First-principles study on the negative/zero area compressibility in Ag₃BO₃ with “wine-rack” architecture

INTRODUCTION

It is well-recognized that materials would contract along three dimensions when being squeezed under hydrostatic pressure^[1]. However, very few materials would abnormally expand or remain constant along one or two directions as hydrostatic pressure increases, which is termed negative or zero compressibility^[2-5]. These counterintuitive pressure-responding behaviors bring great application prospects in many fields, such as shock wave absorption^[6,7], body armor^[8,9], ultrasensitive pressure detectors^[3,5,10], deep-sea optical cable^[11,12], etc. The exploration of these anomalous mechanical materials has been one of the most advanced branches of mechanical functional materials. From the viewpoint of occurring dimensions, the anomalous compressibility can be categorized into negative/zero linear compressibility (NLC/ZLC)^[13,14], negative/zero area compressibility (NAC/ZAC)^[11,15,16], and negative/zero volume compressibility (NVC/ZVC)^[17-19]. Due to the restrictions imposed by thermodynamical laws, the NVC behavior can only occur alongside phase transitions^[17,18], which are forbidden in dynamically stable systems. Meanwhile, ZVC only appears in highly dense-packed structure materials, such as diamond^[19], in which an extremely difficult synthesis condition is required. Therefore, the materials with one/two-dimensional anomalous compressibility become the ideal choice for the application. Compared with NLC or ZLC, NAC/ZAC has a greater advantage in the device where highly dimensional pressure-driven expansion is required, with the example of shock-resistant optical windows^[11] and ultra-precise ferroelectric pressure sensors^[3]. However, due to the same reason, NAC/ZAC is much more scarcely occurring in comparison with NLC/ZLC, and the exploration of NAC/ZAC materials is a very challenging but highly desired issue.

In the past decades, much great effort has been devoted to the investigation of the structure-property relationship of NAC/ZAC, and several structural models have been proposed to elaborate the microscopic mechanism, such as “wine-rack” models^[7,20,21], “helical chain” structures^[3], and Lifshits mechanisms^[22], which have strongly promoted the exploration of abnormal compressibility materials. Among them, the “wine-rack” motif, established on the bar-hinge model, is the most prevailing mechanism of these fascinating mechanical behaviors, which has led to the discovery of anomalous compressibility in dozens of materials, including Ag₃Co(CN)₆^[23], KMn[Ag(CN)₂]₃^[24], methanol monohydrate^[25], MIL-53(Al)^[20], MIL-122(In)^[7], etc. Nonetheless, almost all these ever-discovered negative compressibility (NC) originating from the “wine-rack” motif only occur along one direction, and no NAC material has been detected by this structural prototype. In fact, the detailed strain-stress analysis on the building block bar of “wine-rack” architecture shows that the rotation direction of the bar strongly depends on its initial orientation in the structure: when the angle between bars and principal axis (i-axis) is smaller than 45°, the bar would rotate toward the principal axis and result in the NC along this direction [Figure 1A]; otherwise, it would rotate to the vertical axis and lead to NC along the vertical directions (j-axis) [Figure 1B]. Therefore, controlling the alignment angle of the bars in “wine-rack” structures is the key to modulating the direction of NC, which may promote the occurrence of NAC.

Figure 1. The mechanical analysis of the bar in “wine-rack” structures for the two conditions of the angle between the bar and principal axis (i-axis). (A) The condition for the angle smaller than 45°, in which the hydrostatic pressure results in a torque to drive the bar rotate toward the i-axis; (B) The condition for an angle larger than 45°, in which the hydrostatic pressure results in a torque to drive the bar rotate to be away from the i-axis. The rotation direction of the bar is highlighted by yellow arrows.

