.745 eV; B36, −0.912–1.066 eV). This is confirmed by our density of states (DOS) calculations (Fig. 10). We focus on the partial DOS of the Zr sites in Zr2 Cu in that the pseudoatom character and its evolution is related to the interaction between Zr atoms. The DOS of both ambient and high pressure Zr2 Cu is dominated by d-states while s- and p-states contribute a small part. Compared to the high pressure phase, ambient Zr2 Cu takes a highly structured DOS featured by sharp bonding and antibonding peaks and a prominent pseudogap. The bonding peaks located around −0.73 and −1.23 eV are dominated by the hybridized dxz + yz states with all the other states. The
FIG. 9. Isosurface plot of the band-specific charge density (orbitals) of band 31–36 for Zr2 Cu at 0 GPa (left side, isosurface value 0.033 e/Å3 ) and 60 GPa (right side, isosurface value 0.035 e/Å3 ).
latter peak is from B31 and B32, which are responsible for the pseudoatom character. This is consistent with the fact that the dxz + yz states favor the pseudoatom bonding by their orbital spatial distribution characteristics. In addition, the innerhybridized-peak-stabilized character of the pseudoatom bonding renders it a considerable covalent nature. Upon the phase transition, the highly structured DOS collapses to a broad featureless one without prominent peaks and pseudogap, which might be responsible for the collapse of pseudoatom character of B31 and B32. Furthermore, the situation near the Fermi level changes dramatically. For ambient DOS, the Fermi level is dominated by dxz + yz states with other states taking almost the same small values, while for the high pressure DOS, the Fermi level is instead dominated by the dx2 − y2 states and gets a higher filling density. The filling of these dx2 − y2 states favors the straight bonding by their orbital spatial distribution, corresponding to the formation of the straight Zr-Zr bonding character of B35 and B36. The increased filling of the DOS near the Fermi level results from the increased crossing Fermi level of related bands. This increased interaction will also be manifested in the Fermi surface. Following a compression, the electronic band extremum may shift through the Fermi level. In this case, new elements appear on the Fermi surface and change its topology. Such a topology change, referred as electronic topological transition (ETT), can lead to anomalies in transport, thermodynamic, and vibrational properties, as well as in the lattice structure, as has been intensively studied in IPTs of hcp metals like Zn,46 Cd,47 Os,48 etc. As shown in Fig. 11, for Zr2 Cu, apart from B35 and B36 which contribute most to the Fermi surface, B37 also contributes a small part with small electron pockets. Upon the phase transition B37 gets completely above the Fermi level and results in an ETT. The interaction of B35 and B36 with the Fermi level is relatively complex, which results in numerous ETTs during compression. The most notable is that a pipe-like piece develops
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cumulating process of valence electrons. To the best of our knowledge, this is the first time an IPT is reported to be associated with a pressure-induced collapse of interstitial-electron bonding, in a way opposite to the aforementioned ones. IV. CONCLUSION
FIG. 11. Fermi surface for Zr2 Cu under pressure. From top to bottom: band 35, band 36, band 37. From left to right: 0 GPa, 30 GPa, 40 GPa, and 60 GPa.
around the zone central axis in both bands at about 30 GPa (a little lower than 30 GPa for B35 and higher for B36). It is an “electron pipe” corresponding to the bowl-like arc right below Fermi level around the GZ path in Fig. 8. The emergence and extension of these new pieces of Fermi surface, corresponding to the extra filling states at Fermi level in the DOS plots, stabilize the high pressure phase and destabilize the low pressure phase. So does the fading away of originally existing pieces. It should be mentioned that IPTs are relatively rare and exotic phenomena hence have gained considerable interest and an increasing number of cases of IPTs are emerging. An IPT can occur purely due to subtle changes of electronic structures. A well known example is the case in Ce where the variable chemical valence as a result of the shift between localized and itinerant states in 4f electrons is responsible for the IPT with a large volume collapse of 15%.49 While in many cases, notable symmetry-retaining structural changes are involved such as reorientation of polyhedrons50 and large anisotropy in compressibility45 for sake of finding the optimal atomic packing with respect to pressure-induced densification. Such transitions are often accompanied by changes in bonding structures. IPTs with novel unique features are also emerging, such as the cases reported in albite NaAlSi3 O8 51 and Al-rich NiAl alloys,52 which are associated with the change of the ordering of atoms or defects. Since it often manifests subtle and anomalous changes in electronic structures and hence in physical and chemical properties of materials, IPTs are an intriguing topic in condensed matter physics and the understanding of it is of significant scientific importance to itself and other fields. For instance, the polyamorphic transition in CeAl53 and CaAl54 metallic glasses are approved to be dominated by their components Ce and Ca, respectively, both of which undergo a pressure-induced IPT with a large volume collapse. The IPT in Zr2 Cu shows some special characteristics. It is a transition between phases with axial ratios marking different subgroups of MoSi2 -type phases and shows a remarkable analogy to a simple bcc-to-fcc structural phase transition. The transition is associated with a pressure-induced collapse of pseudoatom bonding of the ambient phase and the formation of the atomic-sandwiches-like structure of high pressure phase. Similar pressure-driven electron density topological changes have been reported for IPTs in fcc Ca44 and Ba8 Si46 clathrates55 and for high-pressure structural phase transitions in Al,56 Si,57 alkali metals (Li,58 Na,59 K60 ), and alkali earth metal Ca.61 But these are all related to an interstitially ac-
