Charge density wave and large non-saturating magnetoresistance in YNiC$_2$ and LuNiC$_2$

TL;DR

This study investigates the charge density wave phenomena and large non-saturating magnetoresistance in YNiC₂ and LuNiC₂, revealing temperature-dependent phase transitions, Fermi surface nesting effects, and significant magnetoresistance enhancements.

Contribution

It provides new insights into the charge density wave transitions and magnetoresistance behavior in YNiC₂ and LuNiC₂, extending the understanding of their electronic properties.

Findings

Charge density wave transitions at 318 K and 275 K in YNiC₂.

Large magnetoresistance reaching 470% in YNiC₂ at 1.9 K and 9 T.

Fermi surface nesting leads to high mobility carriers without magnetic order interference.

Abstract

We report a study of physical properties of two quasi-low dimensional metals YNiC$_2$ and LuNiC$_2$ including the investigation of transport, magnetotransport, galvanomagnetic and specific heat properties. In YNiC$_2$ we reveal two subsequent transitions associated with the formation of weakly coupled charge density wave at $T_{CDW}$ = 318 K, and its locking in with the lattice at $T_1$ = 275 K. These characteristic temperatures follow the previously proposed linear scaling with the unit cell volume, demonstrating its validity extended beyond the lanthanide-based $R$NiC$_2$. We also find that, in the absence of magnetic ordering able to interrupt the development of charge density wave, the Fermi surface nesting leads to opening of small pockets, containing high mobility carriers. This effect gives rise to substantial enhancement of magnetoresistance, reaching 470 % for YNiC$_2$ and 50 % for LuNiC$_2$ at $T$ = 1.9 K and $B$ = 9 T.

Tables

Table 1. Table 1: Thermodynamic parameters: relative increase of specific heat Δ ​ C p C p ​ ( T ) Δ subscript 𝐶 𝑝 subscript 𝐶 𝑝 𝑇 \frac{\Delta C_{p}}{C_{p}(T)} , entropy Δ ​ S Δ 𝑆 \Delta S and enthalpy Δ ​ H Δ 𝐻 \Delta H at transition temperatures T C ​ D ​ W subscript 𝑇 𝐶 𝐷 𝑊 T_{CDW} and T 1 subscript 𝑇 1 T_{1} in YNiC 2 .

	$\frac{\Delta C_p}{C_p\left(T\right)}$ (%)	$\Delta S$ (mJ mol-1K-1)	$\Delta H$ (J mol-1)
$T_{CDW}$	1.1	30.6	9.4
$T_1$	2.9	77.8	21.3

Equations

$${\rho }_{xx}={\rho }_{xx0}+A{T}^p,$$

$$MR=\frac{{\rho }_{xx}(B)-{\rho }_{xx}(B=0)}{{\rho }_{xx}(B=0)}\cdot 100\%,$$

$$MR=f({\omega }_c\tau )=f\left(\frac{B}{{\rho }_{xx}(B=0)}\right)$$

$$\frac{{\rho }_{yx}}{B}=\frac{1}{e}\frac{n_h{\mu }_h^2-n_e{\mu }_e^2+(n_h-n_e){\mu }_e^2{\mu }_h^2{B}^2}{(n_h{\mu }_h+n_e{\mu }_e{)}^2+(n_h-n_e{)}^2{\mu }_h^2{\mu }_e^2{B}^2}$$

$$\frac{{\rho }_{yx}}{B}=\frac{1}{e}\frac{1}{n_h-n_e}=\frac{1}{e}\frac{1}{n_{eff}}$$

$${\sigma }_{xy}(B)=\frac{{\rho }_{yx}}{{\rho }_{yx}^2+{\rho }_{xx}^2}$$

$${\sigma }_{xy}(B)=\sum _i\frac{{\sigma }_i{\mu }_{i}B}{1+{\mu }_i^2{B}^2}$$

$${\mu }_{ext}=\frac{1}{B_{ext}}$$

$${\sigma }_{xy}(B)=n_{xy}e{\mu }_{xy}^{2}B\left(\frac{1}{1+{\mu }_{xy}^2{B}^2}+C_{xy}\right)$$

