Towards the Theory of Isotope Effect of the London Penetration Depth in Cuprates
I. E. Lyubin, M. V. Eremin, I. M. Eremin, H. Keller
备注：
3 pages, 2 figures, corrected version of paper in Conference Proceedings of 3rd International Conference "Fundamental Problems of High Temperature Superconductivity" (FPS'08), P.N.Lebedev Physical Institute of the Russian Academy of Sciences, Moscow-Zvenigorod, October 13-17, 2008. Corrected typos, reference [3] modified, formula (5) modified
The expressions for the superfluid density have been discussed in both weak and strong coupling approaches. The numerical calculations of temperature and isotope composition dependencies have been performed for a number of High-Tc compounds. The tight-binding parameters and corresponding Fermi surfaces are taken in accord to the available photoemission data.
Based on the recent developed real-space theory of superconductivity (arXiv:0910.5511 and arXiv:1001.5067), we study the physical nature of the Meissner effect and London penetration depth in conventional and non-conventional superconductors. It is argued that they originate from an exactly the same reason of the real-space quasi-one-dimensional periodic dynamic charge stripes in the superconductors. The fundamental relationship between the London penetration depth and the superconducting electron density is qualitatively determined.
We report measurements of AC susceptibility and hence the in-plane London penetration depth on the same samples of Bi:2212 and Bi(Y):2212 for many values of the planar hole concentration/CuO2 unit (p). These support the scenario in which the pseudogap weakens the superconducting response only for p less than approximately 0.19.
The London model describes strongly type-2 superconductors as massive vector field theories, where the magnetic field decays exponentially at the length scale of the London penetration length. This also holds for isotropic multi-band extensions, where the presence of multiple bands merely renormalises the London penetration length. We show that, by contrast, the magnetic properties of anisotropic multi-band London models are not this simple, and the anisotropy leads to the inter-band phase differences becoming coupled to the magnetic field. This results in the magnetic field in such systems ha…
The mechanism of formation of dome-like phase diagrams and the features of magnetic field penetration in iron pnictides with hetero- and isovalent doping are considered within the framework of the previously proposed model, which assumes the local character of doping and the cluster structure of resulted superconducting phase. It is shown that the proposed model, despite its simplicity and neglect of the features of the electronic structure, makes it possible not only to accurately calculate the positions of superconducting domes on the phase diagrams of specific HTSC compounds, but also to ex…
Recent measurements of the superconducting penetration depth in Ba(Fe_1-x Co_x)2As2 appeared to disagree on the magnitude and curvature of Δλ_{ab}(T), even near optimal doping. These measurements were carried out on different samples grown by different groups. To understand the discrepancy, we use scanning SQUID susceptometry and a tunnel diode resonator to measure the penetration depth in a single sample. The penetration depth observed by the two techniques is identical with no adjustments. We conclude that any discrepancies arise from differences between samples, either in growth or crystal …
Cho et al. [Phys. Rev. B, 84, 174502 (2011)] have reported on the temperature dependence of the London penetration depth deduced from Tunnel Diode Oscillator (TDO) measurements in optimally doped Fe(Se,Te) single crystals. According to their analysis, these measurements chould suggest a nodeless two-gap pairing symmetry with strong pair breaking effects. However, to reach this conclusion, the authors fit the temperature dependence of the superfluid density with a two band {\it clean} limit model which is incompatible with the presence of strong pair breaking effects, deduced from the $T^n$ tem…
Single crystals of K0.80Fe1.76Se2.02 were suscessfully grown from a ternary solution. We show that although crystals form when cooling a near stoichiometric melt, crystals are actually growing out of a ternary solution that remains liquid to at least 850 C. We investigated their chemical composition, anisotropic magnetic susceptibility and resistivity, specific heat, thermoelectric power, London penetration depth and flux penetration via magneto-optical imaging. Whereas the samples appear to be homogeneously superconducting at low temperatures, there appears to be a broadened transtion range c…
