In the previous four decades, a two-coil mutual inductance (MI) technique has been widely employed in characterizing magnetic penetration depth, $λ$, of superconducting films. However, the conventional methods used to obtain $λ$ are not applicable to small-sized films with common shapes, which limits the application of the MI technique in superconductivity research. Here, we first employed the fast wavelet collocation (FWC) method to a two-coil system and then proposed the possibility of directly obtaining the absolute $λ$ of polygonal superconducting films with arbitrary sizes. To verify its accuracy and advantages, we extracted the $λ$ values of square NbN films with different sizes using the FWC and conventional flux leakage subtraction (FLS) methods. Notably, the FLS method fails for a $5\times 5 \, \rm mm^2$ film, which is attributed to the significant current peak at the film edge. In contrast, the absolute $λ$ extracted using the FWC method was independent of the film size. Finally, we established the applicability of the FWC method to large coil spacings, which may pave the way for integrating high-accuracy $λ$ measurements with the ionic liquid gating technique.
A method is presented to measure the absolute value of the London penetration depth, $λ$, from the frequency shift of a resonator. The technique involves coating a high-$T_c$ superconductor (HTSC) with film of low - Tc material of known thickness and penetration depth. The method is applied to measure London penetration depth in YBa2Cu3O{7-δ} (YBCO) Bi2Sr2CaCu2O{8+δ} (BSCCO) and Pr{1.85}Ce{0.15}CuO{4-δ}$ (PCCO). For YBCO and BSCCO, the values of $λ(0)$ are in agreement with the literature values. For PCCO $λ\approx 2790$ Å, reported for the first time.
We present an experimental approach using magnetic force microscopy for measurements of the absolute value of the magnetic penetration depth $λ$ in superconductors. $λ$ is obtained in a simple and robust way without introducing any tip modeling procedure via direct comparison of the Meissner response curves for a material of interest to those measured on a reference sample. Using a well-characterized Nb film as a reference, we determine the absolute value of $λ$ in a Ba(Fe$_{0.92}$Co$_{0.08}$)$_{2}$As$_{2}$ single crystal and a MgB$_2$ thin film through a comparative experiment. Our apparatus …
Absolute measurement of penetration depth of superconducting thin films using microwave stripline resonators
Tata Institute of Fundamental Research（塔塔基础研究所）；School of Physical Sciences, UM-DAE Centre for Excellence in Basic Sciences, University of Mumbai（孟买大学物理科学学院，UM-DAE基础科学卓越中心）
Zero-field electron spin resonance (ESR) of dilute Gd ions substituted for Y in the cuprate superconductor YBa$_2$Cu$_3$O$_{\rm 6+y}$ is used as a novel technique for measuring the absolute value of the low temperature magnetic penetration depth $λ(T\to 0)$. The Gd ESR spectrum of samples with $\approx 1%$ substitution was obtained with a broadband microwave technique that measures power absorption bolometrically from 0.5 GHz to 21 GHz. This ESR spectrum is determined by the crystal field that lifts the level degeneracy of the spin 7/2 Gd$^{3+}$ ion and details of this spectrum provide informa…
We present a local probe study of the magnetic superconductor, ErNi$_2$B$_2$C, using magnetic force microscopy at sub-Kelvin temperatures. ErNi$_2$B$_2$C is an ideal system to explore the effects of concomitant superconductivity and ferromagnetism. At 500 mK, far below the transition to a weakly ferromagnetic state, we directly observe a structured magnetic background on the micrometer scale. We determine spatially resolved absolute values of the magnetic penetration depth $λ$ and study its temperature dependence as the system undergoes magnetic phase transitions from paramagnetic to antiferro…
We report magnetic force microscopy (MFM) measurements on the iron-based superconductor $\mathrm{Ba_{1-x}K_xFe_2As_2}$. By measuring locally the Meissner repulsion with the magnetic MFM tip, we determine the absolute value of the in-plane magnetic penetration depth ($λ_{ab}$) in underdoped, optimally-doped, and overdoped samples. The results suggest an abrupt increase of $λ_{ab}$ as doping is increased from $x_\mathrm{opt}$, which is potentially related to the presence of a quantum critical point. The response of superconducting vortices to magnetic forces exerted by the MFM tip for $x=0.19$ a…
The first penetration field H_{J}(T) of Josephson vortices is measured through the onset of microwave absorption in the locked state, in slightly overdoped $\rm{Bi_{2}Sr_{2}CaCu_{2}O_{8+δ}}$ single crystals (T_{c} ~ 84 K). The magnitude of H_{J}(T) is too large to be accounted for by the first thermodynamic critical field H_{c1}(T). We discuss the possibility of a Bean-Livingston barrier, also supported by irreversible behavior upon flux exit, and the role of defects, which relates H_{J}(T) to the c-axis penetration depth $λ_{c}(T)$. The temperature dependence of the latter, determined by a ca…