$$C_p=\gamma T+\beta T^3$$

$${\Theta }_D={\left(\frac{12{\pi }^{4}nR}{5\beta }\right)}^{\frac{1}{3}}$$

$$\frac{\Delta C_p}{\gamma T_{CDW}}=1.43$$

Full text

I Introduction

The large variety of unique physical phenomena offered by quasi low-dimensional systems arouse the unfading interest of researchers exploring the field of condensed matter physics. Large anisotropy of the electronic structure is a precursor for the Peierls transition towards charge density wave (CDW) with electronic carrier condensation and Fermi surface nestingMonceau (2012); Grüner (2000, 1988). Low dimensionality has been suggested to play a crucial role in high temperature superconductivity while charge density wave has been found to emerge in all the phase diagrams of the cuprate superconductorsCaprara et al. (2017); Caplan and Orgad (2017); Laliberté et al. (2018); Cyr-Choinière et al. (2018). This fact additionally amplifies the interest in the interactions between various types of electronic, quantum, and magnetic orderingJacques et al. (2016); Chang et al. (2012, 2016); Graf et al. (2004); Gruner et al. (2017); Cho et al. (2018).

Recently, extensive attention has been devoted to the family of ternary carbides $R$NiC2, where $R$ is rare earth element. This group of materials offers the rare opportunity to tune the magnetic ground state via replacement of the rare earth atom (the nickel atoms carry no magnetic moment). Most of the lanthanide-based members of this family are antiferromagnets ($R$ = Ce, Nd, Gd, Tb, Dy, Ho, Er and Tm) with Néel temperature varying from 3.4 K for HoNiC2 to 25 K for TbNiC2Schäfer et al. (1997); Onodera et al. (1995, 1998); Hanasaki et al. (2011); Roman et al. (2018a); Steiner et al. (2018); SmNiC2 is a ferromagnet with $T_{C}$ = 17.5 K Onodera et al. (1998), PrNiC2 shows a weak magnetic anomaly at $T^{*}$ = 8 K, LaNiC2 is a noncentrosymmetric superconductor with $T_{c}$ = 2.9 KWiendlocha et al. (2016); Lee et al. (1996); Pecharsky et al. (1998); Quintanilla et al. (2010); Landaeta et al. (2017) while LuNiC2 was recently reported as a plain paramagnet down to 1.9 KSteiner et al. (2018). Furthermore, most of the $R$NiC2 compounds (with the exception of $R$ = La and Ce) show charge density wave with Peierls temperature ($T_{CDW}$) not only higher than $T_{N}$, $T_{C}$ and $T^{*}$, but exceeding 300 K for the late lanthanides ($R$ = Ho - Lu)Murase et al. (2004); Ahmad et al. (2015); Michor et al. (2017); Laverock et al. (2009); Shimomura et al. (2016). So far, the charge density wave has never been found in $R$NiC2 compounds, with $R$ outside of the lanthanide group. The electronic structure calculations however, have revealed the resemblance between the Fermi surface topology of YNiC2Hase and Yanagisawa (2009) and lanthanide-based $R$NiC2Laverock et al. (2009); Steiner et al. (2018) showing CDW. Alas, the fermiology of this compound has not been discussed in the terms of nesting.

Remarkably, magnetic ordering has been found to mutually interact with the CDW; in SmNiC2 the ferromagnetic transition destroys the charge density waveShimomura et al. (2009); Hanasaki et al. (2012); Kim et al. (2012), while in NdNiC2 and GdNiC2 antiferromagnetism only partially suppresses the CDW, and both entities coexist below $T_{N}$Yamamoto et al. (2013); Kolincio et al. (2016a); Hanasaki et al. (2017); Kolincio et al. (2017). On the other hand, this magnetic anomaly has been found to enhance the nesting properties in PrNiC2 Yamamoto et al. (2013); Kolincio et al. (2017) and some signatures of a constructive influence of CDW on AFM were recently observed in GdNiC2 Kolincio et al. (2016a); Hanasaki et al. (2017), Nd1-xGdxNiC2Roman et al. (2018b), and Nd1-xLaxNiC2 Roman despite the clear competition between these types of ordering. For the majority of the the $R$NiC2 family members, the CDW state is influenced (mostly suppressed) by magnetism, and therefore the Fermi surface nesting is partially or even completely disturbed.

In this paper we study the physical properties of paramagnetic YNiC2 and LuNiC2 in order to explore the consequences of a pure and fully developed Peierls transition in $R$NiC2 in the absence of magnetic ordering. We report for the first time the charge density wave in YNiC2 with $T_{CDW}$ = 318 K followed by a putative lock-in transition at $T_{1}$ = 275 K. We also observe the large positive and remarkably linear magnetoresistance, which at the lowest temperatures reaches 470 % for YNiC2 and 50 % for LuNiC2 with no signs of saturation up to $B$ = 9 T. By detailed analysis of magnetotransport and galvanomagnetic properties we find that this effect stems from the multiband character of the electrical conductivity and existence of the high mobility pockets remaining in the Fermi surface after uninterrupted, yet imperfect nesting characteristic for quasi-2D metals undergoing a Peierls transitionMonceau (2012); Grüner (2000, 1988).