In conclusion, using first-principles calculations, we predict a pressure-induced reversible, first-order, and bcc-tofcc analogous isosymmetric transition in tetragonal Zr2 Cu with its axial ratio increasing remarkably from ∼3.5 to ∼4.2 through a 4 GPa intermediate hysteresis region. The bcc-like and fcc-like features of Zr2 Cu are well characterized by the quite distinct trends of pressure dependences of numerous properties at low and high pressure ranges. Inside the hysteresis region, Zr2 Cu exhibits anomalous softening properties including low mechanical stability and Young’s modulus, both well characterized by the striking basin-like feature of Bain path, and dramatically high elastic anisotropy. The transition is densification-driven in terms of its bcc-to-fcc-like character. Electron density distribution illustrates the pseudo BaFe2 As2 -type structure of low pressure Zr2 Cu, where the tetrahedral interstitial electron density maxima serve as the Fe-position pseudoatoms. This special structure upon the IPT undergoes an electron density topological change and collapses to an atomic-sandwich-like structure. A comprehensive analysis based on electronic band, band-specific contribution to the charge density (orbitals) and density of states reveals the origin of the intriguing bonding structure and its pressure evolution. The pseudo BaFe2 As2 -type structure is demonstrated to be shaped by electronic states below Fermi level with obvious dxz + yz nature, while the high pressure straight Zr-Zr bonding is accommodated by states around Fermi level with dx2 − y2 dominant features. The Fermi surface manifestation of the IPT and the bonding changes is also demonstrated. Besides the main subject of pressure-induced IPT and bonding structure evolution in Zr2 Cu in present work, we sense that the special pseudo BaFe2 As2 -type bonding structure revealed here might be responsible for its intriguing properties like good hydrogen storage ability and exotic oxidation behaviors, which needs further theoretical and experimental investigations. Tetragonal Zr2 Cu is stable under the whole concerned pressure range according to both elastic stability criterion and phonon dispersion analysis. However, this IPT could be metastable and it is possible that Zr2 Cu transforms to other structures with a symmetry change, which needs further clarifying work such as high pressure tests using x-ray diffraction measurements in diamond anvil cells like in previously reported IPT cases.62, 63 Our work also implies the possibility of occurrence of similar IPTs in other MoSi2 -type phases with a shift of axial ratio marking different subgroups. On the other hand, it is reported recently that an IPT occurs in Sb2 O3 64 while does not in its analogue As2 O3 . So, it is doubtful whether an IPT occurs in tetragonal Zr2 Cu’s analogues like Zr2 Pd, Ti2 Cd, Ti2 Pd, and so on. Further, similar IPTs may also occur in other types of intermetallics associated with degraded group-subgroup relations and similar theoretical and experimental investigations are needed.
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ACKNOWLEDGMENTS
This work was supported by the NBRPC (Grant No. 2010CB731600), National Science Foundation of China (NSFC) (Grant Nos. 51171160/51002130/51171163/ 51271161/51071138/51131002). 1 N.
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THE JOURNAL OF CHEMICAL PHYSICS 139, 234504 (2013)
Pressure-induced pseudoatom bonding collapse and isosymmetric phase transition in Zr2 Cu: First-principles predictions Jinliang Ning, Xinyu Zhang,a) Suhong Zhang, Na Sun, Limin Wang, Mingzhen Ma, and Riping Liua) State Key Laboratory of Metastable Materials Science and Technology, Yanshan University, Qinhuangdao 066004, People’s Republic of China
(Received 10 August 2013; accepted 27 November 2013; published online 19 December 2013) The structural evolution of tetragonal Zr2 Cu has been investigated under high pressures up to 70 GPa by means of density functional theory. Our calculations predict a pressure-induced isosymmetric transition where the tetragonal symmetry (I4/mmm) is retained during the entire compression as well as decompression process while its axial ratio (c/a) undergoes a transition from ∼3.5 to ∼4.2 at around 35 GPa with a hysteresis width of about 4 GPa accompanied by an obvious volume collapse of 1.8% and anomalous elastic properties such as weak mechanical stability, dramatically high elastic anisotropy, and low Young’s modulus. Crystallographically, the tetragonal axial ratio shift renders this transition analogous to a simple bcc-to-fcc structural transition, which implies it might be densification-driven. Electronically, the ambient Zr2 Cu is uncovered with an intriguing pseudo BaFe2 As2 -type structure, which upon the phase transition undergoes an electron density topological change and collapses to an atomic-sandwich-like structure. The pseudo BaFe2 As2 -type structure is demonstrated to be shaped by hybridized dxz + yz electronic states below Fermi level, while the high pressure straight Zr-Zr bonding is accommodated by electronic states near Fermi level with dx2 − y2 dominant features. © 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4846995] I. INTRODUCTION
