Critical behavior and magnetocaloric effect in VI$_3$

TL;DR

This study investigates the critical magnetic behavior and magnetocaloric effect in layered VI$_3$ ferromagnetic crystals, revealing critical exponents, phase transition nature, and potential for spintronic applications.

Contribution

It provides the first detailed critical and magnetocaloric analysis of VI$_3$, establishing its second-order transition and near two-dimensional critical behavior.

Findings

Critical exponents indicate a second-order phase transition.

Maximum magnetic entropy change is around 2.64 J/kg·K at T_c.

Transition near a 3D to 2D critical point.

Abstract

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.

Equations

$$M_s(T)=M_0(-\epsilon {)}^{\beta },\epsilon <0,T<T_c,$$

$${\chi }_0^{-1}(T)=(h_0/m_0){\epsilon }^{\gamma },\epsilon >0,T>T_c,$$

$$M=D{H}^{1/\delta },T=T_c,$$

$$(H/M{)}^{1/\gamma }=a\epsilon +b{M}^{1/\beta },$$

$$M(H,\epsilon )={\epsilon }^{\beta }{f}_{\pm }(H/{\epsilon }^{\beta +\gamma }),$$

$$m=f_{\pm }(h).$$

$$\frac{H}{M^{\delta }}=k(\frac{\epsilon }{H^{1/\beta }}),$$

$$\Delta S_M(T,H)={\int }_0^H{\left(\frac{\partial S}{\partial H}\right)}_{T}dH={\int }_0^H{\left(\frac{\partial M}{\partial T}\right)}_{H}dH,$$

$$\Delta S_M(T_i,H)=\frac{{\int }_0^{H}M(T_i,H)dH-{\int }_0^{H}M(T_{i+1},H)dH}{T_i-T_{i+1}}.$$

$${\theta }_{-}=(T_{peak}-T)/(T_{r1}-T_{peak}),T<T_{peak},$$

$${\theta }_{+}=(T-T_{peak})/(T_{r2}-T_{peak}),T>T_{peak},$$

Full text

I INTRODUCTION

Layered intrinsically ferromagnetic (FM) semiconductors hold great promise for both fundamental physics and applications in spintronic devices.McGuire0 ; McGuire ; Huang ; Gong ; Seyler CrI3 has recently attracted much attention since the long-range magnetism persists in monolayer with $T_{c}$ of 45 K.Huang Intriguingly, the magnetism in CrI3 is layer-dependent, from FM in monolayer, to antiferromagnetic (AFM) in bilayer, and back to FM in trilayer.Huang In van der Waals (vdW) heterostructures formed by an ultrathin CrI3 and a monolayer WSe2, the WSe2 photoluminescence intensity strongly depends on the relative alignment between photoexcited spins in WSe2 and the CrI3 magnetization.Zhong The magnetism in ultrathin CrI3 could also be controlled by electrostatic doping, which provides great opportunities for designing magneto-optoelectronic devices.Jiang ; Huang1 Very recently, the two-dimensional (2D) ferromagnetism has also been predicted in VI3 monolayer with a calculated Tc of 98 K, higher than that in CrI3.He

Bulk CrI3 and VI3 belong to a well-known family of transition metal trihalides MX3 (X = Cl, Br and I).Juza ; Dillon When compared to CrI3, in which the chromium has a half filled t2g level yielding S = 3/2, the vanadium in VI3 has two valence electrons that half fill two of the three degenerate t2g states yielding S = 1.Son ; Kong ; Tian Bulk VI3 is an insulating 2D ferromagnet with $T_{c}$ = 55 K and crystallizes in a layered structure.Trotter ; Handy ; Wilson Each V ion is centered in an octahedron of I ions, form a honeycomb lattice within the $ab$ plane [inset in Fig. 1(a)], similar with CrI3. There is a structural transition at $\sim$ 80 K above $T_{c}$, however, the detailed symmetry of the high- or low-temperature structure is still not settled. Tian et al. describes analysis of single crystal x-ray diffraction (XRD) data and concludes that the high temperature structure is monoclinic, and the low temperature structure is trigonal,Tian while Son et al. describes powder XRD and arrives at the inverse conclusion,Son calling for further study. Density functional theory (DFT) calculations suggest that the VI3 not only hosts the long-range ferromagnetism down to a monolayer but also exhibits Dirac half-metallicity, of interest for spintronic applications.He

The magnetocaloric effect (MCE) in the FM vdW materials gives additional insight into the magnetic properties. Bulk CrI3 exhibits anisotropic $-\Delta S_{M}^{max}$ with the values of 4.24 and 2.68 J kg*-1* K*-1* at 5 T for out-of-plane and in-plane fields, respectively,YuLIU however little is known about VI3.

