Abstract
Magnetoelastic distortions are commonly detected across magnetic longrange ordering (LRO) transitions. In principle, they are also induced by the magnetic shortrange ordering (SRO) that precedes a LRO transition, which contains information about shortrange correlations and energetics that are essential for understanding how LRO is established. However these distortions are difficult to resolve because the associated atomic displacements are exceedingly small and do not break symmetry. Here we demonstrate highmultipole nonlinear optical polarimetry as a sensitive and mode selective probe of SRO induced distortions using CrSiTe_{3} as a testbed. This compound is composed of weakly bonded sheets of nearly isotropic ferromagnetically interacting spins that, in the Heisenberg limit, would individually be impeded from LRO by the MerminWagner theorem. Our results show that CrSiTe_{3} evades this law via a twostep crossover from two to threedimensional magnetic SRO, manifested through two successive and previously undetected totally symmetric distortions above its Curie temperature.
Introduction
Ferromagnetic (FM) semiconductors belonging to the transition metal trichalcogenide family have recently been shown to be promising starting materials for realizing monolayer ferromagnets by exfoliation^{1,2}. However predicting the viability of the ferromagnetic longrange ordered state in the 2D limit relies on first understanding how FM longrange ordering (LRO) is established in the 3D bulk crystals, which is often unclear. A case in point is CrSiTe_{3}^{3,4,5,6,7,8}, which consists of ABC stacked sheets of Cr^{3+} (spin3/2) moments arranged in a honeycomb network (Fig. 1). Each Cr atom is coordinated by six Te atoms that form an almost perfect octahedron^{7}, giving rise to a near isotropic (Heisenberg) spin state and dominant FM nearest neighbor exchange interactions (J_{ab } < 0) owing to the near 90° Cr–Te–Cr bond angle. This is corroborated by inelastic neutron scattering experiments^{7} on bulk CrSiTe_{3}, which report a relatively feeble easyaxis (Ising) anisotropy strength (D/J_{ab} < 2 %). According to the MerminWagner theorem^{9}, LRO should be forbidden in a strictly 2D Heisenberg system. Therefore the finite value of the Curie temperature (T_{c} ~ 31 K) in bulk CrSiTe_{3} (Fig. 1) must either be driven by the weak spin anisotropy or by a weak interlayer coupling that mediates a crossover from 2D to 3D character. A dimensional crossover can in principle be uncovered by tracking the spatial anisotropy of shortrange spin correlations using magnetic neutron and Xray scattering techniques. However the requirement of nearly ideal bulk crystals and the difficulty of detecting and integrating diffuse magnetic scattering is restrictive and currently renders these techniques inoperable on exfoliated nanoscale thick sheets. Hence this mechanism is yet to be verified in CrSiTe_{3} or related compounds^{1}.
An alternative route to measuring shortrange spin correlations is through their effects on the crystal lattice. The magnetic energy of an insulating system is given by the thermal expectation value of its magnetic Hamiltonian \({\cal{H}}_m = J_{ij}\mathop {\sum }\nolimits_{i,j} \vec S_i \cdot \vec S_j\) which contains the shortrange spin correlator \(\langle\vec S_i \cdot \vec S_j\rangle\) as well as the exchange interaction J_{ij} between spins at sites i and j. Upon onset of magnetic shortrange ordering (SRO), it may be energetically favorable for the system to readjust the distances and bonding angles between atoms that mediate J_{ij} in order to lower its magnetic energy, at the expense of some gain in elastic energy. Measuring such magnetoelastic distortions therefore yields information about spin correlations along various directions and, for simple low dimensional Hamiltonians, can even provide quantitative values of the \(\langle\vec S_i \cdot \vec S_j\rangle\) function^{10}, which is difficult to obtain by neutron scattering because only a limited range of its spatiotemporal Fourier components are accessed. However, SRO induced distortions are extremely hard to resolve because they are minute by virtue of \(\langle\vec S_i \cdot \vec S_j\rangle\) being small, and because they generally do not break any lattice symmetries. A suitable probe must therefore be sensitive to and able to distinguish between different totally symmetric distortions (i.e., different basis functions of the totally symmetric irreducible representation). This suggests that examining the nonlinear high rank tensor responses of a crystal may be a promising approach.
