Signature of elementary excitations in Se-substituted layered triangular-lattice

Geometrical spin frustration in a layered triangular-lattice system prohibits the long-range magnetic order and could give rise to unusual spin-disordered states of matter in which spins are strongly correlated and highly entangled with low-energy excitations. Here we investigate the low-temperature magnetic and specific heat properties of the layered triangular-lattice system CuCr(S 1-x Se x ) 2 within the entire Se-for-S substitution range. In magnetization measurements, the x = 0 phase exhibits a sharp cusp-like antiferromagnetic transition at 38 K, while for all the Se-substituted samples a broad maximum along with a spin-glass-like freezing at low temperatures is seen. The Curie-Weiss temperature systematically increases from − 122 K for x = 0 to + 15 K for x = 1. These observations suggest that the magnetic ground state is determined by the competition between antiferromagnetic direct exchange interactions and ferromagnetic super-exchange interactions. For x = 0, the magnetic contribution to heat capacity exhibits a sharp jump at the magnetic transition temperature indicative of 3D long-range magnetic order, while for the x > 0 samples it increases smoothly across the magnetic transition range following a power-law (~ T 2.1 ) behavior. All these results are in line with an existence of unusual gapless spin-liquid like excitations originating from the spin frustrations and competing nearest-neighbor interactions in CuCr(S,Se) 2 .


Introduction
Exotic magnetic ground states are expected for layered triangularlattice transition metal compounds; in these materials the geometrical frustration of spins in a triangular lattice and the competing magnetic interactions with distant neighbours give rise to a variety of exciting magnetic, electrical and thermal properties [1,2].The ternary chromium chalcogenides of the MCrX 2 type (M = Li, Na, K, Cu or Ag; X = S, Se) form one of such material class, and thus an interesting playground for creating novel physical properties.A major source of fascination so far has been the magneto-elastic coupling at the antiferromagnetic (AFM) phase transition seen for these compounds [3][4][5].
In the MCrX 2 structure consisting of hexagonal CdI 2 -type CrX 2 layers, the magnetic Cr 3+ ions form a triangular lattice which is well separated by the non-magnetic chalcogenides and the M + ions located in the van der Waals gaps.The triangular geometric arrangement introduces strong spin effects and frustration on the Cr 3+ magnetic moment [6][7][8][9].The relief of this frustration through lattice distortions seems to be the source of the spontaneous multiferroic properties and polarization also reported for this system [10,11].
For the MCrX 2 system, several different magnetic structures have been reported, such as a 120 ○ spin structure for LiCrS 2 [12], a commensurate magnetic structure for KCrS 2 , and a helix structure with an in-plane orientation for NaCrS 2 [4].The Curie-Weiss temperature (θ CW ) related to the dominating intralayer interactions has been found to vary with the Cr-Cr distance from a large negative value of −250 K for LiCrS 2 to a large positive value of + 250 K for KCrSe 2 [3,4,13,14].The half-filled t 2g orbitals of trivalent chromium are responsible for the antiferromagnetic (AFM) nature.In case of X = Se, the direct magnetic Cr-Cr exchange interaction gets weaker, and the indirect perpendicular ferromagnetic (FM) Cr-X-Cr interaction starts to dominate.The weak interlayer coupling results in the relatively low magnetic ordering temperature of T N = 20-55 K, while the interactions among the Cr atoms within the layer are strong [3,4,15].Indeed, for the X = Se compounds, within the layers the Cr atoms are FM coupled, while between the layers they are AFM coupled.
Here we selected the CuCr(S,Se) 2 system with Cr(III) and Cu(I) for our systematic study on the spin-lattice effects.The two end members, CuCrS 2 and CuCrSe 2 , are narrow bandgap p-type AFM semiconductors; the bandgap decreases by ca.70 % when sulfur is replaced by selenium.The magnetic exchange is expected to be stronger in the Cr-S-Cr pathway compared to the interlayer exchange through the Cr-S-Cu-S-Cr bonds.However, a three-dimensional magnetic ordering at T N ≤ 40 K into a complex helical structure with an incommensurate magnetic propagation vector was observed in neutron powder diffraction, which is possible only with the interlayer exchange interactions with the same magnitude of intralayer exchange along the crystallographic transition from rhombohedral to monoclinic geometry [10].This crystallographic transition is also evident from the specific heat measurements which show a peak at T ≤ 40 K, as reported for CuCrS 2 [10,16,17].A kind of maximum different from the sharp peak seen for CuCrS 2 has been observed for CuCrSe 2 at T N = 55 K, which could be due to magnetic ordering without lattice distortion [16].However, as compared to 3D long-range non-collinear magnetic ordering along the distortion, CuCrSe 2 shows short-range ordering that extends up to much higher temperature than T N = 55 K [7,16].A recent study found even-oddlayer-dependent FM in CuCrSe 2 , confirming that the intralayer interactions are strongly ferromagnetic [18].
Despite their isomorphic crystal structures, the magnetic ground states are different for CuCrS 2 and CuCrSe 2 , due to the dominance of AFM direct exchange interactions in CuCrS 2 and FM super-exchange interactions in CuCrSe 2 .Hence the CuCr(S 1-x Se x ) 2 system provides us with an interesting platform to explore the effects of competing magnetic interactions on various physical properties.We will present evidence of an exotic magnetic ground state for the Se-substituted CuCrS 2 samples.The magnetic and heat capacity properties indicate the absence of long-range magnetic order, and the magnetic heat capacity follows a power law T α (α ~ 2.1) behaviour which implies elementary gapless spin-liquid like excitations.