Recent measurements of the doping dependence of the London penetration depth λ(x) at low temperatures in clean samples of isovalent BaFe_2[As_(1-x)P_x]_2 at T<<Tc [Hashimoto et al., Science 336, 1554 (2012)] revealed a peak in λ(x) near optimal doping x=0.3. The observation of the peak at T<<Tc, points to the existence of the quantum critical point (QCP) beneath the superconducting dome. We associate such a QCP with the onset of a spin- density-wave order and show that the renormalization of λ(x) by critical magnetic fluctuations, gives rise to the observed feature. We argue that the case of p…
We show on a few examples of one-band materials with spheroidal Fermi surfaces and anisotropic order parameters that anisotropies $γ_H$ of the upper critical field and $γ_λ$ of the London penetration depth depend on temperature, the feature commonly attributed to multi-band superconductors. The parameters $γ_H$ and $γ_λ$ may have opposite temperature dependencies or may change in the same direction depending on Fermi surface shape and on character of the gap nodes. For two-band systems, the behavior of anisotropies is affected by the ratios of bands densities of states, Fermi velocities, aniso…
The London penetration depth $λ$ is the basic length scale for electromagnetic behavior in a superconductor. Precise measurements of $λ$ as a function of temperature, field, and impurity scattering have been instrumental in revealing the nature of the order parameter and pairing interactions in a variety of superconductors discovered over the past decades. Here we recount our development of the tunnel-diode resonator technique to measure $λ$ as a function of temperature and field in small single crystal samples. We discuss the principles and applications of this technique to study unconvention…
We have characterized an electron-irradiated Ba1-xKxFe2As2 (x = 0.53) single-crystal using two different experimental techniques: magneto-optic measurements and microwave measurements. The crystal has been measured before as well as after the 2.5 MeV electron irradiation process. After irradiation it was annealed in a number of steps, between 90 deg C and 180 deg C, and measured after each annealing step. Most microwave measurements were performed by means of a copper cavity, taking advantage of the TE011 and TM110 modes, allowing for the determination of the London penetration depths changes …
Recently, compressed H$_2$S has been shown to become superconducting at 203 K under a pressure of 155 GPa. One might expect fluctuations to dominate at such temperatures. Using the magnetisation critical current, we determine the ground-state London penetration depth, $λ_0$=189 nm, and the superconducting energy gap, $Δ_0$=27.8 meV, and find these parameters are similar to those of cuprate superconductors. We also determine the fluctuation temperature scale, $T_{\textrm{fluc}}=1470$ K, which shows that, unlike the cuprates, $T_c$ of the hydride is not limited by fluctuations. This is due to it…
We present a simple derivation of an expression for the superfluid density $ n_s \propto 1/λ^2 $ in superconductors with the tight binding energy dispersion. The derived expression is discussed in detail because of its distinction from the known expressions for ordinary superconductors with parabolic energy dispersion. We apply this expression for the experimental data analysis of the isotope effect in London penetration depth parameter $ λ$ in the BiSrCuO and YBaCuO family compounds near optimal doping, taking into account the orthorhombic distortion of crystal structure, and estimate the iso…
Unconventional London penetration depth in Ba(Fe0.93Co0.07)2As2 single crystals
在几种Ba（Fe $_{0.93}$ Co $_{0.07}$）$_2$ As $_2$单晶中测量了伦敦穿透深度$λ(T)$。热力学，电磁和结构表征测量证实，这些晶体是优良的品质。所观察到的$λ(T)$的低温变化遵循幂律，$Δλ(T) \sim T^n$与$n=2.4 \pm 0.1$，表明至少在$0.02T_c$以下存在正常准粒子。这与最近对NdFeAsO $_{1-x}$ F $_x$和SmFeAsO $_{1-x}$ F $_x$的单晶的穿透深度测量相反，其指示各向异性但无节点间隙。我们建议，在电子结构的Ba（Fe $_{0.93}$ Co $_{0.07}$）$_2$作为$_2$的三维字符可能会导致各向异性的$s-$波隙与点节点，将解释所观察到的$λ(T)$。
A simple relation is established between the zero-$T$ penetration depth $λ(0)$ and the slope of $λ^{-2}(T)$ near $T_c$, similar to Helfand-Werthamer's relation for $H_{c2}(0)$ and the slope of $H_{c2}(T)$ at $T_c$ for the isotropic s-wave case with non-magnetic scattering.\cite{HW} When the scattering parameter $ρ=\hbar v/2πT_c\ell$ ($v$ is the Fermi velocity and $\ell$ is the mean-free path) varies from 1 to 10, the coefficient of proportionality between $λ^{-2}(0)$ and $T_c (dλ^{-2}/dT)_{T_c}$ changes from 0.43 to 0.38. Combining this relation with the Rutgers thermodynamic identity, one can…
