Electrical resistivity of the heavy-fermion-filled cage compound ${\text{Ce}}_3{M}_4{\mathrm{Sn}}_{13}(M=\text{Co},\text{Rh},\text{Ru})$ under high pressure

The effect of pressure on electrical resistivity of heavy-fermion compounds ${\mathrm{Ce}}_3{M}_4{\mathrm{Sn}}_{13}$, where $M=\mathrm{Co}$, Rh, Ru, and the ${\mathrm{La}}_3{M}_4{\mathrm{Sn}}_{13}$ counterparts is studied in the framework of the fully relativistic full potential local orbital method. The experiment shows that the electrical resistivity of ${\mathrm{Ce}}_3{\mathrm{Co}}_4{\mathrm{Sn}}_{13}$ and ${\mathrm{Ce}}_3{\mathrm{Rh}}_4{\mathrm{Sn}}_{13}$ increases with pressure, a similar pressure effect is obtained for isostructural La-based reference metals, while opposite behaviors under pressure are documented for ${\mathrm{Ce}}_3{\mathrm{Ru}}_4{\mathrm{Sn}}_{13}$ and ${\mathrm{La}}_3{\mathrm{Ru}}_4{\mathrm{Sn}}_{13}$. The contrasting pressure dependent effects of ${\mathrm{Ce}}_3{\mathrm{Co}}_4{\mathrm{Sn}}_{13}$ and ${\mathrm{Ce}}_3{\mathrm{Ru}}_4{\mathrm{Sn}}_{13}$ are discussed. In order to clarify the various phenomena the band-structure calculations under applied pressure were performed. Here, we show that the resistivity increase with pressure arises from the formation of interband distances at the Fermi level in ${\mathrm{Ce}}_3{\mathrm{Co}}_4{\mathrm{Sn}}_{13}$, this pseudo-gap-like effect is also pressure dependent, while in ${\mathrm{Ce}}_3{\mathrm{Ru}}_4{\mathrm{Sn}}_{13}$ the opposite change of resistivity results from the suppression of spin fluctuations under pressure.

Figure 1

The unit-cell structure of ${\mathrm{Ce}}_3{\mathrm{Co}}_4{\mathrm{Sn}}_{13}$ highlighting the arrangement of the $\text{Sn}1{\left(\text{Sn}\right)}_{12}$ cages and corner-sharing $\text{Co}{\left(\text{Sn}2\right)}_6$ trigonal prisms. Ce atoms are shown as green balls, Co atoms are red balls, Sn2 are gray, and Sn1 are dark green balls.

Figure 2

Difference charge density map $\mathrm{\Delta }{\rho }_e$ in $\text{electron}/{\text{Å}}^3$ for ${\mathrm{Ce}}_3{\mathrm{Co}}_4{\mathrm{Sn}}_{13}$ is compared with $\mathrm{\Delta }{\rho }_e$ obtained recently for ${\mathrm{Ce}}_3{\mathrm{Rh}}_4{\mathrm{Sn}}_{13}$ [16] and for ${\mathrm{Ce}}_3{\mathrm{Ru}}_4{\mathrm{Sn}}_{13}$ [17], and comparison of $\mathrm{\Delta }{\rho }_e$ in the plane $\left(00\frac{1}{4}\right)$. The white dotted line sets the direction of the charge distribution between the metal $M$ and Sn2 atoms located in the plane.

Figure 3

A comparison of the calculated charge density $\mathrm{\Delta }{\rho }_e$ plotted along line representing the nearest-neighbor $\text{Sn}1-M-\text{Sn}1$ bonding in the series ${\mathrm{Ce}}_3{M}_4{\mathrm{Sn}}_{13}$, where $M=\text{Co}$, Rh (the $\mathrm{\Delta }{\rho }_e$ data from Ref. [16]), Ru (cf. Ref. [17]).

Figure 4

Calculated charge density $\mathrm{\Delta }{\rho }_e$ plotted on the line between Sn1 and Sn2 atoms (cf. Fig. 1) for different ${\mathrm{Ce}}_3{M}_4{\mathrm{Sn}}_{13}$ compounds, where $M=\text{Co}$, Rh [16], Ru [17].

