{"id":122117,"date":"2025-08-05T23:23:15","date_gmt":"2025-08-05T23:23:15","guid":{"rendered":"https:\/\/www.europesays.com\/us\/122117\/"},"modified":"2025-08-05T23:23:15","modified_gmt":"2025-08-05T23:23:15","slug":"anisotropic-non-fermi-liquid-and-dynamical-planckian-scaling-of-a-quasi-kagome-kondo-lattice-system","status":"publish","type":"post","link":"https:\/\/www.europesays.com\/us\/122117\/","title":{"rendered":"Anisotropic non-Fermi liquid and dynamical Planckian scaling of a quasi-kagome Kondo lattice system"},"content":{"rendered":"

The temperature dependences of the R(\u03c9) and \u03c31(\u03c9) spectra obtained from the Kramers-Kronig analysis of the R(\u03c9) spectra of CeRhSn along the a- and c-axes is shown in Fig. 1<\/a>a, b, respectively. The significant axial dependence of the spectra reflects the anisotropy of the electronic state. The 4f spin-orbit doublet at \u210f\u03c9 ~300 and 700\u2009meV, namely mid-IR peaks, originating from the strong c-f hybridization24<\/a>, appears in both directions, even at 300\u2009K (As shown in Fig. S1, a similar double peak structure appears in LaRhSn where no c-f hybridization occurs, but the energy positions are shifted from those of CeRhSn and the intensity ratio for E\u2225a and E\u2225c is opposite to that of CeRhSn. Additionally, the mid-IR peak in CeRhSn grows up at low temperatures. Therefore, the mid-IR peak is concluded to originate from the c-f hybridization). On the other hand, a clear anisotropic Drude response appears at \u210f\u03c9\u2009\u2264100\u2009meV, which is consistent with the anisotropic electrical resistivity20<\/a>. As LaRhSn has a weak anisotropy in the electrical resistivity, the temperature dependences of R(\u03c9) and \u03c31(\u03c9) spectra for E\u2225a and E\u2225c are also very similar as shown in Fig. S1a<\/a> (See Supplementary information). This suggests that the axial dependence of this Drude structure is attributed to the anisotropic magnetic interactions.<\/p>\n

Fig. 1: Polarized reflectivity and optical conductivity spectra of CeRhSn.<\/b>\"figure<\/a><\/p>\n

a<\/b> Temperature-dependent polarized reflectivity [R(\u03c9)] spectra of CeRhSn in the photon energy \u210f\u03c9 range of 5\u20131000\u2009meV. Inset: Wide-range R(\u03c9) spectra up to 30\u2009eV at 300\u2009K. b<\/b> Temperature-dependent optical conductivity [\u03c31(\u03c9)] spectra of CeRhSn with E\u2225a (bottom) and E\u2225c (top). The peaks at \u210f\u03c9\u2009~\u200915\u2009meV in both axes originate from phonons.<\/p>\n

Valence fluctuation observed with mid-IR peaks<\/p>\n

Firstly, the mid-IR peaks are compared to the band structure calculations. Figure 2<\/a>a shows the band structure along high symmetry points of CeRhSn shown in Fig. 2<\/a>c. In the right, the density of states (DOS) by the LDA calculations is presented. The high DOS in the range of 0\u2009eV (= Fermi energy; EF) \u2013 0.6\u2009eV originates from the Ce 4f unoccupied states. The calculated band structure is regarded as fully itinerant; therefore, the itinerant character can be checked by comparing the experimental \u03c31(\u03c9) spectra to the band calculation25<\/a>. In Fig. 2<\/a>d, the \u03c31(\u03c9) spectra at 10\u2009K with mid-IR peaks at about 0.2\u20130.3 and 0.6\u20130.7\u2009eV are compared to the calculated \u03c31(\u03c9) spectra. Significant peaks at ~0.1 and ~0.5\u2009eV correspond to the experimentally observed mid-IR peaks, although the energy is shifted. A similar shift in energy was reported in the itinerant superconductor CeRh2As227<\/a>. Therefore, in addition to the appearance of the mid-IR peaks at 300\u2009K, the good correspondence of the mid-IR peaks to the calculated \u03c31(\u03c9) spectra confirms the strong c-f hybridization strength24<\/a>,28<\/a>, which is consistent with the results of previous photoelectron experiments29<\/a>,30<\/a>. It should be noted that recent DFT+DMFT calculations on CeRhSn more reproduce the mid-IR peaks31<\/a>, which also suggests the strong c-f hybridization intensity.<\/p>\n

