Critical behavior and magnetocaloric effect in VI 3

Layered van der Waals ferromagnets are promising candidates for designing new spintronic devices. Here we investigated the critical properties and magnetocaloric effect connected with ferromagnetic transition in layered van der Waals VI$_3$ single crystals. The critical exponents $\beta = 0.244(5)$ with a critical temperature $T_c = 50.10(2)$ K and $\gamma = 1.028(12)$ with $T_c = 49.97(5)$ K are obtained from the modified Arrott plot, whereas $\delta = 5.24(2)$ is obtained from a critical isotherm analysis at $T_c = 50$ K. The magnetic entropy change $-\Delta S_M(T,H)$ features a maximum at $T_c$, i.e., $-\Delta S_M^{max} \sim$ 2.64 (2.27) J kg$^{-1}$ K$^{-1}$ with out-of-plane (in-plane) field change of 5 T. This is consistent with $-\Delta S_M^{max}$ $\sim$ 2.80 J kg$^{-1}$ K$^{-1}$ deduced from heat capacity and the corresponding adiabatic temperature change $\Delta T_{ad}$ $\sim$ 0.96 K with out-of-plane field change of 5 T. The critical analysis suggests that the ferromagnetic phase transition in VI$_3$ is situated close to a three- to two-dimensional critical point. The rescaled $\Delta S_M(T,H)$ curves collapse onto a universal curve, confirming a second-order type of the magnetic transition and reliability of the obtained critical exponents.


I. INTRODUCTION
Layered intrinsically ferromagnetic (FM) semiconductors hold great promise for both fundamental physics and applications in spintronic devices. [1][2][3][4][5] CrI 3 has recently attracted much attention since the long-range magnetism persists in monolayer with T c of 45 K. 3 Intriguingly, the magnetism in CrI 3 is layer-dependent, from FM in monolayer, to antiferromagnetic (AFM) in bilayer, and back to FM in trilayer. 3 In van der Waals (vdW) heterostructures formed by an ultrathin CrI 3 and a monolayer WSe 2 , the WSe 2 photoluminescence intensity strongly depends on the relative alignment between photoexcited spins in WSe 2 and the CrI 3 magnetization. 6 Furthermore, the magnetism in ultrathin CrI 3 could be controlled by electrostatic doping, providing great opportunities for designing magneto-optoelectronic devices. 7,8 Recently, the two-dimensional (2D) ferromagnetism has also been predicted in VI 3 monolayer with a calculated T c of 98 K, higher than that in CrI 3 . 9 Bulk CrI 3 and VI 3 belong to a well-known family of transition metal trihalides MX 3 (X = Cl, Br and I). 10,11 When compared to CrI 3 , in which the chromium has a half filled t 2g level yielding S = 3/2, the vanadium in VI 3 has two valence electrons that half fill two of the three degenerate t 2g states yielding S = 1. [12][13][14] Bulk VI 3 is an insulating 2D ferromagnet with T c = 55 K that crystallizes in a layered structure of BiI 3 with space group R3 below 79 K whereas at higher temperatures VI 3 adopts layered monoclinic C2/m crystal structure with van der Waals bonds along the c-axis. 14-17 Density functional theory (DFT) calculations suggest that the VI 3 not only hosts the long-range ferromagnetism down to a monolayer but also exhibits Dirac half-metallicity, of interest for spintronic applications. 9 The magnetocaloric effect (MCE) in the FM vdW materials gives additional insight into the magnetic properties. Bulk CrI 3 exhibits anisotropic −∆S max M with values of 4.24 and 2.68 J kg −1 K −1 at 5 T for H//c and H//ab, respectively, 18 however little is known about VI 3 .
In the present work we focus on the nature of the FM transition in bulk VI 3 single crystals. We have investigated the critical behavior by the modified Arrott plot and a critical isotherm analysis, whilst the magnetocaloric effect was studied by the heat capacity and magnetization measurements near T c . Critical exponents β = 0.244(5) with T c = 50.10(2) K, γ = 1.028 (12) with T c = 49.97(5) K, and δ = 5.24 (2) at T c = 50 K, suggest that the magnetic transition is second-order and that it is situated near a tricritical point from two-to three-dimensional. This is further confirmed by the scaling analysis of magnetic entropy change −∆S M (T, H), in which the rescaled −∆S M (T, H) collapse on a universal curve independent on temperature and field.

