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Symmetry plays an important role in determining the physical properties in condensed matter physics, as the symmetry operations of any physical property must include the symmetry operations of the point group of the crystal. As a consequence, crystallographic polarity and chirality are expected to have an impact on the Cooper pairing in a superconductor. While superconductivity with crystallographic polarity and chirality have both been found in a few crystalline phases separately; however, their coexistence and material realizations have not been studied. Here, by utilizing transport, Raman scattering, and transmission electron microscopy, we unveil a unique realization of superconductivity in single-crystalline Mo 3 Al 2 C (superconducting T c =8 K) with a polar charge-density-wave phase and well-defined crystallographic chirality. We show that the intriguing charge density wave order leads to a noncentrosymmetric-nonpolar to polar transition below T *=155K via breaking both the translational and rotational symmetries. Superconductivity emerges in this polar and chiral crystal structure below T c =8 K. Our results establish that Mo 3 Al 2 C is a superconductor with crystallographic polarity and chirality simultaneously, and motivate future studies of unconventional superconductivity in this category. The coexistence of crystallographic polarity and chirality is rare in a superconductor. Here, the authors establish that Mo 3 Al 2 C is a superconductor with crystallographic polarity and chirality simultaneously.

The kagome metals of the family A V 3 Sb 5 , featuring a unique structural motif, harbor an array of intriguing phenomena such as chiral charge order and superconductivity. CsV 3 Sb 5 is of particular interest because it displays a double superconducting dome in the region of the temperature-pressure phase diagram where charge order is still present. However, the microscopic origin of such an unusual behavior remains an unsolved issue. Here, to address it, we combine high-pressure, low-temperature muon spin relaxation/rotation with first-principles calculations. We observe a pressure-induced threefold enhancement of the superfluid density, which also displays a double-peak feature, similar to the superconducting critical temperature. This leads to three distinct regions in the phase diagram, each of which features distinct slopes of the linear relation between superfluid density and the critical temperature. These results are attributed to a possible evolution of the charge order pattern from the superimposed tri-hexagonal Star-of-David phase at low pressures (within the first dome) to the staggered tri-hexagonal phase at intermediate pressures (between the first and second domes). Our findings suggest a change in the nature of the charge-ordered state across the phase diagram of CsV 3 Sb 5 , with varying degrees of competition with superconductivity. Kagome metals can house a complex interplay of competing phenomenon and there has been significant investigation into how to engineer the resultant properties, as well as understand the underlying physics. Here, the authors investigate the competition between charge density wave order and superconductivity for CsV 3 Sb 5 using a combination of high-pressure and muon spin rotation measurements.

Kagomé metals are widely recognized, versatile platforms for exploring topological properties, unconventional electronic correlations, magnetic frustration, and superconductivity. In the RV6Sn6 family of materials (R = Sc, Y, Lu), ScV6Sn6 hosts an unusual charge density wave ground state as well as structural similarities with the AV3Sb5 system (A = K, Cs, Rb). In this work, we combine Raman scattering spectroscopy with first-principles lattice dynamics calculations to reveal phonon mixing processes in the charge density wave state of ScV6Sn6. In the low temperature phase, we find at least four new peaks in the vicinity of the V-containing totally symmetric mode near 240 cm−1 suggesting that the density wave acts to mix modes of P6/mmm and R3¯m symmetry - a result that we quantify by projecting phonons of the high symmetry state onto those of the lower symmetry structure. We also test the stability of the short-range ordered density wave state under compression and propose that both physical and chemical pressure quench the effect. We discuss these findings in terms of symmetry and the structure-property trends that can be unraveled in this system.

The sensitivity of materials properties, particularly those of perovskite oxides, to epitaxial strain has been exploited to great advantage to create materials with new or enhanced properties. Although it has certainly been recognized that mismatch in the thermal expansion coefficients of the bulk and substrate material will contribute to the misfit strain, the significance of this contribution for ferroelectric perovskite thin-films has not been systematically explored. We use first-principles density functional theory and the example of ferroelectric PbTiO\\(_3\\) thin-films on various substrates to show that ignoring the thermal expansion of the substrate (that is, assuming that the in-plane lattice parameter of the film remains roughly constant as a function of temperature) results in ferroelectric transition temperatures and structural trends that are completely qualitatively different from calculations in which thermal expansion mismatch is properly taken into account. Our work suggests that the concept of a misfit strain defined as a single number is particularly ill-defined for PbTiO\\(_3\\) and invites further study of the interplay between thermal expansion mismatch and structural and functional properties in other thin-film materials.

Symmetry plays an important role in determining the physical properties in condensed matter physics, as the symmetry operations of any physical property must include the symmetry operations of the point group of the crystal. As a consequence, crystallographic polarity and chirality are expected to have an impact on the Cooper pairing in a superconductor. While superconductivity with crystallographic polarity and chirality have both been found in a few crystalline phases separately; however, their coexistence and material realizations have not been studied. Here, by utilizing transport, Raman scattering, and transmission electron microscopy, we unveil a unique realization of superconductivity in single-crystalline Mo3Al2C (superconducting Tc=8K) with a polar charge-density-wave phase and well-defined crystallographic chirality. We show that the intriguing charge density wave order leads to a noncentrosymmetric-nonpolar to polar transition below T*=155K via breaking both the translational and rotational symmetries. Superconductivity emerges in this polar and chiral crystal structure below Tc=8K. Our results establish that Mo3Al2C is a superconductor with crystallographic polarity and chirality simultaneously, and motivate future studies of unconventional superconductivity in this category.