II Experimental

The polycrystalline samples of YNiC2 and LuNiC2 were synthesized by arc melting of elemental precursors (no excess of Y or Lu was added) followed by annealing at 900o C, according to the procedure described in detail in ref. Kolincio et al. (2017). Powder x-ray diffraction (pXRD) was conducted with PANalytical X’Pert PRO-MPD diffractometer using Kα line of Cu spectrum. Experimental data was analyzed with the use of Fullprof softwareRodríguez-Carvajal (1993).

All the physical properties measurements were performed with a commercial Quantum Design Physical Properties Measurement System (PPMS) in the temperature range from 1.9 to 400 K and magnetic field up to 9 T. Thin ($\phi=37\mu m$) Pt wires serving as electrical contacts for four-probe transport and Hall measurements were spark-welded to the polished samples surfaces. The magnetic field for Hall and magnetoresistance measurements was oriented perpendicularly to the current direction. The Hall signal was measured with the reversal of the direction of the magnetic field and corrected for the parasitic longitudinal resistance component via antisymmetrization of the measured data. Specific heat measurements were conducted with a standard relaxation method on flat samples with polished surfaces. Apiezon N grease was used as a heat conducting medium for measurements in temperature range 1.9 K - 300 K. Since Apiezon N reveals a glass transition above 300 K, Bunting et al. (1969); Kreitman (1971), the data collection at elevated temperatures was performed with Apiezon L.

III Results and discussion

III.1 X-ray diffraction

The quality and phase purity of polycrystalline YNiC2 and LuNiC2 samples were confirmed with pXRD. The analysis of the obtained diffraction patterns, shown in Fig. 1 revealed that the observed peaks for both compounds are successfully indexed in the orthorombic CeNiC2-type structure with space group Amm2 (#38). Only for LuNiC2, additional reflections corresponding to Lu4Ni2C5 impurity phase were detected. From the comparison of the highest peaks corresponding to main and impurity phases respectively, we have estimated the relative amount of Lu4Ni2C5 to 9%. The pXRD results for both $R$NiC2 compounds were analyzed by LeBail refinement which revealed lattice constants of YNiC2: $a$ = 3.5733(1) Å, $b$ = 4.5082(1) Å, $c$ = 6.0351(1) Å  and of LuNiC2: $a$ = 3.4468(1) Å, $b$ = 4.4734(2) Å, $c$ = 5.9787(2) Å. These structural parameters are in good agreement with previous reportsJeitschko and Gerss (1986); Steiner et al. (2018).

III.2 Transport and magnetoresistance

The thermal dependence of the resistivity of YNiC2 and LuNiC2 is depicted respectively in panels a and b of Fig. 2. At high temperatures, the transport properties of YNiC2 show a regular metallic behavior with the resistivity lowering as the temperature is decreased. At $T$ = 318 K a local minimum followed by a hump is observed in $\rho_{xx}$($T$). At lower temperatures, the resistivity returns to the metallic character with $\frac{d\rho_{xx}}{dT}<0$. Such behavior is typical of a Peierls transition in quasi-2D metals, where the nesting is not complete and a certain number of free carriers remain in the Fermi surface. One unusual feature here, however, is the sharp decrease of the resistivity at temperatures slightly below the maximum of $\rho_{xx}$($T$) indicated by an arrow in the Fig. 2a. This stands in contrast to a smooth crossover observed for canonical Peierls transitionsMcMillan (1977). To determine the character of this additional anomaly, we have performed slow heating and cooling temperature sweeps in the vicinity of the CDW transition (0.1 K/min). The results, shown in the lower inset of Fig. 2a reveal the presence of a narrow thermal hysteresis opening approximately at $T_{1H}\simeq$ 285 K and closing at $T_{1L}\simeq$ 265 K. The transition temperature is determined as the average of $T_{1H}$ and $T_{1L}$, giving $T_{1}$ = 275 K. Such a first order transition is expected for the preexisting incommensurate CDW locking in with the lattice and transforming into a commensurate modulationMcMillan (1975) as reported beforehand for the late lanthanide-based $R$NiC2Shimomura et al. (2016); Roman et al. (2018a). Interestingly both the Peierls and presumed lock-in transition temperatures stand in agreement with the previously proposed linear scaling with the unit cell volumeRoman et al. (2018a), as shown in Fig. 2c. One must, however, remember that despite the analogy with the other $R$NiC2, an x-ray diffuse scattering experiment performed with single crystals is required to deliver an ultimate evidence of the lock-in nature of the transition seen at $T_{1}$ = 275 K. An additional feature observed in the low temperature resistivity curve is a decrease of $\rho_{xx}$ at $T$ = 4 K (upper inset of Fig. 2a), reminiscent of the onset of a superconducting transition. The absence of any signature of the bulk superconductivity in specific heat at this temperature allows us to attribute this anomaly to the presence of a trace amount of YC2 impurityGulden et al. (1997), not detected by x-ray diffraction. To estimate the genuine value of $\rho_{xx}$ for YNiC2 at lowest $T$, we have fitted the temperature range 5 K $<T<$ 50 K with the power law:

[TABLE]

(where $\rho_{xx0}$ is the residual resistivity and exponent $p$ depends on the prevailing scattering mechanism) and extended the obtained function to lowest temperature.

For LuNiC2 (see Fig. 2c) the character of the conductivity is metallic up to 400 K, in agreement with previous reports of a similar metal-metal CDW transition at $T_{CDW}\simeq$ 450 KRoman et al. (2018a); Steiner et al. (2018), which is beyond the upper temperature limit of the PPMS, thus not revealed by our current measurement. Interestingly, despite that the CDW modulation wavevector determined previously for LuNiC2 Steiner et al. (2018) differs from the vectors reported for $R$ = Pr, Nd, Sm, Gd and Tb Yamamoto et al. (2013); Shimomura et al. (2009, 2016), the Peierls temperature corresponding to Lu bearing compound (reported in referencesRoman et al. (2018a); Steiner et al. (2018)) obeys the scaling proposed for magnetic $R$NiC2 Roman et al. (2018a) as well, which is shown in Fig. 2c.

An external magnetic field has a negligible impact on the transport properties in the normal state of YNiC2, typical for a casual metal. On the other hand, the application of $B$ significantly increases the electrical resistivity in the CDW state of this material. This is also true, however, to a lesser extent, for the CDW state of LuNiC2. The magnetic field dependence of magnetoresistance defined as:

[TABLE]

is presented in Fig. 3a and b for YNiC2 and LuNiC2, respectively. For YNiC2, the MR curve for $T$ = 1.9 K where the zero field value of $\rho_{xx}$ was estimated from equation 1 is close to the one measured for $T$ = 5 K, above the superconducting transition and where the $\rho_{xx}$ is already close to the residual value. This result confirms that, the MR($B$) curve for $T$ = 1.9 K is not contaminated by any spurious contribution that could stem from the superconducting impurity phase. At low field, the electrical resistance of both compounds follows a MR$\sim B^{q}$ dependence on magnetic field. MR is subquadratic ($q\simeq$ 1.4 at $T$ = 1.9 K) for YNiC2 and approximately parabolic ($q\simeq$ 2.2 K at $T$ = 1.9 K ) for LuNiC2. As the magnetic field is raised, the character of MR$(B)$ curves for both compounds evolves towards a linear manner, without any signs of saturation up to 9 T. The magnitude of the magnetoresistance increases as the temperature is lowered and at $T$ = 1.9 K and $B$ = 9 T reaches 470 % for the former and 50 % for the latter compound, respectively. Typically, the linear magnetoresistance term is described by one of two prevailing models: the classical approach of Parish and Littlewood based on mobility fluctuations in inhomogeneous materialParish and Littlewood (2003, 2005) or Abrikosov’s quantum model used beyond the quantum critical limit, when only a single Landau level is occupiedAbrikosov (1969, 1998, 2000). In a CDW metal however, the linear MR can also originate from the fluctuations of the charge density wave order parameter enhancing the scattering in certain regions of the Fermi surface Sinchenko et al. (2017); Frolov et al. (2018) or from sharp curves on the carrier path due to nesting induced reduction of FS Feng et al. (2019). Another possible scenario is that the linearity in MR originates from the complicated geometry of the Fermi surface (which is additionally modified by nesting), containing both closed and open orbits, which in a polycrystalline sample contribute to MR with saturating and non-saturating signals Lifshitz et al. (1957) respectively.

