Chemical and hydrostatic pressure effect on charge density waves of SmNiC2

Using a first-principles density functional theory method, we have investigated the chemical and the hydrostatic pressure effects on the charge density wave (CDW) properties of the quasi-one-dimensional (1D) compound SmNiC2. With increasing pressure, the relative 1D anisotropy of the electronic structure along a direction is enhanced because of its Ni chain structures. From the analysis of the Fermi surface and the generalized susceptibility, we also find that the Fermi surface nesting is enhanced along the modulation vector q1 = (0.5, 0.52, 0) but is suppressed along qR = (0.5, 0.5, 0.5) under pressure. The enhancement of 1D anisotropy of SmNiC2 under pressure is responsible for increasing CDW strength along q1. We suggest that this quantitative analysis could be used for analysis of the pressure effect on CDW materials.


Introduction
SmNiC 2 , one of the RNiC 2 (R: rare earth atom) compounds, has an orthorhombic CeNiC 2 type structure (space group 38, Amm2) [1][2][3][4][5][6]. It contains Ni and Sm one-dimensional (1D) chains along a direction, as shown in figure 1(a). Such chains are connected with a zig-zag shape along diagonal directions. As a result, each Ni(Sm) chain is surrounded by three Sm(Ni) chains with a trigonal prism shape. C-C dimers locate between prisms to make a three-dimensional (3D) bulk network.
This compound shows the competition between the charge density wave (CDW) and ferromagnetism (FM) [1][2][3][4][5][6][7][8]. Its CDW phase is developed below T CDW = 148 K and it completely disappears at T C = 17.5 K with the FM transition. At the CDW transition, resistivity shows a sharp inflection by opening the band gap [3,4]. From the x-ray diffraction experiment, the satellite peaks appear below T CDW along two independent vectors by the lattice modulation: one dominant CDW peak at q 1 = (0.5, 0.52, 0) and another weak CDW peak at q R = (0.5, 0.5, 0.5) [4]. This CDW phase is interpreted as commensurate with lattice modulation of Ni chains along a direction, but the frustrated 3D inter-chain coupling leads to the incommensurability along b direction [5]. Theoretical band structure calculation captures the origin of the CDW phase from Fermi surface (FS) nesting [9]. Temperature dependent thermal diffuse scattering pattern shows a Kohn anomaly, which indicates the suppression in the phonon frequency along q 1 and q R vector by the CDW transition [4]. A high-resolution photoemission spectroscopy experiment shows the imperfect FS nesting feature by the opening of pseudo-gap below T CDW [7]. Below T C , the local magnetic moment of Sm 4f orbital is large enough to modify the electronic structure and eventually destruct the nesting condition [4,6]. The CDW phase could be switched to an FM phase under magnetic field near T C ; therefore, this compound has a potential as a spintronic device with a giant magneto resistance [6].
Under pressure, SmNiC 2 also has several interesting phenomena. According to a recent report of the transport properties of SmNiC 2 , T CDW increases with pressure, unlike typical CDW materials. In addition, another CDW transition occurs at lower temperature (T CDW2 ), which decreases with pressure [8]. This is also observed by the chemical pressure with other RNiC 2 systems (NdNiC 2 and GdNiC 2 ) [3]. However, there has been little understanding of - this phenomena. Within the CDW phase, a FM quantum criticality has been observed with the pressure near 3.8 GPa [8]. Near the quantum criticality, a variety of magnetic phases comes out and shows the interplay with the CDW phase. Although such complicated magnetic phases have also been observed in other RNiC 2 , [10][11][12][13][14] the FM quantum criticality is a unique property of SmNiC 2 . Pressure is an important parameter for the control of not only the CDW phase but also magnetism or superconductivity. In particular, there have been many pressure experiments for transition metal chalcogenides [15][16][17] and some rare earth compounds [18][19][20], including SmNiC 2 [8], in order to investigate the competition between the CDW and magnetism or superconductivity. However, there has been little theoretical analysis to verify the competition mechanism and pressure dependent order parameter changes.
In this paper, the pressure effect on the electronic structure of quasi 1D SmNiC 2 has been investigated by using a first-principles density functional theory (DFT) method. The electronic band structures and the electrical conductivity show that SmNiC 2 is an electrically quasi 1D system with Ni chains along a direction. Under chemical or hydrostatic pressure, relative 1D anisotropy increases mainly due to the fact that the lattice constant along the Ni chain direction suppresses. By using the calculation of the FSs and the generalized susceptibility (χ 0 (q)), we reveal that the CDW along q 1 vector is enhanced under pressure with increasing 1D anisotropy, while the CDW along q R is suppressed.