Experimental details
For this physical-property characterization study, polycrystalline CuCr(S 1-x Se x ) 2 samples with x = 0, 0.1, 0.2, 0.5, 0.8, 0.9, and 1, were synthesized through solid state reaction from appropriate amounts of elemental precursors, Cu (99.999 %), Cr (99.0 %), S (99.999 %), and Se (99.999 %) in sealed quartz ampoules under vacuum.The ampoules were slowly heated to 900 ○ C and kept there for 24 h, followed by slow cooling to room temperature.The thus obtained reacted charges were grounded for homogenization in an argon-filled glove box and then pressed into pellets under 8 kPa of pressure.The pellets were again sealed in quartz ampoules under vacuum for a second heat treatment at 900 ○ C for 24 h, and finally slowly cooled to room temperature.The phase-purity and structural evolution of the samples were confirmed with X-ray diffraction (XRD; PANanalytical X'Pert PRO MPD Alpha-1; Cu Kα1 radiation); the results for the entire CuCr(S 1-x Se x ) 2 series were recently published elsewhere [19].In short, all the CuCr(S 1- x Se x ) 2 samples were found single-phasic of the expected rhombohedral crystal structure (space group R3m) with lattice parameters linearly increasing with the Se content x according to Vegard's law [20], indicating the complete Se-for-S solid solubility.
The samples were systematically characterized for the temperature and magnetic field dependent magnetization (M) in both field-cooling (FC) and zero-field-cooling (ZFC) modes using a vibrating-sample magnetometer (VSM) attached with a physical property measurement system (PPMS; Quantum Design; equipped with 9 T magnet).The magnetic susceptibility data were fitted to the Curie-Weiss equation χ = C m /(T − θ CW ), where C m is Curie constant and θ CW is the asymptotic Curie-Weiss temperature, to calculate the effective magnetic moment as The mean-field approximation was used to understand the in-plane interaction (J 1 ) and the interplane interactions (J 2 ).It should be noted that the mean-field approximation is valid when the inter-plane interaction J 2 dominates the in-plane interaction J 1 , and in our samples this condition holds only in the Se-rich samples.Hence these considerations should be taken only as a qualitative attempt to understand the behaviors of J 1 and J 2 with increasing Se-substitution level in CuCr(S 1-x Se x ) 2 .The in-plane exchange interaction J 1 is the sum of AFM direct exchange Cr-Cr interactions and FM super-exchange Cr-X-Cr interactions, and interplane exchange interaction J 2 is antiferromagnetic.The sum of J 1 and J 2 was estimated from k B θ CW = Low-temperature specific heat capacity measurements were carried out with the same PPMS equipment.The temperature dependence of heat capacity C p (T) was analyzed using the Debye law; the magnetic contribution (C mag ) was estimated by subtracting the lattice contribution (C l ) from the overall C P (T).The C l in Debye model is given by where N is number of atoms per formula unit, R is the universal gas constant (8.314J/mol K) and θ D is Debye temperature.The magnetic entropy (S mag ) was estimated by integrating

