Dimensional crossover in a layered ferromagnet detected by spin correlation driven distortions

Magneto-elastic distortions are commonly detected across magnetic long-range ordering (LRO) transitions. In principle, they are also induced by the magnetic short-range ordering (SRO) that precedes a LRO transition, which contains information about short-range 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 high-multipole nonlinear optical polarimetry as a sensitive and mode selective probe of SRO induced distortions using CrSiTe3 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 Mermin-Wagner theorem. Our results show that CrSiTe3 evades this law via a two-step crossover from two- to three-dimensional magnetic SRO, manifested through two successive and previously undetected totally symmetric distortions above its Curie temperature.


Supplementary Note 1. Mathematical expressions for EQ induced RA-SHG patterns
Expressions for the bulk EQ derived RA-SHG intensity I from a crystal with 3 � point group symmetry for the four experimental polarization configurations are given below, where θ = 10° is the experimental angle of incidence. The RA-SHG data were simultaneously fit to all four expressions to extract the values of the susceptibility tensor elements.   (1) (4)

Supplementary Note 2. Best fits to alternative SHG processes
Supplementary Figure 1. Best fits of the RA-SHG data (circles) at T = 200 K to the following SHG processes (blue curves): (from top to bottom) surface ED, bulk MD (MMM), bulk MD (EEM), bulk MD (MEE) and bulk EQ. Simultaneous fits were made to the data in four different polarization configurations: PP, SP, SS and PS. The intensity in the SS and PS channels is ~ 50 % smaller than in the PP and SP channels, which is the reason for the larger scatter in those data sets.
We examined all of the leading order contributions to SHG allowed in a centrosymmetric crystal: the surface electric-dipole (ED) process given by = , three bulk magnetic-dipole (MD) processes given by = , = and = , and a bulk electric-quadrupole (EQ) process given by = ∇ , where P and M are the induced polarization and magnetization, and E and H are the incident electric and magnetic fields respectively. Representative best fits of each of these processes to our RA-SHG data are shown in Supplementary Figure 1. Clear discrepancies are observed for the surface ED and bulk MMM and MEE processes and are therefore immediately ruled out. Discrepancies in the bulk EEM fits also exist but are more subtle. Most notably, the small lobes in both PP and SP polarizations are not adequately captured over most of the temperature range of the experiment. Although these discrepancies in the bulk EEM fits are difficult to notice in the polar plots shown in Supplementary Figure

Supplementary Note 3. High temperature background subtraction procedure and fits
To obtain the curves shown in Figure 3 of the main text, we first extracted the temperature dependence of each susceptibility tensor element by fitting the expressions given in Supplementary Note 1 to the raw RA-SHG data. For each tensor element independently, we then subtracted a weakly linear background using the data above 150 K where the shapes of the RA-SHG patterns have ceased changing (i.e. far above where magnetic correlations start to affect our data). In Supplementary Figure 3 we explicitly show that over the temperature range used for estimating the background, there is indeed no measureable change in shape of the RA-SHG patterns. The direct fit results before any background subtraction or normalization is performed are shown in Supplementary To understand the temperature dependence of the short-range spin correlations in CrSiTe 3 , we calculated the nearest neighbor correlations from a classical Heisenberg model, treating the spins as vectors of length S = 3/2 and using the exchange parameters from Ref. [7]. To calculate the correlations, we used a large-N approximation, expected to be reasonably accurate above the critical temperature. Specifically, the fixed length constraint is implemented on average via a Lagrange multiplier λ(T), which is self-consistently determined at each temperature. The spin correlations were then obtained as the Fourier transform of the inverse exchange interaction matrix, shifted appropriately by λ(T). This approach gives T c = 33.8 K, which agrees excellently with experiment and serves as a further self-consistency check of the dimensional crossover picture. Supplementary Figure 6 shows the calculated temperature dependence of the nearest neighbor intralayer and nearest neighbor interlayer spin correlator, which shows a sub-linear temperature dependence that is consistent with that of the xxxz, yyyz and zzzz susceptibility tensor elements shown in Figure 3 of the main text. Supplementary Figure 7. a, Change in the susceptibility tensor elements due to a displacement along the g 2 normal coordinate calculated using the hyper-polarizable bond mode including only the Cr-Te bonds. b, Schematic of the displacement δ of the Cr atoms.

Supplementary Note 6. Effect of g distortion on Cr-Te bond contribution to ∆
In Figure 4b of the main text, we showed how a displacement along the g 2 normal coordinate affects the interlayer Cr-Cr bond contribution to . Here we show how this distortion affects the intralayer Cr-Te bond contribution to through a modification of the � associated with the Cr-Te bonds. As shown in Supplementary Figure 7, it is again the zzzz element that is primarily affected, consistent with our experiments (Fig. 3b). However, in this case the xxxz, yyyz, xxzz, zzxx and xxyy elements are also affected to a smaller degree, which we cannot resolve experimentally. Therefore we conclude that the Cr-Cr bond contribution dominates over the Cr-Te contribution for the g 2 distortion.

Supplementary Note 7. Details of impulsive stimulated Raman scattering measurements
To independently verify the presence of a structural distortion at T 3D , we performed impulsive stimulated Raman scattering (ISRS) measurements on CrSiTe 3 to track the behavior of Raman active phonons as a function of temperature. In our experiment, the sample was excited using an ultrashort (< 100 fs) optical pump pulse at a wavelength of 1200 nm, which excites coherent phonons through the ISRS mechanism. Phonon vibrations were then resolved in the time-domain by measuring the instantaneous polarization rotation angle θ of a reflected optical probe pulse at a wavelength of 800 nm as a function of time-delay t. As shown in Supplementary Figure 8, traces of θ(t) exhibit clear oscillations due to coherent phonons and are dominated by two frequency components at ~3.7 THz and ~2.8 THz, which correspond to the frequencies of the A g and E g modes respectively. The amplitudes of the modes were determined by the peak intensities of the Fourier transform (inset Supplementary Figure 8) and tracked as function of temperature. A clear anomaly in the amplitude of the E g mode was observed at T 3D (inset Fig. 3b). No clear anomaly at T 3D was found in the A g mode, likely indicating a weaker nonlinear coupling to the g 2 distortion.
Supplementary Figure 8. Time-resolved polarization rotation of CrSiTe 3 acquired at T = 15 K showing a beat pattern coming from the E g and A g phonon modes. The two frequency components are more clearly resolved in the Fourier transform (inset).