Optical second harmonic generation (SHG), a frequency doubling of light produced by its nonlinear interactions with a material, is governed by high rank (>2) susceptibility tensors that are sensitive to many degrees of freedom in a crystal. Traditionally SHG has been exploited as a symmetry sensitive probe because the leading order electric dipole susceptibility necessarily vanishes if the system possesses a center of inversion. This makes SHG particularly powerful for studying surfaces of centrosymmetric crystals^{11}, and for identifying bulk symmetry breaking phase transitions through the appearance of additional, high symmetry forbidden, tensor elements^{12,13,14,15}. In principle, SHG can also be utilized to study symmetry preserving distortions^{16} by examining their subtle effects on the existing symmetry allowed tensor elements. However this potential capability is highly underexplored, in part due to the technical demand of simultaneously tracking small changes across an entire set of allowed tensor elements.
Recently we managed to surmount this challenge by developing a rotating scattering plane based SHG polarimetry technique^{17}. In these experiments, linear (either P or S) polarized light of frequency ω is focused obliquely onto the surface of a bulk single crystal. The intensity of either the P or S component of reflected light at frequency 2ω is then measured as a function of the angle (φ) that the scattering plane is rotated about the caxis (Fig. 2a), which allows a multitude of SHG susceptibility tensor elements to be sampled. By collecting these rotational anisotropy (RA) patterns with different polarization combinations, a complete set of SHG susceptibility tensor elements can typically be uniquely determined. Here we apply this technique to track the magnitudes of all of the symmetry allowed SHG susceptibility tensor elements of CrSiTe_{3} as a function of temperature. Evidence of previously undetected structural distortions are observed above T_{c} at T_{2D} ~ 110 K and T_{3D} ~ 60 K. Using a hyperpolarizable bond model, we are able to attribute the distortions at T_{2D} and T_{3D} to displacements along different totally symmetric normal mode coordinates, which are consistent with an onset of intralayer and interlayer spin correlations respectively.
Results
SHG polarimetry results
The full temperature evolution of the RA patterns acquired from the (001) surface of CrSiTe_{3} under select polarization geometries is displayed in Fig. 2b. We first note that a finite weak SHG intensity is present at all temperatures despite previous work showing that CrSiTe_{3} always retains a centrosymmetric structure with \(\bar 3\) point group symmetry^{18}. This suggests that the SHG originates from a higher multipole process such as electric quadrupole (EQ) radiation, which is governed by a fourth rank susceptibility tensor \(\chi _{ijkl}\) that has only eight independent nonzero elements (\(\chi _{xxxz}\), \(\chi _{xxyy}\), \(\chi _{xxzz}\), \(\chi _{yxxx}\), \(\chi _{yyyz}\), \(\chi _{zzxx}\), \(\chi _{zzxy}\), \(\chi _{zzzz}\)) after accounting for the symmetries of the \(\bar 3\) point group and the degeneracy of the incident electric fields^{19}. Expressions for the RA SHG intensity \(I\left( {2\omega } \right) \propto \left {\hat e_i^{2\omega }\left( \varphi \right)\chi _{ijkl}\hat e_j^\omega \left( \varphi \right)\kappa _k\left( \varphi \right)\hat e_l^\omega \left( \varphi \right)} \right^2I_0^2\left( \omega \right)\) derived under these conditions (here \(\hat e\) are the polarization directions, \(\kappa\) is the incident wave vector and \(I_0\) is the incident intensity; see Supplementary Note 1) indeed produce excellent fits to our set of RA patterns at any given temperature and allow us to uniquely determine the values of \(\chi _{ijkl}\) at each temperature. In contrast, other possible allowed SHG processes such as surface electric dipole or bulk magnetic dipole radiation cannot reproduce the RA data and are thus treated as negligibly small (see Supplementary Note 2).