The issue of whether the quantum critical point (QCP) is hidden inside unconventional superconductors is a matter of hot debate. Although a prominent experiment on London penetration depth has demonstrated the existence of the QCP in the isovalent-doped iron-based superconductor BaFe$_2$(As$_{1-x}$P$_x$)$_2$, with the observation of a sharp peak in the penetration depth in the vicinity of the disappearance of magnetic order at zero temperature, the nature of such an emerging QCP remains unclear. Here, we provide a unique picture to understand well the phenomena of the QCP based on the framewor…
Irradiation with 1.4 GeV $^{208}$Pb ions was used to induce artificial disorder in single crystals of iron-arsenide superconductor Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$ and to study its effect on the temperature-dependent London penetration depth and transport properties. Study was undertaken on overdoped single crystals with $x$=0.108 and $x$=0.127 characterized by notable modulation of the superconducting gap. Irradiation with doses 2.22$\times10^{11}$$\textit{d}$/cm$^2$ and 2.4$\times10^{11}$$\textit{d}$/cm$^2$, corresponding to the matching fields of $B_ϕ = $6 T and 6.5 T, respectively, suppresse…
The breakdown magnetic field is the key parameter which determines the performance of superconducting radio-frequency cavities. This is the maximum field up to which the Meissner state remains stable and in uniform material, it is approximately given by the thermodynamic field. There are several recent suggestions to use nonuniform structures to enhance the breakdown field. One of possible realizations of such structure is depth profile of the scattering rate which, in the first approximation, modifies the London penetration depth but does not change the thermodynamic field. In this paper, we …
We reply to the Comment by T. Klein, P. Rodière and C. Marcenat (arXiv:1205.2566), on our paper, "Precision global measurements of London penetration depth in FeTe$_{0.58}$Se$_{0.42}$", Phys. Rev. B 84, 174502 (2011). Our work was motivated by Klein \emph{et. al}, "Thermodynamic phase diagram of Fe(Se$_{0.5}$Te$_{0.5}$) single crystals in fields up to 28 Tesla", Phys. Rev. B 82, 184506 (2010). In their paper, Klein et al. have attributed a factor of five difference in the value of the London penetration depth obtained from their tunnel diode resonator (TDR) measurements and calculated from the…
London penetration depth, $λ(T)$, was measured in single crystals of K$_{1-x}$Na$_x$Fe$_2$As$_2$, $x$=0 and 0.07, down to temperatures of 50~mK, $\sim T_c/50$. Isovalent substitution of Na for K significantly increases impurity scattering, with $ρ(T_c)$ rising from 0.2 to 2.2 $μΩ$cm, and leads to a suppression of $T_c$ from 3.5~K to 2.8~K. At the same time, a close to $T$-linear $Δλ(T)$ in pure samples changes to almost $T^2$ in the substituted samples. The behavior never becomes exponential as expected for the accidental nodes, as opposed to $T^2$ dependence in superconductors with symmetry i…
We study the effects of anisotropic order parameters on the temperature dependence of London penetration depth anisotropy $γ_λ(T)$. After MgB$_2$, this dependence is commonly attributed to distinct gaps on multi-band Fermi surfaces in superconductors. We have found, however, that the anisotropy parameter may depend on temperature also in one-band materials with anisotropic order parameters $Δ(T,k_F)$, a few such examples are given. We have found also that for different order parameters, the temperature dependence of $Δ(T)/Δ(0)$ can be represented with good accuracy by the interpolation suggest…
We report tunnel diode resonator measurements of in-plane London penetration depth, $λ(T)$, in optimally-doped single crystals of Fe(Te$_{0.58}$Se$_{0.42}$) with $T_c\sim$ 14.8 K. Systematic measurements were carried out for six samples with different size and surface roughness. The power-law behavior, $Δλ(T) = A T^n$ was found for all samples with the average exponent $n_{avg} = 2.3 \pm 0.1$ and the pre-factor $A_{avg} = 1.0 \pm 0.2$ nm/K$^{2.3}$. The average superfluid density is well described by the self-consistent two-gap $γ$ model resulting in $Δ_{_{I}}(0)/k_B T_c$ = 1.93 and $Δ_{_{II}}(…
