Concurrent broken inversion and time reversal symmetry in VAgP2Se6
VAgP2Se6 belongs to the layered vdW metal phosphorus trichalcogenides family MM’(PX3)2 (M, M’ = metal; X = S, Se), which can be considered as a salt of (M, M’)2+ and P2X64− ions.48 The layered material VAgP2Se6 crystallizes in the noncentrosymmetric monoclinic structure with the space group C2 (no. 5), with each layer built of the stacking of PSe3 - V/Ag - PSe3 along the c-axis, as shown in the crystal structure schematics in Fig. 1a. We have successfully synthesized VAgP2Se6 single crystals with chemical vapor transport (CVT) and horizontal flux growth as an alternative method (see methods). Figure 1b presents the XRD pattern along the c-axis of the flux grown VAgP2Se6 single crystal. The sharp (00l) reflections from the surface of an as-grown single crystal are indexed with the previously reported structure and demonstrate excellent crystallinity.49 No binary phase or secondary phase was observed. With the flux method, we have synthesized relatively larger crystals (inset of Fig. 1b), which can be easily exfoliated into 2D thin layers. Our optical and electrical measurements show that VAgP2Se6 is a semiconductor with an optical bandgap of 2.31 eV (Supplementary Figs. 1 and 2), comparable to the previously reported bandgap of 2.14 eV.49
To verify the symmetry of VAgP2Se6, we performed the optical second harmonic generation (SHG). For point group 2, the crystal physics coordinates (Z1, Z2, Z3) are defined as ([010]×[001], [010], [001]). The crystal was oriented such that Z1 and Z2 were parallel to the lab X-axis and Y-axis, respectively. Figure 1c shows the experimental geometry and the collected polar plots on VAgP2Se6. Since SHG response originates from the second-order nonlinear optical process and exists only in materials that lack inversion symmetry, our observation of SHG response confirms the noncentrosymmetric structure in VAgP2Se6. The response was decomposed into components parallel to the lab X and Y and fitted to a model based on point group 2 in the crystal physics coordinates:
$${I}_{X}\propto {\left({d}_{11}{\text{cos}}^{2}\psi -{d}_{22}\text{sin}2\psi -{d}_{11}{\text{sin}}^{2}\psi \right)}^{2}$$
$${I}_{Y}\propto {({-d}_{22}{\text{cos}}^{2}\psi -{d}_{11}\text{sin}2\psi +{d}_{22}{\text{sin}}^{2}\psi )}^{2}$$
.
The good fitting suggests that the structure of VAgP2Se6 agrees with the previously reported structure with C2 space group. Its point group, 2, belongs to one of the five polar-chiral point symmetry groups (1, 2, 3, 4, and 6), and its structural chirality is highlighted in Supplementary Fig. 3.
Furthermore, previous literature suggests that VAgP2Se6 is an intrinsic ferromagnet with the Curie temperature of 19 K and the spin easy-axis oriented along the c-axis.49 The broken TRS of flux-growth VAgP2Se6 bulk crystal is verified in our magnetization measurements. Figure 1d and 1e present the temperature dependence of magnetic susceptibility of the flux-grown VAgP2Se6, measured with H//c and H//ab configurations under field-cooled (FC) and zero-field-cooled (ZFC) histories. These data, together with the magnetic hysteretic behavior probed in isothermal magnetization (Fig. 1f and 1g), reveal a ferromagnetic transition at Tc = 15 K, which is consistent with a prior report of FM ordering in the CVT-grown VAgP2Se6.49 The isothermal magnetization data also shows steep magnetic polarization, with the magnetic moments tending to saturate above 0.25T. The saturated magnetic moment is 1.5 µB/f.u for H//c (see the inset to Fig. 5c), larger than that for H//ab (1.0 µB/f.u., see the inset to Fig. 5d.), indicating the c-axis is the spin-easy axis. We also measured the magnetic properties of the CVT-grown VAgP2Se6 and found its Tc is ~ 18 K, slightly higher than that of the flux-grown samples. Consequently, the VAgP2Se6 crystal concurrently hosts chirality, broken inversion symmetry and time-reversal symmetry, providing a unique platform to explore the emergent quantum phenomena. For example, in VAgP2Se6, the combination of time-reversal symmetry breaking and the chiral crystal structure may lead to rare properties such as magnetochiral dichroism (MchD), which features a difference in the optical absorption of unpolarized light based on the relative orientation of the magnetic field and the light propagation directions.50,51
Group theory analysis of the potential domain structure