Method
To find the optimized crystal structure (both lattice constants and atomic positions) of RNiC 2 (R=Nd, Sm, Gd), Vienna ab initio simulation package (VASP) is used [21]. Because the aim of this work is describing FS nesting at high temperature, we consider the crystal structure with perfectly localized 4f orbital. To achieve the aim of our study, we have chosen a pseudo-potential that contains non-f orbital. The projector augmented wave method with the PBE-GGA is used for the DFT calculation [22]. The cut-off energy in the planewave expansion of the valence states is set to 500 eV. The 15 × 15 × 15 Monkhost-pack k mesh is used in the full Brillouin zone (BZ). With the total energy of each volume (−8% to + 4% changes from the optimized volume), a Birch-Murnaghan equation of state is used to estimate the pressure for corresponding volume [23,24]. Table 1 shows the lattice parameters of SmNiC 2 form −6 to 12 GPa and those of GdNiC 2 and NdNiC 2 at ambient pressure.
Since 4f states are treated as a core state in VASP, a WIEN2k package was used for calculating the electronic and magnetic structures of RNiC 2 to describe the 4f orbital of rare earth atoms more precisely [25]. This package uses a full potential L/APW + lo methods based on the DFT. As the non-overlapping radius of muffin-tin, 2.5 (rare earth), 1.95 (Ni), 1.28 (C) in atomic unit were used. The 10 × 10 × 10 mesh is used for the self-consistent charge density calculation. We consider a dense k-mesh for precise description of the conductivity [26] (36 × 37 × 37 k-mesh) and the χ 0 (q) (100 × 100 × 100 k-mesh). In the description of high temperature non-magnetic (NM) phase, a 4f orbital was treated as core state by the opencore method [25]. In the low temperature FM calculation, a GGA+U+SO (U eff = 8 eV) method was used in order to consider the on-site Coulomb interaction (U) and the spin-orbit (SO) interaction.
The generalized susceptibility (Lindhard response function) χ 0 (q) is calculated directly from the energy eigenvalue using where f (ε) is the Fermi-Dirac distribution function, and ε n,k and ε n ,k+q are the energy eigenvalues in the first BZ with band indices n and n . Because the purpose of this work is to examine the static FS nesting feature, ω was set to 0. The real part of χ 0 (q) was taken for the description of the FS nesting strength [9,27,28]. We have only considered the bands which cross the Fermi level because these bands have a main contribution to χ 0 (q) and the others have featureless contribution to χ 0 (q). In equation (1) we have employed the approximation of constant matrix element. Although it cannot describe the detail strength of the FS nesting correctly, it is very useful to check the size of the FS nesting vector. Indeed, our calculation correctly captures the CDW vectors measured in experiment. There are severe numerical problems in obtaining an exact value of χ 0 (q) due to the singular value in the denominator.
To solve this problem, one can use imaginary part of χ 0 (q) and sizable δ for the broadening effect [27,28]. Instead, we set δ = 0 and averaged χ 0 (q) using large number of random k-points (∼10 9 ) where the eigenvalue of the each random k-point is obtained by linear interpolation of eigenvalues from 100 × 100 × 100 k-mesh in the full BZ. We have confirmed that this produces a more precise function of χ 0 (q).

Results and discussion
The CDW is a periodic charge and lattice modulation that can be realized when the electron-phonon interaction induces an energy gap, overcoming the lattice strain energy [29,30]. The CDW vector (2k F ) of a metallic system can usually be identified by inspecting the FS nesting, which can be understood as the realization of quasi 1D system showing Peierls instability [31]. Pressure is an important factor to control the FS nesting and the strength of the CDW phase. As pressure increases, the weaker nesting condition in system usually leads to the destruction of the CDW phase. In some cases, however, the CDW phase could be enhanced by the anisotropic lattice modulations [16,[32][33][34]. There is little theoretical explanation on the mechanism of enhancing CDW strength. We show that the enhancement of the CDW strength with the pressure effect on SmNiC 2 by the calculation of crystal structure changes, the electronic structures, the FSs and the generalized susceptibility χ 0 (q).  the Fermi level, mainly along a direction, namely -X , S * -Y * and U * -Z * lines 5 . For this reason, quasi 1D electronic structures of SmNiC 2 along a direction is expected. In particular, the Ni 3d orbital has a dominant contribution along -X lines. The Ni atom is located in the center of the Sm trigonal prism. In general, the crystal split pattern of trigonal prism is (d x y , d x2−y2 ) < d z2 < (d x z , d yz ). However the shape of Sm 6 Ni prism is distorted from a perfect triangle prism and is pressed along c direction, which is higher the d yz orbital (along a direction) above d x z , d z2 orbital, leading to the split pattern