From the raw RA data (Fig. 2b) we can clearly discern the bulk threefold rotational symmetry of CrSiTe_{3} and, as expected, we observe no change in symmetry as a function of temperature. Yet the absolute and relative intensities of the various features do undergo changes upon cooling, which must encode symmetry preserving distortions. Most notably, there is a dramatic increase of intensity below T_{c} that, as we will show later on, arises from LRO induced magnetoelastic distortions that have previously been detected by optical absorption^{20}, Raman scattering^{20}, and Xray diffraction^{18}, and are also captured by our dilatometry measurements (Fig. 1a). Surprisingly however, we find that the RA patterns continue to subtly evolve even far above T_{c}. In SP polarization geometry for example (Fig. 2c), representative RA patterns at 140, 80, and 40 K have qualitatively different shapes, indicating that the magnitude of the \(\chi _{ijkl}\) elements change nonuniformly with temperature.
Temperature dependence of the nonlinear susceptibility
The temperature (T) dependence of each of the eight individual \(\chi _{ijkl}\) elements was extracted through the aforementioned fitting procedure (Fig. 2c). From every \(\chi _{ijkl}\left( T \right)\) curve, we subtracted a high temperature background using the data above 150 K, where the shapes of the RA patterns have ceased evolving (see Supplementary Note 3). Figure 3 shows the complete set of background subtracted curves \({\mathrm{\Delta }}\chi _{ijkl}\left( T \right)\) that have all been normalized to their low temperature values. Three distinct sets of behavior are clearly resolved. Below a characteristic temperature T_{2D} ~ 110 K, the xxxz and yyyz elements alone start to grow in tandem (Fig. 3a). Then below a second characteristic temperature T_{3D} ~ 60 K, a solitary zzzz element begins to grow (Fig. 3b). The temperature dependence of the former and latter set of elements are sublinear above T_{c} and scale with classical calculations of the nearest neighbor intralayer and interlayer spin correlators respectively (see Supplementary Note 5). By contrast, below T_{c} the remaining five elements turn up with an order parameter like temperature dependence indicative of a phase transition (Fig. 3c), with a critical exponent twice that reported^{7} for the magnetization (2β ≈ 0.3). Since magnetoelastic distortions scale like the square of the magnetic order parameter, this further confirms that \(\chi _{ijkl}\) is probing the lattice degrees of freedom. This also shows that \(\chi _{ijkl}\) is a timereversal invariant itensor^{19}, which naturally explains why our measurements are insensitive to magnetic domains^{21}.
Microscopic origin of the susceptibility change
To understand the microscopic origin of the features in \({\mathrm{\Delta }}\chi _{ijkl}\left( T \right)\), we appeal to a simplified hyperpolarizable bond model^{22}, which treats the crystal as an array of charged anharmonic oscillators centered at the chemical bonds and constrained to only move along the bond directions. The nonlinear polarizability of each oscillator is calculated by solving classical equations of motion, and then appropriately summed together to form the total nonlinear susceptibility. Recently an expression for the EQ SHG susceptibility was derived using this model^{23} and was found to take the form \(\chi _{ijkl} \propto \mathop {\sum }\nolimits_n \alpha _\omega \alpha _{2\omega }\left( {\hat b_n \otimes \hat b_n \otimes \hat b_n \otimes \hat b_n} \right)_{ijkl}\), where \(\alpha _\omega\) and \(\alpha _{2\omega }\) are the firstorder (linear) and secondorder (hyper) polarizabilities, \(\hat b_n\) is a unit vector that points along the n^{th} bond, and all bond charges are assumed equal. Using this expression, we investigated how distortions along each of the four totally symmetric normal mode coordinates allowed in the \(\bar 3\) point group (i.e., the four basis functions \(A_g^1\), \(A_g^2\), \(A_g^3\) and \(A_g^4\) of its totally symmetric irreducible representations) change the individual \(\chi _{ijkl}\) elements.