Magnetic properties
The magnetization versus temperature data measured at 1 kOe for representative CuCr(S 1-x Se x ) 2 samples are presented in Fig. 1(a), and the inverse magnetic susceptibility versus temperature plots for the same samples are shown in Fig. 1(b).For x = 0, a sharp cusp-like AFM transition is seen at T N = 38 K, while the Se-substituted x = 0.1, 0.5, and 0.8 samples show broad maxima around 32, 22 and 28 K, respectively.For x = 0, the magnetization is reversible under the ZFC and FC conditions, while for the x = 0.1, 0.5, and 0.8 samples a splitting between the ZFC and FC curves is seen below 13, 18 and 25 K, respectively, indicating a spin-glass (SG) like freezing of magnetic moments.In the temperature range of 100-300 K, the magnetization data follow the Curie-Weiss law for all samples, see the dotted lines in Fig. 1(b); the obtained θ CW .C m and μ eff values for different samples are shown in Table 1.Importantly, for all samples the μ eff value is very close to the effective moment of 3.9 μ B / Cr, as expected (total spin 3/2 in d 3 configuration with the quenched orbital contribution).Our previous spin polarized electronic band structure calculations gave significantly smaller values, i.e. 3.28 μ B , 3.37 μ B , and 3.50 μ B for x = 0, 0.5 and 1, respectively [19].This could be due to the strong hybridization of the Cr 3d orbitals with the p orbitals of surrounding chalcogen atoms.With increasing Se content, θ CW systematically increases from −122 K for x = 0 to −71 K for x = 0.8, +22 K for x = 0.9, and finally to + 15 K for x = 1, signifying the domination of FM super-exchange interaction over the AFM direct exchange interaction.The values calculated through mean field approximation for the in-plane and interplane magnetic interactions (Table 1) clearly indicate that the in-plane interaction J 1 is strongly AFM for x = 0 and with increasing Se content its strength decreases and finally it turns FM for x = 1.The interplane interaction J 2 remains AFM for all the samples with its strength decreasing with increasing x.
The M versus B isotherms presented for the x = 0 sample in Fig. 1(c) are all linear up to 5 T both in the paramagnetic and AFM regimes.For the x = 0.5 sample (Fig. 1(d)) the linear dependency is seen only above 50 K; the small FM-like hysteresis loop seen below 50 K shown in supplementary information (SI) Fig. S1 is presumably related to the SG like freezing observed in this phase.The dc magnetization at different applied fields (0.1 T, 1 T and 2 T) is measured for SG phase (x = 0.5) and presented in SI in Fig. S2.The peak temperature ZFC magnetization and the splitting temperature ZFC and FC magnetization moves to lower temperature with increasing applied magnetic field and could a typical characteristic of SG phase.

Heat capacity
Heat capacity data measured for the CuCr(S 1-x Se x ) 2 samples are presented in Fig. 2. For x = 0, the sharp peak seen in C p (T) perfectly coincides with the magnetic transition temperature measured for this sample at T N = 38 K.It also corresponds to the structural transition from rhombohedral to monoclinic symmetry previously observed for this phase in neutron and synchrotron diffraction studies [10,17].Moreover,

