Spin-induced negative thermal expansion and spin–phonon coupling in van der Waals material CrBr3

The two-dimensional van der Waals (vdW) magnets retaining magnetic order in atomically thin limit demonstrate challenging physical phenomena and they are considered as prospective building blocks for construction of advanced spintronics and nanoelectronics devices. Here, we present experimental evidence for negative thermal expansion of lattice volume and vdW layers and strong spin–phonon coupling effects, caused by formation of the long-range ferromagnetic order in the vdW material CrBr3. The neutron and X-ray diffraction measurements revealed anomalous temperature variation of lattice parameters and interatomic distances and angles in the vicinity of Curie temperature (TC). A pronounced rise of the frequencies of the most of the observed vibrational modes and unusual reversal broadening of their full widths at half maximum below TC was found from Raman spectroscopy measurements.


INTRODUCTION
Recent discovery of magnetism in the atomically thin twodimensional (2D) van der Waals (vdW) materials has put them into focus of extensive scientific research due to importance of such materials for development of 2D spintronic and nanoelectronic devices as well as study of challenging physical phenomena like exotic quantum phases, topological spin excitations, etc. [1][2][3][4][5][6] .
The family of cleavable layered semiconductors CrX 3 (X = Cl, Br, I) is of particular interest due to ferromagnetic intralayer coupling and comparable magnetic ordering temperatures in bulk and atomically thin forms 5,7,8 . The CrBr 3 crystallizes in the rhombohedral BiI 3 structure of R3 symmetry and retains this structure at low temperatures. The CrI 3 and CrCl 3 crystallize in the monoclinic AlCl 3 structure of C2/m symmetry and undergo a structural phase transition to the rhombohedral R3 structure below 210 and 240 K, respectively [8][9][10] . In the bulk form of CrBr 3 and CrI 3 , both the intralayer and interlayer exchange couplings are ferromagnetic (FM), resulting in the formation of the FM ground state below T C = 37 and 61 K. In the CrCl 3 the interlayer exchange coupling becomes antiferromagnetic (AFM), leading to appearance of the AFM ground state below T N = 17 K 7-10 . Considering the 2D fewlayered forms of these materials, in the case of CrBr 3 the intralayer and interlayer couplings remain ferromagnetic and the magneticordering temperature in the monolayer limit decrease quite slightly to 27 K. In contrast, CrI 3 demonstrates a change of interlayer exchange coupling towards antiferromagnetic one and the magnetic ordering temperature is reduced more significantly to 45 K in the monolayer limit. In CrCl 3 , the interlayer coupling retains antiferromagnetic nature and the magnetic-ordering temperature remains about the same, T N = 16 K in the two-layer limit 7 .
In the few-layer forms of CrX 3 and heterostructures containing these materials a number of emerging phenomena, including electric field and stacking-dependent control of magnetism, giant tunneling magnetoresistance, giant nonreciprocal second harmonics generation, magnetic proximity effect, important for development of broad range spintronic devices, was observed [11][12][13][14][15][16][17] . These phenomena arise from a complex interplay of various degrees of freedom associated with the charge carriers and atomic layers. One of the factors, playing important role in such interplay, is the spin-lattice coupling. A pronounced spin-lattice coupling was recently found in the vdW material Cr 2 Ge 2 Te 6 18 . In bulk CrCl 3 , an anomaly in magnetic susceptibility at the lowtemperature structural phase transition was observed, pointing to a presence of spin-lattice coupling 10 . A perfect model system to search for emergent physical phenomena, associated with the spin-lattice coupling in the CrX 3 family, is CrBr 3 due to absence of structural phase transitions at low temperatures and similarity of magnetic order in bulk and few-layer forms. Using a combination of the X-ray, neutron powder diffraction, and Raman spectroscopy, we revealed a negative thermal volume expansion (NTE) phenomenon in CrBr 3 below the Curie temperature with a coefficient α V = −1.9 × 10 −5 K −1 . The relevant coefficient, characterizing linear thermal expansion of the 2D vdW layers in CrBr 3 , α l = −1.6 × 10 -5 K −1 , is also negative and comparable by the order of magnitude with one of graphene at low temperatures. The observed anomalies in the temperature behavior of lattice parameters, interatomic distances and angles as well as vibrational modes frequencies and linewidths provide clear evidence for a strong coupling between spin and lattice degrees of freedom in ferromagnetic vdW magnets of CrX 3 family and related materials.