On the other hand, the reduced symmetry of VAgP2Se6 may lead to potential domain formation in the crystal. Applying the group theory is a fundamental method to analyze and determine the possible orientational domain variants. The space group of the room temperature phase of the VAgP2Se6 crystal is C2. Among the vdW metal phosphorus trichalcogenides crystal family, we found the highest symmetry space group is the \(P\stackrel{-}{3}1c\) (number 163) in CuBi(PSe3)2, ErAg(PSe3)2, GaAg(PSe3)2, InAg(PSe3)2 and InCu(PSe3)2.52 Assuming the parent structure of the VAgP2Se6 possesses the same space group symmetry, the maximal subgroup chain from \(P\stackrel{-}{3}1c\) to C2 is:
$$P\stackrel{-}{3}1c \left[12\right]\supset C2/c\left[4\right]\supset C2\left[2\right]$$
1
The number in the brackets means the number of symmetry operations of the corresponding group. Notably, the space group C2/c is also common in the same family of crystals.52 The C2 is a t-subgroup of \(P\stackrel{-}{3}1c\), with an index of it = 6.53–55 The index it equals the ratio between the orders (total number of operations) of the corresponding point groups of space group \(P\stackrel{-}{3}1c\) and C2.56 The theoretical analysis suggests the number of the potential orientational (twin) domain variants is six. Further coset decomposition of the subgroups is performed with respect to Eq. (1) to analyze the loss of symmetry elements and understand the types of domain boundaries (see method), and three distinct types of domain boundaries are identified.57 The resulting operations of each variant (coset) are listed in Table 1, and the corresponding transformation matrix of each operation is listed in the Supporting Information (Supplementary Table 1). The schematics of each domain variants in the analysis are listed in the Supporting Information (Supplementary Fig. 4).
Atomic scale observation of the domain structure in VAgP2Se6
We performed AC-STEM and acquired atomically resolved annular dark-field (ADF) STEM images to examine the nanostructure of the VAgP2Se6 crystals. The ADF-STEM images show a variety of orientational domains forming in the crystals, and the domains stack along the [001] direction. The domains and domain boundaries observed in the atomic-resolution STEM images can be directly interpreted with the group theory symmetry analysis. As illustrated in Fig. 2, We have managed to experimentally image all six orientational domain variants in the ADF-STEM images as predicted by group theory analysis. The schematics of each domain variant are shown in Fig. 2a. The atomically resolved ADF-STEM images of the six domain variants, designated as \({D}_{1}\), \({D}_{1}^{{\prime }}\), \({D}_{2}\), \({D}_{2}^{{\prime }}\), \({D}_{3}\), and \({D}_{3}^{{\prime }}\), are shown in Fig. 2b-g. It is worth noting that such a complicated domain structure along the z-axis direction in van der Waal’s crystals is not often observed and directly imaged at the atomic scale.39,46
The \({D}_{1}\), \({D}_{2}\), and \({D}_{3}\) variants are shown in Fig. 2b-d, with the schematics of the crystal structure superimposed. The \({D}_{1}\) variant represents the crystal structure viewed from [100] zone axis, where adjacent layers stack in the vertical direction; on the other hand, the \({D}_{2}\) and \({D}_{3}\) variants represent the structure viewed from [\(\stackrel{-}{1}10\)] and [\(\stackrel{-}{1}\stackrel{-}{1}0\)] zone axis, where the layers stacking directions ([001]) are tilted to the right or left side, respectively. After a close examination of the crystal structure, we found that \({D}_{1}\), \({D}_{2}\), and \({D}_{3}\) domain variants are featured by an in-plane rotation of roughly 120o around the [001] axis.
In contrast, the \({D}_{1}^{{\prime }}\), \({D}_{2}^{{\prime }}\), and \({D}_{3}^{{\prime }}\) domain variants feature a reversal of the [010] axis (chirality switching) when compared, respectively, with \({D}_{1}\), \({D}_{2}\), and \({D}_{3}\) domain variants. To highlight the crystal structural change across the domain walls, Fig. 2e-g show two sets of domain walls between \({D}_{1}\) & \({D}_{1}^{{\prime }}\), \({D}_{2}\) & \({D}_{2}^{{\prime }}\), and \({D}_{3}\) & \({D}_{3}^{{\prime }}\), respectively. When comparing the domain variants \({D}_{1}\) & \({D}_{1}^{{\prime }}\) in the atomically resolved ADF-STEM images ([100] zone axis), the most distinctive feature is a switch between the Ag (brighter) and V (darker) atomic column positions across the domain boundaries, which is caused by the reversal of the [010] axis. The same observation also applies to \({D}_{2}\) & \({D}_{2}^{{\prime }}\), and \({D}_{3}\) & \({D}_{3}^{{\prime }}\).
