Magnetic Phase Diagram of van der Waals Antiferromagnet TbTe3

Terbium tritelluride, TbTe3, orders antiferromagnetically in three steps at TN1 = 6.7 K, TN2 = 5.7 K, and TN3 = 5.4 K, preceded by a correlation hump in magnetic susceptibility at T* ~8 K. Combining thermodynamic, i.e., specific heat Cp and magnetization M, and transport, i.e., resistance R, measurements we established the boundaries of two commensurate and one charge density wave modulated phases in a magnetic field oriented along principal crystallographic axes. Based on these measurements, the magnetic phase diagrams of TbTe3 at H‖a, H‖b and H‖c were constructed.


Introduction
Transition metal-based van der Waals compounds attract attention in the field of spintronics since they represent a natural crystal-perfect alternative to quasi-two-dimensional magnetic films obtained by epitaxial methods [1]. Also, they are of interest in fundamental research, providing a new platform for the study of low-dimensional magnetism. Recently, the halides and chalcogenides of rare-earths RE have emerged as important objects to reveal the effects of magnetic anisotropy in the processes of the long-range order formation in van der Waals compounds.
The layered YbCl 3 can be considered as a model van der Waals system to investigate Heisenberg interactions on a honeycomb net. It has been shown that this material reaches the Neel-type ground state with a reduced moment and anisotropic in-plane bonddependent coupling preceded by a short-range magnetic order. It satisfies the prerequisites of the Kitaev-Heisenberg model, lying on the border of Kitaev spin liquid from the antiferromagnetic side [2]. The rare-earth chalcohalides also belong to the van der Waals class of Kitaev spin liquid candidates. YbOCl has been investigated numerically in this respect, evidencing an exclusively rich magnetic phase diagram in terms of anisotropic exchange interactions J ± and J zz . Along with Neel, collinear, stripy, zigzag, and 120-AFM phases, this diagram hosts the spin-disordered one related presumably to the Kitaev physics [3].
The rare-earth chalcohalides with large magnetic moments and Ising-type anisotropy behave differently compared to Yb compounds with effective moment J eff = 1/2. Thus, DyOCl reaches Neel order at low temperatures with magnetic moments oriented along the easy axis in the plane of the van der Waals layer. A moderate external field applied along this axis generates a spin-flip transition transforming DyOCl into a ferromagnet [4].
The rare-earth tellurides RETe 3 are known as a family of van der Waals compounds hosting the charge density waves and experiencing antiferromagnetic order at low temperatures [5]. Recently, the effect of a magnetic field on thermodynamic properties in GdTe 3 has been investigated. Two antiferromagnetic transitions at T N1 = 11.5 K and T N2 = 9.7 K Materials 2022, 15, 8772 2 of 6 and an anomaly at T 1 = 7 K were found in both magnetic susceptibility χ and specific heat C p . It was conjectured that the anomaly at T 1 might originate from the incommensurate charge density wave in the van der Waals coupled layers of GdTe 3 [6].
