Scale-invariant magnetoresistance in a cuprate superconductor

TL;DR

This study reveals that the metallic state in cuprate superconductors exhibits a scale-invariant, linear magnetoresistance up to 80T, linking it to quantum criticality and strange metal behavior.

Contribution

It provides direct evidence of scale-invariant magnetoresistance in cuprates, connecting high-field behavior to quantum criticality and the strange metal state.

Findings

Magnetoresistance is linear in magnetic field up to 80T.

High-field slope of magnetoresistance is temperature-independent.

Linear-in-temperature resistivity correlates with quantum criticality.

Abstract

The anomalous metallic state in high-temperature superconducting cuprates is masked by the onset of superconductivity near a quantum critical point. Use of high magnetic fields to suppress superconductivity has enabled a detailed study of the ground state in these systems. Yet, the direct effect of strong magnetic fields on the metallic behavior at low temperatures is poorly understood, especially near critical doping, $x=0.19$. Here we report a high-field magnetoresistance study of thin films of \LSCO cuprates in close vicinity to critical doping, $0.161\leq x\leq0.190$. We find that the metallic state exposed by suppressing superconductivity is characterized by a magnetoresistance that is linear in magnetic field up to the highest measured fields of $80$T. The slope of the linear-in-field resistivity is temperature-independent at very high fields. It mirrors the magnitude and doping evolution of the linear-in-temperature resistivity that has been ascribed to Planckian dissipation near a quantum critical point. This establishes true scale-invariant conductivity as the signature of the strange metal state in the high-temperature superconducting cuprates.

Figures

b,d and Figure S3). For all other samples the high-field magnetoresistance was only measured up to $55$T – not high enough for $β$ to reach saturation at low temperatures. The comprehensive magnetoresistance dataset at multiple temperatures in these samples is redundant enough for reliable estimate of the saturation value of the field-slope $β(p)$. The simplest method using no modeling at all takes for $β$ the value of $dρ/dB$ at the highest field for temperatures in the interval between $20$K and $30$K ($β_{4}$ in Figure S4). In this temperature interval the magnetoresistance is linear-in-fie

B). The saturation value of $β(p)$ is shown in Fig. 2C along with $α(p)$ in natural energy units. Both $α(p)/k_{B}$ and $β(p)/μ_{B}$ decrease monotonically with doping in this doping range and evolve at a similar rate. The weak doping dependence of $β(p)$ and $α(p)$ approaching critical doping is in apparent contrast to the rapid increase in the Hall coefficient [Taillefer-lsco-2016 ](#bib.bib38); [Boebinger2009 ](#bib.bib28); [Boebinger-BLSCO-nature-2003 ](#bib.bib27) and the divergence of the effective mass [Ramshaw-mass-Science2015 ](#bib.bib21) as the critical doping is approached at low t

Full text

High-temperature superconductivity in the cuprates is born directly out of a “strange” metallic state that is characterized by linear-in-temperature resistivity up to the highest measured temperatureslinear-BiSCCO ; Anderson-science ; Hussey-pheno ; Zaanen-review . In conventional metals, current is carried by long-lived electronic quasiparticles, which requires the scattering length not to be significantly shorter than the de Broglie wavelength IoffeRegel ; Mott1972 ; bad-metals ; Hussey2004-MIR . In contrast, the resistivity in the strange metal state of the cuprates does not saturate or exhibit a crossover at the temperature where the inferred quasiparticle scattering length is comparable to the electronic wavelength. This behavior is sometimes referred to as “Planckian dissipation”, which suggests that the transport relaxation rate, $ℏ/τ$, (where $ℏ$ is the reduced Planck constant and $τ$ is the relaxation time) is limited directly by the thermal energy scale $k_{B}T$ (where $k_{B}$ is the Boltzmann constant and $T$ is absolute temperature), rather than by quasi-particle interactions and lattice disorderVarma1989 ; Phillips2004 ; Mackenzie2013 ; Zaanen-planckian ; Zaanen-review ; AjiVarma ; Zaanen-string ; Kivelson2017 ; Hussey2009 . This calls into question the very existence of quasi-particles in the strange metal state. More important, it indicates scale-invariant dynamics (i.e., the lack of an intrinsic energy scale). This behavior is observed in both classes of high-Tc superconductors –- the cuprates and the pnictides Fisher2014 ; Hayes2016 –- but its microscopic origin and implications for superconductivity have yet to be fully understood.

