Planckian Dissipation and non-Ginzburg-Landau Type Upper Critical Field in Bi2201

Fig. 1(a) and (b) show optical images of our Bi2201 sample with a Hall bar structure after the micro-fabrication. In Fig. 1(b), the dark-green area denotes the Bi2201 sample, and the yellow areas represent the gold electrodes. The sizes of the sample and gold electrodes are also denoted in Fig. 1(b). The thickness of the Bi2201 sample is $d=1.5$ $\mu$m measured by an atomic force microscope (AFM). We mark all the sizes here for determining the longitudinal resistivity and Hall coefficient. Since the electrodes have certain sizes, we use the central point to calculate related quantities. The well measured sizes allow us to precisely determine the absolute values of the resistivity and Hall coefficient.

Fig. 1(c) shows the temperature dependence of the in-plane resistivity of Bi2201 at zero field in a wide temperature region, and zero resistance state is achieved at $T_{c}^{zero}$ = 6.7 K. In Fig. 1(d), we present the resistive transitions below 15 K under different magnetic fields. According to previous studies, with the nominal composition x = 0.05 and the $T_{c}$ value, we judge that the hole doping level of the present sample is in the region $0.14<p<0.16$, namely it is in the slightly underdoped region but close to the optimal doping point LuoHQ . The transition near the onset temperature is very rounded, indicating a strong superconducting fluctuation. This rounded resistive transition from temperatures above $T_{c}$ is a quite common effect in hole-doped cuprates, and is proved to arise from strong superconducting fluctuations by the observation of, for example, strong Nernst effect in La_2-xSr_xCuO_4-δ and La-doped Bi2201 WangYYPRB , and strong excess conductivity in YBa₂Cu₃O_7-δ Leridon . Furthermore, this seemingly broad transition occurs mainly due to the gradual dropping down of resistivity in the temperature region far above $T_{c}^{zero}$, which occurs also in the original single crystals LuoHQ . This is not due to inhomogeneity as occurring in conventional superconductors. The basic reason is that the transition near zero resistance is sharp, which would not be the case if the sample was inhomogeneous. Thus it is hard to define an onset transition temperature $T_{c}^{onset}$ and determine the transition width. The inset in Fig. 1(c) shows the temperature dependence of magnetization measured on one sample taken from the same batch under a field of 1 Oe. The onset transition of magnetization is about 7.1 K, corresponding roughly to the middle transition point of resistivity at zero field. One can see that, the resistivity exhibits a roughly $T$-linear dependence from 300 K down to 20 K. Below 20 K, there is a slight upward curvature of resistivity, but still a linear-like temperature dependence can be seen even down to 2 K after applying a magnetic field of 15 T. This slight upward curvature is induced by the elastic scattering of impurities, which will be addressed below. It is found that a magnetic field of 15 T can already kill the zero resistance state and recovers about 88% normal state resistivity at 400 mK. This allows us to explore the most $H-T$ phase diagram.

Although the normal states of some cuprate superconductors exhibit anomalous behaviors, such as the $T$-linear dependence of resistivity and pseudogap effect, the quasi-particle features were still observed both in the normal state Sato ; TailleferPRB2021 and superconducting state Hoffman ; GuQQSTM . Thus in the following, we use the Drude formula to extract the scattering rate, as described in reference Planckian . As adopted by many researchers, we assume the Drude formula can still be used to describe the quasiparticle scattering and transport properties in normal state, thus the resistivity can be written as,

Here $e$ is the charge of an electron, $n$ is the carrier density and $m^{*}$ is the effective mass of the quasiparticles, and the latter two parameters should be determined from experiment. The temperature dependence of resistivity is then reflected by the scatting rate $1/\tau$. In principle, the value of $m^{*}/n$ can be determined directly from the optical conductivity LCCOGreene , while due to the small Drude weight and very strong phonon contributions in the sample, we didn’t successfully get credible data of $m^{*}/n$. Alternatively, we can get the charge carrier density from the Hall effect measurements. Fig. 2(a) shows the Hall resistivity as a function of magnetic field at different temperatures. It is found that, above 20 K, the Hall resistivity changes linearly with magnetic field. At 10 K, the Hall resistivity shows a non-linear dependence of magnetic field, this may be induced by the involvement of vortex motion. We can get the Hall coefficient $R_{H}$ by linearly fitting to the Hall resistivity data in Fig. 2(a). The obtained Hall coefficient $R_{H}$ is plotted as a function of temperature in Fig. 2(b). One can see that there is a slight change of Hall coefficient $R_{H}$ above 20 K. The charge carrier density $n$ is obtained from the Hall coefficient $R_{H}$ based on the formula $R_{H}=1/ne$. Since the temperature dependence of the carrier density $n$ is weak, we take the average of the carrier densities between 20 K and 300 K and get $n_{A}=3.4\times 10^{21}$ cm^-3. This value is close to the previously reported one in La-doped Bi2201 Bi2212 .