It shall be noted that, such a large positive magnetoresistive effect has never been reported for any members of the $R$NiC2 family. Typically, in $R$NiC2 compounds exhibiting Peierls transition, MR shows a negative sign due to partial or complete suppression of charge density wave, induced either by magnetic field or magnetic orderingYamamoto et al. (2013); Hanasaki et al. (2012); Kolincio et al. (2016a, 2017); Hanasaki et al. (2017); Lei et al. (2017). In YNiC2 and LuNiC2 however, no signatures of magnetism has been found above $T$ = 1.9 K.

The magnetoresistance can be discussed in the framework of the semiclassical Kohler approachPippard (2009). The prediction of this model is that on the condition of uniform scattering over the whole Fermi surface and a single type of electronic carriers with constant concentration, all the plots:

[TABLE]

(where $\tau$ and $\omega_{c}$ are relaxation time and cyclotron frequency, respectively) superimpose into a single line. The plot of MR as a function of $\frac{B}{R_{0}}$ ($R_{0}$ is the zero field resistance) for YNiC2 is shown in Fig. 3c. The deviations from Kohler scaling, shown in Fig. 3c, are weak yet visible. Upon decreasing the temperature, the plots are subsequently moved higher on the vertical scale (faster increase of MR($\frac{B}{R_{0}}$)) which is observed in a wide temperature range above $T$ = 40 K. For $T<$ 40 K, curves are pressed slightly lower (slower $\frac{B}{R_{0}}$ growth), as depicted in the inset of Fig. 3c. At lowest temperatures, where the zero field resistance is already close to the residual value, the curves superimpose (within the experimental resolution). As shown in Fig. 3d, for LuNiC2 the divergence from Kohler’s scaling is even less pronounced but still visible. The strongest effect is seen in the temperature range $T>\frac{T_{CDW}}{2}$, where the CDW gap is expected not to be completely open yetGrüner (1988). This violation of the Kohler rule can be attributed both to the reconstruction of the Fermi surface due to nesting and to presence of more than one type of carriers in the CDW stateMcKenzie et al. (1998); Yasuzuka et al. (2005). We suggest that a stronger manifestation of the deviation from the MR scaling could be observed at temperatures in the close vicinity of $T_{CDW}$ as in tungsten bronzes also showing Peierls transition Kolincio et al. (2016b). This range is, however, beyond the scope of our experimental equipment.

III.3 Hall effect

To explore the evolution of carrier concentrations, we have examined the Hall effect for both compounds. The thermal dependence of the Hall resistivity ($\rho_{yx}$) is depicted in Fig. 3e (YNiC2) and 3f (LuNiC2). For YNiC2, $\frac{\rho_{yx}}{B}$ is almost temperature independent above $T_{CDW}$. At this characteristic temperature, the Hall resistivity shows an abrupt downturn, indicating the loss of free electrons due to the CDW condensation. The presumed lock-in transition is indicated by a kink in $\frac{\rho_{yx}}{B}(T)$. At lower temperatures, the Hall resistance shows a minimum and then returns to less negative values. Previously, such an effect was observed in magnetic $R$NiC2, and was attributed both to the suppression of charge density wave by the magnetic ordering and to the onset of the anomalous component of the Hall effectKim et al. (2012); Kolincio et al. (2016a, 2017); Lei et al. (2017). Due to the absence of long range magnetism in YNiC2, these two terms appear to be irrelevant in this case. At temperatures below $T_{1}$, the $\frac{\rho_{yx}}{B}(T)$ curves do not superimpose into a single line which suggests that in the CDW state, $\rho_{yx}$ is not linear with $B$.

For LuNiC2 the Peierls temperature $T_{CDW}\simeq$ 450 KRoman et al. (2018a); Steiner et al. (2018), thus at 400 K, which is the maximum temperature limit of our experiment, the system is already in the charge density wave state. All the curves reveal a kink at $T\simeq$ 355 K. Its origin is not clear, however, while this weak anomaly is not detected by other measurements, it might result from the experimental artifact instead of being truly intrinsic to the sample. Another scenario is, that this anomaly originates from the Lu4Ni2C5 impurity phase. Similarly to YNiC2, the sign of $\rho_{yx}$ is negative in the whole temperature range, indicating the dominance of electrons. This is not the only similarity between the $\frac{\rho_{yx}}{B}$ curves for both compounds. Here we also find that for LuNiC2 the Hall resistivity is also driven to more negative values as the free electrons are condensed in the CDW state, which is followed by the return of $\rho_{yx}$ to close to zero at lower temperatures. We find that the $\frac{\rho_{yx}}{B}$ superimpose at temperatures above approximately 250 K. At lower temperatures, the plots do not coincide with each other, indicating a nonlinearity of $\rho_{yx}(B)$ also in LuNiC2. Similarly to the case of YNiC2, further temperature decrease leads to the upturn of the Hall resistivity, which also cannot be attributed to magnetic ordering. A plausible scenario to explain these features is the existence of more than one type of electronic carrier, originating from unnested pockets remaining in the Fermi surface after imperfect nesting, a situation characteristic of quasi-2D metals showing charge density waveMonceau (2012); Grüner (2000).