Various aspects of charge density wave transitions in RE tritellurides are discussed in [7]. Within this family, TbTe 3 stands out as material evidencing three successive magnetic phase transitions at T N1 = 6.6 K, T N2 = 5.6 K, and T N3 = 5.4 K ascribed to the interplay of magnetism with charge density waves [8]. It possesses an orthorhombic structure described by space group Cmcm with lattice parameters a = 4.298 Å, b = 25.33 Å, and c = 4.303 Å at 300 K [9]. A slight difference in a and c lattice constants originates from the formation of a charge density wave with propagation vector q c = (0, 0, 0.296) at T c = 330 K [10]. It contains Tb-Te bilayers alternating with double tellurium layers Te −1/2 along the b-axis shown in the left panel of Figure 1. The Tb-Te bilayer constitutes a square lattice with Tb-Tb distances along the rungs d r~5 .1 Å and along the diagonals d d~4 .3 Å, as shown in the right upper and lower panels of Figure 1. Competing exchange interactions between rare earth magnetic dipoles are J d1 ≈ J d2 J r due to the difference in distances. In accordance with neutron diffraction data [9], the ordering vectors of antiferromagnetic structures below T N1 (AF1) and T N2 (AF2) are equal to q m1 = (1/2, 1/2, 0) and q m2 = (0, 0, 1/2). The deviations of these structures from exact commensurability are discussed in Ref. [10]. Herein, we represent the study of transport, magnetic, and thermal properties of TbTe 3 aimed at the determination of magnetic phase diagrams along principal crystallographic axes. the easy axis in the plane of the van der Waals layer. A moderate external field applied along this axis generates a spin-flip transition transforming DyOCl into a ferromagnet [4].
The rare-earth tellurides RETe3 are known as a family of van der Waals compounds hosting the charge density waves and experiencing antiferromagnetic order at low temperatures [5]. Recently, the effect of a magnetic field on thermodynamic properties in GdTe3 has been investigated. Two antiferromagnetic transitions at TN1 = 11.5 K and TN2 = 9.7 K and an anomaly at T1 = 7 K were found in both magnetic susceptibility χ and specific heat Cp. It was conjectured that the anomaly at T1 might originate from the incommensurate charge density wave in the van der Waals coupled layers of GdTe3 [6].
Various aspects of charge density wave transitions in RE tritellurides are discussed in [7]. Within this family, TbTe3 stands out as material evidencing three successive magnetic phase transitions at TN1 = 6.6 K, TN2 = 5.6 K, and TN3 = 5.4 K ascribed to the interplay of magnetism with charge density waves [8]. It possesses an orthorhombic structure described by space group Cmcm with lattice parameters a = 4.298 Å, b = 25.33 Å, and c = 4.303 Å at 300 K [9]. A slight difference in a and c lattice constants originates from the formation of a charge density wave with propagation vector qc = (0, 0, 0.296) at Tc = 330 K [10]. It contains Tb-Te bilayers alternating with double tellurium layers Te −1/2 along the b-axis shown in the left panel of Figure 1. The Tb-Te bilayer constitutes a square lattice with Tb-Tb distances along the rungs dr ~5.1 Å and along the diagonals dd ~4.3 Å, as shown in the right upper and lower panels of Figure 1. Competing exchange interactions between rare earth magnetic dipoles are Jd1 ≈ Jd2 ≫ Jr due to the difference in distances. In accordance with neutron diffraction data [9], the ordering vectors of antiferromagnetic structures below TN1 (AF1) and TN2 (AF2) are equal to qm1 = (1/2, 1/2, 0) and qm2 = (0, 0, 1/2). The deviations of these structures from exact commensurability are discussed in Ref. [10]. Herein, we represent the study of transport, magnetic, and thermal properties of TbTe3 aimed at the determination of magnetic phase diagrams along principal crystallographic axes.