RESULTS AND DISCUSSION
Neutron and X-ray diffraction The neutron diffraction patterns of CrBr 3 measured at selected temperatures are shown in Fig. 1a. They are consistent with the rhombohedral crystal structure of R3 symmetry in the whole studied temperature range of 6-300 K. In this structure (Fig. 1b), Cr 3+ ions arrange in the honeycomb magnetic lattice and they are surrounded by edge-sharing octahedra formed by Br − ions. The Br-Cr-Br layers are stacked along the c-axis and bonded by the vdW forces. The obtained lattice parameters (in hexagonal setting) at ambient conditions, a = 6.3019(2) and c = 18.332(1) Å are consistent with previous studies 8, 19 .
On cooling below 35 K, an appearance of additional magnetic scattering contribution to intensities of the peaks (110), (012), and (101) was detected (Fig. 1a), evidencing formation of the FM ground state. The temperature dependence of the Cr 3+ -ordered magnetic moments (Fig. 1c) evaluated from the neutron diffraction data, can be well fitted in the framework of the molecular field approach where B s is the Brillouin function, S is the spin of the system (S = 3/ 2) and M 0 is the ordered magnetic moment at T = 0. The calculated values of the magnetic ordering temperature T C = 36 (2) K and M 0 = 2.74(8) μ B were obtained. The M 0 value is consistent with one of 2.83 μ B , determined from the polarized neutron diffraction study 20 . The ordered magnetic moments are oriented along the c-axis, in accordance with the easy spin axis direction 7,21 .
The temperature dependences of the lattice parameters and volume normalized to the ambient temperature values obtained from neutron and X-ray diffraction measurements are shown in Fig. 2. The thermal expansion of CrBr 3 lattice is strongly anisotropic with the pronounced variation of the c lattice parameter. The temperature dependence of the a lattice parameter demonstrates anomalous behavior. Its thermal expansion changes from conventional positive character in the temperature range above T C to unexpected negative character below T C . This also results in negative thermal expansion of the unit cell volume for T < T C (Fig. 2). The average volume thermal expansion coefficients (α V = (1/V)(dV/dT)) are α V = 3.6 × 10 −5 K −1 for T > T C and α V = −1.9 × 10 −5 K −1 for T < T C , respectively. The latter value is about twice larger in comparison with one for the frustrated antiferromagnetic spinel material ZnCr 2 Se 4 , α V = −1.08 × 10 −5 K −1 for 21-40 K 22 .
It should be noted that information about thermal expansion behavior is of special importance for fabrication of heterostructures involving vdW materials and their practical applications. The difference in thermal expansion of neighboring layers may result in appearance of strains, significantly affecting the physical properties. The corresponding average linear thermal expansion coefficients of the a lattice parameter (α l = (1/a)(da/dT)), characterizing thermal expansion of the 2D vdW layers in CrBr 3 , are α l = 0.7 × 10 −5 K −1 for T > T C and α l = −1.6 × 10 −5 K −1 for T < T C , respectively. The graphene, one of the most popular 2D materials for design of heterostructures, also demonstrates negative thermal expansion over a wide range of temperatures with α l =  The interatomic intralayer and interlayer Cr-Cr distances decrease slightly on cooling in the temperature range above T C and they also demonstrate opposite increasing trend for T < T C (Fig. 3). In the bromium octahedral units forming inside the vdW layers around chromium ions (Fig. 1b), there are two nonequivalent types of interatomic distances, three pairs of Cr-Br1 and three pairs of Cr-Br2 ones. Both distances have comparable values of 2.501(2) and 2.495(2) Å at ambient temperatures and they are weakly reduced on cooling downwards the Curie temperature. Below T C , a pronounced anisotropic distortion of the octahedral units occurs, resulting in reduction of the Cr-Br1 distances to 2.452(2) Å and enlargement of the Cr-Br2 distances to 2.513(2) Å at T = 6 K, respectively (Fig. 3). The temperature behavior of the Cr-Br-Cr interatomic angles, mediating ferromagnetic superexchange interactions, demonstrates growing trend on cooling in the temperature range above T C and anomaly in the vicinity of T C , caused by rapidly enhanced temperature variation rate. The anomalous behavior of the thermal expansion in the plane, perpendicular to the magnetic easy axis c, interatomic distances and angles in the vicinity of T C are caused by a complex interplay between the spin and lattice degrees of freedom in CrBr 3 .