Planar coordination groups, such as [BO₃]^3-, [CO₃]^2-, and [NO₃]^-, can be considered to be dimensionless along the direction perpendicular to the plane, with which a small angle is possible generated so as to satisfy the structure requirement of the ZAC/NAC governed by “wine-rack” motif. Here, the [BO₃] system is chosen to perform the structural screening. By connecting with the adjacent polyhedral units, [BO₃] groups have constructed many crystals with anomalous pressure-responding properties, such as KBe₂BO₃F₂^[16], LiBO₂^[5], AEB₂O₄^[26], etc. In this work, by adopting the first-principles simulations, we predict that two allotropes of Ag₃BO₃, Ag₃BO₃-I and -II, with “wine-rack” architecture in the [BO₃] system exhibit the ZAC and NAC behaviors, respectively. The relationship between the geometrical configuration and the ZAC/NAC-occurring direction of the “wine-rack” structure is systematically discussed. It is revealed that NAC/ZAC in these two structures is attributed to the dominant effect from the bar rotation under pressure, and the opening angle within the bars determines whether NAC or ZAC occurs.

MATERIALS AND METHODS

The first-principles calculation was performed by the CASTEP^[27], a program based on the plane-wave pseudopotential density functional theory (DFT)^[28]. The functionals developed by Ceperley, Alder, Perdew, and Zunger (CA-PZ) in the form of local density approximation (LDA)^[29,30] were chosen to model the exchange-correlation energy. The ultrasoft pseudopotentials^[31] were used to describe the effective interaction between the atomic cores and valence electrons. Plane-wave energy cutoff was set to 400 eV, and the Monkhorst k-point mesh spanning less than 0.03 ų was adopted. To get pressure-dependent crystal structures, the hydrostatic pressure was exerted from 0 to 10 GPa, and crystal structures were geometrically optimized by Broyden-Fletcher-Goldfarb-Shanno (BFGS) scheme^[32], and the convergence criterion for energy, maximum force, maximum stress, and maximum displacement were set to 5 × 10^-6 eV/atom, 0.01 eV/Å, 0.02 GPa, and 5 × 10^-4 Å, respectively. The validity of first-principles calculation has been verified in some other anomalous mechanical materials, such as LiBO₂^[5] and [C(NH₂)₃][Cd(HCOO)₃]^[12]. The number of steps for each strain and the maximum strain amplitude were set to 6 and 0.002 to calculate elastic constants. Finally, based on the theoretical cell parameters, the PASCal^[33] program developed by Cliffe et al. was used to fit the compressibility along the mechanical principal axes^[33].

RESULTS AND DISCUSSION

The structures of Ag₃BO₃ were first determined by Jansen et al. and have two phases with the trigonal space group of R32 and R-3c, termed as Ag₃BO₃-I and Ag₃BO₃-II, respectively^[34,35]. As shown in Figure 2A and B, Ag₃BO₃-I and Ag₃BO₃-II share a similar atomic configuration. In both two structures, one boron atom was coordinated with three oxygen atoms to construct the planar [BO₃] triangles arranged parallel to the (a, b) plane. Silver atoms are bonded with two oxygen atoms to form two-fold coordinated bars of O-Ag-O. By sharing the ligand oxygen atoms, the O-Ag-O bars are further connected with the adjacent interlayer [BO₃], giving rise to a “wine rack”-like structure. Accordingly, the lengths of the a- and c-axes are determined by the sum of the projection of the O-Ag-O bar along the a-axis combined with the size of [BO₃] groups and the projection of the O-Ag-O bar along the c-axis, respectively. Interestingly, the O-Ag-O bars manifest the different configurations in these two structures. In Ag₃BO₃-I, the O-Ag-O bar is prominently distorted, and the angle of $$\angle$$O-Ag-O is 133.79°, while the counterpart in Ag₃BO₃-II manifests a more linear configuration and the angle of $$\angle$$O-Ag-O is 176.41°. Meanwhile, Ag₃BO₃-I has one symmetrically independent oxygen atom (O1), while two (O1 and O2) are contained in Ag₃BO₃-II due to the different space groups.

Figure 2. The crystal structure of AgBO₃ and Ag₃Co(CN)₆ (A) AgBO₃-I; (B) AgBO₃-II. (C) Ag₃Co(CN)₆-I; (D) Ag₃Co(CN)₆-II. The blue, green, red, grey, and brown balls are represented by Ag, B, O, N, and C atoms, respectively.