Scale-invariant transport is commonly associated with metallic quantum criticality. A characteristic energy scale is continuously tuned by an external parameter and vanishes when the tuning parameter crosses a critical value Zaanen-review . For hole dopings below the critical point, $p<0.19$, the Hall effect in  $\text{La}_{2-x}\text{Sr}_{x}\text{CuO}_{4}$ (19) and quantum oscillations in  $\text{YBa}_{2}\text{Cu}_{3}\text{O}_{6+\delta}$ Taillefer-QO ; Ramshaw-mass-Science2015 provide evidence for a small carrier pocket, believed to be associated with a charge density waveCDW-Xray-Hayden-2012 ; CDW-Xray-Damascelli-2014 ; CDW-LSCO-Hayden . By contrast, above the critical doping, $p>0.19$, quantum oscillations in  $\text{Tl}_{2}\text{Ba}_{2}\text{Cu}\text{O}_{6+\delta}$ Vignolle-over-2008 indicate a large hole-like Fermi surface, in agreement with band structure calculations OKA . Measurements of Hall resistivity Boebinger2009 ; Boebinger-BLSCO-nature-2003 ; Taillefer-lsco-ybco-2016 ; Hussey-pheno , the upper-critical magnetic field Taillefer-Hc2 and the quasiparticle effective mass Taillefer-QO ; Ramshaw-mass-Science2015 ; Vignolle-over-2008 , as well as the zero-temperature collapse of a line of phase transitionsBourges ; Kerr ; ultrasound ; dichroism ; secondharmonic , suggest a quantum critical point near $p=0.19$. At this doping the linear-in-temperature resistivity extends to the lowest temperatures Hussey2009 ; Boebinger1996 ; Zaanen-review and therefore one might expect to access the anomalous behavior in the strange metal state in the broadest range of magnetic fields.

Magnetic fields have been instrumental in the study of both conventional and correlated metals because they couple directly to the charge carriers. Previous studies of the cuprates have made use of magnetic fields as a way of suppressing superconductivity to reveal the normal ground state properties through the magnetoresistance and quantum oscillations Taillefer-QO ; Ramshaw-mass-Science2015 ; Vignolle-over-2008 ; Hussey2009 ; Proust2016 ; LSCO-logT ; Boebinger1996 ; Boebinger-BLSCO-nature-2003 ; Boebinger2009 ; Taillefer-lsco-2016 ; Taillefer-lsco-ybco-2016 . The linear-in-temperature resistivity, however, suggests a strong interaction between the metallic state and the critical fluctuations associated with the quantum critical point. What has been missing is a study of how the magnetic field affects these fluctuations and thus the metallic state. To this end, we studied the electrical transport of  $\text{La}_{2-x}\text{Sr}_{x}\text{CuO}_{4}$ in high magnetic fields for a range of compositions near the critical doping, $x≈0.19$. We found a scale-invariant response to the magnetic field that is distinct from the well-understood response of charged quasi-particles to the Lorentz force in conventional metals Abrikosov ; Pippard . Strikingly, linear-in-field resistivity at high fields, together with linear-in-temperature resistivity at high temperatures, emerges as an intrinsic characteristic of the strange metal state in a cuprate superconductor.

Figure 1 shows the in-plane resistivity ($ρ$) of a thin-film  $\text{La}_{2-x}\text{Sr}_{x}\text{CuO}_{4}$ cuprate sample at $p=0.190$ Bozovic2009 ; Bozovic2016 ; Bozovic2009 ; Bozovic2015 ; Takagi1989 ; Liang-doping ; valueofp in magnetic fields aligned along the crystallographic c-axis up to $80$ T. Linear-in-temperature resistivity down to the superconducting transition temperature, $T_{c}=38.6$ K (Figure 1E), indicates close proximity to the critical doping. Figure 1A shows that the magnetoresistance below $40$ K is linear in magnetic field over the entire normal-state field range. To quantify this observation we define the field-slope, $β(B,T)=dρ(B,T)/dB$. We observe that at $70$ T, $β(B,T)$ saturates below $25$ K (Figures 1, B and C, and Figure S3) which suggests that linear-in-field resistivity is an intrinsic property of the strange metal state. The saturation value of $β$ at low temperature and high fields in natural energy units is $β/μ_{B}=5.2$ $μΩ$cm/meV where $μ_{B}$ is the Bohr magneton. This is comparable in magnitude to the temperature-slope, $α(T)=dρ(T)/dT$, which is $11.8$ $μΩ$cm/meV in $α/k_{B}$ energy units.