For simplicity, we considered only the nearest neighbor intralayer Cr–Te bonds and the nearest neighbor interlayer Cr–Cr bonds, which is reasonable because the states accessed by our photon energy (2ħω = 3 eV) are predominantly composed of Cr and Te orbitals^{24,25}. Remarkably, our hyperpolarizable bond model shows that under a small distortion along the \(A_g^1\) normal coordinate δ, which we implement by changing the \(\hat b_n\) while keeping the \(\alpha _\omega \alpha _{2\omega }\) values constant, only the xxxz and yyyz elements are affected (Fig. 4a), in perfect agreement with our observations below T_{2D} (Fig. 3a). Since motion along \(A_g^1\) deforms the Te octahedra and can bring the Cr–Te–Cr bond angle closer to 90° to strengthen J_{ab}, it is natural to associate this distortion with the development of FM inplane spin correlations. This is further supported by neutron scattering experiments^{7}, which show a rise in magnetic diffuse scattering around T_{2D} (inset Fig. 3a) indicative of a growing inplane correlation length \(\xi _{ab}\).
To uncover a mechanism that would exclusively affect the zzzz element below T_{3D} (Fig. 3b), we note that the distortion along the \(A_g^2\) normal coordinate involves a pure outofplane displacement of the Cr atoms. Although this motion does not change the \(\hat b_n\) of the interlayer Cr–Cr bonds since they remain parallel to the zaxis, it will change their polarizabilities by virtue of their altered bond length. Assuming that it is these Cr–Cr bonds that primarily contribute to the observed changes at T_{3D} (see Supplementary Note 6), our model indeed shows that tuning either \(\alpha _\omega\) or \(\alpha _{2\omega }\) of the Cr–Cr bond will exclusively affect the zzzz element (Fig. 4b). This naturally suggests an association of the \(A_g^2\) distortion with the development and enhancement of FM interlayer spin correlations, and hence an identification of T_{3D} as the 2D to 3D dimensional crossover temperature. Independent evidence for a structural distortion at T_{3D} was also found via anomalies in the E_{g} and E_{u} phonons using impulsive stimulated Raman scattering (see Supplementary Note 7) and infrared absorption measurements^{20} respectively (inset Fig. 3b), which likely arise from their nonlinear coupling to the \(A_g^2\) distortion.
As a further consistency check, we note that one expects interlayer correlations to onset when \(\xi _{ab}\) grows to a size where the total interlayer exchange energy becomes comparable to the temperature. In a mean field approximation, this condition is expressed as \(T = N\left( T \right)J_cS\left( {S + 1} \right)/3k_B\), where N is the number of inplane correlated spins of magnitude S that are interacting with the next layer, J_{c} is the interlayer Cr–Cr exchange and \(k_B\) is Boltzmann’s constant. Using the values of \(\xi _{ab}\left( T \right)\) and J_{c} determined from neutron scattering^{7}, we find a solution to the mean field equation at T ~ 70 K (see Supplementary Note 8), which is reasonably close to T_{3D}. Displacements along the remaining two \(A_g^3\) and \(A_g^4\) normal coordinates are found from our model to affect all eight of the tensor elements (Fig. 4c, d) and are therefore not measurably induced at either T_{2D} or T_{3D}. It is possible that they occur below T_{c} where we observe all elements to change (Fig. 3), but details of LRO induced distortions are outside the scope of this work.
Discussion
Our EQ SHG data and analysis taken together provide a comprehensive picture of how the quasi2D Heisenberg ferromagnet CrSiTe_{3} evades the MerminWagner theorem via a multiple stage process to establish longrange spin order (Fig. 4e), and shows that interlayer interactions are vital to stabilizing LRO at such high temperatures. More generally, our results demonstrate that the nonlinear optical response is a highly effective probe of shortrange spin physics and their associated totally symmetric magnetoelastic distortions, which are typically unresolvable by capacitance dilatometry^{10} (Fig. 1a) or lower rank optical processes like linear reflectivity and Raman scattering due to their limited degrees of freedom (see Supplementary Note 9), and are challenging to detect by diffraction based techniques limited to picometer resolution^{18}. This technique will be particularly useful for studying anisotropic or geometrically frustrated magnetic systems, which tend to display interesting shortrange spin correlations. It will also be useful for uncovering magnetic ordering mechanisms in monolayer or few layer ferromagnetic and antiferromagnetic nanoscale flakes and devices^{26,27,28,29}, which are often unclear because of their inaccessibility by neutron diffraction. We anticipate that access to this type of information may offer new strategies to control magnetism based on manipulating SRO induced distortions through chemical synthesis, static perturbations or even outofequilibrium excitations^{30}.
