Relaxation time spectrum of low-energy excitations in one- and two-dimensional materials with charge or spin density waves

The long-time thermal relaxation of (TMTTF)${}_2\mathrm{Br},{\mathrm{Sr}}_{14}{\mathrm{Cu}}_{24}{\mathrm{O}}_{41}$ and ${\mathrm{Sr}}_2{\mathrm{Ca}}_{12}{\mathrm{Cu}}_{24}{\mathrm{O}}_{41}$ single crystals at temperatures below 1 K and magnetic field up to 10 T is investigated. The data allow us to determine the relaxation-time spectrum of the low-energy excitations caused by the charge-density wave or spin-density wave (SDW). The relaxation time is mainly determined by a thermal activated process for all investigated materials. The maximum relaxation time increases with increasing magnetic field. The distribution of barrier heights corresponds to one or two Gaussian functions. The doping of ${\mathrm{Sr}}_{14-x}{\mathrm{Ca}}_x{\mathrm{Cu}}_{24}{\mathrm{O}}_{41}$ with Ca leads to a drastic shift of the relaxation-time spectrum to longer time. The maximum relaxation time changes from 50 s $\left(x=0\right)$ to 3000 s $\left(x=12\right)$ at 0.1 K and 10 T. The observed thermal relaxation at $x=12$ clearly indicates the formation of the SDW ground state at low temperatures.

Figure 1

Schematic picture of the experimental setup.

Figure 2

A constant power of the sample heater is switched off at $t=0$ after the waiting time $t_w=2t_{\text{in}}$ and the temperature relaxes from $T_1=102$ mK to $T_0=92$ mK. The inner relaxation is finished after the time $t_{\text{in}}$ (arrows). The thermal relaxation follows Eq. (4) at longer time $t>t_{\text{in}}$ (straight lines). The quasiequilibrium relaxation time ${\tau }_{\text{eq}}$ and the end of the inner relaxation $t_{\text{in}}$ depend strongly on the magnetic field.

Figure 3

The data of Fig. 2 in the semilog presentation. The straight lines of Fig. 2 (quasiequilibrium relaxation) are shown by the dashed curves. More than 90% of the measured initial temperature difference is caused by the inner relaxation process. The solid lines are calculated by Eqs. (2, 3, 4) with a Gaussian distribution function. Two fitting parameters of this distribution function ${\tau }_m$ (the position of the maximum) and $w$ (the width of the distribution) are given in Table 1 and Fig. 5 for $T_0=0.092$ K. For zero field, two additional fitting curves show the cases of $1.1w$ (blue) and $0.9w$ (red).

Figure 4

The measured inner thermal relaxation $\mathrm{\Delta }{T}_i/\mathrm{\Delta }{T}_0$ in the presentation $d(\mathrm{\Delta }{T}_i/\mathrm{\Delta }{T}_0)/d(lnt)$ as a function of time (blue points) for the data shown in Figs. 2 and 3. This gives roughly the distribution function of relaxation time (see the text). The numerical calculation of $\mathrm{\Delta }{T}_i/\mathrm{\Delta }{T}_0$ with Eq. (2) yields a Gaussian distribution (dashed curve) with nearly the same position of the maximum ${\tau }_m$.

Figure 5

Equation (2) yields a good fit to the inner thermal relaxation $\mathrm{\Delta }{T}_i\left(t\right)$ when a Gaussian distribution function of relaxation time is used. The obtained fitting parameters ${\tau }_m$ (the position of the maximum) and $w$ (the width of the distribution) are shown in Figs. 5 and 5 for the data of Fig. 3 $(T_0=0.092$ K and different magnetic field). Figure 5 also shows the time $t_{\text{in}}$ (for $t>t_{\text{in}}$ the inner relaxation can be neglected, see Figs. 2 and 3). ${\tau }_m$ and $t_{\text{in}}$ are proportional to $H^2$ at high magnetic field, i.e., with increasing magnetic field the distribution function is shifted to higher values without change of the distribution width. The error corresponds to the size of symbols.

Figure 6

The heat capacity of ${\mathrm{Sr}}_{14}{\mathrm{Cu}}_{24}{\mathrm{O}}_{41}$ as a function of the temperature for different magnetic fields measured at very short time $(t=30$ ms) and long time $\left(t>t_{\text{in}}\right)$. The CDW does not contribute to the short-time heat capacity. The solid lines show the total equilibrium heat capacity at different magnetic field. Below 0.3 K it is determined by a magnetic field independent quasilinear term (dashed line) and a Schottky contribution. The Schottky energy is proportional to $H^2$.

Figure 7

The end of the inner thermal relaxation $t_{\text{in}}$ (see also Fig. 8) follows the Arrhenius law [Eq. (1)] with a magnetic field independent activation energy $E_a/k_B=0.2$ K. The absolute value depends weakly on the magnetic field. Two Gaussian distribution functions of relaxation time are necessary to calculate the inner relaxation $\mathrm{\Delta }{T}_i\left(t\right)$ [according to Eq. (2)] with the maximum at ${\tau }_{ms}$ and ${\tau }_{ml}$ (see also Fig. 9). Both parameters are magnetic field independent, i.e., the increase of $t_{\text{in}}$ at fixed temperature and increasing magnetic field is a consequence of an increasing width. The temperature dependence of ${\tau }_{ml}$ corresponds to Eq. (1) with the same activation energy as for $t_{\text{in}}$. The temperature dependence of the shorter ${\tau }_{ms}$ deviates from Eq. (1) at low temperatures. The solid line is obtained when we include in addition a tunneling process [see Eqs. (10, 11, 12)].