In addition, the domain boundary between any two orientational domains can be classified into three distinct types of boundaries (see methods for derivation), as shown in Fig. 3. The first type is the interface between \({D}_{1}\)/\({D}_{2}\), \({D}_{1}\)/\({D}_{3}\), \({D}_{2}\)/\({D}_{3}\), \({D}_{1}^{{\prime }}\)/\({D}_{3}^{{\prime }}\), \({D}_{1}^{{\prime }}\)/\({D}_{2}^{{\prime }}\), and \({D}_{2}^{{\prime }}\)/\({D}_{3}^{{\prime }}\), which features a 120° rotation around the (001)m plane normal (the subscript m stands for monoclinic C2) across the boundary. An experimental ADF-STEM image of the type I boundary between \({D}_{2}^{{\prime }}\) & \({D}_{3}^{{\prime }}\) is shown in Fig. 3a. The second type is the interface between \({D}_{1}\)/\({D}_{1}^{{\prime }}\), \({D}_{2}\)/\({D}_{2}^{{\prime }}\), and \({D}_{3}\)/\({D}_{3}^{{\prime }}\). Two domains across the second type of boundaries possess the same directions for the [100]m axis and [001]m axis but an opposite direction for the [010]m axis. An experimental ADF-STEM image of the type II boundary between \({D}_{3}^{{\prime }}\) & \({D}_{3}\) is shown in Fig. 3b. The third type is the interface between \({D}_{1}\)/\({D}_{2}^{{\prime }}\), \({D}_{1}\)/\({D}_{3}^{{\prime }}\), \({D}_{2}\)/\({D}_{3}^{{\prime }}\), \({D}_{2}\)/\({D}_{1}^{{\prime }}\), \({D}_{3}\)/\({D}_{1}^{{\prime }}\), and \({D}_{3}\)/\({D}_{2}^{{\prime }}\), which features a 120° rotation around the (001)m plane normal and the reversed [010]m axis across the boundary. An experimental ADF-STEM image of the type III boundary between \({D}_{3}^{{\prime }}\) & \({D}_{1}\) is shown in Fig. 3c. The second and third types of domain boundaries feature chirality switching across the domain boundaries. We further constructed the crystal models of three types of boundaries based on the interpretation from group theory analysis and performed the ADF-STEM image simulations (Fig. 3d-f). The simulated ADF-STEM images show remarkable agreement with the experimental ADF-STEM images.
Altered domain structure with different synthesis methods
Furthermore, there are alterations in overall domain structures for the VAgP2Se6 crystals synthesized with different methods. To examine the crystal structures at the atomic level, we performed AC-STEM on two TEM specimens, one extracted from the crystal synthesized with CVT and the other from the crystal grown with the flux method. Two representative ADF-STEM images acquired from different regions in the CVT-grown sample are shown in Fig. 4a and 4b. The ADF-STEM images acquired from the flux-grown sample are shown in Fig. 4c and 4d. For visualizing the domain structures, each domain variant is marked with filled or shaded bars with different colors on the right side of the STEM images in Fig. 4a-4d. \({D}_{1}\) & \({D}_{1}^{{\prime }}\) and \({D}_{3}\) & \({D}_{3}^{{\prime }}\) domain variants occur more frequently than \({D}_{2}\) & \({D}_{2}^{{\prime }}\) in STEM images taken from the CVT-grown crystal (Fig. 4a and 4b). On the other hand, in the flux-grown crystal (Fig. 4c and 4d), the different domain variants randomly appear and stack along the [001] axis. Additionally, the density of the domain boundaries is higher, and the domain size is smaller in the crystal synthesized with the flux method. The averaged domain size measured in these STEM images is around 1.79 nm (2.6 layers) for CVT-grown crystal and around 0.85 nm (1.2 layers) for flux-grown crystal. Consequently, the average size of the superstructure formed by any two adjacent domains is around 3.58 nm and 1.70 nm for CVT and flux-grown crystals, respectively.
The observation of the complicated domain structures further motivated an analysis of the reciprocal space through electron diffraction patterns. The selected area electron diffraction (SAED) patterns of two samples synthesized with the CVT and flux methods were acquired and analyzed, as shown in Fig. 5a and 5b. Numerous extra reflections appeared in different locations of the SAED patterns indicating the complicated yet distinct microstructures for both samples. For example, Fig. 5c and 5d show the magnified SAED patterns around [020] reflections (from the yellow dashed box in Fig. 5a and 5b) from the CVT and flux sample, respectively. In the SAED pattern acquired from the CVT-grown crystal (Fig. 5c), discrete reflections connected by the faint streaks were observed, suggesting the formation of the planar defects with the (001) interface.