Methods
Sample growth and characterization
The CrSiTe_{3} crystals used in this study were grown using a Te selfflux technique^{20}. High purity Cr (Alfa Aesar, 99.999%), Si (Alfa Aesar 99.999%), and Te (Alfa Aesar 99.999%) were weighed in a molar ratio of 1:2:6 (Cr:Si:Te) and loaded into an alumina crucible sealed inside a quartz tube. The quartz ampoule was evacuated and backfilled with argon before sealing. Platelike crystals up to 5 mm thick with flat, highly reflective surfaces were then removed from the reaction crucible. Xray diffraction (XRD) data collected on crushed crystals using an Emperyan diffractometer (Panalytical) confirmed the correct \(R\bar 3\), space group 148, CrSiTe_{3} phase. Measurements of the temperature and fielddependence of the magnetization were carried out using a Magnetic Property Measurement System (MPMS, Quantum Design). The samples were mounted with the field applied parallel to the abplane of the crystals for magnetization and susceptibility measurements. The thermal expansion coefficient was measured using the Quantum Design dilatometer option in a PPMS DynaCool. Dilation was measured along the caxis; sample thickness in this direction was 0.41 mm. Data were collected under a ramp rate of 0.1 K/min.
RA SHG measurements
Incident light with <100 fs pulse width and 800 nm center wavelength was derived from a ti:sapph amplified laser system (Coherent RegA) operating at 100 kHz. Specular reflected secondharmonic light at 400 nm was selected using shortpass and narrow bandpass filters and measured with a twodimensional EMCCD camera (Andor iXon Ultra 897). Both the sample and detector remained fixed while the scattering plane is rapidly mechanically spun about the central beam axis. The angle of incidence was fixed at 10°. A detailed description of the RA SHG apparatus used can be found in ref. ^{17}. The fluence of the beam was maintained at ~340 μJ cm^{−2} with a spot size of ~30 μm FWHM. The close agreement between the T_{c} values measured using RA SHG and magnetic susceptibility indicates negligible average heating by the laser beam. Each complete RA pattern was acquired with a 5 min exposure time. Samples (~1 mm × 2 mm × 0.1 mm) were cleaved prior to measurement and immediately pumped down in an optical cryostat to a pressure better than 10^{−6} Torr.
Data availability
The datasets generated are/or analyzed during the current study are available from the corresponding author on reasonable request.
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Acknowledgements
This work was supported by ARO MURI Grant No. W911NF16–1–0361. D.H. also acknowledges support for instrumentation from the David and Lucile Packard Foundation and from the Institute for Quantum Information and Matter, an NSF Physics Frontiers Center (PHY1733907). A.R. acknowledges support from the Caltech Prize Fellowship. The MRL Shared Experimental Facilities are supported by the MRSEC Program of the NSF under Award No. DMR 1720256; a member of the NSFfunded Materials Research Facilities Network. S.D.W. acknowledges support from the Nanostructures Cleanroom Facility at the California NanoSystems Institute (CNSI). We thank Tom Hogan for performing the dilatometry measurements and Liangbo Liang, David Mandrus, Jan Musfeldt, Kai Xiao, and Houlong Zhuang for helpful discussions.
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A.R. and D.H. conceived the experiment. A.R. performed the optical measurements. A.R., D.H., and L.B. analysed the data. L.B. performed the classical Heisenberg model calculations. E.Z. and S.D.W. prepared and characterized the sample. A.R. and D.H. wrote the manuscript.
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Ron, A., Zoghlin, E., Balents, L. et al. Dimensional crossover in a layered ferromagnet detected by spin correlation driven distortions. Nat Commun 10, 1654 (2019). https://doi.org/10.1038/s41467019096633
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DOI: https://doi.org/10.1038/s41467019096633
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