Now let us turn to the effective mass of Bi2201. Generally speaking, it is quite hard to obtain the effective mass of a correlated metal. Usually, one can estimate the effective mass from measurements of quantum oscillation, optical reflectivity or specific heat. Here we use specific heat to determine the effective mass $m^{*}$ by using the relation effective-mass $\gamma_{n}=(\pi N_{A}k_{B}^{2}/3\hbar^{2})a^{2}m^{*}$, where $a$ is the lattice constant, $N_{A}$ is the Avogadro’s number. In the paper of Legros et al. Bi2212 , concerning the effective mass of Bi2201, they referred to the La-doped Bi-2201 with $T_{c}$ = 19 K in the overdoped regime specific-heat . About the La-free Bi2201, there have been some published results of specific heat, but mainly concerning the low lying quasiparticle excitations in superconducting state Bi2201SH1 ; Bi2201SH2 . Thus there have been no reported values about $\gamma_{n}$ until a recent report Girod . In this paper, except for a collection and analysis of specific heat for La-doped Bi2201 and La_2-xSr_xCuO₄, the authors present one set of data for the La-free Bi2201 sample with doping level close to ours (they called Bi2201 #1). They found that the $C/T$ exhibits a saturation and slight upturning in double logarithmic plot of $C/T$ vs. $T$ in low-$T$ region. This slight upturning makes it difficult to precisely determine the $\gamma_{n}$ value, a gross estimate of $\gamma_{n}$ at 0.65 K is about 13 $\pm$ 1 mJ mol^-1K^-2. For comparison, we took a La-free Bi2201 sample with similar doping level and $T_{c}$ as the one for our transport measurements to measure the specific heat at low temperatures. The measured data at 9 T down to 1.89 K are shown in the inset of Fig. 3(a). It is found that the data can be roughly fitted with the Debye model in low-$T$ region. If we use a linear extrapolation of the low-$T$ data in the form of $C/T$ vs. $T^{2}$, as highlighted by the solid line in low-$T$ region in the inset of Fig. 3(a), we get an extrapolated value of $\gamma$ (9T) = 7.4 mJ mol^-1K^-2 at $T=0$ K. The values of $C/T$ of our results are grossly consistent with the data of Girod et al. Girod at temperatures above 2 K. Although the upper critical field at $T=0$ K is higher than 9 T in the present system, from the resistivity data, one can see that a field of 9 T has already killed the zero resistance state above 0.4 K, thus even $\gamma$ will still increase with field beyond 9 T, but that increase should be limited. Thus, for the La-free Bi2201 near the optimal doping point with $T_{c}$ of about 6.7-10 K (note: different people define $T_{c}$ with different criterions, with doping level in the region 0.13 $<p<$ 0.16), the $\gamma_{n}$ may locate in the region of 8-13, or 10.5 $\pm$ 2.5 mJ mol^-1K^-2 (the upper bound was quoted from Fig.6 of Girod et al. Girod ). By taking this value we can estimate $m^{*}=7.35\pm 1.75m_{0}$ via the equation between $\gamma_{n}$ and $m^{*}$, where $m_{0}$ is the bare electron mass.

Fig. 3(a) displays the resistivity data measured at different time. The measurements of Round 2 (abbreviated as R2) and R3 were carried out half a year after the first measurement R1. The measurement of R4 was carried out nine months after the first measurement. There is only a slight change of resistivity in the low temperature region with measurements spanned in about nine months, but the transition part overlaps very well, so the Bi2201 sample is quite stable. The resistivity plotted in Fig. 1(c) is R4 displayed in Fig. 3(a). Using $n_{A}=3.4\times 10^{21}$ cm^-3 determined from the Hall effect measurements, and $m^{*}=7.35m_{0}$, we calculate the scatting rate as a function of temperature for our Bi2201 sample from the R4 displayed in Fig. 3(a) via Eq. 1. Shown in Fig. 3(b) are the scattering rates derived from the data R4 (open dark symbols) together with that measured under 15 T (blue circles) at low temperatures. Perhaps due to the slightly different electric connection details in different equipments, there is a little difference between these two sets of data, thus the data measured under 15 T was off-set up of about 3% in order to have a smooth connection at 16 K and above. We should note that the residual resistance ratio RRR = $\rho(300K)/\rho(0K)$ $\approx$ 3.33 in present sample is not big. This could be induced by the larger out-of-plane disorder effect induced by a larger mismatch of the ionic radius between Sr²⁺ and Bi³⁺. Since the magnetoresistance at higher temperatures, for example at 16 K, is very small, we believe the 3% mismatch of resistivity under 15 T (measured in Pekin University by another PPMS) is not due to the disorder scattering. And the temperature dependent scattering rate is intrinsic and reflects the electronic properties of the system.