To obtain a more detailed picture of the electronic parameters, we have examined the magnetic field dependence of $\rho_{yx}$. The results of field sweeps at constant temperatures, shown in the Fig. 4a for YNiC2 and 4b for LuNiC2 reveal a visible deviation of Hall signal from linearity. In the absence of long range magnetic interactions or ordering, this effect is a clear manifestation of the multiband character of electrical conductivityAkiba et al. (2017); Li et al. (2016); Liu et al. (2017); Luo et al. (2015); Wang et al. (2014). In the two-band model, the Hall resistivity is expressed with equation (4)Hur (1972):

[TABLE]

where $n_{h}$, $n_{e}$, $\mu_{h}$ and $\mu_{e}$ are respectively concentrations and mobilities corresponding to two (hole and electron) conduction channels. The direct $\rho_{yx}$ fit with eq. (4) gives four dependent parameters, which may lead to misguiding conclusionsRotella et al. (2015). However, the high field limit of this equation gives an approximate measure of the effective carrier concentration $n_{eff}$Sun et al. (2014), which will be discussed in section D:

[TABLE]

III.4 Multiband conductivity

More detailed information can be extracted by transforming components of resistivity tensor $\rho_{yx}$ and $\rho_{xx}$ to obtain Hall conductivity $\sigma_{xy}$ via the following equation:

[TABLE]

In the multiband system, $\sigma_{xy}$ is a superposition of the terms originating from subsequent contributing bands. Equation (6) can be then rewritten asLin et al. (2016):

[TABLE]

Hall conductivity is commonly used to determine the electronic parameters, since the extremum of $\sigma_{xy}(B)$ is a direct measure (or at least a good approximation in a multiband system) of the dominant mobility $\mu_{ext}$ calculated from the inverse of the magnetic field $B_{ext}$, at which $\sigma_{xy}$ peaksLiang et al. (2014):

[TABLE]

The Hall conductivity for both compounds is negative in the whole temperature range and at low temperatures shows a minimum, which for YNiC2 is visibly sharper than for LuNiC2. The position of this minimum shifts from high fields to lower values of $B$ as temperature is lowered. For YNiC2, the $B_{ext}$ is clearly defined, while the broad extremum seen in LuNiC2 precludes the precise determination of the peak position. Since the direct fitting of $\sigma_{xy}$ with equation 7 assuming one hole and one electron bands once again requires using four dependent parameters, for further analysis we have used an approachTakahashi et al. (2011); Ishiwata et al. (2013), in which we have assumed the existence of a single band with high mobility carriers and the remaining band(s) to show significantly lower mobility:

[TABLE]

Equation 9 allows the estimation of the mobility $\mu_{xy}$, and concentration $n_{xy}$ of this single ’fast’ band (pocket), while other ’slower’ bands contribute to $C_{xy}$ parameter. The typical fits are shown by solid lines in panels c and d of Fig. 4 respectively. We have found that $\sigma_{xy}$ can be reasonably well-described with equation (9) despite of the fact that the zero field values of $\rho_{xx}$ can be significantly increased due to the polycrystalline character of the samples.

The parameters derived from this procedure, as well as the values of $n_{eff}$ and $\mu_{ext}$, are summarized in Fig. 4e for YNiC2 and 4f, for LuNiC2. The mobilities $\mu_{ext}$ and $\mu_{xy}$ coincide with each other for YNiC2, and both quantities reach very large values of $7\cdot 10^{3}$ cm2 V*-1* s*-1* at $T$ = 1.9 K. The electronic mobility $\mu_{xy}$ of LuNiC2 is twice as small as in the case YNiC2, yet still considerable. The coincidence of $\mu_{xy}$ and $\mu_{ext}$ is an additional argument for the correctness of the value calculated from $\sigma_{xy}$. It shall be, however, noted that, next to the increase of the residual resistivity, the polycrystalline samples character is expected also to substantially lower the electronic mobility in comparison with the single crystal.