Both Ag₃BO₃-I and Ag₃BO₃-II have a similar “wine-rack” structure constructed by the O-Ag-O bars, as the CN-Ag-CN ones in Ag₃Co(CN)₆ [Figure 2C and D]. However, different from the hexagonal configuration of [CoC₆] in Ag₃Co(CN)₆, the planar [BO₃] groups in Ag₃BO₃-I and Ag₃BO₃-II, serving as the hinges of the “wine-rack” motif, are almost aligned along the (a, b) plane. Such a planar hinge configuration induces a more slant alignment of the bars with respect to the c-axis, and the angles between the O-Ag-O bar and c-axis in Ag₃BO₃-I and in Ag₃BO₃-II were 59.35° [Figure 2A] and 60.59° [Figure 2B], respectively, in comparison with the value of 31.8° [Figure 2C] and 40.56° [Figure 2D] in Ag₃Co(CN)₆-I and Ag₃Co(CN)₆-II^[23]. According to the abovementioned mechanical analysis, it is anticipated that the conversion of the inclination angle across 45° from Ag₃Co(CN)₆ to Ag₃BO₃ might reverse the inclination direction of the bars and result in the expansion along the (a, b) plane under pressure.

To verify these postulations, first-principles calculations based on DFT are implemented to investigate the compressive behavior of Ag₃BO₃-I and Ag₃BO₃-II. In the first step, to confirm the validity of the DFT strategy on “wink-rack”-type structures, the calculation on the compressibility of Ag₃Co(CN)₆ was performed. As plotted in Figure 3A and B, and Supplementary Figure 1, regardless of a higher theoretical phase-transition pressure (3 GPa) than the observed values (0.19 GPa), both the NLC behavior and phase transition from P-31m to C2/m space groups in Ag₃Co(CN)₆ is reproduced [Supplementary Table 1]. This confirms the feasibility of our calculating methods. As shown in Figure 3C and D, no discontinuity of the enthalpy and the cell parameters emerge as pressure mounts from 0 to 10 GPa, indicating the absence of phase transitions and the high mechanical stability of both the two phases. In the pressure range of 0-10 GPa, the enthalpy of Ag₃BO₃-II is always smaller than that of Ag₃BO₃-I. Since the enthalpy difference between the two phases is 51.48 meV per atom, corresponding to the elementary excitation at 199.1 K per vibration freedom, one cannot distinguish which is more stable at room temperature (300 K). According to the theoretical cell parameters, it is observed that the a/b-axis of Ag₃BO₃-I decreases from 9.97276 to 9.88381 Å (by -0.089%), and the c-axis decreases from 3.01461 to 2.84532 Å (by -5.6%) as pressure increases 0-10 GPa [Figure 3D and Supplementary Figure 2,Supplementary Table 2]. The compressibilities, fitted by the PASCal program^[33], are 0.97 (1) TPa^-1 and 4.95 (11) TPa^-1 [Supplementary Figure 3] along the principal mechanical X₃- and X₁-axes (Transformation matrix was shown in Supplementary Table 3). The linear compressibility along the X₁-axis is of a similar magnitude to that of diamond (0.71 TPa^-1), so Ag₃BO₃-I is a material with extremely limited ZAC. Different from Ag₃BO₃-I, the a/b-axis of Ag₃BO₃-II abnormally expands from 9.94640 to 9.96232 Å (by 0.016%), and the c-axis decreases from 12.33943 to 11.83234 Å (by -4.11%) with the pressure increasing from 0 to 4 GPa [Figure 3D and Supplementary Figure 2, Supplementary Table 2]. The compressibilities along X₁ and X₃-axes are fitted to be -0.24 (22) TPa^-1 and 9.26 (79) TPa^-1, respectively [Supplementary Figure 3, Supplementary Table 3], which demonstrates that Ag₃BO₃-II presents a NAC behavior with an area compressibility of -0.48 (44) TPa^-1 within (a, b) plane [Supplementary Figure 4]. The observation of NAC in Ag₃BO₃-II is the first case in the “wine-rack” system. Above 4 GPa, as the pressure increases sequentially to 10 GPa, the a/b-axis and c-axis are inversed to shrink from 9.96232 to 9.93624 Å (by -0.26%) and from 11.83234 to 11.39554 Å (by -3.7%), and the linear compressibility along the principal mechanical X₃- and X₁-axis in this pressure range are 0.53 (2) TPa^-1 and 5.51 (3) TPa^-1 respectively. Accordingly, analogous to Ag₃BO₃-I, ZAC was also shown in Ag₃BO₃-II, with an area compressibility of 1.06 (4) TPa^-1 within the (a, b) plane. To make a comparison with Ag₃BO₃-I, the compressibility of Ag₃BO₃-II in the whole pressure range (0~10 GPa) was fitted as well, and the linear compressibilities of X₃- and X₁-axes were 0.09 (14) TPa^-1 and 6.56 (21) TPa^-1, respectively [Supplementary Figure 3 and Supplementary Table 3], both of which are similar with the ones of Ag₃BO₃-I. Due to the similar cell parameter change in response to pressure, Ag₃BO₃-I and Ag₃BO₃-II share a similar mechanical property, especially for the bulk modulus, 101.55 GPa and 87.37 GPa, respectively, as listed in Supplementary Tables 4 and 5.