Measurements of London penetration depth are a sensitive tool to study multi-band superconductivity and it has provided several important insights to the behavior of Fe-based superconductors. We first briefly review the "experimentalist - friendly" self-consistent Eilenberger model that relates the measurable superfluid density and structure of superconducting gaps. Then we focus on the \BaFe -derived materials, for which the results are consistent with 1) two distinct superconducting gaps; 2) development of strong in-plane gap anisotropy with the departure from the optimal doping; 3) appearan…
The upper critical field $H_{c2}$ and its anisotropy are calculated for order parameters with line nodes at equators, $k_z=0$, of the Fermi surface of uniaxial superconductors. It is shown that characteristic features found in Fe-based materials -- a nearly linear $H_{c2}(T)$ in a broad $T$ domain, a low and increasing on warming anisotropy $γ_H= H_{c2,ab}/ H_{c2,c}\, $ -- can be caused by competing effects of the equatorial nodes and of the Fermi surface anisotropy. For certain material parameters, $γ_H(T)-1$ may change sign on warming in agreement with recorded behavior of FeTeS system. It i…
Superconducting gap structure was probed in type-II Dirac semimetal PdTe$_2$ by measuring the London penetration depth using tunnel diode resonator technique. At low temperatures, the data for two samples are well described by weak coupling exponential fit yielding $λ(T=0)=230$~nm as the only fit parameter at a fixed $Δ(0)/T_c\approx 1.76$, and the calculated superfluid density is consistent with a fully gapped superconducting state characterized by a single gap scale. Electrical resistivity measurements for in-plane and inter-plane current directions find very low and nearly temperature-indep…
The half-Heusler compounds YPdBi and TbPdBi, while having similar band structure, exhibit different magnetic properties. YPdBi is a diamagnet, while TbPdBi shows antiferromagnetic order below 5.5 K. Both are superconductors with T${_c}\approx$1 K for YPdBi and T${_c}\approx$1.75 K for TbPdBi. Such a contrast in properties between these two compounds opens a question about the effects of band structure or magnetic correlations on superconductivity. Using the combination of a tunnel diode oscillator (TDO) and a commercial dilution refrigerator, we measured temperature dependent magnetic penetrat…
The London penetration depth is evaluated for isotropic materials for any transport and pair-breaking Born scattering rates. Besides known results, a number of new features are found. The slope $|dρ/dθ|$ of the normalized superfluid density $ρ=λ^2(0)/λ^2(θ)$ at the transition $θ=T/T_c=1$ has a minimum near the value of the pair-breaking parameter separating gapped and gapless states. The low-$T$ exponentially flat part of $ρ$ for the s-wave materials is suppressed by increasing pair breaking. For strong $T_c $ suppression by magnetic impurities the "Homes scaling" $λ^{-2}(0) \propto σT_c$ with…
We study the effect of Coulomb repulsion between oxygen holes on the London penetration depth $λ$ based on the concept of spin-polaron nature of Fermi quasiparticles in cuprates superconductors. It is shown that for the generally accepted values of the parameters of the spin-fermion model, taking into account the Coulomb interaction, both the one-site Hubbard $U_p$ and interaction between the holes on next-nearest-neighbor oxygen ions $V_2$, allows one to achieve a much better agreement of the calculated temperature dependencies of the value $λ^{-2}$ with the experimental data in La$_{2-x}$Sr$…
Magnetic susceptibility of non-ellipsoidal samples is a long-standing problem in experimental studies of magnetism and superconductivity. Here the quantitative description of the Meissner-London response (no Abrikosov vortices) of right circular cylinders in an axial magnetic field is given. The three-dimensional adaptive finite-element modeling was used to calculate the total magnetic moment, m, in a wide range of London penetration depth, λ, to sample size ratios. By fitting the numerical data, the closed-form universal magnetic susceptibility is formulated involving only sample dimensions a…
The optimum design of high-sensitivity Superconducting Quantum Interference Devices (SQUIDs) and other devices based on thin HTS films requires accurate inductance modeling. This needs the London penetration depth $λ$ to be well defined, not only at 77 K, but also for any operating temperature, given the increasingly widespread use of miniature low-noise single-stage cryocoolers. Temperature significantly affects all inductances in any active superconducting device and cooling below 77 K can greatly improve device performance, however accurate data for the temperature dependence of inductance …