Figure 5

The total and spin-resolved density of states within LSDA approximation at different pressure calculated for ${\mathrm{Ce}}_3{\mathrm{Co}}_4{\mathrm{Sn}}_{13}$ (a) and for ${\mathrm{Ce}}_3{\mathrm{Ru}}_4{\mathrm{Sn}}_{13}$ (b). The inset to panel (a) and (b) displays the details in pressure dependent total DOS near the Fermi energy for the respective compounds. The total DOSs show different energy shifts near ${\epsilon }_F$ for the bands calculated for ${\mathrm{Ce}}_3{\mathrm{Co}}_4{\mathrm{Sn}}_{13}$ (a) and for ${\mathrm{Ce}}_3{\mathrm{Ru}}_4{\mathrm{Sn}}_{13}$ (b); the effect is discussed in Sec. 3b.

Figure 6

The total and spin-resolved density of states within LSDA approximation under the pressure 0 and 2.6 GPa, calculated for ${\mathrm{La}}_3{\mathrm{Ru}}_4{\mathrm{Sn}}_{13}$. The inset displays the details in pressure dependent total DOS near the Fermi energy. The sharp maximum in DOS at ${\epsilon }_F$ can cause spin waves experimentally observed in ${\mathrm{La}}_3{\mathrm{Ru}}_4{\mathrm{Sn}}_{13}$ (discussion in Sec. 3b).

Figure 7

Atomic partial $4f$ density of states for Ce (a), $3d$ states for Co (b), and Sn2 $4s$ states (c) obtained for ${\mathrm{Ce}}_3{\mathrm{Co}}_4{\mathrm{Sn}}_{13}$ at different pressure; the figure compares the data obtained for $P=0$ and 2.7 GPa. The inset displays Sn2 $4s$ conduction states for energies $|\epsilon -{\epsilon }_F|\approx k_B{T}_D$.

Figure 8

The pressure dependence of the hybridization matrix element $V_{fs}={[\mathrm{\Delta }/\pi N\left({\epsilon }_F\right)]}^{1/2}$, where $\mathrm{\Delta }\approx 140$ meV for ${\mathrm{Ce}}_3{\mathrm{Co}}_4{\mathrm{Sn}}_{13}$ and $\sim 70$ meV for ${\mathrm{Ce}}_3{\mathrm{Ru}}_4{\mathrm{Sn}}_{13}$ (cf. Ref. [15]), and $J_{fs}$ coupling between $f$ electron and conduction-band states vs $P$.

Figure 9

The energy-band structure of ${\mathrm{Ce}}_3{\mathrm{Co}}_4{\mathrm{Sn}}_{13}$ for spin-up and spin-down channels along the high symmetry directions.

Figure 10

The energy-band structure of ${\mathrm{Ce}}_3{\mathrm{Ru}}_4{\mathrm{Sn}}_{13}$ for the spin-up channel. The band structure for spin-up and spin-down channels is the same.

Figure 11

Electrical resistivity for ${\mathrm{Ce}}_3{\mathrm{Co}}_4{\mathrm{Sn}}_{13}$ (a) (these $\rho$ vs $P$ data were previously presented in Ref. [12]) and ${\mathrm{Ce}}_3{\mathrm{Ru}}_4{\mathrm{Sn}}_{13}$ (b) under applied pressure. The inset to panel (b) displays the resistivity of ${\mathrm{La}}_3{\mathrm{Ru}}_4{\mathrm{Sn}}_{13}$ vs pressure (cf. Ref. [38]). For ${\mathrm{Ce}}_3{\mathrm{Co}}_4{\mathrm{Sn}}_{13}$ and ${\mathrm{Ce}}_3{\mathrm{Rh}}_4{\mathrm{Sn}}_{13}$ (cf. Ref. [12]) the effect is observed rather weakly above $T_D$ (where $T_D$ is the temperature of the crystallographic distortion), while at the low-temperature range the effect is significant.

Figure 12

Electrical resistivity $\rho$ at $T=6$ K vs pressure, normalized to $\rho$ at $T=6$ K and $P=0$ for the series of ${\mathrm{Ce}}_3{M}_4{\mathrm{Sn}}_{13}$ and the La counterparts. The $\rho \left(T\right)$ vs $P$ data were taken for ${\mathrm{Ce}}_3{\mathrm{Rh}}_4{\mathrm{Sn}}_{13}$ from Ref. [12], while for respective La-reference compounds from Refs. [38, 39].