As seen in Fig. 4e and f, for both compounds, the concentration of the carriers originating from the high mobility band increases as temperature is lowered. The growth of $n_{xy}$ is concomitant with the decrease of the effective carrier concentration $n_{eff}$ below Peierls temperature. This is consistent with the nesting picture: while the majority of electrons are removed from the conducting band and condenses towards CDW, the parallel opening of unnested pockets results in the increase of the high mobility carriers. Interestingly, while the results of the $\sigma_{xy}$ analysis suggests the electron origin of the carriers described with concentration $n_{xy}$, the upturn of Hall resistivity and $n_{eff}$ at lowest temperatures can possibly be caused by the existence of holes with not so large mobility, thus contributing only to $C_{xy}$ parameter in equation (9). The temperature interval in which this effect is observed corresponds to the range in which the a turnover of deviations from Kohler scaling is observed in YNiC2 (inset of Fig. 3c).

The $C_{xy}$ parameter serves as an estimate of the ratio of the conductivities stemming from ’slow’ to ’fast’ bands respectively. Thermal dependence of $C_{xy}$ for both compounds is shown in figure 5. For both compounds, the values of this parameter show values close to unity at high $T$ and decrease as temperature is lowered. $C_{xy}$ reaches $\simeq$ 0.001 for YNiC2 and $\simeq$ 0.1 for LuNiC2. Small upturn is seen at low temperatures at low temperatures, which can be associated with the existence of an additional band as suggested above. The relatively low values of $C_{xy}$, especially in the former compound, underline the major role played by the carriers originating from the ’fast’ pocket in the terms of electronic transport and show that the used approximate model can be used to describe the properties of YNiC2 and LuNiC2.

The presence of both electron and hole pockets in the CDW state of LuNiC2 is also consistent with the results of band structure calculationsSteiner et al. (2018). Owing to the similarities between the Fermi surfaces of YNiC2 Hase and Yanagisawa (2009) and other $R$NiC2 showing CDW, it is reasonable to assume the relevance of the same scenario for Y bearing compound as well.

The high mobility of carriers contained in these pockets is then likely responsible for the high magnitude of MR in both compounds. Opening of such pockets was reported in a number of quasi-2D CDW materials showing strong, yet imperfect Fermi surface nesting, leading to the enhancement of magnetoresistanceRötger et al. (1994); Hess et al. (1996); Yasuzuka et al. (1999); Chen et al. (2017), themopowerRhyee et al. (2009); Rhyee and Kim (2015) and galvanothermomagnetic propertiesBel et al. (2003); Kolincio et al. (2016c).

This result supports the scenario of strong Fermi surface reconstruction in YNiC2 and LuNiC2, which is possible due to the absence of any competing magnetic ordering which was responsible for the CDW suppression in majority of $R$NiC2 family members Yamamoto et al. (2013); Hanasaki et al. (2012); Kolincio et al. (2016a, 2017); Hanasaki et al. (2017); Lei et al. (2017).

III.5 Specific heat

To complement the results of transport, magnetotransport, and Hall experiments, and to further characterize the CDW transition in YNiC2, we have measured the specific heat $C_{p}$. Fig. 6a depicts the temperature dependence of the specific heat capacity $C_{p}(T)$ in the temperature range 1.9 - 300 K. At 300 K, $C_{p}$ reaches approximately 80% of the value expected by Dulong-Petit law (3nR $\sim$ 100 J mol*-1* K*-1*), suggesting that the Debye temperature for YNiC2 exceeds 300 K.

No anomalies have been detected at low temperatures, which confirms the absence of bulk superconductivity or magnetic ordering. The specific heat data plotted as $\frac{C_{p}}{T}$ vs. $T^{2}$ presented in Fig. 6b has been fitted to the equation (10) with both sides divided by $T$.

[TABLE]

where the first and second terms represent electronic and lattice contributions, respectively. The fit revealed values of Sommerfeld coefficient $\gamma$ = 1.65(1) mJ mol*-1K-2* and $\beta$ = 0.326(4) mJ mol*-1K-4*, the latter corresponds to the Debye temperature $\Theta_{D}$ = 620 K according to:

[TABLE]

where R = 8.314 J mol*-1K-1* and $n$ is the number of atoms per formula unit ($n$ = 4 for YNiC2). This value is larger than the $\Theta_{D}$ = 456 K reported previously for YNiC2 Long et al. (2001). The Debye temperature found here is also larger than the value reported for LaNiC2 ($\Theta_{D}$ = 445 K) Prathiba et al. (2016). Such behavior can be reasonably explained by a mass relationship: for molar mass of Y smaller than La, one expects higher $\Theta_{D}$.