Intermetallics based on the “early” and “late” transition metal elements (ET-LT) exhibit numerous intriguing phenomena such as martensitic transformation and shape memory effect (B2 phases NiTi1 and ZrCu2 ), quasicrystal formation,3 hydrogen absorption capability,4, 5 superconductivity (C16 phases like Zr2 Co6 ), good glass forming ability, charge density waves,7 and so on. Zr2 Cu as a typical ET-LT intermetallic compound has attracted considerable scientific and industrial interest. It has been considered as potential fuel element cladding materials for nuclear power reactors and its anomalous oxidation behavior has been reported with a decrease of the oxygen consumption rate in the temperature range 890– 975 ◦ C and a Zr-preferred selective oxidation.8 As a potential hydrogen storage material, the hydrogenation behavior of Zr2 Cu has raised a lot of research interest.4, 9, 10 Similarly, Zr2 Cu also plays a critical role during the formation and devitrification processes of ZrCu-based metallic glasses since it is one of the main competing crystalline phases. In spite of its technological interest, the knowledge of polymorphic transformation in Zr2 Cu is of fundamental importance and remains incomplete. The prevailing crystal structure for Zr2 Cu is the MoSi2 type, with unit cell parameters a = b = 3.2204, and c = 11.1832 Å, space group I4/mmm, first determined by Nevitt.11 A metastable NiTi2 -type “big cube” Zr2 Cu is reported to form by ball milling and annealing, but only in one source.12 The amorphous state in Zr2 Cu alloy can be easily obtained by rapid quenching, and occasionally a) Authors to whom correspondence should be addressed. Electronic ad-
dresses: [email protected] and [email protected]. 0021-9606/2013/139(23)/234504/9/$30.00
also by ball milling13 and electron irradiation.14 A metastable fcc solid solution of Zr2 Cu can form during the crystallization of amorphous Zr2 Cu subject to high energy ball milling13 or electron irradiation.14 In very few sources,15 a low temperature structure different from the prevailing C11b structure is proposed. It is described to be a superstructure of the high temperature C11b type, but no detailed crystallographic data are given. The application of high pressure is another effective way to modify both electronic structures and atomic arrangements of materials. High pressure research on Zr2 Cu and other ET-LT intermetallics is fundamental both itself and to the understanding of high pressure behavior of ET-LT-based metallic glasses, typified by Zr41 Ti14 Cu12.5 Ni10 Be22.5 .16 It has been reported that high pressure has a complex effect on ZrTiCuNiBe metallic glasses and results in numerous phenomena such as pressure-tuned glass forming ability,17 primary precipitate phase, and precipitate sequence.18 As both the glass formation and devitrification are processes dominated by phase competing, detailed information about the pressure-driven phase transition of related competing crystalline phases is necessary. However, such investigations are still dormant. In this paper, we focus on the possibility of a special pressure-induced phase transition in Zr2 Cu when pondering deeply over the nature of its MoSi2 -type structure. This structure can be described as a superstructure of the bcc structure with three subcells stacked along [001] direction. On the basis of their axial ratios, MoSi2 -type phases can be classified into subgroups such as large groups (with rich members) with c/a ratios of either ∼2.45 or ∼3.45 and smaller ones (with poor members) with c/a ratios at 3.0 and 4.24.19 In general,
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both electronic structure and atomic packing play a role in determining the structure (mainly c/a) of these phases. At ambient conditions, Zr2 Cu takes an axial ratio of 3.473, corresponding to a distorted bcc superstructure, while another two phases with similar compositions, Ti2 Ag with c/a = 4.014 and CdTi2 with c/a = 4.684,20 are more like an fcc lattice. The intriguing subgrouping of MoSi2 -type phases and drastic c/a oscillation of the three above-mentioned C11b phases with similar compositions might imply the feasibility of drastic axial ratio shifts between different subgroups under certain conditions. Given the isostructural feature of such structural changes, they should be tagged as the “isosymmetric phase transitions (IPTs).” Though temperature21 or magnetic field22 is shown to be driving forces to induce the transition, most reported IPTs are induced by pressure, which in addition is more ready to theoretical calculations. By virtue of the plane wave pseudopotential method, we managed to examine our hypothesis and identify the pressure-induced isosymmetric phase transition in Zr2 Cu with an axial ratio shift from a subgroup (∼3.45) to another (∼4.24). IPTs are phenomena relatively exotic and rare due to their symmetry-retaining character. Such transitions are usually accompanied by discontinuities in pressure dependences of macroscopic properties, such as volume collapses, elastic anomalies, and magnetic transitions. For the case of IPT in Zr2 Cu, our subsequent calculations probing into the pressure dependences of structural, elastic, and electronic properties reveal that it is pressure-induced densification-driven, analogous to a bcc-to-fcc transition and results from the collapse of an intriguing pseudoatom bonding. II. COMPUTATION METHOD
The present calculations are implemented in the CASTEP code23 based on density functional theory (DFT) framework.24 The Vanderbilt ultrasoft pseudopotential25 with the cutoff energy 450 eV is employed to treat the ionelectron interaction and the original valence configurations are 3d10 4s1 and 4s2 4p6 4d2 5s2 for Cu and Zr, respectively. For the calculations of structural properties without external pressure, both the local density approximation (LDA)26 and the Perdew–Burke–Ernzerh form27 of the generalized gradient approximation (GGA) are adopted to describe the exchange and correlation potentials for electron–electron interaction. And the rest of the calculations are performed within GGA. The sampling over the Brillouin zone is treated by Monkhorst–Pack mesh28 with the k-points separation set as 0.025/Å. During the geometry optimization, the structure was relaxed within the Broyden–Fletcher–Goldfarb–Shanno minimization methods,29 until the average force on each atom is less than 0.01 eV/Å and the energy change is less than 5.0 × 10−6 eV/atom. The elastic coefficients are derived by applying two necessary strain patterns and calculating the resultant stress when optimizing the internal degrees of freedoms. Three negative and three positive amplitudes are applied for each strain pattern with the maximum strain value of 0.003. The convergence tolerance when optimizing the atomic internal freedoms is set as follows: maximum ionic displacement within 1 × 10−4 Å, ionic Hellmann–Feynman forces