In conventional metals, magnetoresistance originates from the motion of electron quasi-particles around the Fermi surface under the action of the Lorentz force Abrikosov ; Pippard . For a given Fermi surface morphology the strength of magnetoresistance is controlled by the product of the cyclotron frequency, $ω_{c}=eB/m^{*}$ (where $m^{*}$ is quasiparticle mass), and the quasi-particle relaxation time $τ$. Magnetoresistance generally decreases in conventional metals as τ decreases with increasing temperature. This is in contrast to what we observe in  $\text{La}_{2-x}\text{Sr}_{x}\text{CuO}_{4}$ at $p=0.190$ (Figure 1). At $80$ T, and between $4$ and $25$ kelvin we observe nearly a factor of $2$ increase in resistivity, suggesting a factor of $2$ decrease in $τ$ (Figs. 1, A and D) ybco-comment , and yet the strength of the magnetoresistance [$dρ(T)/dB$] at $80$ T between $4$ and $25$ kelvin is independent of temperature (Fig. 1 B and C). This indicates that at very high magnetic fields the transport relaxation rate is set directly by the magnetic field through $ℏ/τ∝μ_{B}B$. A mechanism other than the traditional picture of orbiting quasi-particles must therefore underlie the high-field magnetoresistance in  $\text{La}_{2-x}\text{Sr}_{x}\text{CuO}_{4}$ . One conclusion is that the magnetic field directly affects the dynamics of critical fluctuations that are responsible for the relaxation time Zaanen-review ; Zaanen-planckian ; AjiVarma ; Zaanen-string ; Kivelson2017

The smooth evolution of the temperature-slope $α(p)$ across the critical doping Hussey2009 ; Ando2004 is another indication of a lack of well-defined quasi-particles in the strange metal phase at high temperatures in contrast to the divergence of quasi-particle effective mass approaching the critical doping at low temperatures Ramshaw-comment . The doping evolution of the magnitude of $β(p)$ may provide further insight into the character of transport in the strange metal state. We measured the ab-plane resistivity in magnetic fields along the c-axis up to $55$ T in  $\text{La}_{2-x}\text{Sr}_{x}\text{CuO}_{4}$ over the range of dopings $p=0.161$ to $p=0.184$ (Figure 2). All samples in this doping range exhibit linear-in-temperature resistivity at high temperatures (Fig. 2B). The saturation value of $β(p)$ is shown in Fig. 2C along with $α(p)$ in natural energy units. Both $α(p)/k_{B}$ and $β(p)/μ_{B}$ decrease monotonically with doping in this doping range and evolve at a similar rate. The weak doping dependence of $β(p)$ and $α(p)$ approaching critical doping is in apparent contrast to the rapid increase in the Hall coefficient Taillefer-lsco-2016 ; Boebinger2009 ; Boebinger-BLSCO-nature-2003 and the divergence of the effective mass Ramshaw-mass-Science2015 as the critical doping is approached at low temperature and high magnetic fields. This again indicates that, despite the observation of quantum oscillations at low temperatures [in  $\text{YBa}_{2}\text{Cu}_{3}\text{O}_{6+\delta}$ Taillefer-QO ; Ramshaw-mass-Science2015 and  $\text{Tl}_{2}\text{Ba}_{2}\text{Cu}\text{O}_{6+\delta}$ Vignolle-over-2008 ], the high-field, high temperature magnetoresistance in cuprates has a non–quasi-particle origin.