The dashed lines plotted in Fig. 3(b) are the fitting curves of scattering rate in different temperature regions using the formula $1/\tau=aT+bT^{2}+c$, where $a$, $b$ and $c$ are the fitting parameters. In order to see the fitting more clearly at low temperatures, we enlarged the view by taking a semi-logarithmic scale (right-bottom inset of Fig. 3(b)). One can see a little difference between the data and the fitting curves at low temperatures below about 5 K, which may be induced by the presence of small amount of elastic scattering. The global fittings look quite good and the fitting parameters are given in Table I. In the upper-left inset of Fig. 3(b), we show the contributions of the $T$-linear and quadratic temperature dependence arising from the fitting to the data between 2-300 K, and one can see that the quadratic term is much smaller than the linear term, especially below 200 K. It is clear that the fitting parameters are quite close to each other when we fit the data in temperature regions between 16-300 K or 2-300 K. For temperatures ranging from 2 K to 300 K, the linear term can give us $\alpha=0.925$ from the fit. Taking the different Hall coefficient values at 20 K and 300 K into consideration, we find that $0.77<\alpha<1.16$. Therefore, our refined fitting results confirm the Planckian dissipation over a wide temperature range Davison ; Hartnoll ; SKY . We need to notice that, in recent publications Barisic ; Greven , specially in the system HgBa₂CuO_4+δ, it was shown that either the longitudinal resistivity $\rho(T)$ or the Hall angle $cot(\Theta_{H})=\rho(T)/\rho_{H}\propto m^{*}/\tau$ exhibits a quadratic temperature dependence in low-$T$ region, suggesting a Fermi liquid feature. We tried to plot our data either in $\rho(T)$ or $cot(\Theta_{H})$ versus $T^{2}$, but we didn’t see a quadratic temperature dependence in low-$T$ region. Although the curve $cot(\Theta_{H})\propto m^{*}/\tau$ versus $T$ shows a little curvature in the high temperature region, the data in low-$T$ region (up to about 65 K) can still be fitted with a linear relation. Thus we believe the scattering rate of present Bi2201 sample does not have a quadratic temperature dependence in low-$T$ region, which is different from the HgBa₂CuO_4+δ.

Fig. 4(a) shows the temperature dependencies of the critical field at which the resistivity has reached 1$\%$, 10$\%$, 50$\%$, 70$\%$, 90$\%$ and 95$\%$ of $\rho_{n}(T)$. Here the normal state resistivity $\rho_{n}(T)$ is obtained by following the normal state extrapolation linear line measured between 10 and 20 K under a high magnetic field. The same data are shown in Fig. 4(b) in a semi-logarithmic way. We choose the field at which the resistivity has reached 95$\%\rho_{n}$ as the onset point of upper critical field $H_{c2}(T)$, thus it is called as $H_{c2}^{on}(T)$. In the following we focus on fitting the $H_{c2}^{on}(T)$ curve according to the formula expected by the Ginzburg-Landau (GL) theory, since this line is the usually defined upper critical field at which the droplet of superconductivity starts to form. One can see that the $H_{c2}^{on}(T)$ displayed in Fig. 4(a) does not saturate in the zero temperature approach, which violates the expectation by the GL theory. The critical fields determined with other criterions of lower resistivity ratio $\rho(T)/\rho_{n}(T)$ should have involved the flux flow, thus its divergent behavior at low temperatures is understandable. In order to demonstrate that the $H_{c2}^{on}(T)$ curve strongly deviates from the behavior expected by the GL theory, we calculated the $H_{c2}(T)$ curve based on the Werthamer-Helfand-Hohenberg (WHH) theory WHH . In the dirty limit which is the case for the Bi2201 system, the upper critical field can be described by the WHH theory via the function WHH

Here, $t=T/T_{c}$, $\bar{h}=4\mu_{0}H_{c2}/(\pi^{2}H^{\prime}T_{c})$ with $H^{\prime}=\mu_{0}|dH_{c2}/dT|_{T_{c}}$; $\lambda_{so}$ is the parameter representing the strength of the spin-orbit interaction; $\alpha_{0}$ is the one reflecting the strength of the spin paramagnetic effect. The red solid lines displayed in Fig. 4(a) and (b) are the fitting results with the parameters $\alpha_{0}=0$ and $\lambda_{so}=0$ according to the WHH theory. The fitting result has no way to be consistent with the experimental results. The fitting quality is equally poor when we adjust the parameters $\alpha_{0}$ and $\lambda_{so}$ (not shown here). The difficulty for the fitting here is that all the theoretical curves should show a negative curvature and flattened feature in the low temperature limit, but the experimental data all show a positive curvature and a clear divergent behavior. We should note that, in the cuprates, this kind of upward curvature of $H_{c2}(T)$ in low-$T$ region has been observed in many other systems TlBaCuO ; AndoPRB ; Osofsky ; Sebastian ; WenHHPRL1999 . It was argued that this kind of divergence of $H_{c2}(T)$ may have involved the vortex motion WenHHPRL1999 ; WenHHPRL2000 .