Figure 3. The calculated enthalpy and variation of cell parameters in Ag₃Co(CN)₆ and Ag₃BO₃vs. pressure. (A) Pressure-dependent enthalpy for the low and high-pressure phases of Ag₃Co(CN)₆. (B) Pressure-dependent cell parameters of Ag₃Co(CN)₆. (C) Pressure-dependent enthalpy in Ag₃BO₃-I and Ag₃BO₃-II. (D) The relative change of pressure-dependent cell parameters in Ag₃BO₃-I and Ag₃BO₃-II. By considering the different numbers (Z value) of chemical formulas in the two phases of AgBO₃ and Ag₃Co(CN)₆, the enthalpies of AgBO₃-I and Ag₃Co(CN)₆-I are multiplied by the factor of 4 and 2, respectively.

To shed light on the microscopic mechanism of anomalous compressibility behavior within the (a, b) plane of Ag₃BO₃, the most intuitive way is to trace the bond length and angle evolution with respect to pressure. The size of the a/b-axis is determined by the size of [BO₃] groups and the O-Ag-O bar projection along them [Supplementary Figure 5] and can be expressed by the following formula:

in which γ and θ are the angle $$\angle$$O-Ag-O and the angle between the O-Ag-O bar and (a, b) plane, respectively, and l_Ag-O is the bond length between Ag and O. Due to the restriction of symmetry, the Ag-O1 is equivalent to Ag-O2 in Ag₃BO₃-I. In Ag₃BO₃-I, as the pressure increases from 0 to 10 GPa, the length of B-O bond in planar [BO₃] groups and Ag-O bond in O-Ag-O bars shrink from 1.36648 to 1.35554 Å (by -0.8%) and from 2.14276 to 2.10956 Å (by -1.5%), respectively [Figure 4A and Supplementary Table 6]. Meanwhile, the angle $$\angle$$O-Ag-O in the O-Ag-O bar decreases from 133.792° to 132.190° (by -1.2%). All of these structural modifications lead to the contraction of the a/b-axis. However, the angle between the O-Ag-O bar and (a, b) plane decreases from 30.654° to 29.457° (by -3.9%), which expands their projection onto the a/b-axis. These two effects subtlely counterbalance with each other and give rise to the invariant size of the a/b-axis in response to pressure, i.e., the ZAC within the (a, b) plane [Figure 4B]. For Ag₃BO₃-II, in the pressure range of 0 to 4 GPa, the length of B-O bonds in [BO₃] groups, Ag-O1 and Ag-O2 bonds in the O-Ag-O bar experience contraction from 1.36840 to 1.36267 Å (by -0.4%), from 2.09401 to 2.08012 Å (by -0.7%), and from 2.09607 to 2.08375 Å (by -0.6%), respectively [Figure 4C and Supplementary Table 7]. However, different from Ag₃BO₃-I, the angle of $$\angle$$O-Ag-O only shrinks from 176.406° to 176.210° (by -0.1%), which does not contribute to the contraction of the O-Ag-O bar. Due to the almost absence of $$\angle$$O-Ag-O decreasing, the overall contraction effect of the a/b-axis is suppressed by the expansion induced by the decrease of the angle between the O-Ag-O bar and (a, b) plane (from 29.410° to 28.286°, by -3.8%), giving rise to the net expansion response, i.e., NAC of the (a, b) plane [Figure 4D]. Further, as the pressure mounts from 4 to 10 GPa, the contractions of B-O, Ag-O1, and Ag-O2 bond are enhanced and change from 1.36267 to 1.35485 Å (by -0.6%), from 2.08012 to 2.06086 Å (by -0.9%), and from 2.08375 to 2.06665 Å (by -0.8%), respectively. Similar to the behavior in the range of 0~4 GPa, the angle of $$\angle$$O-Ag-O experiences a very tiny change (from 176.210° to 176.106°, less than -0.1%). However, the modification of the angle between the O-Ag-O bar and the (a, b) plane is weakened in comparison with the value of 3.8% from 0 to 4 GPa and only decreases from 28.286° to 27.414° (by -3.1%). This makes the competition between the contraction and expansion of the a/b-axis return to the balance and leads to the ZAC response in the pressure range of 4~10 GPa, as Ag₃BO₃-I. In “wine-rack” architecture, the opening degree of the framework determines the magnitude of pressure-induced expansion effects^[36]. Benefited from its larger structural openness endowed by the O-Ag-O chain closer to the linear configuration than Ag₃BO₃-I, at low pressure (0~4 GPa), the prominent expansion effect from inclination of O-Ag-O chain leads to the manifestation of NAC in Ag₃BO₃-II, as elucidated in Figure 4D. With the structural densification under pressure, such an expansion effect is gradually weakened, and its counterbalance with the contraction effect triggers the occurrence of ZAC from 4 to 10 GPa. Therefore, the different pressure responses between Ag₃BO₃-I and Ag₃BO₃-II, as well as that between low-pressure and high-pressure phases of Ag₃BO₃-II, could be attributed to the structural openness degree determined by the $$\angle$$O-Ag-O angle and pressure-induced structure densification.

Figure 4. The structural modification of Ag₃BO₃-I and Ag₃BO₃-II under pressure. (A) The relative change of bond length and angle in Ag₃BO₃-I; (B) The schematic of ZAC in Ag₃BO₃-I; (C) The relative change of bond length and angle in Ag₃BO₃-II; (D) The schematic of NAC in Ag₃BO₃-II.

CONCLUSIONS

In summary, through a quantified elaboration, the transition across 45° for the angle between the bar and principal axis governs the occurring direction of expansion effect in the “wine-rack” architecture. Accordingly, through a structural screening in the planar coordination system, the “wine-rack” structure-driven NAC and ZAC were theoretically predicted in Ag₃BO₃-I and Ag₃BO₃-II. Analysis of bond lengths and bond angles ascribes to the occurrence of abnormal compressibility to the competitive effect between the shrinkage effect from bond length/angle contraction and the expansion effect from inclination angles between the O-Ag-O bar and (a, b) plane. By virtue of large structure openness generated by linear O-Ag-O coordination, the strong expansion effect promotes the occurrence of the NAC of Ag₃BO₃-II in the low-pressure range of 0~4 GPa, while the ZAC in Ag₃BO₃-I and Ag₃BO₃-II in the pressure range of 4~10 GPa is ascribed to the subtle counterbalance between these shrinkage and expansion effects. The work establishes a quantified relation between the atomic coordination and expansion response in “wine-rack” architecture under pressure. It will bring continuous discovery of the material with anomalous mechanical properties.