In the present work we focus on the nature of the FM transition in bulk VI3 single crystals. We have investigated the critical behavior by the modified Arrott plot and a critical isotherm analysis, whilst the magnetocaloric effect was also studied by heat capacity and magnetization measurements near $T_{c}$. Critical exponents $\beta$ = 0.244(5) with $T_{c}$ = 50.10(2) K, $\gamma$ = 1.028(12) with $T_{c}$ = 49.97(5) K, and $\delta$ = 5.24(2) at $T_{c}$ = 50 K, suggest that the magnetic transition in VI3 is of second-order and that it is situated near a critical point from three- to two-dimensional. This is further confirmed by the scaling analysis of magnetic entropy change $-\Delta S_{M}(T,H)$, in which the rescaled $-\Delta S_{M}(T,H)$ collapse on a universal curve independent on temperature and field.

II Experimental details

Bulk VI3 single crystals were fabricated by the 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, placed inside a multi-zone furnace and then reacted over a period of 7 days with the source zone at 650 ∘C, the middle growth zone at 550 ∘C, and the third zone at 600 ∘C. The crystal structure was characterized by powder x-ray diffraction (XRD) in the transmission mode at 28-D-1 beamline of the National Synchrotron Light Source II (NSLS II) at Brookhaven National Laboratory (BNL). Data were collected using a 0.5 mm2 beam with wavelength $\lambda\sim$ 0.1668 Å. A Perkin Elmer 2D detector (200 $\times$ 200 microns) was placed orthogonal to the beam path 990 mm away from the sample. The single crystal XRD were taken with Cu Kα ($\lambda=0.15418$ nm) radiation of Rigaku Miniflex powder diffractometer. The element analysis was performed using an energy-dispersive x-ray spectroscopy (EDS) in a JEOL LSM-6500 scanning electron microscope, confirming a stoichiometric VI3 single crystal. The magnetization data as a function of temperature and field were collected using Quantum Design MPMS-XL5 system. The heat capacity was measured in Quantum Design PPMS-9 system.

III RESULTS AND DISCUSSIONS

III.1 Structure and basic magnetic properties

The as-grown single crystals are shiny black platelets with lateral dimensions up to several millimeters. In the single-crystal XRD scan [Fig. 1(a)], only $(00l)$ peaks are detected, indicating that the plate-shaped surface parallel to the $ab$ plane, and we assign the axis $c^{*}$ is normal to the plane. The layer spacing of VI3 is calculated as 6.67(1) Å, close to the reported value.Son ; Kong ; Tian Rietveld powder diffraction analysis was carried out on data obtained from the raw 2D diffraction data integrated and converted to intensity versus $Q$ using the Fit2d software where $Q=4\pi sin\theta/\lambda$ is the magnitude of the scattering vector.Hammersley The refinement was performed using GSAS-II modeling suite.Toby Figure 1(b) shows the refinement result of synchrotron powder XRD data of VI3 at room temperature (space group $R\overline{3}$). The determined lattice parameters are $a$ = 6.9137(11) Å and $c$ = 19.9023(21) Å.

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 is clearly seen that VI3 exhibits a ferromagnetic transition near $T_{c}$ = 50 K for both magnetic field directions, consistent with the previous reports.Son ; Kong ; Tian The magnetic susceptibility is nearly isotropic in H = 50 kOe, however, significant magnetic anisotropy is observed in low fields. When T $<$ Tc, the divergence of zero-field cooling (ZFC) and field-cooling (FC) curves exhibit a characteristic behavior of possible spin-glass state with the temperature of divergence decreasing with increasing field. Besides this, the magnetic domain creep, i.e., the magnetic domain walls jump from one pinning site to another, can also lead to this kind of irreversible behavior.Tian The evolution of ferromagnetic domain as a function of magnetic field and temperature was further investigated,Kong confirming the ferromagnetism and a small domain-wall-energy in VI3. It should be noted that there is an additional weak anomaly at 80 K for $\mathbf{H\parallel c^{*}}$ [inset in Fig. 2(b)], which is field-independent. A structural phase transition accompanies similar feature in the susceptibility of CrI3,McGuire indicating strong spin-lattice coupling.