The London penetration depth in single crystals of Ba(Fe_{1-x}Co_x)_2As_2 at various doping levels
用隧道二极管谐振器技术测量了Ba（Fe $_{1-x}$ Co $_{x}$）$_{2}$ As $_{2}$单晶的伦敦穿透深度$λ(T)$。测量的$x=$ ~0.038、0.047、0.058、0.074和0.10的掺杂水平的范围从欠掺杂到过掺杂浓度。测量结果表明，参与超导性的载流子密度急剧下降，在欠掺杂制度，但作为温度的函数的渗透深度表现出强大的幂律，$Δλ(T)\sim T^{n}$，所有测量的掺杂，与$n$之间的2和2.5。我们讨论了这些结果的影响和可能的解释，这种强大的行为。
The low temperature variation of the London penetration depth for a number of iron-pnictide and iron-chalcogenide superconductors is nearly quadratic, $Δλ(T) = βT^n$ with $n\approx 2$. The coefficient in this dependence shows a robust scaling, $β\propto 1/T_c^3$ across different families of these materials. We associate the scaling with a strong pair-breaking. The same mechanism have recently been suggested to explain the scalings of the specific heat jump, $ΔC \propto T_c^3$, and of the slopes of the upper critical field, $dH_{c2}/dT\propto T_c$ in these materials. This suggests that thermody…
We study the effect of disorder on the London penetration depth in iron-based superconductors. The theory is based on a two-band model with quasi-two-dimensional Fermi surfaces, which allows for the coexistence region in the phase diagram between magnetic and superconducting states in the presence of intraband and interband scattering. Within the quasiclassical approximation we derive and solve Eilenberger's equations, which include a weak external magnetic field, and provide analytical expressions for the penetration depth in the various limiting cases. A complete numerical analysis of the do…
The isotope coefficient $β$ of the magnetic penetration depth in the superconducting state is studied at T=0 for a $d$-CDW and a nodal metal model. Disregarding superconductivity the Fermi surface of the first model possesses arcs whereas the second model has no arcs. We show that a large increase of $β$ in the pseudogap region is generically incompatible with Fermi arcs in the pseudogap state. Thus only the second model shows a large increase of $β$ with decreasing doping. The required electron-phonon coupling is small and compatible with first-principles calculations based on the local densi…
We present a general theory of the singularity in the London penetration depth at symmetry-breaking and topological quantum critical points within a superconducting phase. While the critical exponents, and ratios of amplitudes on the two sides of the transition are universal, an overall sign depends upon the interplay between the critical theory and the underlying Fermi surface. We determine these features for critical points to spin density wave and nematic ordering, and for a topological transition between a superconductor with $\mathbb{Z}_2$ fractionalization and a conventional superconduct…
Irradiation with Pb ions was used to study the effect of disorder on the in-plane London penetration depth, $λ(T)$, in single crystals of Ba(Fe$_{1-x}$T$_x$)$_2$As$_2$ (T=Co, Ni). An increase of the irradiation dose results in a monotonic decrease of the superconducting transition temperature, $T_c$, without affecting much the transition width. In both Co and Ni doped systems we find a power-law behavior, $Δλ(T) \propto T^n$, with the exponent $n$ systematically decreasing with the increase of disorder. This observation, supported by the theoretical analysis, conclusively points to a nodeless …
Non-exponential London penetration depth in RFeAsO$_{0.9}$F$_{0.1}$ (R=La,Nd) single crystals
测量了RFeAsO $_{0.9}$ F $_{0.1}$（R=La，Nd）单晶（R-1111）的超导穿透深度$λ(T)$。在Nd-1111中，我们发现$λ(T)$在冷却时出现了一个上升，并将其归因于Nd离子的顺磁性，类似于电子掺杂的铜酸盐Nd-Ce-Cu-O的情况。经过顺磁性校正后，发现伦敦穿透深度变化遵循幂律行为，在低温下为$Δλ_L(T)\propto T^{2}$。在非磁性La-1111晶体中发现了与$λ(T)$相同的$T^2$变化。在整个温度范围内的超流密度和渗透深度各向异性的分析是一致的双间隙超导。在此基础上，结合我们以前的工作，我们得出结论：RFeAsO（1111）和BaFe $_2$ As $_2$（122）族磷属元素化物超导体都具有非常规的双能隙超导性。
The in-plane London penetration depth, $λ(T)$, was measured in single crystals of the iron-chalcogenide superconductors Fe$_{1.03}$(Te$_{0.63}$Se$_{0.37}$) and Fe$_{1.06}$(Te$_{0.88}$S$_{0.14}$) by using a radio-frequency tunnel diode resonator. As is also the case for the iron-pnictides, these iron-chalcogenides exhibit a nearly quadratic temperature variation of $λ(T)$ at low temperatures. The absolute value of the penetration depth in the $T \to 0$ limit was determined for Fe$_{1.03}$(Te$_{0.63}$Se$_{0.37})$ by using an Al coating technique, giving $λ(0)\approx560 \pm 20$ nm. The superfluid…
