Magnetic field-temperature competition and quantum criticality in a strange metal

Abstract

Strange metals defy the quasiparticle description of conventional metals, exhibiting a linear in temperature ($T$-linear) resistivity in a broad temperature range. It has become increasingly clear that, together with $T$-linear resistivity, strange metals exhibit a characteristic response in strong magnetic fields, which might point to the quantum critical origin of the strange metal behavior. To explore the effects of strong magnetic fields on the dynamics of quantum fluctuations in a strange metal, here we report the thermodynamic study of electronic density of states on the Fermi surface in CeCoIn$_5$. Using ultrafast nanocalorimeters, we access the electronic density of states at low temperatures and high magnetic fields through two independent thermodynamic probes -- the nuclear spin-lattice relaxation rate and the electronic specific heat -- measured simultaneously on the same crystal. Both thermodynamic probes exhibit magnetic field and temperature competition, characteristic of quantum criticality, indicating that magnetic field acts as a cutoff for the dynamics of quantum critical fluctuations in CeCoIn$_5$. However, at low temperatures and high magnetic fields, the electronic specific heat and the nuclear spin-lattice relaxation rate cannot be understood solely in terms of a critical enhancement of the electronic density of states at the Fermi surface. This indicates that quantum criticality in CeCoIn$_5$ involves both local and itinerant fluctuating critical modes.

Figures (26)