Fig. 2: Mid-IR peaks compared to the band calculations.<\/b>\"figure<\/a><\/p>\n

a<\/b> Band structure and density of states (DOS) of CeRhSn by the LDA calculation with spin-orbit interaction. b<\/b> Crystal structure of CeRhSn with a quasi-kagome Ce lattice in the basal plane. Rh atoms have two different sites, namely Rh1 and Rh2, corresponding to the different densities of states shown in (a<\/b>). c<\/b> The first Brillouin zone and high symmetry points of CeRhSn. d<\/b> Calculated \u03c31(\u03c9) spectra for E\u2225a (red solid line) and E\u2225c (blue dashed line) compared with the experimentally obtained \u03c31(\u03c9) spectra after subtraction of the Drude component shown in Fig. S2<\/a> in the Supplementary information.<\/p>\n

Extended drude analysis<\/p>\n

Next, we discuss the spectral shape of the Drude component using the extended Drude analysis32<\/a>. Figure 3<\/a> indicates the obtained mass enhancement (m*\/m0, where m* and m0 are the effective mass of the quasiparticles and the rest mass of an electron, respectively) and the scattering rate (1\/\u03c4) along both directions. Here, the carrier densities along both axes for the extended Drude analysis were evaluated using the Hall coefficients at the lowest accessible temperature of 6\u2009K20<\/a>, i.e., 6.9\u2009\u00d7\u20091020\u2009cm\u22123 for E\u2225a and 4.2\u2009\u00d7\u20091021\u2009cm\u22123 for E\u2225c. In the low-energy limit, the effective mass increased with decreasing temperature in both directions. However, m*\/m0 for E\u2225c increased continuously on cooling from 300\u2009K, whereas m*\/m0 for E\u2225a is almost unchanged down to 80\u2009K, below which m*\/m0 increased significantly. Also, in 1\/\u03c4, the peak energy at the lowest temperature is about 30\u2009meV for E\u2225a and about 100\u2009meV for E\u2225c. The peak energy corresponds to the energy at which m*\/m0 begins to increase. These facts suggest the different characteristics of quasiparticles depending on the crystal axis. At the lowest temperature, 1\/\u03c4 is proportional to \u03c91 for E\u2225a. On the other hand, for E\u2225c, 1\/\u03c4 is roughly proportional to \u03c91.5, which is not \u03c91 nor \u03c92. It is known that 1\/\u03c4 is proportional to \u03c92 in the normal Fermi liquid state33<\/a>, but it is proportional to \u03c91 if the state is located very near QCP32<\/a>. In the case that the state is slightly shifted from QCP but in an NFL state, 1\/\u03c4 is proportional to \u03c9n with 1\u2009n\u200934<\/a>. Therefore, the power dependence of 1\/\u03c4 on \u03c9 indicates that the state along the a axis is located very near QCP and is slightly shifted from QCP along the c axis, but both axes are in NFL states. This result is consistent with the recently observed anisotropic NFL behavior in the specific heat23<\/a>. In CeRhSn, the T-linear region is below 40\u2009K (~3\u2009meV). Even at 80\u2009K, 1\/\u03c4 is proportional to \u210f\u03c9 as shown in Fig. 3<\/a>a2. This fact suggests that the Planckian form can be applied at temperatures below 80\u2009K.<\/p>\n