Experimental
Single crystals of TbTe 3 were grown by a self-flux technique under a purified argon atmosphere, as described previously [11]. High-quality single crystals were selected for the measurements. Thin plates with a thickness of less than 1 µm were prepared by micromechanical exfoliation from thick plates glued on a sapphire substrate. The quality of selected crystals and the spatial arrangement of crystallographic axes were controlled by X-ray diffraction techniques.
The bridges with a width of 50-80 µm in well-defined, namely (100) and (001), orientations were cut from un-twinned single crystals. Measurements of temperature dependencies of resistance have been performed with a conventional 4-probe configuration. Gold evaporation and indium cold soldering were employed for the preparation of electric contacts. Measurements in the magnetic field were done in superconducting solenoid up to B = 8 T.
Thermodynamic measurements, i.e., magnetization and specific heat, were performed on the thin plates of square shape weighing several mg. Measurements for B b were done using the vibrating sample magnetometer option for Physical Properties Measurement System PPMS 9T "Quantum Design". Measurements for B ac were done by means of Magnetic Properties Measurement System MPMS 7T "Quantum Design".

Thermodynamic Properties
At elevated temperatures, M/H vs. T curve in TbTe 3 follows the Curie-Weiss law χ = χ 0 + C/(T − Θ) with temperature independent term χ 0 = − 5 × 10 −4 emu/mol, Weiss temperature Θ = −3.5 K, and Curie constant C = 12.3 emu K/mol. The value of the Curie constant allows estimating the effective magnetic moment of Tb 3+ ions according to the ratio 8 C = µ eff 2 . It gives the µ eff = 9.92 µB, which is in good correspondence with the tabular magnetic moment of Tb 3+ ions µ calc = 9.7 µB. At lowering temperatures, the χ(T) curve deviates downward from the extrapolation of the Curie-Weiss law and evidences a round hump at T*~8 K. This feature is ascribed routinely to correlation effects in low dimensional magnetic systems [12]. Specific heat C p in TbTe 3 evidences sharp singularities at T < T*.
The temperature dependences of both magnetization M and specific heat C p are shown in the left panel of Figure 2. The close inspection of the low-temperature portions of M/H vs. T and C p vs. T curves allows the revealing of these singularities. As shown in the right panel of Figure 2, there are three features on M/H curves at T N1 = 6.7 K, T N2 = 5.7 K, and T N3 = 5.3 K most pronounced in the Fisher specific heat d[(M/H)T]/dT. This function is closely similar to the magnetic specific heat in the region of transition [13]. These three anomalies are also seen in the C p vs. T curve, which mainly consists of the contribution from the magnetic subsystem in the region of transition sharpest of them is detected at T N2 .

Experimental
Single crystals of TbTe3 were grown by a self-flux technique under a purified argon atmosphere, as described previously [11]. High-quality single crystals were selected for the measurements. Thin plates with a thickness of less than 1 μm were prepared by micromechanical exfoliation from thick plates glued on a sapphire substrate. The quality of selected crystals and the spatial arrangement of crystallographic axes were controlled by X-ray diffraction techniques.
The bridges with a width of 50-80 μm in well-defined, namely (100) and (001), orientations were cut from un-twinned single crystals. Measurements of temperature dependencies of resistance have been performed with a conventional 4-probe configuration. Gold evaporation and indium cold soldering were employed for the preparation of electric contacts. Measurements in the magnetic field were done in superconducting solenoid up to B = 8 T.
Thermodynamic measurements, i.e., magnetization and specific heat, were performed on the thin plates of square shape weighing several mg. Measurements for B||b were done using the vibrating sample magnetometer option for Physical Properties Measurement System PPMS 9T "Quantum Design." Measurements for B||ac were done by means of Magnetic Properties Measurement System MPMS 7T "Quantum Design."

Thermodynamic Properties
At elevated temperatures, M/H vs. T curve in TbTe3 follows the Curie-Weiss law χ = χ0 + C/(T − Θ) with temperature independent term χ0 = − 5 × 10 −4 emu/mol, Weiss temperature Θ = −3.5 K, and Curie constant C = 12.3 emu K/mol. The value of the Curie constant allows estimating the effective magnetic moment of Tb 3+ ions according to the ratio 8 C = µeff 2 . It gives the µeff = 9.92 μB, which is in good correspondence with the tabular magnetic moment of Tb 3+ ions µcalc = 9.7 μB. At lowering temperatures, the χ(T) curve deviates downward from the extrapolation of the Curie-Weiss law and evidences a round hump at T* ~8 K. This feature is ascribed routinely to correlation effects in low dimensional magnetic systems [12]. Specific heat Cp in TbTe3 evidences sharp singularities at T < T*. The temperature dependences of both magnetization M and specific heat Cp are shown in the left panel of Figure 2. The close inspection of the low-temperature portions of M/H vs. T and Cp vs. T curves allows the revealing of these singularities. As shown in the right panel of Figure 2, there are three features on M/H curves at TN1 = 6.7 K, TN2 = 5.7 K, and TN3 = 5.3 K most pronounced in the Fisher specific heat d[(M/H)T]/dT. This function is closely similar to the magnetic specific heat in the region of transition [13]. These three anomalies are also seen in the Cp vs. T curve, which mainly consists of the contribution from the magnetic subsystem in the region of transition sharpest of them is detected at TN2.   keeping their appearance. There are two sharp peaks at T N3 and T N2 and a shoulder at T N1 on all these curves, corresponding to the succession of three magnetic phase transitions.
Selected temperature dependences of M/H taken along principal crystallographic axes in TbTe3 at various magnetic fields are shown in Figure 3. Full measurements at H//b were possible only at the lowest magnetic field µ0H = 0.01 T; at higher fields, the fieldinduced mechanical readjustment of the sample prevented reliable measurements. At increasing magnetic field, the M/H vs. T curves monotonically shift to lower temperatures, essentially keeping their appearance. There are two sharp peaks at TN1 and TN2 and a shoulder at TN3 on all these curves, corresponding to the succession of three magnetic phase transitions. The field dependences of magnetization taken at various temperatures along principal axes of TbTe3 are shown in Figure 4. M(H) curves measured along the a and c axes look identical. Both curves are close to zero in weak magnetic fields and evidence spin-flip transition at µ0H = 2.33 T followed by the tendency to saturation. M(H) curve measured along the b-axis grows almost linear in the external magnetic field and evidences a sequence of small jumps at µ0HC1 = 2.9 T, µ0HC2 = 3.8 T, and µ0HC3 = 5.9 T attributed to the crossings of the phase boundaries. These data allow suggesting that the magnetic moments of Tb 3+ ions are oriented within the ac plane.