Results of the detailed measurements of $C_{p}(T)$ above room temperature are shown in the inset of Fig. 6a. The Peierls transition is signaled by a small maximum of $C_{p}(T)$ at $T$ = 310 K, being in rough agreement with the transition temperature $T_{CDW}$ established from resistivity measurements. The relative increase of specific heat at the charge density wave formation temperature denotes $\frac{\Delta C_{p}}{C_{p}(T_{CDW})}\simeq 1.1\%$, thus is at the same order of magnitude as in canonical CDW systems as NbSe3Tomić et al. (1981), K0.9Mo6O17Escribe-filippini et al. (1984), or tungsten bronzesChung et al. (1993).

The mean-field weak coupling description of the Peierls transition predicts the specific heat jump of:

[TABLE]

In the case of YNiC2, the equation (12) gives the value of 1.79, slightly larger than the BCS prediction, indicating the relevance of a weak coupling scenario.

Visibly stronger and sharper anomaly accompanies the presumed lock-in crossover at $T_{1}$ = 275 K. Here the specific heat increases by $\frac{\Delta C_{p}}{C_{p}(T_{1})}\simeq 2.9\%$, with $C_{p}(T_{1})$ estimated from the background. The magnitude of this anomaly is noticeably larger than for the features typically observed at the incommensurate-commensurate CDW transformationCraven and Meyer (1977); Kuo et al. (2004).

The entropy $\Delta S$ and enthalpy $\Delta H$ of both anomalies were estimated from the excess specific heat at each transition by integrating the $\frac{\Delta C_{p}}{T}dT$ of and $\Delta C_{p}dT$ respectively, after evaluating and subtracting the background values of $C_{p}$. The integrated regions are highlighted by light violet color in Fig. 6a.

The results of the integration of $C_{p}$ excess accompanying the phase transitions are summarized in Tab. 1. While the size of $\Delta C_{p}$ step at $T_{CDW}$ stands in agreement with the BCS predictions as well as with the values found in other materials exhibiting a weakly coupled charge density wave, we find an unusualy low value of $\Delta S$ accompanying this transition. This can be imposed by the high Peierls temperature, resulting in a large denominator of $\frac{\Delta C_{p}}{T}$ and thus small result of the integral. The value of enthalpy however, does not diverge from the typically observed values in CDW systemsEscribe-filippini et al. (1984); Wang et al. (2006). In agreement with the comparison of $\Delta C_{p}$ jump, for the crossover at $T_{1}$, the values $\Delta S$ and $\Delta H$, are significantly larger than for the Peierls transition at $T_{CDW}$. This result is unexpected, since typically the lock-in transition is not associated with the opening of a new electronic gap, next the one already existing in the CDW state. The sharp peak shape of this anomaly can suggest a large role played by CDW order parameter fluctuations McMillan (1977); Kuo et al. (2001); Kwok et al. (1990). The detailed analysis of crystal structure, as well as of the phonon spectra, performed on a single crystal is required to elucidate this issue.

IV conclusions

We have examined the physical properties of polycrystalline YNiC2 and LuNiC2. The former compound shows at $T_{CDW}$ = 318 K Peierls transition with signatures of BCS - mean field weak coupling scenario, followed by presumed lock-in crossover at $T_{1}$ = 275 K. The temperatures corresponding to these anomalies, revealed by transport, Hall effect and specific heat measurements, are found to obey the linear scaling with the unit cell volume, observed previously with lanthanide-based $R$NiC2 compounds. Both studied materials show large magnetoresistance in the CDW state, reaching 470 % for YNiC2 and 50 % for LuNiC2 at $T$ = 1.9 K and $B$ = 9 T. To discuss its origin, we have combined the analysis of thermal and magnetic field depencence of Hall effect and magnetoresistance. We have found that the effect standing behind such strong magnetoresistive features in YNiC2 and LuNiC2 is the existence of pockets, including at least one with high mobility carriers, remaining in the Fermi surface after nesting, caused by fully developed CDW transition not interrupted by competing orders such as magnetism or superconductivity.

Acknowledgements.

The authors gratefully acknowledge the financial support from National Science Centre (Poland), grant number: UMO-2015/19/B/ST3/03127. The authors would also like to thank to M. Hirshberger and C. Zhang (both RIKEN), Alain Pautrat (CRISMAT), Helen Walker (ISIS), Nathan Runyon, and Jesse Sprowes for their helpful advice.