It is well known that the transport relaxation rate is linear-in-temperature, $ℏ/τ∝k_{B}T$, in the fan-shaped region of the temperature-doping plane (Figure 3, magenta) emerging from the critical point Ando2004 . Our results (Figure 2) suggest that an analogous fan-shaped region exists in the magnetic field–doping plane (Figure 3, blue) where the relaxation rate is linear-in-field, $ℏ/τ∝μ_{B}B$. This extends a quantum critical region in field, temperature, and doping where the transport relaxation rate is set by the dominant energy scale, $ℏ/τ∝\max\{k_{B}T,μ_{B}B\}$, as illustrated in Figure 3 subdominant .

These measurements establish the linear magnetoresistance at very high fields as a fundamental property of the strange metal state in the cuprates. A linear dependence on an external energy scale is not the only possible outcome of scale invariance near quantum critical point: in principle, any power-law dependence is possible. It is therefore striking that the temperature and field dependence of the resistivity in  $\text{La}_{2-x}\text{Sr}_{x}\text{CuO}_{4}$ assumes the simplest possible form. Both the cuprates and the pnictides Hayes2016 , exhibit this simple form of scale invariance, revealing another universal characteristic of high-temperature superconductors.

Acknowledgments: We thank J. Analytis, J.-H. Chu, N. Doiron-Leyraud, A. Finkel’stein, I. Fisher, S. Hartnoll, I. Hayes, S. Kivelson, J. Paglione, L. Taillefer, C. Varma, and J. Zaanen for discussions, and the entire 100 T operations team at the NHMFL Pulsed Field Facility for their support during the experiment. A.S. acknowledges the hospitality of the Aspen Center for Physics. The high-field resistivity measurements were performed in the 60 T long-pulse and 100 T magnet systems at the NHMFL Pulsed Field Facility, which is supported by NSF grant DMR-1157490 and the U.S. Department of Energy, Basic Energy Sciences (DOE/BES) “Science at 100 T” grant. Molecular beam epitaxy synthesis, lithography, and characterization of the samples were done at BNL, which is supported by DOE/BES, Materials Sciences and Engineering Division. X.H. was supported by the Gordon and Betty Moore Foundation’s EPiQS Initiative through grant GBMF4410. Aspen Center for Physics is supported by NSF grant PHY-1066293.

Appendix A Supplementary Materials

A.1 Sample Preparation

High-quality single-crystal thin film  $\text{La}_{2-x}\text{Sr}_{x}\text{CuO}_{4}$ were grown by the combinatorial molecular beam epitaxy (COMBE) technique Bozovic2015 . The substrates are $1$ cm $×1$ cm $×1$ mm LaSrAlO4 single crystals cut perpendicular to the crystallographic [001] direction. A copper-oxide superconductor film (of $25$ nm thickness for the highest $2$ dopings and $16.5$ nm for the rest) is patterned into a high-aspect-ratio ($300$ $µ$m × $9.8$ mm) strip with a strontium gradient along the longer edge of the strip. The Sr content changes by 4% from one end to the other. Current leads are attached to the opposite ends of the strip. A total of $62$ voltage leads, separated from one another by $300µ$m, are patterned on the sides of the strip, allowing longitudinal voltage measurements on segments along the strip with different strontium content.

The high magnetic field measurements were done at the Pulsed Field Facility of the National High Magnetic Field Laboratory, at Los Alamos National Laboratory. Two different magnet systems were used for our measurements. The large gradient samples were measured in the $60$ T Controlled Magnet Waveform system with a $32$ mm size bore, that achieved $55$ T tesla pulses of several seconds duration during our magnet run. We have then chosen a sample closest to the critical doping ($x≈0.19$) for the measurements in the $100$ T Multi-shot Magnet system. This magnet system has a smaller bore ($10$ mm) and shorter pulse (few microseconds), therefore a smaller segment of the sample (about $3$ mm $×3$ mm) was cut out for measurements.