Transport Properties
Temperature dependences of in-plain resistance at various orientations of the external magnetic field with respect to crystal axes and electric current I are shown in Figure 5. At the lowest magnetic fields, all R(T) dependences show a drop between TN3 and TN2. This tendency is gradually suppressed by the magnetic field. The kinks at R(T) curves marked by arrows correspond to the phase transitions identified in thermodynamic measurements. This means that, despite the assumption that the electric current I presumably flows through the double layers of tellurium, the processes occurring in the terbiumtellurium bilayers have a significant effect on the scattering of current carriers. Note that The field dependences of magnetization taken at various temperatures along principal axes of TbTe 3 are shown in Figure 4. M(H) curves measured along the a and c axes look identical. Both curves are close to zero in weak magnetic fields and evidence spin-flip transition at µ 0 H = 2.33 T followed by the tendency to saturation. M(H) curve measured along the b-axis grows almost linear in the external magnetic field and evidences a sequence of small jumps at µ 0 H C3 = 2.9 T, µ 0 H C2 = 3.8 T, and µ 0 H C1 = 5.9 T attributed to the crossings of the phase boundaries. These data allow suggesting that the magnetic moments of Tb 3+ ions are oriented within the ac plane.
Selected temperature dependences of M/H taken along principal crystallographic axes in TbTe3 at various magnetic fields are shown in Figure 3. Full measurements at H//b were possible only at the lowest magnetic field µ0H = 0.01 T; at higher fields, the fieldinduced mechanical readjustment of the sample prevented reliable measurements. At increasing magnetic field, the M/H vs. T curves monotonically shift to lower temperatures, essentially keeping their appearance. There are two sharp peaks at TN1 and TN2 and a shoulder at TN3 on all these curves, corresponding to the succession of three magnetic phase transitions. The field dependences of magnetization taken at various temperatures along principal axes of TbTe3 are shown in Figure 4. M(H) curves measured along the a and c axes look identical. Both curves are close to zero in weak magnetic fields and evidence spin-flip transition at µ0H = 2.33 T followed by the tendency to saturation. M(H) curve measured along the b-axis grows almost linear in the external magnetic field and evidences a sequence of small jumps at µ0HC1 = 2.9 T, µ0HC2 = 3.8 T, and µ0HC3 = 5.9 T attributed to the crossings of the phase boundaries. These data allow suggesting that the magnetic moments of Tb 3+ ions are oriented within the ac plane.

Transport Properties
Temperature dependences of in-plain resistance at various orientations of the external magnetic field with respect to crystal axes and electric current I are shown in Figure 5. At the lowest magnetic fields, all R(T) dependences show a drop between TN3 and TN2. This tendency is gradually suppressed by the magnetic field. The kinks at R(T) curves marked by arrows correspond to the phase transitions identified in thermodynamic measurements. This means that, despite the assumption that the electric current I presumably flows through the double layers of tellurium, the processes occurring in the terbiumtellurium bilayers have a significant effect on the scattering of current carriers. Note that

Transport Properties
Temperature dependences of in-plain resistance at various orientations of the external magnetic field with respect to crystal axes and electric current I are shown in Figure 5. At the lowest magnetic fields, all R(T) dependences show a drop between T N3 and T N2 . This tendency is gradually suppressed by the magnetic field. The kinks at R(T) curves marked by arrows correspond to the phase transitions identified in thermodynamic measurements. This means that, despite the assumption that the electric current I presumably flows through the double layers of tellurium, the processes occurring in the terbium-tellurium bilayers have a significant effect on the scattering of current carriers. Note that the effect of normal carriers scattering on magnetic orders depends strongly on the direction of the applied electric field (see Figure 5a,b) and becomes much more pronounced under the application of a magnetic field. As can be seen, the peculiarity corresponding to the AF1 transition appears in R(T) curves only in the magnetic field and in the form of an increase of resistance in contrast to results of [5], where all three transitions have been identified only from derivative dR(T)/dT and appear as decrease of resistance. the effect of normal carriers scattering on magnetic orders depends strongly on the direction of the applied electric field (see Figure 5a,b) and becomes much more pronounced under the application of a magnetic field. As can be seen, the peculiarity corresponding to the AF1 transition appears in R(T) curves only in the magnetic field and in the form of an increase of resistance in contrast to results of [5], where all three transitions have been identified only from derivative dR(T)/dT and appear as decrease of resistance.