Fig. 3: Extended drude analysis.<\/b>\"figure<\/a><\/p>\n

Mass enhancement (m*\/m0, a1<\/b>) and scattering rate (\u210f\/\u03c4, a2<\/b>) as functions of photon energy and temperature for E\u2225a. \u210f\/\u03c4 values at \u210f\u03c9\u2009=\u20090\u2009eV evaluated by the electrical resistivity and Hall coefficient data20<\/a> are plotted with an open circle (80\u2009K) and an open square (6\u2009K) in (a2<\/b>). a3<\/b> \u210f\/(\u03c4kBT) as a function of \u210f\u03c9\/kBT obtained from (a2<\/b>), but the residual damping of \u210f\/\u03c4\u2009=\u20093.3\u2009meV obtained from the electrical resistivity and the Hall coefficient is subtracted. The formula expected from the Planckian dissipation, \\(\\hslash \/(\\tau {k}_{{\\rm{B}}}T)={[1+{(\\hslash \\omega \/{k}_{{\\rm{B}}}T)}^{2}]}^{1\/2}\\), is also plotted as a dashed line. c1<\/b>, c2<\/b> Same as (a1<\/b>, a2<\/b>), but for E\u2225c. Note that the negative values and approaching zero of m*\/m0 in (a1<\/b>) and (c1<\/b>) have no meaning because they appear in the photon energy regions outside of Drude components as shown in Fig. 1<\/a>b.<\/p>\n

In Planckian metals of NFL, \u210f\/\u03c4 will follow \\({[{(\\hslash \\omega )}^{2}+{({k}_{{\\rm{B}}}T)}^{2}]}^{1\/2}\\), i.e., \\(\\hslash \/(\\tau {k}_{{\\rm{B}}}T) \\sim {[1+(\\hslash \\omega \/{k}_{{\\rm{B}}}T)]}^{1\/2}\\). The experimental \u210f\/(\u03c4kBT) spectra subtracted by the residual damping \u210f\/\u03c4(0)\u2009=\u20093.3\u2009meV evaluated from the extrapolated values of the electrical resistivity and Hall coefficient to 0\u2009eV20<\/a> are plotted as a function of \u210f\u03c9\/kBT in Fig. 3<\/a>a3. The slope in the region of \u210f\u03c9\/kBT\u2009\u2264\u200910 is scaled with \u210f\u03c9\/kBT. The ideal Planckian formula \\(\\hslash \/(\\tau {k}_{{\\rm{B}}}T)={[1+{(\\hslash \\omega \/{k}_{{\\rm{B}}}T)}^{2}]}^{1\/2}\\) is also plotted in the figure. The formula can explain the slope, suggesting that the E\u2225a of CeRhSn is Planckian.<\/p>\n

For E\u2225a, as shown in Fig. 3<\/a>a2, the \u210f\/\u03c4 at \u210f\u03c9\u2009=\u20090eV decreases with decreasing temperature, which is the definition of Planckian metals, but that at \u210f\u03c9\u2009~\u200910\u2009meV shows the opposite behavior, i.e., the slope (d\u03c4\u22121\/d\u03c9) is strongly suppressed with increasing temperature. This result is in contrast to the parallel slope of \u210f\/\u03c4(\u03c9) at different temperatures in a high-Tc cuprate13<\/a>. However, as shown in Fig. 3<\/a>a3, the constant slope of d(\u03c4\u22121T\u22121)\/d(\u03c9\/T) at different temperatures in the region of \u210f\u03c9\/kBT\u2009\u2264\u200910 following the Planckian scaling might be a property of heavy fermion materials.<\/p>\n

Dynamical Planckian scaling<\/p>\n

Finally, we discuss whether the temperature dependence of the Drude component can be explained with DPS. If DPS is realized, the temperature dependence of the \u03c31(\u03c9) spectrum has a relationship11<\/a>,14<\/a>:<\/p>\n

$${\\sigma }_{in}(\\omega )\\cdot {T}^{\\alpha }=f(\\hslash \\omega \/{k}_{{\\rm{B}}}T),$$<\/p>\n

\n (1)\n <\/p>\n

where \u03c3in(\u03c9) is the real-part intrinsic optical conductivity and f(x) is a function and \u03b1\u2009=\u20091 for DPS.<\/p>\n