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within 0.002 eV/Å and difference on the total energy within 1 × 10−6 eV/atom. The Fermi surface calculations are carried out by means of the plane wave pseudopotential method, as implemented in the Quantum espresso package.30 The electron–electron interactions are also treated with the PBE form of GGA in ultrasoft pseudopotentials. The electronic wave function was expanded with a plane-wave basis set with a kinetic energy cutoff of 60 Ry. The augmentation of charges was expanded up to 480 Ry. A Gaussian smearing of 0.02 Ry has been applied. The lattice-dynamics are calculated using the density functional perturbation theory (DFPT) through the Quantum-ESPRESSO package.30
III. RESULTS AND DISCUSSIONS A. Structural evolution under pressure
We first examine the ground state structure of Zr2 Cu. The MoSi2 -type structure, which has been described as a superstructure of the bcc or CsCl structure with three subcells stacked along [001] direction, is illustrated by the inset of Fig. 1(b). The detailed geometry optimizing results, including Wyckoff positions of atoms and lattice parameters, are listed in Table I, showing good agreement with available experiments11 and previous calculations.31 Meanwhile, compared with LDA, GGA gives slightly larger lattice parameters, yet closer to the experimental results. Thus, we choose GGA for subsequent calculations. We then investigate the variation of the crystal geometry during both compression and decompression processes. The applied external hydrostatic pressure is increased from 0 to 70 GPa in discrete upward steps to simulate the compression process, and then reverts back to zero in discrete downward steps to model the decompression process. At every step, the crystal geometry obtained from the last step is fully optimized again and is used as the initial structure for the next compression (decompression) step. For the sake of studying phase transitions, symmetry constraints are not imposed. No noticeable symmetry deviation from the original I4/mmm space group occurs at any concerned pressure. So, at hydrostatic pressures (0–70 GPa) and low temperatures Zr2 Cu’s tetragonal symmetry can survive. The pressure evolution of the reduced lattice parameters (a/a0 , c/c0 ), axial ratio (c/a), and the unit-cell volume of tetragonal Zr2 Cu are illustrated in Fig. 1. A hysteresis loop about 4 GPa is pronounced in both Figs. 1(a) and 1(b). Though both lowand high-pressure phases are tetragonal, there is an evident volume change (1.8%) upon the phase transition. A volume discontinuity and hysteresis loop is a necessary feature for reversible thermodynamic first-order transitions. Thus, a firstorder isosymmetric phase transition in Zr2 Cu can now be determined. This IPT share several features of that in sillimanite reported by Oganov.32 It is reversible, first-order and exhibits negative linear compressibility along the c-axis. It also exhibits the “preparation” behavior that the lattice parameters of both low- and high-pressure phases tend to merge towards the transition. So our example is another case standing for the view that in cases where the symmetries of the phases are
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FIG. 2. Pressure dependence of elastic constants.
between phases with a distorted bcc lattice and a distorted fcc lattice. B. Pressure dependence of elastic properties
As an IPT is often accompanied or indicated by discontinuities or anomalies in the pressure dependence of elastic properties, the elastic parameters of Zr2 Cu as a function of pressure are calculated and discussed. The full set of elastic constants calculated in stress-strain method33 are shown in Fig. 2. Generally, it can be seen that all elastic constants increase with pressure, but the anomalous tendency is remarkable in the hysteresis region, in contrast to the relatively simple trends at low and high pressure regions. Additionally, according to the mechanical stability criteria for a tetragonal crystal,34, 35 FIG. 1. Pressure evolution of (a) the reduced lattice parameters and (b) axial ratio and (c) the unit-cell volume of tetragonal Zr2 Cu. (b) Inset is the plot of unit cell of Zr2 Cu.
C44 > 0, C66 > 0, C11 > |C12 | , and
identical, both structures show a tendency to approach each other in the vicinity of the transition.32 Furthermore, we highlight the remarkable structural character of this transition. It is a transition between phases with low c/a ratio (∼3.45) and high c/a ratio (∼4.24) at their respective pressure ranges. Considering the bcc superstructure nature of the MoSi2 -type structure, it is natural to compare this isosymmetric phase transition to a simple bcc-to-fcc structural phase transition. It can be looked on as a transition
F = C11 + C12 −
2 2C13 > 0. C33
(1)
Zr2 Cu is stable in the concerned pressure range (0–70 GPa), which is confirmed by the calculated phonon dispersions in Fig. 3 where no imaginary frequencies appear in any branch of the dispersions. However, note that the F index is extremely low in the hysteresis region (see Fig. 5), which indicates its weak stability against tetragonal distortion and is consistent with the abrupt change of axial ratio. Then some derivative elastic parameters under compression are obtained, including the single crystal Young’s modulus E (reflecting the resistance of materials against uniaxial
TABLE I. Crystallographic data of Zr2 Cu. Unit cell internal parameters Type Present (GGA) Present (LDA) Previous calc.a Expt.b a b
Zr(4e) 0 0 0 0
0 0 0 0
0.346 0.347 0.348 0.34
Lattice parameters (Å) Cu(2a) 0 0 0 0
0 0 0 0
0 0 0 0
a
c
3.229 3.153 3.2224 3.2204
11.193 10.930 11.094 11.1832
Reference 31. Reference 11.