Electronic structures
). The d yz orbital crosses the Fermi level.
The quasi 1D property of the electronic structure could be analyzed more quantitatively using the calculation of the electrical conductivity for each direction. Figure 2(c) shows the relative electrical conductivity, calculated by the Boltzmann transport equation [26], as a function of pressure. At the ambient pressure, it shows quite an anisotropic resistivity with σ c /σ a ∼ 0.24(ρ c /ρ a ∼ 4.18), σ b /σ a ∼ 0.36(ρ b /ρ a ∼ 2.81). This is well matched with both our band structure analysis and the experimental results [4]. We also explore the change of the electrical properties under the chemical and hydrostatic pressure (see figure 2(c)). Under chemical pressure, denoted by open circles, electric anisotropy is enhanced from the reduction of both σ b /σ a and σ c /σ a . However, under hydrostatic pressure (line) the σ b /σ a ratio is decreased while the σ c /σ a ratio is increased. The increase of conductivity 5  along c direction for the hydrostatic pressure could be caused by a rare earth 5d orbital. Since the atomic size of rare earth atoms decreases by chemical pressure, the overlap between rare earth 5d orbitals perpendicular to a direction is reduced, despite the shortened lattice constants. Also, the most important orbital for the CDW phase is the Ni 3d orbital [5] and the contribution of rare earth 5d orbital is negligible to develop the CDW phase. In the following two sections, we will discuss if the pressure behaviors of CDW phase are similar to both chemical and hydrostatic cases. This is responsible for increasing quasi 1D property of Ni chain.

Fermi surfaces
In connection with the electrical properties, we have analyzed the CDW phase in terms of the calculated FSs and χ 0 (q). As shown in figure 4(a), the two parallel FSs along a * direction show quasi 1D behavior consistent with the electrical properties. Figures 4(b) and (c) show the twodimensional (2D) cross sections of the FS (black) in the NM phase. The strong FS nesting is found with wave vectors q 1 = (0.5, 0.52, 0) and q R = (0.5, 0.5, 0.5), which correspond to the experimental CDW vector (blue arrow) [4]. The FS nesting occurs mainly near the central part of BZ around the -X symmetry line, where the Ni 3d orbital gives a dominant contribution (see figure 2(b)). For detailed analysis of the nesting directions, the calculation of χ 0 (q) is needed.
For the FS of the FM phase, as we mentioned above, there occurred a band splitting near the Fermi level. This small band splitting leads to split FSs (red), depending on the different spin characters, as shown in figure 4. Although the split FSs looks still well nested, like the NM phase, the inter-band FS nesting is substantially suppressed in the FM phase. The interband scattering between different spins is negligible under the external charge potential of FM ordering. Therefore, FS nesting vector is split and the overall FS nesting feature should be suppressed as each FS has different spin configurations. The detailed analysis will be shown in the calculation of χ 0 (q). We also investigated the change of the FSs with increasing pressure. The FSs under various pressures are shown in the figure 5 for ab plane (for q 1 vector) and diagonal part of ab, ac plane (for q R vector). Because the chemical and hydrostatic pressures have similar contribution to the FS, a similar effect on the CDW phase is expected. As the pressure increases, the overall FSs become flatter and more 1D like along a direction. This is consistent with the increased conductivity along a direction with the pressure. Despite the flattened FS, the overall FS nesting feature near -X line still holds for all the pressure ranges. For this reason, one can predict that the FS nesting feature maintains for all the pressure ranges of our interest. This corresponds to the experimental results where T CDW does not vanish for high pressure over 6 GPa or chemical pressure, such as GdNiC 2 and NdNiC 2 compounds [3,8].

