Calorimetric measurement of nuclear spin-lattice relaxation rate in metals

The quasiparticle density of states in correlated and quantum-critical metals directly probes the effect of electronic correlations on the Fermi surface. Measurements of the nuclear spin-lattice relaxation rate provide one such experimental probe of quasiparticle mass through the electronic density of states. By far the most common way of accessing the spin-lattice relaxation rate is via nuclear magnetic resonance and nuclear quadrupole resonance experiments, which require resonant excitation of nuclear spin transitions. Here we report nonresonant access to spin-lattice relaxation dynamics in AC-calorimetric measurements. The nuclear spin-lattice relaxation rate is inferred in our measurements from its effect on the frequency dispersion of the thermal response of the calorimeter-sample assembly. We use fast, lithographically defined nanocalorimeters to access the nuclear spin-lattice relaxation times in metallic indium from 0.3 to 7 K and in magnetic fields up to 35 T.

Figure 1

Thermal impedance spectroscopy with lithographic nanocalorimeters. (a) Sketch of the components of the nanocalorimeter. Colored elements indicate the heat bath (purple), chromium leads (orange), and calorimeter platform (blue), which contains the heater and the thermometer as well as the $1\times 1$ mm SiN membrane (black). The distance markers $x=0,L,s,h$ on the lead (highlighted in orange) illustrate the notation used in the discussion of the thermal susceptibility of a quasi-one-dimensional object in Appendix pp4. ${\kappa }_{\mathrm{L}\mathrm{B}}$ and ${\kappa }_{\mathrm{C}\mathrm{L}}$ indicate the thermal conductance of the lead-bath contact and lead-platform contact, respectively, used in the discussion in Appendix pp2. (b) Thermal conductance of the calorimeter-heat-bath heat link ${\kappa }_{\mathrm{C}\mathrm{B}}$ and heat capacity of the calorimeter platform $C_{\mathrm{C}}$. The ratio of $C_{\mathrm{C}}/{\kappa }_{\mathrm{C}\mathrm{B}}$ is an indicator of the characteristic time of the calorimeter platform, ranging from 1 to 2 ms between 0.1 and 1 K. (c) The heat flow diagram of the calorimeter-sample assembly. The sample is thermally coupled to the calorimeter platform via a thin layer of grease with contact conductance ${\kappa }_{\mathrm{C}\mathrm{S}}$. (d) Optical image of the nanocalorimeter. The gold-capped chromium leads are $400\mu \mathrm{m}$ long, $35\mu \mathrm{m}$ wide, and 60 nm thick. The $1\times 1{\mathrm{mm}}^2$ SiN membrane is 150 nm thick [5, 6].

Figure 2

The thermal impedance of the calorimeter-sample assembly in the complex plane of $\zeta$. (a) Measured thermal impedance $\zeta \left(\omega \right)$ at 35 T at temperatures in the range from 0.7 to 8 K. The upper half, $\mathrm{Im}\zeta >0$, shows the observed thermal impedance. The lower half, $\mathrm{Im}\zeta <0$, is added as a guide for the eye to represent the thermal impedance at negative frequencies, $\zeta (-\omega )={\zeta }^{*}\left(\omega \right)$. (b) The normalized thermal impedance $\zeta \left(\omega \right)/\zeta (\omega =0)$, where $\zeta (\omega =0)=1/{\kappa }_{\mathrm{C}\mathrm{B}}$. The upper half of the plot shows the observed normalized thermal impedance. The lower half shows the fit to Eq. (1) with fitting parameters discussed in Fig. 3.

Figure 3

Fitting the thermal impedance of the calorimeter-sample assembly. (a) The temperature dependence of the fitting parameters ${\tau }_{\mathrm{S}}=C_{\mathrm{S}}/{\kappa }_{\mathrm{C}\mathrm{B}}, {\tau }_{\mathrm{C}}=C_{\mathrm{C}}/{\kappa }_{\mathrm{C}\mathrm{B}}$, and ${\tau }_{\mathrm{N}}=C_{\mathrm{N}}/{\kappa }_{\mathrm{C}\mathrm{B}}$; the nuclear spin-lattice time $T_1$; and a dimensionless ratio $\nu ={\kappa }_{\mathrm{C}\mathrm{B}}/{\kappa }_{\mathrm{S}\mathrm{C}}$, determined by the fit to Eq. (1). The solid circles represent measurements in the resistive magnet at 35 T on a 15 nmol size sample. The open circles represent measurements in the superconducting magnet at 12 T on a 7.8 nmol sample. The nuclear Schottky for the 12 T measurements is scaled by a factor ${(35/12)}^2$. Thick gray lines represent the characteristic times $m_1, m_2$, and $m_3$ of the calorimeter-sample assembly determined by the observed thermal impedance via $\zeta \left(\omega \right)={\sum }_{i=1,2,3}{A}_i/[-i\omega +1/m_i]$. The dashed cyan line indicates the nuclear spin-lattice time $T_1$ measured in NQR experiments [12, 13]. Solid lines tracing ${\tau }_{\mathrm{S},\mathrm{N},\mathrm{C}}$ and $\nu$ are guides for the eye. (b) and (c) Frequency dependence of the polar components (amplitude and phase) of the observed thermal impedance in the frequency range of 10 mHz to 1 kHz. (d) and (e) Frequency dependence of the polar components of $\zeta \left(\omega \right)$ in Eq. (1) with the best-fit parameters from (a). The values of parameters ${\tau }_{\mathrm{C},\mathrm{S},\mathrm{N}}$ and $T1$ are indicated by markers at $f=1/\left(2\pi {\tau }_{\mathrm{C},\mathrm{S},\mathrm{N}}\right)$ on top of 7.3 K and at 0.7 K frequency scans.

Figure 4

Temperature and magnetic field dependence of the spin-lattice relaxation rate and nuclear and lattice heat capacity. (a) Magnetic field dependence of the nuclear Schottky $C_{\mathrm{N}}$ (in red) and electronic specific heat $C_{\mathrm{S}}$ (in purple) at 0.7 K. The magnetic field axis scales as field squared. The shaded region around the line represents the error bars (Appendix pp3). (b) The nuclear spin-lattice relaxation rate $1/T_{1}T$ vs field. (c) Temperature dependence of the nuclear Schottky $C_{\mathrm{N}}$ (per mol) and the electron ($+$ phonon) specific heat $C_{\mathrm{S}}$ (per mol) at 35 T (solid circles) and 12 T (open circles). For the sake of comparison, the nuclear Schottky at 12 T is scaled up by a factor of ${(35/12)}^2$. The dashed blue line indicates heat capacity measured in Ref. [14], $c_S=(1.69\mathrm{mJ}/\mathrm{mol}{\mathrm{K}}^2)T+(1.43\mathrm{mJ}/\mathrm{mol}{\mathrm{K}}^4)T^3$. The dashed red line indicates the expected magnitude of the nuclear Schottky in indium metal, $c_N=(0.015\mathrm{mJ}\mathrm{K}/\mathrm{mol}{\mathrm{T}}^2)B^2/T^2$. (d) Temperature dependence of the nuclear spin-lattice relaxation rate $1/T_{1}T$ at 35 T (solid circles) and 12 T (open circles).