We use first-principles theory to show that the ingredients assumed to be essential to the occurrence of negative thermal expansion (NTE) -- rigid unit phonon modes with negative Gr\"uneisen parameters -- are neither sufficient nor necessary for a material to undergo NTE. Instead, we find that NTE in PbTiO\\(_3\\) involves a delicate interplay between the phonon properties of a material (Gr\"uneisen parameters) and its anisotropic elasticity. These unique insights open new avenues in our fundamental understanding of the thermal properties of materials, and in the search for NTE in new materials classes.

In this Tutorial, we describe the use of the quasiharmonic approximation and first-principles density functional theory (DFT) to calculate and analyze the thermal expansion of insulating solids. We discuss the theory underlying the quasiharmonic approximation, and demonstrate its practical use within two common frameworks for calculating thermal expansion: the Helmholtz free energy framework and Gr\"uneisen theory. Using the example of silicon, we provide a guide for predicting how the lattice parameter changes as a function of temperature using DFT, including the calculation of phonon modes and phonon density of states, elastic constants, and specific heat. We also describe the calculation and interpretation of Gr\"uneisen parameters, as well as how they relate to coefficients of thermal expansion. The limitations of the quasiharmonic approximation are briefly touched on, as well as the comparison of theoretical results with experimental data. Finally, we use the example of ferroelectric PbTiO\\(_3\\) to illustrate how the methods used can be adapted to study anisotropic systems.

Spontaneous symmetry breaking-the phenomenon where an infinitesimal perturbation can cause the system to break the underlying symmetry-is a cornerstone concept in the understanding of interacting solid-state systems. In a typical series of temperature-driven phase transitions, higher temperature phases are more symmetric due to the stabilizing effect of entropy that becomes dominant as the temperature is increased. However, the opposite is rare but possible when there are multiple degrees of freedom in the system. Here, we present such an example of a symmetry-ascending phenomenon in a magnetic kagome metal FeGe by utilizing neutron Larmor diffraction and Raman spectroscopy. In the paramagnetic state at 460K, we confirm that the crystal structure is indeed hexagonal kagome lattice. On cooling to TN, the crystal structure changes from hexagonal to monoclinic with in-plane lattice distortions on the order of 10^(-4) and the associated splitting of the double degenerate phonon mode of the pristine kagome lattice. Upon further cooling to TCDW, the kagome lattice shows a small negative thermal expansion, and the crystal structure becomes more symmetric gradually upon further cooling. Increasing the crystalline symmetry upon cooling is unusual, it originates from an extremely weak structural instability that coexists and competes with the CDW and magnetic orders. These observations are against the expectations for a simple model with a single order parameter, hence can only be explained by a Landau free energy expansion that takes into account multiple lattice, charge, and spin degrees of freedom. Thus, the determination of the crystalline lattice symmetry as well as the unusual spin-lattice coupling is a first step towards understanding the rich electronic and magnetic properties of the system and sheds new light on intertwined orders where the lattice degree of freedom is no longer dominant.

We present a technique for partitioning the total energy from a semi-local density functional theory calculation into contributions from individual electronic states in a localized Wannier basis. We use our technique to reveal the key role played by the \\(s\\) and \\(p\\) orbitals of the apical oxygen atoms in a curious elastic anomaly exhibited by ferroelectric PbTiO\\(_3\\) under applied stress, which has so far gone unexplained. Our technique enables new insights into the chemical origins of the mechanical properties of materials, or any property given by an energy derivative.

Elucidating the microscopic mechanisms responsible for the charge density wave (CDW) instability of the AV\\(_3\\)Sb\\(_5\\) (A=Cs, K, Rb) family of kagome metals is critical for understanding their unique properties, including superconductivity. In these compounds, distinct CDW phases with wave-vectors at the \\(M\\) and \\(L\\) points are energetically favorable, opening the possibility of tuning the type of CDW order by appropriate external parameters. Here, we shed light on the CDW landscape of CsV\\(_3\\)Sb\\(_5\\) via a combination of first-principles calculations and phenomenology, which consists of extracting the coefficients of the CDW Landau free-energy expansion from density functional theory. We find that while the main structural distortions of the kagome lattice in the staggered tri-hexagonal CDW phase are along the nearest-neighbor V-V bonds, distortions associated with the Sb ions play a defining role in the energy gain in this and all other CDW states. Moreover, the coupling between ionic displacements from different unit cells is small, thus explaining the existence of multiple CDW instabilities with different modulations along the c-axis. We also investigate how pressure and temperature impact the CDW phase of CsV\\(_3\\)Sb\\(_5\\). Increasing pressure does not change the staggered tri-hexagonal CDW ground state, even though the \\(M\\)-point CDW instability disappears before the \\(L\\)-point one, a behavior that we attribute to the large nonlinear coupling between the order parameters. Upon changing the temperature, we find a narrow regime in which another transition can take place, toward a tri-hexagonal Star-of-David CDW phase. We discuss the implications of our results by comparing them with experiments on this compound.