• Figure 1: Thermal impedance spectroscopy.a. Heat flow diagram of the calorimeter-sample assembly that underlies the thermal impedance of Eq. (\ref{['eq:theR']}). The nuclear spin subsystem represents indium $^{115,113}$In and cobalt $^{57}$Co nuclei. b. Color-enhanced optical image of the calorimeter platform with the mounted CeCoIn$_5$ sample. c. Sketch of lithographically defined nanocalorimeter showing its major components; thermal bath (280 $\mu$m silicon wafer, in purple), calorimeter platform containing thermometer and heater (in blue), the thermally insulating membrane (150 nm SiN$_x$), and gold-capped chromium leads (about 60 nm thick) Tagliati2012Willa2017Khansili2023. The three internal components of the calorimeter platform, the ac-heater, the thermometer and the offset heater, are shown as color-coded legend. d. Complex thermal impedance $\zeta(\omega)$ showing single-pole and multi-pole structure in the complex plane. The multi-pole $\zeta(\omega)$ describes a multi-timescale response. e. Example of the observed thermal impedance of CeCoIn$_5$ in 12 T field parallel to $c$-axis for several temperatures from 0.15 K to 3.10 K. The multi-pole structure at low temperatures accounts for multiple timescales of the calorimetry-sample assembly.
• Figure 2: Crossover temperature $T_{\alpha}(B \parallel c)$ for fields along the $c$-axis determined by the nuclear magnetic resonance (NMR), electronic specific heat and Thermal Impedance Spectroscopy (TISP) measurements.a Temperature dependence of electronic $C/T$ from Ref. Bianchi2003 for fields along the $c$-axis. The upward arrow indicates the crossover temperature $T_{\alpha}(B \parallel c)$. b Temperature dependence of the nuclear spin lattice relaxation rate $1/T_1T$ from Ref. Sakai2011 for fields along the $c$-axis. All solid curves are guides for the eye. c Temperature dependence of the electronic $C/T$ obtained by TISP for fields along the $c$-axis. d Temperature dependence of $1/T_1T$ obtained by TISP for fields along the $c$-axis. Arrow denotes the crossover temperature $T_{\alpha}(B \parallel c)$. e. Field dependence of the crossover temperature $T_{\alpha}(B \parallel c)$ (filled red circles $\bullet$) determined from panel d. Filled grey markers represent the superconducting boundary determined by AC-calorimetry measurements (Extended Data Fig. 2). $\square$ markers indicate $T_{\alpha}(B \parallel c)$ determined by earlier specific heat meaurements Bianchi2003, $\diamond$ indicate $T_{\alpha}(B \parallel c)$ from thermal expansion measurements Zaum2011, $\triangle$, from magnetoresistance measurements Ronning2005. Solid lines and dashed curves are guides for the eye. See also Supplementary Fig. 4. f. Crystal structure of CeCoIn$_5$ field angle convention.
• Figure 3: Temperature dependence of $1/T_1T$ for several fields and three field orientations: $B\!\parallel\!c$, $B\!\parallel\!ab$, and $45^{\circ}$ between them. a,b,c$1/T_1T$ for magnetic fields from 2 to 12 T, shifted vertically for clarity (the shift factors are indicated in the legend, unshifted curves are shown in Extended Data Fig. 3). Filled (upward) arrows denote the crossover temperature $T_{\alpha}(B,\theta)$. Open (downward) arrows denote the superconducting transition $T_{\text{c}}(B,\theta)$ determined from specific heat in Fig. 4. All solid curves are guides for the eye. d,e,f The field dependence of the crossover temperature $T_{\alpha}(B,\theta)$ for three field orientations (filled red circles) determined from panels a-c. Open blue markers represent $B_{c2}(T,\theta)$ determined from specific heat in Fig. \ref{['Fig:4']}. Open grey markers represent AC-calorimetry measurements (Extended Data Fig. 2). All solid lines and dashed curves are guides for the eye. See also Supplementary Fig. 4. g.$1/T_1T$ for several magnetic field orientations at 12 T, shifted vertically for clarity (the shift factors are indicated in the legend, unshifted curves are shown in Extended Data Fig. 3). Filled (upward) arrows denote the crossover temperature $T_{\alpha}(12\,\mathrm{T},\theta)$. e. Angular dependence of the crossover temperature $T_{\alpha}(B,\theta)$ at 12 T. Each data point corresponds to the crossover temperature obtained from panel g. The grey curve is fit to the lowest-angular-harmonic in the tetragonal crystal structure of CeCoIn$_5$ (see Supplementary Fig. 5).
• Figure 4: Temperature dependence of electronic $C/T$ and electronic entropy $S$ for several fields and three different field orientations.a,b,c Electronic $C/T$ for magnetic fields from 0 to 12 T. Each curve is shifted vertically for clarity. The vertical offset is indicated in the legend (see Extended Data Fig. 4 for curves without offset). Filled (downward) arrows denote the crossover temperature $T_{\alpha}(B,\theta)$ determined from $1/T_1T$ in Fig. (\ref{['Fig:2']}). All solid curves are guides for the eye. d,e,f Electronic entropy $S$ for selected fields obtained by integrating electronic $C/T$ curves. The horizontal gray line in panel i represents the value $({1}/{3})R$ln2. All solid curves are guides to the eye.
• Figure 5: Comparison of electronic $C/T$ and $1/T_1T$ behavior across the crossover temperature $T_{\alpha}$.a, b. Comparison between $1/T_1T$ and the electronic $C/T$ across the crossover temperature for a magnetic field of 12 T applied along the $c$-axis (panel a) and $ab$-plane (panel b). While $1/T_1T$ saturates below $T_{\alpha}$, the electronic $C/T$ continues to increase, indicating the presence of excess entropy. The dotted line marks the crossover value of electronic $C/T$; the portion of electronic $C/T$ below this value is the excess specific heat, $(C/T)_{\text{excess}}$, indicated by vertical gray arrows at the lowest measured temperature. c,d. Ratio of electronic $C/T$ to $(1/T_1T)^{1/2}$ for fields along the $c$-axis (panel c) and at 12 T for different angles (panel d). The corresponding crossover temperatures $T_{\alpha}$ are indicated by upward arrows. In conventional metals, the electronic $C/T$ scales with the density of states at the Fermi level, $N_0$, while the nuclear spin-lattice relaxation rate follows $1/T_1T \propto N_0^2$. Thus, this ratio provides a meaningful way to compare the behaviors of electronic $C/T$ and $1/T_1T$. The dotted curve serves as a guide to the eye.