II. EXPERIMENTAL DETAILS
Bulk VI 3 single crystals were fabricated by chemical vapor transport method starting from an intimate mixture of vanadium powder (99.95 %, Alfa Aesar) and anhydrous iodine beads (99.99 %, Alfa Aesar) with a molar ratio of 1 : 3. The starting materials were sealed in an evacuated quartz tube and then placed inside a multizone furnace. Materials were reacted over a period of 7 days with source zone at 650 • C, middle growth zone at 550 • C, and third zone at 600 • C. The x-ray diffraction (XRD) data were taken with Cu K α (λ = 0.15418 nm) radiation of Rigaku Miniflex powder diffractometer. The element analysis was performed using an energydispersive x-ray spectroscopy in a JEOL LSM-6500 scanning electron microscope, confirming a stoichiometric VI 3 single crystal. The magnetization data as a function of temperature and field were collected using Quantum Design MPMS-XL5 system in temperature range from 10 to 90 K with a temperature step of 1 K around T c and field up to 5 T. The applied field (H a ) has been corrected for the internal field as H = H a − N M , where M is the measured magnetization and N is the demagnetization factor. The corrected H was used for the analysis of the critical behavior. The magnetic entropy change −∆S M from the magnetization data was estimated using a Maxwell relation.

III. RESULTS AND DISCUSSIONS
The crystal structure of VI 3 forms in monoclinic AlCl 3 type with space group C2/m at room temperature [inset in Fig. 1], 12-14 similar to that of CrI 3 . The V ions are arranged in a honeycomb network and are located at the centers of distorted edge-sharing octahedra of six I anions. The I-V-I triple layers of composition VI 3 are stacked along the c axis and there are vdW gaps between them. The as-grown single crystals are shiny black platelets with lateral dimensions up to several millimeters, as shown in the inset of Fig. 1. In the single-crystal x-ray diffraction (XRD) scan [ Fig. 1], only (00l) peaks are detected, indicating that the crystal surface is normal to the c axis with the plate-shaped surface parallel to the ab plane.
Figures 2(a) and 2(b) present the temperature dependence of dc magnetic susceptibility measured in the fields ranging from 100 Oe to 50 kOe applied in the ab plane and along the c axis, respectively. It can be clearly seen that VI 3 exhibits a ferromagnetic transition at low temperature for both field directions. The magnetic susceptibility is nearly isotropic in 10 and 50 kOe, however, significant magnetic anisotropy is observed in 100 Oe and 1 kOe at low temperature. When T < T c , the divergence of zero-field cooling (ZFC) and field-cooling (FC) curves exhibit a characteristic behavior of possible spinglass or cluster-glass state with the temperature of divergence decreasing with increasing field. An additional weak anomaly at 80 K is also observed for H//c, which is field-independent. A structural phase transition accompanies similar feature in the susceptibility of CrI 3 , 2  Near T c the second order phase transition is governed by magnetic equation of state and is characterized by critical exponents β, γ and δ that are mutually related. 20 Spontaneous magnetization M s and inverse initial susceptibility χ −1 0 , below and above T c can be used to obtain β and γ whereas δ is the critical isotherm exponent. Hence, from magnetization: where ε = (T − T c )/T c is the reduced temperature, and M 0 , h 0 /m 0 and D are the critical amplitudes. Isothermal magnetization in the temperature range from 40 to 60 K with a temperature step of 1 K is shown in Fig. 3(a). The Arrott plot involves meain-field critical exponents β = 0.5 and γ = 1.0. 22 Based on this, magnetization isotherms M 2 vs H/M should be a set of parallel straight lines and the isotherm at the critical temperature T c should pass through the origin. As shown in Fig. 3(b), all curves in the Arott plot of VI 3 are nonlinear, with a downward curvature, demonstrating that the Landau mean-field model is not applicable to VI 3 . However, it is possible to estimate the order of the magnetic transition through the slope of the straight line based on Banerjee ′ s criterion. 23 First (second) order phase transition corresponds to negative (positive) slope. Therefore, the downward slope reveals a second-order PM-FM transition in VI 3 .  Fig. 3(h). It is clearly seen that the N S of 2D Ising model shows the largest deviation from unity. The N S of 3D Ising model is close to N S = 1 mostly above T c , while that of tricritical mean field model is the best below T c , suggesting a 3D magnetic behavior in bulk  Fig. 4(a), such fitting yields δ = 5.24 (2). The Widom scaling law gives δ = 1 + γ/β. From β and γ obtained with the modified Arrott plot, δ = 5.21 (4), which is very close to that obtained from critical isotherm analysis.