Here, the \u03c31(\u03c9) spectra in Fig. 1<\/a>b comprise carriers\u2019 contributions and interband transitions. The carrier component can be classified as \u201cheavy\u201d and \u201clight\u201d quasiparticles with and without strong electron correlation. On the other hand, the second contribution originates from the interband transition from the valence band to the c-f hybridization band with the Ce 4f spin-orbit splitting. Under these assumptions, the \u03c31(\u03c9) spectrum is decomposed into two Drude (\u03c3in(\u03c9) and \u03c3BG(\u03c9) for the heavy and light quasiparticle components, respectively) and two Lorentz (corresponding to the Ce 4f5\/2 amd 4f7\/2 states) components as shown in Fig. S2<\/a>. The individual contributions from heavy and light quasiparticles manifest in the several effective masses appearing in quantum oscillations35<\/a>. In \u03c31(\u03c9) spectra, the evidence of the light quasiparticles appears as a background of the Drude structure36<\/a>,37<\/a>. Then, we derive the heavy quasiparticles\u2019 intrinsic optical conductivity \u03c3in(\u03c9) spectra from the \u03c31(\u03c9) spectra by subtracting the light quasiparticles\u2019 background \u03c3BG(\u03c9) and two Lorentzians. The DPS plot of \u03c3in(\u03c9) is shown in Fig. 4<\/a>a, b. (The DPS plot for all temperatures for E\u2225a is shown in Fig. S3<\/a> in Supplementary information. In E\u2225a, the standard deviation (SD) from a straight line to \\(\\log {\\sigma }_{in}(\\omega ){T}^{\\alpha }\\) vs \\(\\log \\hslash \\omega \/{k}_{{\\rm{B}}}T\\) at different \u03b1 values is plotted in the inset of Fig. 4<\/a>. The minimum SD appears at \u03b1\u2009~\u20091.02, which is nearly equal to 1. The result is consistent with the case of YbRh2Si214<\/a>, suggesting a universal behavior in heavy fermion materials. The DPS is realized at temperatures below 80\u2009K for E\u2225a. However, for E\u2225c, the data are not scaled even with \u03b1\u2009=\u20091.5 and 2. Therefore, it could be concluded that the heavy quasiparticles only for E\u2225a follow the DPS.<\/p>\n

Fig. 4: Dynamical Planckian scaling plot.<\/b>\"figure<\/a><\/p>\n

\u03c3in(\u03c9) \u22c5 T\u03b1 as a function of \u210f\u03c9\/kBT as shown in Eq. (1<\/a>) for E\u2225a (a<\/b>) and E\u2225c (b<\/b>). The intrinsic heavy quasiparticles’ \u03c3in(\u03c9) spectra were obtained by subtracting the background spectra due to light quasiparticles and interband transitions from the original \u03c31(\u03c9) spectra. The broad solid line in (a<\/b>) indicates a guide to the eye to indicate the scaling. DPS requires \u03b1\u2009=\u20091, but different \u03b1 values are adopted for E\u2225c in (b<\/b>). The inset of (a<\/b>) shows the standard deviation (SD) of \\(\\log [{\\sigma }_{in}(\\omega ){T}^{\\alpha }]\\) vs \\(\\log [\\hslash \\omega \/{k}_{{\\rm{B}}}T]\\) from a straight line as a function of \u03b1.<\/p>\n

The maximum electrical resistivity along the a-axis appears at about 80 K, below which the coherent Kondo lattice is realized20<\/a>. The fact suggests that the DPS for E\u2225a appears in the coherence state. It would be due to the geometrical frustration of the quasi-kagome structure. However, in the temperature regions where the DPS holds in Fig. 4<\/a>a, the relation between \\(\\log [{\\sigma }_{in}(\\omega )\\cdot T]\\) and \\(\\log [\\hslash \\omega \/{k}_{{\\rm{B}}}T]\\) is a straight line with the relation of \\({\\sigma }_{in}(\\omega )\\cdot T\\propto {(\\hslash \\omega \/{k}_{{\\rm{B}}}T)}^{-1.8}\\). In high-Tc cuprates, \u03c31(\u03c9, T) \u22c5 T is a universal function of \u03c9\/T based on the Drude formula12<\/a>. The formula for \u210f\u03c9\/kBT\u2009\u226b\u20091 becomes \\({\\sigma }_{1}(\\omega ,T)\\cdot T\\propto {(\\hslash \\omega \/{k}_{{\\rm{B}}}T)}^{-2}\\). The relation of \\({\\sigma }_{in}(\\omega )\\cdot T\\propto {(\\hslash \\omega \/{k}_{{\\rm{B}}}T)}^{-1.8}\\) for CeRhSn is almost consistent with the simple formula. However, the order is inconsistent with the value of YbRh2Si2, where \\({\\sigma }_{in}(\\omega )T\\propto {(\\hslash \\omega \/{k}_{{\\rm{B}}}T)}^{-1}\\)14<\/a>. These results imply that the heavy fermion systems of CeRhSn and YbRh2Si2 follow the DPS but have individual scaling, which may provide a meaningful result. Further data should be accumulated to establish a universal DPS in heavy-fermion systems.<\/p>\n