Isothermal magnetization at $T$ = 2 K [Fig. 2(c)] shows saturation moments of $M_{s}\approx$ 0.72 $\mu_{B}$/V and 0.95 $\mu_{B}$/V for $\mathbf{H\parallel ab}$ and $\mathbf{H\parallel c^{*}}$, respectively. The value is smaller than the expected saturated moment of 2 $\mu_{B}$ for V3+ ion. The difference of saturation magnetization for the two directions is also unusual, which may be due to anisotropic $g$ factor with unquenched orbital angular moment, calling for further neutron scattering and/or electron spin resonance studies.Son ; Kong ; Tian The coercive field is about 15 kOe for $\mathbf{H\parallel c^{*}}$, much larger than that of 1.5 kOe for $\mathbf{H\parallel ab}$, suggesting a hard ferromagnet behavior and the easy $c^{*}$ axis. The coercive field is significantly larger than that in CrI3 with fully filled Cr3+ orbitals. Son et al. proposed that the smaller saturated moment in V3+ driven by the smaller number of $d$-orbital spin and the larger magnetic anisotropy coming from the partially filled $t_{2g}$ $d$-band of V3+ would lead to the larger coercive field in VI3 when compared with CrI3.Son Ac susceptibility was further measured with zero field cooling at oscillated ac field of 3.8 Oe and frequency of 499 Hz. Three distinct peaks in the real part $\chi^{\prime}(T)$ along the $c^{*}$ axis [Fig. 2(d)], one strong peak for both directions corresponding the PM-FM transition at $T_{c}$ = 50 K and two additional peaks above $T_{c}$, as well as the weak anomalies at the same temperatures in the $ab$ plane, indicating a complex multiple-step magnetic ordering in VI3.

III.2 Critical behavior

To determine the accurate $T_{c}$, we first considered the well-known Arrott plot.Arrott1 Magnetization isotherms along the easy $c$ axis were measured in the vicinity of $T_{c}$ [Fig. 3(a)]. The Arrott plot involves the mean-field critical exponents $\beta$ = 0.5 and $\gamma$ = 1.0.Arrott1 Based on this, magnetization isotherms $M^{2}$ vs $H/M$ should be a set of parallel straight lines and the isotherm at $T_{c}$ should pass through the origin. As is seen, all curves in the Arrott plot of VI3 are nonlinear [Fig. 3(b)], with a downward curvature, demonstrating that the mean-field model does not work for VI3. Based on Banerjee′s criterion,Banerjee we can estimate the order of the magnetic transition through the slope of the straight line. First (second) order phase transition corresponds to negative (positive) slope. Therefore, the downward slope reveals a second-order PM-FM transition in VI3.

In the vicinity of $T_{c}$ the second order phase transition is governed by magnetic equation of state and is characterized by critical exponents $\beta$, $\gamma$ and $\delta$ that are mutually related.Stanley Spontaneous magnetization $M_{s}$ and inverse initial susceptibility $\chi_{0}^{-1}$, below and above $T_{c}$ can be used to obtain $\beta$ and $\gamma$ whereas $\delta$ is the critical isotherm exponent. Hence, from magnetization:

[TABLE]

[TABLE]

[TABLE]

where $\varepsilon=(T-T_{c})/T_{c}$ is the reduced temperature, and $M_{0}$, $h_{0}/m_{0}$ and $D$ are the critical amplitudes.Fisher For the original Arrott plot, $\beta$ = 0.5 and $\gamma$ = 1.0.Arrott1 In a more general case, the Arrott-Noaks equation of state provides modification of Arrott plot:Arrott2

[TABLE]

where $\varepsilon=(T-T_{c})/T_{c}$ and $a$ and $b$ are fitting constants. Since the mean-field model does not work, we adopt the modified Arrott plot in order to better understand the nature of the PM-FM transition in VI3.

Figures 3(c)-3(g) exhibit the modified Arrott plots using possible exponents from 2D Ising ($\beta=0.125,\gamma=1.75$), 3D Ising ($\beta=0.325,\gamma=1.24$), 3D Heisenberg ($\beta=0.365,\gamma=1.386$), 3D XY ($\beta=0.345,\gamma=1.316$), and tricritical mean-field ($\beta=0.25,\gamma=1.0$) models.Kaul ; Khuang ; LeGuillou The modified Arrott plot should be a set of parallel lines in the high field region with the same slope $S(T)=dM^{1/\beta}/d(H/M)^{1/\gamma}$. The model which fits the data best is selected via the normalized slope [$NS=S(T)/S(T_{c})$] that compares with the ideal value of unity. Plot of $NS$ vs $T$ for different models is also presented in Fig. 3(h). It is clearly seen that the $NS$ of 2D Ising model shows the largest deviation from unity. The $NS$ of 3D Ising model is close to $NS=1$ mostly above $T_{c}$, while that of tricritical mean field model is the best below $T_{c}$.