As shown in Fig. 5c, we identified three pairs of reflections 020/02\(\stackrel{-}{1}\), \(\stackrel{-}{1}\stackrel{-}{1}0\)/\(\stackrel{-}{1}\stackrel{-}{1}\stackrel{-}{1}\) and \(1\stackrel{-}{1}1\)/\(1\stackrel{-}{1}0\), and they were pointed by the red, green, and blue arrows. These three pairs of reflections originated from the diffraction from [100], [\(\stackrel{-}{1}10\)] and [\(\stackrel{-}{1}\stackrel{-}{1}0\)] zone axes, respectively. The appearance of these three zone axes is attributed to the presence of \({D}_{1}\) & \({D}_{1}^{{\prime }}\), \({D}_{2}\) & \({D}_{2}^{{\prime }}\), and \({D}_{3}\) & \({D}_{3}^{{\prime }}\) domains. As a result, the appearance of these reflections agrees with the existence of domain variants with different crystallographic orientations within the sample. In Fig. 4c, the intensity of the 020/02\(\stackrel{-}{1}\) and \(1\stackrel{-}{1}1\)/\(1\stackrel{-}{1}0\) reflections is much stronger than that of the \(\stackrel{-}{1}\stackrel{-}{1}0\)/\(\stackrel{-}{1}\stackrel{-}{1}\stackrel{-}{1}\) reflections. The variations in the reflection intensity suggest that the crystallographic variants have dissimilar volume fractions. Since \(\stackrel{-}{1}\stackrel{-}{1}0\)/\(\stackrel{-}{1}\stackrel{-}{1}\stackrel{-}{1}\) reflections originate from \({D}_{2}\) and \({D}_{2}^{{\prime }}\) domain variants, the low intensity indicates the infrequent occurrence of \({D}_{2}\) & \({D}_{2}^{{\prime }}\) variants in the CVT-grown crystal. This interpretation agrees with the observation of the ADF-STEM image of the CVT-grown crystal (Fig. 4a and 4b). In addition, the faint streaking and elongated superlattice reflections are observed between the Bragg peaks. Both the faint streaking and the superlattice reflections suggest the formation of the superstructures. The elongated shape of superlattice reflections indicates the variation in the size of the superstructures along the [001] direction.
In contrast, for the flux-grown crystal, the presence of a row of near-continuous (strong streaking) and modulated intensity along the 02l (l is an integer) reflections is characteristic of its SAED pattern (Fig. 5d). The near-continuous intensity indicates the formation of very fine domains along the (001) interfaces. The reason is that the width of the reciprocal space lattice in the z*-direction is inversely proportional to the domains (supercells) size in the z-direction in the real space,58,59 which agrees with the fine domains with the averaged size of 0.85 nm (1.2 layers) stacking along the z-direction in flux-grown crystal shown in the ADF-STEM image analysis (see Fig. 4c and d).
Furthermore, the extra superlattice reflections are also observed along the g = (001) direction, as shown in Fig. 5d and 5e. The locations of the extra reflections in the reciprocal space are different for the two crystals, which also suggests the varied periodicities for the superstructures along the [001] direction. To further understand the superstructure periodicity, the intensity line profile along the [001] direction is plotted as shown in Fig. 5g and 5h. There are two sets of superlattice reflections for the SAED pattern from the CVT-grown crystal (Fig. 5g). One set of superlattice reflections is closer to the Bragg reflections (0.24 nm− 1), and the other set with lower intensity is at the (0, 0, l + \(\frac{1}{2}\)) (0.76 nm−1). For the SAED pattern from the flux-grown crystal (Fig. 5h), there is one set of superlattice reflections at 0.70 nm− 1 away from the Bragg reflections. We performed electron diffraction simulations and discovered that the measured distances between these peaks are directly linked to the periodicity of the superstructure (see Supplementary Fig. 5 for electron diffraction simulations). The average superstructure sizes in the real space can be calculated by taking the inverse of the distance between superlattice spots and Bragg spots in the reciprocal space. We find the averaged periodicity of 4.17 nm and 1.43 nm for the superstructures in the CVT and flux-grown specimens, respectively. The superstructure periodicity measurement in the reciprocal space agrees with the measurement from atomically resolved STEM images.