Figure S1A shows the superconducting transition temperatures for nominal strontium compositions, $x$, in the range from $x≈0.16$ to $x≈0.19$. Here, $T_{c}$ is defined as the maximum in the temperature-derivative of resistivity (Figure S1B). For the compositions studied in this work, Tc tracks closely the empirical formula $T_{c}/T_{c,max}=1−b(x−0.16)^{2}$ Takagi1989 ; Liang-doping , where $T_{c,max}=42.1$ K is the maximum critical temperature for thin-film  $\text{La}_{2-x}\text{Sr}_{x}\text{CuO}_{4}$ and $b=100$ is obtained from best fit to the data (shown in red in Figure S1 A). The value of hole-doping p for each sample is obtained from Tc using best fit formula as described above. The actual difference between nominal value of x (nominal strontium content) and the inferred value of hole doping $p$ is in fact quite small small, less than 0.005 (Figure S1A).

The half-width at half-maximum of the temperature derivative of the resistivity curve is around 1 K for all samples studied in this work (Figure S1B for two samples). All samples show small residual resistance (less than $10$ $µ\textohm$cm - Figure S1c) as determined by extrapolation of the linear-in-temperature fit in the range between $150$ K and $300$ K.

Figure S2 shows the resistivity plotted as a function of $B^{2}$ for multiple compositions and temperatures, a superset of the data presented in Figure 2 of the main text. In the $p=0.190$ sample, a $B^{2}$-magnetoresistance over the entire field range ($80$ T) is observed only at the highest measured temperatures ($140$ K and $180$ K). All lower temperatures the magnetoresistance shows clear negative curvature, a direct consequence of a slower-than-$B^{2}$ increase at high magnetic fields.

A.2 Running-window slope analysis

For the $p=0.190$ sample where we have measured magnetoresistance up to $80$ T, the field-slope $β$ saturates at low temperatures well before the field limit is reached (Figure 1b,d and Figure S3). For all other samples the high-field magnetoresistance was only measured up to $55$T – not high enough for $β$ to reach saturation at low temperatures. The comprehensive magnetoresistance dataset at multiple temperatures in these samples is redundant enough for reliable estimate of the saturation value of the field-slope $β(p)$. The simplest method using no modeling at all takes for $β$ the value of $dρ/dB$ at the highest field for temperatures in the interval between $20$K and $30$K ($β_{4}$ in Figure S4). In this temperature interval the magnetoresistance is linear-in-field in the broadest field range in all measured samples: superconductivity becomes important at lower temperatures while the competition between magnetic field and temperature kicks in at higher temperatures. Fig. 2C of the main text uses the values of $β$ estimated using this simple method.

Estimates of saturation values of $β$ (for samples measured up to $55$ T) using a more quantitative analysis supports this qualitative estimate. Figure S3 illustrates three different quantitative estimates. $β_{1}(T)$ is obtained through the value of $dρ/dB$ at $80$ T (black circles), which for samples other than $p=0.19$ can be well approximated by extending the measured data using an empirical formula $dρ/dB=c_{0}+c_{1}e^{−c_{2}B}$. $β_{2}(T)$ is obtained through the value of $dρ/dB$ at $55$ T (red diamonds). $β_{3}(T)$ is only calculated for the $p=0.19$ sample, via a linear fit for $dρ/dB$ in the field interval $65$ T and $77$ T, for which this quantity is either field-independent or depends weakly on field for all measured temperatures (blue squares). The zero-temperature limit of all βi=1..4(T) can be estimated by fitting each $β_{i=1..4}(T)$ to an Interpolating formula $β_{i}(T)=b_{0}+b_{1}T/\sinh(b_{2}T)$, which captures correctly the behavior at both limits and satisfactorily interpolates the behavior of $β_{i=1..4}(p)$ in the $p=0.19$ sample in the entire field range up to $80$ T. Figure S4 shows the values of $β_{i=1..4}(p)T⟶0$ plotted vs hole-doping $p$.

A.3 Temperature Dependence of Resistivity at High-Field

Figure S5 shows resistivity as a function of temperature for all compositions studied in this work (the plots are identical to Figure 2A of the main text except here each composition is shown on a separate panel). Resistivity as a function of temperature for $p=0.190$ (Figure 1 E of the main text) is not shown here. The red curve shows the zero-field resistivity. The resistivity curves are in good agreement with a previous study in bulk crystals by Cooper et al Hussey2009 . Superposed on each plot is $ρ(T)$ at $55$ T (blue circles) which shows monotonic decrease in resistivity with decreasing temperature for all studied samples Taillefer-lsco-2016 .