Figure 8

A constant power of the sample heater is switched off at $t=0$ after the waiting time $t_w=2t_{\text{in}}$ and the temperature relaxes from $T_1=112$ mK to $T_0=104$ mK. The inner relaxation is finished after the time $t_{\text{in}}$ (the arrow). The thermal relaxation follows Eq. (4) at longer time $t>t_{\text{in}}$ (dashed curve). The solid line shows the result of the calculation $(\mathrm{\Delta }{T}_{\text{eq}}+\mathrm{\Delta }{T}_i)/\mathrm{\Delta }{T}_0$ where $\mathrm{\Delta }{T}_i$ is obtained from Eq. (2) with two Gaussian functions in Fig. 9 (dashed lines).

Figure 9

The measured inner thermal relaxation $\mathrm{\Delta }{T}_i/\mathrm{\Delta }{T}_0$ in the presentation $d(\mathrm{\Delta }{T}_i/\mathrm{\Delta }{T}_0)/d(lnt)$ as a function of time (blue points) for the data shown in Fig. 8. This gives roughly the distribution function of relaxation time (see the text). The numerical calculation of $\mathrm{\Delta }{T}_i/\mathrm{\Delta }{T}_0$ with Eq. (2) yields two Gaussian distributions (dashed curve). The corresponding four fitted parameters are shown. The positions of the peaks do not depend on the magnetic field (see Fig. 7).

Figure 10

The heat capacity of ${\mathrm{Sr}}_2{\mathrm{Ca}}_{12}{\mathrm{Cu}}_{24}{\mathrm{O}}_{41}$ measured at long time $t>t_{\text{in}}$ as a function of the temperature for different magnetic field. Antiferromagnetic ordering is observed at 2.8 K. The heat capacity is determined below the transition by a $T^3$ term (magnons and phonons), quasilinear term with a magnetic field dependent gap, and the Schottky term which is strongly time dependent for $t<t_{\text{in}}$. The values of the gap $\mathrm{\Delta }$ and Schottky energy $E_s$ for different magnetic field are given in the inset. The solid lines show the calculated equilibrium heat capacity, which is determined below 0.3 K by the quasilinear term and the Schottky contribution only.

Figure 11

A constant power of the sample heater is switched off at $t=0$ after the waiting time $t_w=2t_{\text{in}}$ and the temperature relaxes from 160.1 to 149.6 mK. Figures 11 and 11 show the measured thermal relaxation $\mathrm{\Delta }T\left(t\right)/\mathrm{\Delta }{T}_0$ as a function of time in different semilog presentations. The inner relaxation process is finished at $t_{\text{in}}=130$ s. The thermal relaxation follows Eq. (4) at longer time $t>t_{\text{in}}$ with the quasiequilibrium relaxation time ${\tau }_{\text{eq}}=62$ s [see Fig. 11]. An additional thermal relaxation term $\mathrm{\Delta }{T}_i/\mathrm{\Delta }{T}_0$ is observed at $t<t_{\text{in}}$ due to some inner relaxation process. We get a good fit of this term by a stretch exponential law with an exponent $\beta =0.9$. This demonstrates that the width of the distribution function of relaxation time is small since $\beta =1$ corresponds to the case of a Delta function $\delta \left({\tau }_m\right)$. The measured inner thermal relaxation $\mathrm{\Delta }{T}_i/\mathrm{\Delta }{T}_0$ is shown in Fig. 11 (blue points) in the presentation $d(\mathrm{\Delta }{T}_i/\mathrm{\Delta }{T}_0)/d(lnt)$ as a function of time for the data shown in Figs. 11 and 11. This gives roughly the distribution function of relaxation time (see the text). The numerical calculation of $\mathrm{\Delta }{T}_i/\mathrm{\Delta }{T}_0$ with Eq. (2) yields a Gaussian distribution (dashed curve) with nearly the same position of the maximum ${\tau }_m$.

Figure 12

The measured inner thermal relaxation $\mathrm{\Delta }{T}_i/\mathrm{\Delta }{T}_0$ in the presentation $d(\mathrm{\Delta }{T}_i/\mathrm{\Delta }{T}_0)/dlnt$ as a function of time at 10 T and different temperatures. The dashed curves show the distribution function obtained by the calculation of $\mathrm{\Delta }{T}_i$ with Eq. (2). The width of the distribution function is independent from the magnetic field and temperature $\left(w=0.5\right)$. The temperature and magnetic field dependence of ${\tau }_m$ (the position of the maximum) is given in Fig. 13.

Figure 13

The Arrhenius plot of the peak position in the distribution function for different magnetic field. The activation energy increases with increasing magnetic field. Tunneling is observed at low temperatures. The curves are calculated for a relaxation rate determined by tunneling and thermal activated processes.

Figure 14

The distribution function of the relaxation time obtained for the three different materials at nearly the same temperature and high magnetic field. The doping of ${\mathrm{Sr}}_{14}{\mathrm{Cu}}_{24}{\mathrm{O}}_{41}$ with Ca shifts the distribution function to much longer time.