Table 1
Neel temperature (T N ), Curie constant (C m ), spin-glass freezing temperature (T g ), asymptotic Curie temperature (θ CW ), effective magnetic moment (μ eff ), intralayer exchange (J 1 ) and interlayer exchange (J 2 ), and Debye temperature (θ D ) for CuCr(S 1-x Se x ) 2 samples.a previous work has reported a similar behavior of C p (T) for AgCrS 2 and explained it by the latent heat contribution related to structural transition [23].For our CuCr(S 1-x Se x ) 2 system, the sharp transition as observed for the x = 0 sample rapidly smears down upon the Se-for-S substitution such that for the x > 0 samples just a smooth increase in the C p (T) curve across the magnetic transition is seen.The lattice contribution to heat capacity C l (T) follows the Debye law at high temperatures as shown by a red continuous curve in Fig. 2(a).The fitted values of θ D are presented in Table 1 for all the samples.The C mag (T) data are shown in Fig. 2(b).For x = 0, C mag is non-zero below 110 K and rises steadily with decreasing temperature until T N , below which a sharp jump occurs.A similar behavior of C mag (T) for CuCrS 2 was previously reported by A. Karmakar et al. [17].For the Se-for-S substituted samples, C mag is non-zero below 180 K for x = 0.5, below 150 K for x = 0.2 and below 110 K for x = 0.8 and x = 0.9.At lower temperatures, C mag rises slowly and reaches a well-rounded maximum between 30 and 70 K, decreasing slowly and remaining non-zero down to the lowest measurement temperature of 2 K.The absence of sharp change in C mag but a well-rounded maximum around 30 -70 K is suggestive of a gapless ground state.The non-zero value of C mag at high T above T N indicates short-range magnetic ordering in paramagnetic state.For the x > 0 samples the short-range ordering extends to much higher temperatures due to the domination of FM super-exchange interactions in these samples.

Magnetic entropy
Magnetic entropy S mag for the x = 0 sample is small at low temperatures and increases rapidly with temperature when approaching the magnetic transition at 38 K due to the destruction of long-range magnetic order by thermal fluctuations.The 38-K value of ~ 12 J/mole-K for x = 0.0 is close to the entropy for the localized moments of Cr 3+ ions with S = 3/2 [S mag = Rln(2S +1) = 11.53J/mole-K].Above 38 K magnetic entropy continuously increases and reached maximum value ~ 25 J/mole-K around 100 K.For the x > 0 samples, S mag remains high at low temperatures and slowly increases with temperature smoothly across the magnetic transition.For x = 0.2 and 0.5, S mag increases up to 150 K.In other samples (x = 0, 0.8 and 0.9) the rise in S mag is observed up to 100 K.The large value of magnetic entropy in comparison to the localized moments of Cr 3+ in paramagnetic state is unusual and needs further investigation.
From the C mag (T) data presented in log-log scale for the same samples (Fig. 3(b)), linear behavior is seen between 2 and 25 K for the Sesubstituted samples, the slope of the lines being in the range of 2.10-2.25.Thus, C mag for x > 0 follows T 2 (C mag (T) = AT 2 ) with the exponent A being at 7.2x10 -3 J/mole-K 3 (1.27x10-2 J/mole-K 3 ) for x = 0.5 (x = 0.9).Earlier we found that C mag (T) for x = 1 follows a T 2 -dependence at low temperature from 2 to 14 K [24].A similar T 2dependence of C P (T) has been observed for Cu(Cr 1-x V x )S 2 at x = 0.3, in which the long-range magnetic order is suppressed by the lattice defects and rather a spin-glass like freezing is observed at low temperatures below 20 K [25].The T 2 -dependence of C P (T) at low temperatures can be seen in many other systems as well, such as Kagome spin-glass, frustrated antiferromagnets, and diluted 2D antiferromagnets [26].On the other hand, for a typical canonical spin-glass system, C P (T) follows a linear T-dependence below the freezing temperature [26].In general, we expect a T 3 -dependence for conventional long range AFM systems because of 3D magnetic excitations and a sublinear T 2/3 -dependence for the 2D quantum-spin-liquid (QSL) and nearly T 2 -dependence for 3D spin-liquid systems [27][28][29][30][31].The nearly T 2 -dependence in current Sesubstituted samples with non-Heisenberg magnetic interactions clearly suggests existence of low-energy spin-liquid like elementary excitations, originating from the competition between AFM direct exchange and FM super-exchange interactions.The non-Heisenberg magnetic interaction here represent three-way exchange interactions; the in plane (AFM (Cr-Cr) and FM (Cr-X-Cr)), and out of plane (AFM (Cr-X-Cu-X-Cr)) interlayer exchange.