Following the methods of Pramanik and Banerjee,Pramanik the linearly extrapolated $M_{s}$ and $H/M$ are plotted as a function of temperature in Fig. 4(a). The solid lines are fitted lines according to Eqs. (1) and (2). The critical exponents $\beta=0.244(5)$, with $T_{c}=50.10(2)$ K, and $\gamma=1.028(12)$, with $T_{c}=49.97(5)$ K, are obtained. As we can see, the value of $\gamma$ is close to that of tricritical mean-field model ($\gamma=1.0$), while $\beta$ lies between the values of tricritical mean-field ($\beta=0.25$) and 2D XY model ($\beta=0.23$).Bramwell It is summarized that the value of $\beta$ for a 2D magnet should be within a window $0.1\leq\beta\leq 0.25$.Taroni Therefore, the obtained critical exponents suggest that the magnetic transition of VI3 is situated close to a three- to two-dimensional critical point, in contrast to those of CrI3 exhibiting 3D critical behavior and Cr2(Si,Ge)2Te6 showing 2D Ising-type coupled with a long-range interaction.YuL ; GT ; BJLIU ; YULIU ; GTLIN ; JC According to Eq. (3), the $M(H)$ at $T_{c}$ should be a straight line in log-log scale with the slope of $1/\delta$. Such fitting yields $\delta=5.24(2)$ [inset in Fig. 4(a)]. The Widom relation gives $\delta=1+\gamma/\beta$.Widom From $\beta$ and $\gamma$ obtained with the modified Arrott plot, $\delta$ is calculated to be 5.21(4), which is agree with that obtained from critical isotherm analysis.

Scaling analysis can be used to estimate the reliability of the obtained critical exponents. Near phase transition the magnetic equation of state is:

[TABLE]

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 $m\equiv\varepsilon^{-\beta}M(H,\varepsilon)$ and rescaled field $h\equiv\varepsilon^{-(\beta+\gamma)}H$ as

[TABLE]

For the correct scaling relations and correct choice of $\beta$, $\gamma$, and $\delta$, 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

[TABLE]

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 $MH^{-1/\delta}$ vs $\varepsilon H^{-1/(\beta\delta)}$ experimental data collapse into a single curve and the $T_{c}$ is located at the zero point of the horizontal axis.

III.3 Magnetic entropy change

Figure 5(a) shows the temperature dependence of heat capacity $C_{p}$ at different fields. A sharp peak at $T\sim$ 80 K is observed. There is almost no shift when the magnetic field changes, corresponding to the structural transition, consistent with the susceptibility anomaly [inset in Fig. 2(b)]. In contrast, the peak of magnetic order at lower temperature is gradually suppressed when the magnetic field increases. At $T_{c}\sim$ 50 K, the heat capacity change $\Delta C_{p}=C_{p}(T,H)-C_{p}(T,0)$ exhibits a sharp change from negative to positive [Fig. 5(b)]. The entropy $S(T,H)$ = $\int_{0}^{T}C_{p}(T,H)/TdT$ and the magnetic entropy change $-\Delta S_{M}(T,H)=S_{M}(T,H)-S_{M}(T,0)$. The adiabatic temperature change $\Delta T_{ad}$ caused by the field change can be obtained by $\Delta T_{ad}(T,H)=T(S,H)-T(S,0)$, where $T(S,H)$ and $T(S,0)$ are the temperatures in $H\neq 0$ and $H=0$, respectively, at constant total entropy $S(T,H)$. Figures 6(b) and 6(c) show the temperature dependence of $-\Delta S_{M}$ and $\Delta T_{ad}$ estimated from heat capacity with out-of-plane field change. The maxima of $-\Delta S_{M}$ and $\Delta T_{ad}$ increase with increasing field and reach the values of 2.80 J kg*-1* K*-1* and 0.96 K, respectively, with the field change of 5 T. The obtained $-\Delta S_{M}$ and $\Delta T_{ad}$ of VI3 are significantly smaller than those of well-known magnetic refrigerating materials, such as Gd5Si2Ge2, LaF13-xSix, and MnP1-xSix,GschneidnerJr however, comparable with those of Cr(Br,I)3 and Cr2(Si,Ge)2Te6.Xiaoyun ; YuLIU ; YL