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FIG. 3. Phonon dispersion for Zr2 Cu at 20 (a), 35 (b), 50 (d) GPa during compression and 35 GPa (c) during decompression. The notation of the highsymmetry points is as follows: Z (0.5, 0.5, −0.5), (0, 0, 0), X (0, 0, 0.5), P (0.25, 0.25, 0.25), N (0, 0.5, 0). The phonon frequencies are in cm−1 .
tensions), the polycrystalline bulk modulus B (a measure of the resistance of a material against volume change under hydrostatic pressure), and the shear modulus G (describing the resistance to shape change caused by a shearing force) on the basis of the Voigt-Reuss-Hill approximation method.36 Also, the universal elastic anisotropy index AU 37 is obtained by AU = 5
GV BV + − 6. GR BR
(2)
As shown in the upper panel of Fig. 4, inside the hysteresis region, we have observed large discrepancy between the Voigt and Reuss estimates of shear modulus and bulk modulus, where both BR and GR are much lower than BV and GV , re-
FIG. 4. Upper panel: Pressure dependence of the Bulk modulus B and shear modulus G in respective Voigt and Reuss forms. Lower panel: Pressure dependence of the universal elastic anisotropy index AU derived from polycrystalline elastic moduli.
FIG. 5. The pressure dependence of Young’s moduli Ex = Ey , Ez and their ratio Ez /Ex together with the elastic stability factor F.
spectively. This discrepancy results from the high single crystal elastic anisotropy which can be measured by the index AU , as shown in the lower panel of Fig. 4. The calculated Young’s moduli along a-axis (Ex ) and c-axis (Ez ) are plotted together with the F index in Fig. 5. It is notable that, like F, they are also anomalously low in the hysteresis region. This interesting synchronization phenomenon is not a coincidence and will be further discussed later. In addition, their ratio Ez /Ex , much lower than 1 in the hysteresis region, also confirms the high elastic anisotropy discussed above by the index AU . To get further understanding of the phase transition, we extend the application of volume-conserving Bain path to Zr2 Cu. Though typically applied to simple elemental substances38, 39 along tetragonal distortion path connecting bcc and fcc structures, the Bain path model will still be powerful applied on tetragonal complexes or bcc superstructures, as in this work and other reported cases.40 A plot of energy variation with volume-conserving axial ratio, as shown in Fig. 6, may provide a clear picture of the peculiar structural and elastic response of Zr2 Cu to compression.
FIG. 6. The energy variation of Zr2 Cu as a function of axial ratio c/a for volumes: V0, V30, V35, V40, V60. Vx denotes the volume of the unit cell geometry optimized by CASTEP at a hydrostatic pressure x GPa. The energy of equilibrium Zr2 Cu is set as the energy reference.
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The reliability of the geometry optimizing calculations under pressure implemented in the CASTEP code is validated to some extent by that for each of the five energy-c/a curves, the energy minimum falls at the very axial ratio which belongs to the CASTEP-optimized geometry with whose unit cell volume the corresponding Bain path is developed. The c/a ratio at the energy minimum increases with pressure, as demonstrated by the red dotted curve linking each energy minimum marked by red stars, reproducing the trend in Fig. 1(b). It manifests itself that low compression (0–30 GPa) favors a relatively bcc-like structure with c/a ratio of ∼3.5 and high compression (40–70 GPa) prefers a relatively fcc-like structure with c/a ratio of ∼4.2. Frequently, simple metals and some transition metals have a pronounced double-well feature (the “well” here refers to the shape of the energy-c/a curve around an energy minimum) in their Bain path.38 In contrast, the Bain path for Zr2 Cu takes a single-well feature for low and high compression states, and the “well” is deeper former than latter. The V0 Bain path of Zr2 Cu is similar to that of FeAl, CoAl, and NiAl and different from that of TiAl, VAl, and CrAl.40 More notable is the becoming not pronounced of energy minimum for intermediately compressed Zr2 Cu. Instead, a rather wide and plain basin-like bottom is in sight. This anomalous feature relates to the low mechanical stability and low Young’s modulus of Zr2 Cu within the hysteresis region, since intuitively there is almost no energy barrier for tetragonal deformation in terms of the basin-like feature. Now we discuss in detail the correlation between the basin-like feature of Bain path and the low F and low Young’s modulus. Marcus et al. states that F is given by the curvature at the minima of epitaxial Bain path,41 1 d 2 E EBP (a) F = , (3) c0 da 2 a0 and thus is always positive dictated by the mathematical nature of the minima in a curve. As to this work, fairly good correspondence is obtained between the curvatures at minima of the Bain path and F values. It can be easily seen that, for the case of both low and high compressed states, the pronounced well-like feature gives a high curvature around minima of the Bain path and thus leads to a high value of F; for the case of intermediate pressure range, the basin-like feature of Bain path means low curvature around the minima and hence a low F value. Note that the Bain path used here is a volumeconserving one for simplicity. So, we have explicitly revealed the origin of the low F values at the hysteresis region. The same argument goes for the low Young’s modulus since they are calculated by 2 /C33 (C11 − C12 ) C11 + C12 − 2C13 (4a) Ex = Ey = 2 C11 − C13 /C33 and
2 /C33 C11 + C12 − 2C13 , Ez = (C11 + C12 ) /C33
(4b)
and thus are dominated by F, as shown in Fig. 5. Therefore, at the hysteresis region, Zr2 Cu shows an anomalous softening in stark contrast to low (0–30 GPa) and high (40–70 GPa) pressure regions, including weak mechan-
ical stability and low Young’s modulus (both are illuminated by the basin-like feature of Bain path), and dramatically high elastic anisotropy. These findings consolidate our original view that Zr2 Cu undergoes a pressure-induced bcc-to-fcc-like structural phase transition through a series intermediate transition states. The only difference compared with a real bccto-fcc structural phase transition, which can be accomplished through intermediate tetragonal states as demonstrated by the Bain path model, is that the parent phase in this case takes a distorted bcc lattice but not a strict one and the product phase a distorted fcc structure. Both the parent and product phases are tetragonal, the same as the intermediate states. These facts degrade the group-subgroup relations that exist in an exact bcctetragonal-fcc structural phase transition, and thus leaving the phase transition isosymmetric. IPTs can originate from structural changes, but the ones with such close connections to a simple structural phase transition have never been reported. Although similar pressure-induced tetragonal axial ratio fluctuations occur in In and its alloys,42 they are anti-isostructural phase transitions between structures with opposite signs of tetragonal distortion from an fcc structure, different from the case in Zr2 Cu. We might term such IPTs like in Zr2 Cu as quasistructural phase transitions (QPTs). QPTs may occur in systems for which the parent phase and product phase share the symmetry of intermediate states but can be connected to different high-symmetry lattices, i.e., associated with degraded group-subgroup relations between the isosymmetric parent (and product) phase and intermediate phase.