Magnetic phase diagram
From the M(T) curves, we determined the T N values and constructed the magnetic phase diagram for the CuCr(S 1-x Se x ) 2 system, see Fig. 4(a).The phase diagram clearly reveals different magnetic regimes with increasing Se content: with increasing x, T N decreases up to x = 0.5 and then increases again with a further increase in x.For the range 0.1 ≤ x ≤ 0.9 we observed spin-glass like freezing at low temperatures: the freezing temperature was found to first increase from that of 13 K for x = 0.1 to 18 K for x = 0.5, and 25 K for x = 0.8, then sharply decrease to 11 K for x = 0.9.This Se substitution range corresponds to the regime where the spin-liquid like behavior of C mag ∼ T 2 is observed.The S-rich and Se-rich end member phases are identified as strong and weak AFM phases, respectively.
Finally, we show in Fig. 4(b) the monotonic increase of θ CW with increasing nearest neighbor in-plane Cr-Cr distance (which linearly increases with x).For the MCrX 2 compounds, the in-plane spin structure shows four distinct areas depending on θ CW : ferromagnetic or A-type spin structure for θ CW ≥ 90 K, helical spin structure for 0≤ θ CW ≤ 90 K, complex spin structure with monoclinic lattice distortions for −150≤ θ CW ≤ 0 K, and 120 • spin structure for θ CW ≤ −150 K [22].According to this, the x = 0 end member with θ CW = −122 K is of 120 0 spin structure and the other x = 1 end member with θ CW = 15 K is of the helical spin structure.The middle phases (0.1 ≤ x ≤ 0.90) fall within the complex spin structure regime.This change in magnetic structure from x = 0 to x = 1 is caused by the large super-exchange interaction.

Conclusions
We have systematically investigated the low-temperature magnetic and specific heat capacity properties within the layered triangularlattice CuCr(S 1-x Se x ) 2 solid-solution system.The samples appeared antiferromagnetic at low temperatures: the χ −1 showed a sharp cusp-like AFM transition at 38 K for x = 0, and a rather broad maximum for the Se-substituted samples.At high temperatures above 100 K, the inverse magnetic susceptibility followed Curie-Weiss behavior, the asymptotic Weiss temperature systematically increasing with increasing Se-for-S substitution level from −122 K for x = 0 to + 15 K for x = 1.The sign change from the negative to slightly positive reflected the gradual domination of the FM indirect super-exchange interaction over the AFM direct exchange interactions.The magnetic heat capacity exhibited a sharp jump at the magnetic transition temperature of 38 K for the x = 0 sample, while for the Se-substituted samples no sharp changes were seen.For these x > 0 samples, the well-rounded maxima seen between 30 and 70 K was taken as an indication of a gapless ground state.Moreover, the T 2 -dependence of the magnetic heat capacity for these samples in the temperature range from 2 to 25 K pointed towards the presence of unusual spin-liquid like elementary excitations.The competing magnetic interactions in the CuCr(S 1-x Se x ) 2 system is apparently the source of the low-energy spin-liquid like excitations at low temperature.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Fig. 2 .
Fig. 2. (a) Heat capacity versus temperature for CuCr(S 1-x Se x ) 2 samples; red curve is Debye model fit for x = 0 in 100 -300 K range.(b) Magnetic contribution to heat capacity for the same samples.(For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 3 .
Fig. 3. (a) Temperature dependence of magnetic entropy for CuCr(S 1-x Se x ) 2 samples.(b) Logarithmic plots of C mag versus T (dashed line with slope of 2.1 drawn to guide the eye).
G.C.Tewari et al.