Figures 6(a) and 6(b) show the initial isothermal magnetization with the temperature ranging from 4 K to 78 K for $\mathbf{H\parallel c^{*}}$ and $\mathbf{H\parallel ab}$, respectively. The magnetic entropy change

[TABLE]

where $\left(\frac{\partial S}{\partial H}\right)_{T}$ = $\left(\frac{\partial M}{\partial T}\right)_{H}$ is based on Maxwell’s relation.Amaral For magnetization measured at small (H,T) intervals,

[TABLE]

Figures 6(c) and 6(d) give the calculated $-\Delta S_{M}(T,H)$ as a function of temperature in $\mathbf{H\parallel c^{*}}$ and $\mathbf{H\parallel ab}$, respectively. All the $-\Delta S_{M}$ curves exhibit a pronounced peak at $T_{c}$. The maxima $-\Delta S_{M}$ reach 2.64 and 2.27 J kg*-1* K*-1* with out-of-plane and in-plane field change of 5 T, respectively. In view of a large magnetic anisotropy in VI3, the rotational magnetic entropy change $\Delta S_{M}^{R}$ is calculated as $\Delta S_{M}^{R}(T,H)=\Delta S_{M}(T,H_{c})-\Delta S_{M}(T,H_{ab})$. Figure 6(e) shows the temperature-dependent $-\Delta S_{M}^{R}$ of VI3, which is smaller than that of CrI3.YuLIU

The magnetic entropy change is also correlated with the intrinsic magnetic coupling through a series of critical exponents. The maximal magnetic entropy change $-\Delta S_{M}^{max}=aH^{n}$.VFranco ; VFranco1 The relative cooling power $RCP$ is defined as $RCP=-\Delta S_{M}^{max}\times\delta T_{FWHM}$, where $\delta T_{FWHM}$ is the full-width at half maximum, and $RCP=bH^{d}$.VFranco ; VFranco1 Figure 6(f) presents the field dependence of $-\Delta S_{M}^{max}$ and RCP. Fitting of $-\Delta S_{M}^{max}$ and RCP give that $n=0.58(2)$ and $d=1.02(1)$ for out-of-plane field, while $n=0.67(1)$ and $d=1.15(1)$ for in-plane field. As is known, the exponents $n$ and $d$ are correlated with the critical exponents as $n=1+(\beta-1)/(\beta+\gamma)$ and $d=1+1/\delta$.Franco The obtained $n$ is close to that of 3D Ising model ($n$ = 0.569) for out-of-plane field and approaches the value of mean-field model ($n=0.667$) for in-plane field.

The $-\Delta S_{M}$ scaling analysis is assessed from normalizing all the $-\Delta S_{M}$ curves against their maxima $-\Delta S_{M}^{max}$, i.e., $\Delta S_{M}/\Delta S_{M}^{max}$ by temperature $\theta$ rescaling based on:Franco

[TABLE]

[TABLE]

where $T_{r1}$ and $T_{r2}$ are the temperatures of the two reference points that have been selected as those corresponding to $\Delta S_{M}(T_{r1},T_{r2})=\Delta S_{M}^{max}/2$. It could be seen that the $-\Delta S_{M}(T,H)$ in different magnetic fields fall on a single line near $T_{c}$ [Figs. 6(g) and 6(h)]. The well scaling of $-\Delta S_{M}(T,H)$ curves near $T_{c}$ indicate that the magnetic phase transition of VI3 is of second-order. The slight deviation at low temperature is most likely contributed by its magnetic anisotropy effect.

IV CONCLUSIONS

In summary, we have studied the critical behavior and magnetocaloric effect around the FM-PM transition in VI3 single crystal. The PM-FM transition in VI3 is identified to be of second order. The critical exponents $\beta$, $\gamma$, and $\delta$ suggest the ferromagnetic phase transition in VI3 is situated close to a 3D to 2D critical point. 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.

Note added. We became aware of several related works after the completion of our work.JYan ; Elena ; Dolezal

Acknowledgements

This work was funded by the Computation Material Science Program (Y.L. and C.P.). This research used the 28-ID-1 beamline of the National Synchrotron Light Source II, a U.S. DOE Office of Science User Facility operated for the DOE Office of Science by Brookhaven National Laboratory under Contract No. DE-SC0012704.