C. Pressure evolution of electronic properties
As the macroscopic structural and mechanical properties reflect the microscopic bonding structure of materials, to obtain further insight into the relationship between them in Zr2 Cu and shed light on the underling mechanism of the IPT, the electronic structures and their pressure dependences are discussed as follows. Figure 7 shows the calculated total charge density distribution of the bcc-like phase (at 0 GPa) and fcc-like phase (at 60 GPa) during compression, by using contour plots in (100) and (110) planes and a schematic picture of the bonding structure with sticks representing the bond paths. The HohenbergKohn theorem24 states that a system’s ground-state properties are a consequence of its charge density, which can be denoted as a scalar field ρ(r). From this theorem, the relationship between the topology of the charge density and many properties of materials can be rationalized. The topology of this scalar field is given in terms of its critical points (CPs), which are the zeros of the gradient of this field. In Fig. 7, each bond path is given by following the gradient of the charge density from a bond critical point (BCP) toward the bound atoms. A BCP is a (3, −1) CP that has two negative and one positive curvatures in three orthogonal directions and thus usually guarantees a ridge of maximum charge density connecting two nuclei (see details in Ref. 43). For the bcc-like phase, the most interesting character we revealed here is the pseudo BaFe2 As2 -type structure with the same symmetry I4/mmm. There is a considerable charge accumulation in the interstitial of each Zr tetrahedron,
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FIG. 7. The contour plots of total charge density in (100) and (110) planes and schematic picture of the bonding structure of Zr2 Cu at 0 GPa (a) and at 60 GPa (b). The small dark blue balls in the schematic bonding picture of (a) denote the pseudoatoms.
indicated by the local maximum of charge density in the (100) plane contour plot. This charge accumulation serves just like a pseudoatom which is bound to four nearest Zr atoms, similar to the situation in hcp Zr43 and compressed fcc Ca.44 In addition, there is no straight Zr-Zr bonding CPs, but (3, +1) CPs at the boundaries of nearest Zr atoms. This bonding picture renders ambient Zr2 Cu an interesting pseudo BaFe2 As2 -type structure where the tetrahedral interstitial charge density maxima serve as the Fe-position pseudoatoms. The BaFe2 As2 type structure is famous in iron-based superconductors and is also adopted by A2 BH2 hydrides (A = Zr, Ti, Hf; B = Cu, Pd) which are considered as potential hydrogen storage materials. With external hydrostatic pressure increasing towards the IPT pressure, each pseudoatom separates into two which gradually fade out when moving up and down along c-axis toward their respective adjacent Zr (001) layers. In the meantime, the two BCPs in the illustrated (100) plane between the pseudoatom and nearest Zr atoms will also move toward the midpoint of the two Zr atoms until they meet each other near that point. After the transition, each Zr atom now is bound to four nearest Zr atoms in the same (001) plane and four nearest Cu atoms in the adjacent, through approximately straight bonds, as shown in Fig. 7(b). And note that the interaction between interlayer Zr atoms becomes even weaker. Therefore, the exotic pseudo BaFe2 As2 -type structure of ambient Zr2 Cu collapses upon the IPT to a layered structure formed by the periodic stacking of Zr-Cu-Zr atomic sandwiches along caxis. An IPT accompanied by similar topological changes in the charge density was reported in fcc Ca44 but in a somehow opposite way. For the low-pressure phase, the nearest neighboring Ca atoms are connected through bifurcated bond paths with deep minima in the octahedral holes, while for the highpressure one, the atoms bond through non-nuclear maxima in the octahedral holes. In addition, the above electron density topological analysis agrees, in principle, but not in detail, with our previous viewpoint that the IPT in Zr2 Cu is analogous to a simple bccto-fcc structural phase transition. By the Bain path model, a tetragonal axial ratio of about 3 makes the low pressure phase a bcc superstructure, while 4.24 the high pressure phase an
fcc. From the perspective of bonding structure, in the low pressure phase, each atom has eight coordinates (Cu has 8 nearest Zr neighbors and Zr is bound to 4 Cu atoms and 4 pseudoatoms) and is relatively open compared with the high pressure phase in terms of the notable volume collapse upon the IPT and the fact that in the high pressure phase each Cu gets additional 4 Cu coordinates. The special open structure of ambient Zr2 Cu is accommodated by its unique bonding structure. The incipient change of the bonding structure of Zr2 Cu should be the origin of the high anisotropy of compressibility and the negative linear compressibility along the c-axis in the hysteresis region. The formation of the Zr-Zr and Cu-Cu straight bonding in (001) plane and the collapse of the pseudoatom bonding soften C11 and reinforce C33 , as shown in Fig. 2, and thus result in the collapse of the a-axis and the elongation of the c-axis. Similar negative linear compressibility and underlying origin in bonding changes are also reported in AuTe2 45 and Al2 SiO5 .32 To further probe into the bonding evolution of Zr2 Cu in real space, especially the collapse of the pseudoatom bonding, we investigate the evolution of the electronic structure in k space. We calculated the band-specific contribution to the charge density distribution, i.e., the