Raman spectroscopy
In the Raman spectra of CrBr 3 , measured at selected temperatures, the six lines were observed (Fig. 4). The factor group analysis predicts eight Raman active modes, Γ ram = (4A g + 4E g ) for the rhombohedral R3 lattice symmetry (corresponding point group C 3i ) of this material 24,25 . The detected lines were tentatively assigned as 74.0 cm −1 (E g ), 106.3 cm −1 (A g ), 141.3 cm −1 (E g ), 150.4 cm −1 (E g ), 182.4 cm −1 (A g ), 278.0 cm −1 (E g ) phonon modes, according to refs. 24-26 . In the paramagnetic region, the frequencies of observed phonon modes (Fig. 5) demonstrate increasing trend with a temperature lowering, while full-width at half-maximum (FWHM) of relevant phonon peaks decreases. In the anharmonic approach, the corresponding temperature dependences can be described as where ν 0 is the harmonic phonon frequency, FWHM 0 is the phonon broadening induced by disorder, n B is the thermal population Bose factor, A 1 (B 1 ) and A 2 (B 2 ) are the coefficients, described contributions of cubic and quartic anharmonic terms 18,27 .
Below T C , a rapid increase of the phonon frequencies in the energy range above 150 cm −1 occurs compared to thermal behavior expected within the anharmonic approach. The FWHM of all the observed phonon modes reach minimum in the vicinity of T C and demonstrate anomalous reversal broadening in the T < T C range (Fig. 5). Both effects reveal a presence of the strong spin-phonon coupling in CrBr 3 . The spin-phonon coupling is associated with the modification of the magnetic exchange interactions caused by the ionic motions, and the relevant Fig. 3 The evolution of interatomic distances and angles as a function of temperature. The temperature dependences of the interatomic distances and angles in CrBr 3 . The lines are guides to eyes only. Fig. 4 The Raman spectra of CrBr 3 . The Raman spectra, measured at selected temperatures. The inset shows enlarged sections of the spectra in the Raman shift region of 60-90 cm −1 . Fig. 5 The temperature dependences of selected Raman peak frequencies and full-width at half-maximum (FWHM). The solid lines represent the fitting results in the anharmonic approach as described in the text. D.P. Kozlenko et al. phonon frequency shifts can be expressed as where ν 0 is the frequency in the absence of spin-phonon interactions, λ is the spin-phonon coupling constant, and 〈S i ·S j 〉 is the spin-spin correlation function of the neighboring spins 28 .
The value of the spin-phonon coupling constant for a given phonon mode can be evaluated to the first order of magnitude as l $ P i;α ∂J ∂ui;α u i;α , where J is the nearest-neighbor exchange integral, u i,α is the ionic displacement components α = (x,y,z) taken for atoms i contributing to the considered mode 29,30 . The evaluated spin-phonon coupling constants for various Raman modes, calculated using the value of S = 3/2 for Cr 3+ spins, are presented in the Table 1. The largest values of λ = 0.27 and 0.40 cm −1 are found for the E g modes with frequencies of 143.5 and 284.9 cm −1 , respectively. According to the theoretical calculations performed for the FM phase of relevant CrI 3 compound 26 , these phonon modes involve vibrations of the magnetic chromium ions, providing more pronounced coupling effects. In particular, the E g mode at 284.9 cm −1 involves out-ofphase vibrations of two in-plane Cr atoms only, resulting in enhanced sensitivity to the onset of magnetic order. The λ coefficients are comparable with those reported for another vdW ferromagnet Cr 2 Ge 2 Te 6 18 , although about an order of magnitude less compared to antiferromagnetic chromium spinels 31 .
The present results demonstrate a strong and complex interplay between spin and lattice degrees of freedom in the vdW material CrBr 3 . Below the Curie temperature a pronounced negative thermal volume expansion and linear expansion of the 2D vdW layers in CrBr 3 with the coefficients α V = −1.9 × 10 −5 K −1 and α l = −1.6 × 10 −5 K −1 are revealed. The latter value is comparable by the order of magnitude with one of graphene at low temperatures. These effects are also accompanied by anomalous thermal variation of interatomic distances and angles. The associated spin-phonon coupling effects provoke extra rise of the most of the observed vibrational modes frequencies and reversal broadening of their FWHM values, particularly pronounced for the modes involving vibrations of the magnetic chromium ions.

METHODS Samples
The single crystalline CrBr 3 samples were supplied by HQ Graphene.

Neutron diffraction
The neutron diffraction measurements with the powdered samples were performed in the temperature range 6-300 K using the DN-6 diffractometer at the IBR-2 high flux pulsed reactor with the CCR-based refrigerator 32 . The diffraction patterns were collected at the scattering angle of 90°with the resolution of Δd/d = 0.022. The sample volume was about 5 mm 3 . The neutron diffraction patterns were analyzed by the Rietveld method using the Fullprof program 33 .

X-ray diffraction
The X-ray powder diffraction measurements were performed in the temperature range 15-300 K using the Malvern PANalytical Empyrean diffractometer with the Cu K α radiation, λ = 1.541 Å, and the CCR-based refrigerator. The X-ray diffraction patterns were also analyzed by the Rietveld method using the Fullprof program 33 .

Raman spectroscopy
The Raman spectra with the single crystalline CrBr 3 samples were collected using a LabRAM HR Evolution spectrometer (Horiba, France) with a wavelength excitation of 632.8 nm emitted from He-Ne laser, 1800 grating, confocal hole of 200 μm, and ×20 objective. The low-temperature Raman measurements were carried out using low vibration helium refrigerator (Advanced Research Systems, USA) in temperature range 19-300 K.