As shown in Fig. 3<\/a>a3, the \u210f\/(\u03c4kBT) curve can follow the Planckian scaling of \\(\\hslash \/(\\tau {k}_{{\\rm{B}}}T) \\sim {[1+(\\hslash \\omega \/{k}_{{\\rm{B}}}T)]}^{1\/2}\\) and \u210f\/(\u03c4kBT)\u2009~\u20091 at \u210f\u03c9\u2009=\u20090, which is consistent with quantum oscillation data, where \u210f\/(\u03c4kBT)\u2009~\u20091 is observed6<\/a>. The same carriers can probably be observed in quantum oscillations and optical conductivity experiments. On the other hand, thermodynamical properties at EF pronounce \u210f\/(\u03c4kBT)\u2009=\u20090.01\u2009\u2212\u20090.02, where the Planckian dissipation is under debate15<\/a>. The inconsistency in \u210f\/(\u03c4kBT) should be resolved by the results of many further experiments.<\/p>\n

In most three-dimensional Ce-based heavy-fermion materials in the vicinity of QCP, the mid-IR peaks are slightly visible owing to the relatively weak c-f hybridization intensity24<\/a>,38<\/a>,39<\/a>,40<\/a>. However, the mid-IR peaks clearly appear in CeRhSn, being the hallmark of strong valence fluctuation. The simultaneous appearance of the valence fluctuation and DPS owing to the quasi-kagome structure describes that the quantum criticality of CeRhSn is different from usual NFL heavy-fermions like CeCu6\u2212xAux, where AFM correlations are responsible for the quantum criticality41<\/a>.<\/p>\n

Conclusion remarks<\/p>\n

To summerize, polarized optical conductivity measurements and first-principles calculations of the quasi-kagome Kondo lattice material CeRhSn have revealed the anisotropic electronic structure and Drude response. Along the hexagonal a- and c-axes, the 4f spin-orbit doublet showing the strong c-f hybridization appears even at room temperature, indicating the strong c-f hybridization intensity. On the other hand, a renormalized Drude response at \u210f\u03c9\u2009\u2264\u2009100\u2009meV indicates the formation of heavy quasiparticles. Analysis of the Drude structure shows that it follows the DPS only along the a-axis, resulting from the magnetic fluctuations based on the Ce quasi-kagome lattice. These findings support that the quantum criticality of CeRhSn coexists with the valence fluctuation. This work should motivate further investigation to clarify whether the anisotropic DPS commonly describes low-temperature responses of low-dimensional NFL heavy-fermion materials.<\/p>\n","protected":false},"excerpt":{"rendered":"The temperature dependences of the R(\u03c9) and \u03c31(\u03c9) spectra obtained from the Kramers-Kronig analysis of the R(\u03c9) spectra…\n","protected":false},"author":3,"featured_media":122118,"comment_status":"","ping_status":"","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[25],"tags":[2270,834,19533,492,836,159,31064,31065,31066,67,132,68],"class_list":{"0":"post-122117","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-physics","8":"tag-condensed-matter-physics","9":"tag-general","10":"tag-materials-science","11":"tag-physics","12":"tag-quantum-physics","13":"tag-science","14":"tag-structural-materials","15":"tag-surfaces-and-interfaces","16":"tag-thin-films","17":"tag-united-states","18":"tag-unitedstates","19":"tag-us"},"share_on_mastodon":{"url":"https:\/\/pubeurope.com\/@us\/114978737684901020","error":""},"_links":{"self":[{"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/posts\/122117","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/comments?post=122117"}],"version-history":[{"count":0,"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/posts\/122117\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/media\/122118"}],"wp:attachment":[{"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/media?parent=122117"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/categories?post=122117"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/tags?post=122117"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}