orbitals, to find which band(s) contributes to the pseudoatom bonding character and its collapse. In CASTEP, the calculated “orbital” is the square of the absolute value of the wavefunction for a given electronic band, summed over all k-points. A band is defined by the position of its eigenvalue in the ordered list of electronic energies at each k-point. For example, orbital (or band) number 3 is obtained by summation over all k-points of the wavefunctions with the third lowest energy. The sum of the “orbital densities” obtained in this way gives the total electron density of the crystal. We find that mainly 6 bands, band 31 (B31) to B36 highlighted in Fig. 8, are related to the phase transition. The calculated orbitals (Fig. 9) indicate that the pseudoatom of ambient Zr2 Cu is mainly contributed from B31 and B32, of which the charge density distributions have the same pseudoatom bonding character. And the straight bonding character of the high-pressure phase is mainly contributed from B35 and B36. Upon the IPT, both B31 and B32 lose their
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FIG. 8. Band structure of Zr2 Cu at 0 GPa (a) and 60 GPa (b). The six bold and colored bands are bands 31 to 36 from lower to upper. Horizontal dashed line indicates the Fermi level. The notation of the high-symmetry points is as follows: Z (0, 0, 0.5), A (0.5, 0.5, 0.5), M (0.5, 0.5, 0), G (0, 0, 0), R (0, 0.5, 0.5), X (0, 0.5, 0). FIG. 10. Partial DOS of Zr in Zr2 Cu at zero pressure (a) and 60 GPa (b). Vertical dashed line indicates the Fermi level.
pseudoatom characters, at least partially. For B31, there still exists a tetrahedral interstitial accumulation region of charge density, but it becomes spatially extended especially along caxis, which makes it not so typical as a pseudoatom like in the ambient phase. While B32 loses its original character completely and puts on a straight bonding character. So, it seems that the ambient pseudoatom bonding is shaped by lower level bands below the Fermi level (B31, −2.024–−1.002 eV; B32, −1.757–−0.845 eV), while the straight Zr-Zr bonding character is accommodated by upper level bands crossing the Fermi level (B35, −1.078–0.745 eV; B36, −0.912–1.066 eV). This is confirmed by our density of states (DOS) calculations (Fig. 10). We focus on the partial DOS of the Zr sites in Zr2 Cu in that the pseudoatom character and its evolution is related to the interaction between Zr atoms. The DOS of both ambient and high pressure Zr2 Cu is dominated by d-states while s- and p-states contribute a small part. Compared to the high pressure phase, ambient Zr2 Cu takes a highly structured DOS featured by sharp bonding and antibonding peaks and a prominent pseudogap. The bonding peaks located around −0.73 and −1.23 eV are dominated by the hybridized dxz + yz states with all the other states. The
FIG. 9. Isosurface plot of the band-specific charge density (orbitals) of band 31–36 for Zr2 Cu at 0 GPa (left side, isosurface value 0.033 e/Å3 ) and 60 GPa (right side, isosurface value 0.035 e/Å3 ).
latter peak is from B31 and B32, which are responsible for the pseudoatom character. This is consistent with the fact that the dxz + yz states favor the pseudoatom bonding by their orbital spatial distribution characteristics. In addition, the innerhybridized-peak-stabilized character of the pseudoatom bonding renders it a considerable covalent nature. Upon the phase transition, the highly structured DOS collapses to a broad featureless one without prominent peaks and pseudogap, which might be responsible for the collapse of pseudoatom character of B31 and B32. Furthermore, the situation near the Fermi level changes dramatically. For ambient DOS, the Fermi level is dominated by dxz + yz states with other states taking almost the same small values, while for the high pressure DOS, the Fermi level is instead dominated by the dx2 − y2 states and gets a higher filling density. The filling of these dx2 − y2 states favors the straight bonding by their orbital spatial distribution, corresponding to the formation of the straight Zr-Zr bonding character of B35 and B36. The increased filling of the DOS near the Fermi level results from the increased crossing Fermi level of related bands. This increased interaction will also be manifested in the Fermi surface. Following a compression, the electronic band extremum may shift through the Fermi level. In this case, new elements appear on the Fermi surface and change its topology. Such a topology change, referred as electronic topological transition (ETT), can lead to anomalies in transport, thermodynamic, and vibrational properties, as well as in the lattice structure, as has been intensively studied in IPTs of hcp metals like Zn,46 Cd,47 Os,48 etc. As shown in Fig. 11, for Zr2 Cu, apart from B35 and B36 which contribute most to the Fermi surface, B37 also contributes a small part with small electron pockets. Upon the phase transition B37 gets completely above the Fermi level and results in an ETT. The interaction of B35 and B36 with the Fermi level is relatively complex, which results in numerous ETTs during compression. The most notable is that a pipe-like piece develops
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cumulating process of valence electrons. To the best of our knowledge, this is the first time an IPT is reported to be associated with a pressure-induced collapse of interstitial-electron bonding, in a way opposite to the aforementioned ones. IV. CONCLUSION
FIG. 11. Fermi surface for Zr2 Cu under pressure. From top to bottom: band 35, band 36, band 37. From left to right: 0 GPa, 30 GPa, 40 GPa, and 60 GPa.