Generalized susceptibility, χ 0 (q)
The CDW vector, corresponding to the FS nesting direction, is accurately evaluated by calculating χ 0 (q) using equation (1). Figure 6 shows the plot of χ 0 (q) along (0.5, x, 0) and (0.5, 0.5, x) symmetry lines and the 2D contour plot of χ 0 (q) within (0.5, x, y) plane, where 0 x, y 1. The calculated χ 0 (q) has one significant global maximum peak at q 1 = (0.5, 0.52, 0). This is same as the experimental CDW vector q 1 [4] and similar to the previous theoretical work with q 1 = (0.5, 0.56, 0) [9]. Also, there is broad local maximum feature around q R = (0.5, 0.5, 0.5), which is another weak CDW vector measured in the experiment [4]. However, according to the 2D plot of χ 0 (q) shown in figure 6(b), the q R vector is a saddle point within a broad plateau containing local maxima points denoted as X near q R . Compared to the prominent peak at q 1 , the peak at q R has broad and ambiguous nesting behavior. This result is also consistent with the experimental measurement that x-ray satellite peak along q R vector is weak, compared with q 1 vector [4]. The suppression of the CDW phase with the FM ordering [4] can be understood with the suppression of the FS nesting with the splitting of the conduction bands. Although previous studies have not shown clear suppression of the FS nesting from the calculated χ 0 (q) [9], figure 6(a) clearly shows the suppressed and broad hump feature (green dotted line) of χ 0 (q) near q 1 . This result obviously explains that the FS nesting or the CDW phase is destructed by the FM ordering [4]. As already pointed out in the previous section, it is important to consider only the intra-band contribution of χ 0 (q). If we consider both intra-and inter-band contributions, we find similar results to those of the previous studies [9].
Finally, the calculated χ 0 (q)s under various pressure in NM phase along (0.5, x, 0), (0.5, 0.5, x) and (0.5, x, 0.5) symmetry lines are summarized in figure 7. Their overall amplitudes are reduced with the pressure because the DOS at the Fermi level decreases as the pressure increases. In figure 7(a), the FS nesting vector q 1 is gradually shifted from (0.5, 0.5, 0) at −6 GPa to (0.5, 0.55, 0) at 12 GPa as the pressure increases. This shows that χ 0 (q) can capture very small changes of the FSs, as shown in figure 5. Chemical pressure, however, does not affect the direction of the FS nesting vector q 1 ∼ (0.5, 0.52, 0). This is quite different from the previous observations where the nesting vector is significantly changed with chemical substitution [9]. The difference between the chemical and the hydrostatic pressure comes from the different changes of FSs near -X line. Slightly tilted FSs around in figure 5 single peak starts to split from the symmetry lines. The amount of splitting becomes larger with increasing pressure. Thus, one can expect that the FS nesting feature along q R becomes unclear with increasing the pressure, and this leads to the suppression of the CDW phase as increasing pressure.

CDW strength versus dimensionality
RNiC 2 shows different behavior with typical CDW compounds. The T CDW becomes larger by both chemical and hydrostatic pressure even for high pressure [3,8]. Some CDW compounds also show increasing T CDW under pressure [16,[32][33][34], but it only increases at low pressure. Also, it shows a second CDW phase of decreasing transition temperature (T CDW2 ) by pressure. In this compound, our band calculation has found that 1D anisotropy along a direction increases under pressure unlike other typical CDW compounds. In addition, the FS nesting along q 1 vector remains strong, even for high pressure, while the FS nesting along q R vector becomes ambiguous as pressure increases. The enhancement of the CDW strength (T CDW ) along q 1 is responsible for the increased 1D anisotropy along a direction. This would hold for high pressure if there is no structural transition. However, the CDW along q R of second CDW phase shows a complicated feature. Under low pressure, it holds an FS nesting feature with weak nesting along near (0.5, 0.5, 0.5). This corresponds to T CDW2 . As pressure increases further, it becomes more broadened and χ 0 (q) peak becomes broad plateau, which means suppression of the CDW phase. For chemical pressure cases, there is no experimental analysis for the second CDW phase [3] and more experiments, such as x-ray diffraction experiment [4], are needed.

Conclusion
We have investigated the change of the crystal structure, the electronic structures, the FSs, and the χ 0 (q) of SmNiC 2 with the existence of magnetism and pressure. At ambient pressure, the NM and FM calculation explains the experimental observation of the construction and destruction of the CDW phase, respectively. In addition, our results are consistent with recent experimental studies under pressure, where multiple CDW transitions are observed [3,8]. Our electronic structure calculation with pressure suggests that the increased 1D anisotropy along a direction in the electronic structure enhances the CDW strength along q 1 and it increases T CDW . Also, the additional CDW phase along q R is suppressed because pressure induces the broadening of the χ 0 (q) peak near the q R vector.