Arrot-Noaks equation of state provides modification of
Scaling analysis can be used to estimate the reliability of the obtained critical exponents and T c . Near phase transition the magnetic equation of state is: where f + for T > T c and f − for T < T c , respectively, are the regular functions. Eq.(5) can be expressed via rescaled magnetization m ≡ ε −β M (H, ε) and rescaled field h ≡ ε −(β+γ) H as For the correct scaling relations and correct choice of β, γ, and δ, scaled m and h fall on universal curves above T c and below T c , respectively. Figure 4(b) presents the scaled m 2 vs h/m that collapse on two separate branches below and above T c , respectively, confirming proper treatment of the critical regime. The scaling equation of state also takes another form where k(x) is the scaling function. From Eq. (7), all the experimental data should fall into a single curve. This is indeed seen in the inset of Fig. 4(b); the M H −1/δ vs εH −1/(βδ) experimental data collapse into a single curve and the T c is located at the zero point of the horizontal axis.
The change of magnetic entropy is: This can be expressed using Maxwell's relation For magnetization measured at small (H,T) intervals, ∆S M (T, H) is becomes: where T r1 and T r2 are the temperatures of the two reference points that have been selected as those corresponding to ∆S M (T r1 , T r2 ) = ∆S max M /2. It can be seen that the −∆S M (T, H) in different magnetic fields fall on a single line near T c [inset in Fig. 5(b)].
At a second-order transition, 29 maximal magnetic entropy change can be expressed as −∆S max M = aH n . 30 The relative cooling power (RCP) is defined: is the maximum entropy change around T c whereas δT F W HM is the fullwidth at half maximum. 31 The RCP changes in magnetic field as with RCP = bH m . Figure 5 The −∆S M can also be obtained from the heat capacity measurement with out-of-plane fields up to 5 T. The signature of magnetic order at T c = 50 K in H = 0 [ Fig. 6(a)] is suppressed in fields. Heat capacity change Fig. 6(a)] is ∆C p < 0 for T < T c and ∆C p > 0 for T > T c . At T c there is a sharp change from negative to positive. In contrast, the sharp peak at higher temperature (∼ 80 K) is insensitive to external field and there is almost no shift when the field is up to 5 T. This heat capacity anomaly corresponds to the structural transition, in line with the magnetic susceptibility anomaly, indicating strong spin-lattice coupling in VI 3 . The entropy S(T, H) can be deduced by S(T, H) =

IV. CONCLUSIONS
In summary, we have studied the critical behavior and magnetocaloric effect around the FM-PM transition in bulk VI 3 single crystal. The PM-FM transition in VI 3 is identified to be of the second order. The critical exponents β, γ, and δ estimated from the modified Arrott plot follow the tricritical mean-field model. Considering its ferromagnetism can be maintained upon exfoliating bulk crystals down to a single layer, further investigation on the size-dependent properties is of interest.