around the zone central axis in both bands at about 30 GPa (a little lower than 30 GPa for B35 and higher for B36). It is an “electron pipe” corresponding to the bowl-like arc right below Fermi level around the GZ path in Fig. 8. The emergence and extension of these new pieces of Fermi surface, corresponding to the extra filling states at Fermi level in the DOS plots, stabilize the high pressure phase and destabilize the low pressure phase. So does the fading away of originally existing pieces. It should be mentioned that IPTs are relatively rare and exotic phenomena hence have gained considerable interest and an increasing number of cases of IPTs are emerging. An IPT can occur purely due to subtle changes of electronic structures. A well known example is the case in Ce where the variable chemical valence as a result of the shift between localized and itinerant states in 4f electrons is responsible for the IPT with a large volume collapse of 15%.49 While in many cases, notable symmetry-retaining structural changes are involved such as reorientation of polyhedrons50 and large anisotropy in compressibility45 for sake of finding the optimal atomic packing with respect to pressure-induced densification. Such transitions are often accompanied by changes in bonding structures. IPTs with novel unique features are also emerging, such as the cases reported in albite NaAlSi3 O8 51 and Al-rich NiAl alloys,52 which are associated with the change of the ordering of atoms or defects. Since it often manifests subtle and anomalous changes in electronic structures and hence in physical and chemical properties of materials, IPTs are an intriguing topic in condensed matter physics and the understanding of it is of significant scientific importance to itself and other fields. For instance, the polyamorphic transition in CeAl53 and CaAl54 metallic glasses are approved to be dominated by their components Ce and Ca, respectively, both of which undergo a pressure-induced IPT with a large volume collapse. The IPT in Zr2 Cu shows some special characteristics. It is a transition between phases with axial ratios marking different subgroups of MoSi2 -type phases and shows a remarkable analogy to a simple bcc-to-fcc structural phase transition. The transition is associated with a pressure-induced collapse of pseudoatom bonding of the ambient phase and the formation of the atomic-sandwiches-like structure of high pressure phase. Similar pressure-driven electron density topological changes have been reported for IPTs in fcc Ca44 and Ba8 Si46 clathrates55 and for high-pressure structural phase transitions in Al,56 Si,57 alkali metals (Li,58 Na,59 K60 ), and alkali earth metal Ca.61 But these are all related to an interstitially ac-
In conclusion, using first-principles calculations, we predict a pressure-induced reversible, first-order, and bcc-tofcc analogous isosymmetric transition in tetragonal Zr2 Cu with its axial ratio increasing remarkably from ∼3.5 to ∼4.2 through a 4 GPa intermediate hysteresis region. The bcc-like and fcc-like features of Zr2 Cu are well characterized by the quite distinct trends of pressure dependences of numerous properties at low and high pressure ranges. Inside the hysteresis region, Zr2 Cu exhibits anomalous softening properties including low mechanical stability and Young’s modulus, both well characterized by the striking basin-like feature of Bain path, and dramatically high elastic anisotropy. The transition is densification-driven in terms of its bcc-to-fcc-like character. Electron density distribution illustrates the pseudo BaFe2 As2 -type structure of low pressure Zr2 Cu, where the tetrahedral interstitial electron density maxima serve as the Fe-position pseudoatoms. This special structure upon the IPT undergoes an electron density topological change and collapses to an atomic-sandwich-like structure. A comprehensive analysis based on electronic band, band-specific contribution to the charge density (orbitals) and density of states reveals the origin of the intriguing bonding structure and its pressure evolution. The pseudo BaFe2 As2 -type structure is demonstrated to be shaped by electronic states below Fermi level with obvious dxz + yz nature, while the high pressure straight Zr-Zr bonding is accommodated by states around Fermi level with dx2 − y2 dominant features. The Fermi surface manifestation of the IPT and the bonding changes is also demonstrated. Besides the main subject of pressure-induced IPT and bonding structure evolution in Zr2 Cu in present work, we sense that the special pseudo BaFe2 As2 -type bonding structure revealed here might be responsible for its intriguing properties like good hydrogen storage ability and exotic oxidation behaviors, which needs further theoretical and experimental investigations. Tetragonal Zr2 Cu is stable under the whole concerned pressure range according to both elastic stability criterion and phonon dispersion analysis. However, this IPT could be metastable and it is possible that Zr2 Cu transforms to other structures with a symmetry change, which needs further clarifying work such as high pressure tests using x-ray diffraction measurements in diamond anvil cells like in previously reported IPT cases.62, 63 Our work also implies the possibility of occurrence of similar IPTs in other MoSi2 -type phases with a shift of axial ratio marking different subgroups. On the other hand, it is reported recently that an IPT occurs in Sb2 O3 64 while does not in its analogue As2 O3 . So, it is doubtful whether an IPT occurs in tetragonal Zr2 Cu’s analogues like Zr2 Pd, Ti2 Cd, Ti2 Pd, and so on. Further, similar IPTs may also occur in other types of intermetallics associated with degraded group-subgroup relations and similar theoretical and experimental investigations are needed.
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ACKNOWLEDGMENTS
This work was supported by the NBRPC (Grant No. 2010CB731600), National Science Foundation of China (NSFC) (Grant Nos. 51171160/51002130/51171163/ 51271161/51071138/51131002). 1 N.
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