Magnetic Phase Diagram
To establish the boundaries of various magnetic phases in TbTe3 in a magnetic field oriented along principal crystallographic axes, the dM/dT, dM/dH, and dR/dT derivatives were analyzed. The crossings of phase boundaries were identified as the sequences of peaks. The positions of these peaks are shown in the magnetic field-temperature phase diagrams, as shown in Figure 6. Additionally, sometimes unpronounced features are shown by dash lines. Overall, the data obtained in thermodynamic and transport properties agree well and complement each other.

Summary
The boundaries of two commensurate AF1 at TN1 and AF2 at TN2 and incommensurate AF3 at TN3 phases, as defined in neutron scattering measurements in the absence of an external magnetic field, are confirmed in thermodynamic and transport measurements. The application of magnetic field shifts these boundaries to lower temperatures in parallel. In thermodynamic properties, the anomalies TN3 and TN2 are seen as the sharp

Magnetic Phase Diagram
To establish the boundaries of various magnetic phases in TbTe 3 in a magnetic field oriented along principal crystallographic axes, the dM/dT, dM/dH, and dR/dT derivatives were analyzed. The crossings of phase boundaries were identified as the sequences of peaks. The positions of these peaks are shown in the magnetic field-temperature phase diagrams, as shown in Figure 6. Additionally, sometimes unpronounced features are shown by dash lines. Overall, the data obtained in thermodynamic and transport properties agree well and complement each other. the effect of normal carriers scattering on magnetic orders depends strongly on the direction of the applied electric field (see Figure 5a,b) and becomes much more pronounced under the application of a magnetic field. As can be seen, the peculiarity corresponding to the AF1 transition appears in R(T) curves only in the magnetic field and in the form of an increase of resistance in contrast to results of [5], where all three transitions have been identified only from derivative dR(T)/dT and appear as decrease of resistance.

Magnetic Phase Diagram
To establish the boundaries of various magnetic phases in TbTe3 in a magnetic field oriented along principal crystallographic axes, the dM/dT, dM/dH, and dR/dT derivatives were analyzed. The crossings of phase boundaries were identified as the sequences of peaks. The positions of these peaks are shown in the magnetic field-temperature phase diagrams, as shown in Figure 6. Additionally, sometimes unpronounced features are shown by dash lines. Overall, the data obtained in thermodynamic and transport properties agree well and complement each other.

Summary
The boundaries of two commensurate AF1 at TN1 and AF2 at TN2 and incommensurate AF3 at TN3 phases, as defined in neutron scattering measurements in the absence of an external magnetic field, are confirmed in thermodynamic and transport measurements. The application of magnetic field shifts these boundaries to lower temperatures in parallel. In thermodynamic properties, the anomalies TN3 and TN2 are seen as the sharp

Summary
The boundaries of two commensurate AF1 at T N1 and AF2 at T N2 and incommensurate AF3 at T N3 phases, as defined in neutron scattering measurements in the absence of an external magnetic field, are confirmed in thermodynamic and transport measurements. The application of magnetic field shifts these boundaries to lower temperatures in parallel. In thermodynamic properties, the anomalies T N3 and T N2 are seen as the sharp signatures of the first-order transitions. It can be due to the giant magnetostriction of the rare-earth compounds, which transforms λ-type anomaly inherent for a second-order phase transition of purely magnetic origin to a symmetric sharp peak at the magnetostructural phase transition. At the same time, the anomaly at T N1 is seen as a shoulder in both C p (T) and Fisher heat. It is not always straightforward to detect this anomaly under a magnetic field, which usually smears the transitions. In essence, the magnetic phase diagram obtained provides a full description of the magnetic properties of the terbium tritelluride.