The zero temperature value of the in-plane London penetration depth, $λ_{ab}(0)$, has been measured in single crystals of Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$ as a function of the Co concentration, $x$, across both the underdoped and overdoped superconducting regions of the phase diagram. For $x\gtrsim0.047$, $λ_{ab}(0)$ has been found to have values between 120 $\pm$ 50~nm and 300 $\pm$ 50~nm. A pronounced increase in $λ_{ab}(0)$, to a value as high as 950 $\pm$ 50~nm, has been observed for $x\lesssim0.047$, corresponding to the region of the phase diagram where the itinerant antiferromagnetic and …
We present high precision measurements of the penetration depth λof single crystals of κ-(ET)_2Cu[N(CN)2]Br and κ-(ET)_2Cu(NCS)_2 at temperature down to 0.4 K. We find that, at low temperatures, the in-plane penetration depth varies as a fractional power law, λ\sim T^1.5. W hilst this may be taken as evidence for novel bose excitation processes, we show that the data are also consistent with a quasi-linear variation of the superfluid density, as is expected for a d-wave superconductor with impurities. Our data for the interplane penetration depth show similar features and give a direct measure…
We use a magnetic force microscope (MFM) to investigate single vortex pinning and penetration depth in NdFeAsO$_{1-x}$F$_x$, one of the highest-$T_c$ iron-based superconductors. In fields up to 20 Gauss, we observe a disordered vortex arrangement, implying that the pinning forces are stronger than the vortex-vortex interactions. We measure the typical force to depin a single vortex, $F_{\mathrm{depin}} \simeq 4.5$ pN, corresponding to a critical current up to $J_c \simeq 7 \times 10^5$ A/cm$^2$. Furthermore, our MFM measurements allow the first local and absolute determination of the supercond…
The magnetic penetration depth $λ$ has been measured in MgCNi$_{3}$ single crystals using both a high precision Tunnel Diode Oscillator technique (TDO) and Hall probe magnetization (HPM). In striking contrast to previous measurements in powders, $δλ$(T) deduced from TDO measurements increases exponentially at low temperature, clearly showing that the superconducting gap is fully open over the whole Fermi surface. An absolute value at zero temperature $λ(0)=230 $nm is found from the lower critical field measured by HPM. We also discuss the observed difference of the superfluid density deduced f…
Muon spin rotation (mSR) measurements of the in-plane penetration depth lambda_ab have been performed in the electron-doped infinite layer high-Tc superconductor (HTS) Sr0.9La0.1CuO2. Absence of the magnetic rare-earth ions in this compound allowed to measure for the first time the absolute value of lambda_ab(0) in electron-doped HTS using mSR. We found lambda_ab(0)=116(2) nm. The zero-temperature depolarization rate sigma(0)?1/lambda_ab(0)^2=4.6(1) MHz is more than four times higher than expected from the Uemura line. Therefore this electron-doped HTS does not follow the Uemura relation found…
We report the basal-plane critical current and superfluid density of magnetically aligned NdFeAsO$_{0.86}$F$_{0.14}$ powder. This sample has individual crystallite grains permanently oriented with their c axis along the external field. Magnetic irreversibilities at high field suggest strong flux pinning of basal-plane critical currents, with monotonic field dependence and no evidence of the "fishtail" effect. The small particles provide a sensitive indicator of \textit{dc} flux penetration, and allow analysis of the temperature dependence of $ab-$plane London penetration depth $λ_{ab,\mathrm{L…
We use magnetic force microscopy (MFM) and scanning SQUID susceptometry to measure the local superfluid density $ρ_{s}$ in Ba(Fe$_{0.95}$Co$_{0.05}$)$_2$As$_2$ single crystals from 0.4 K to the critical temperature $T_c=18.5$ K. We observe that the penetration depth $λ$ varies about ten times more slowly with temperature than previously published, with a dependence that can be well described by a clean two-band fully gapped model. We demonstrate that MFM can measure the important and hard-to-determine absolute value of $λ$, as well as obtain its temperature dependence and spatial homogeneity. …
We have studied the penetration depth for a model of $YBa_{2}Cu_{3}O_{7}$ involving pairing both in the $CuO_{2}$ planes and in the CuO chains. In this model pairing in the planes is due to an attractive interaction, while Coulomb repulsion induces in the chains an order parameter with opposite sign. Due to the anticrossing produced by hybridization between planes and chains, one obtains a d-wave like order parameter which changes sign on a single sheet of the Fermi surface and has nodes in the gap. We find that our model accounts quite well for the anisotropy of the penetration depth and for …
Single crystals of MgCNi$_3$, with areas sized up to 1 mm$^{2}$, were grown by the self flux method using a cubic anvil high pressure technique. The first critical field \textit{H$_{c1}$}, determined from a zero temperature extrapolation, is around 18 mT. Using the tunnel - diode resonator technique, the London penetration depth was measured with no applied \textit{dc} field and the Campbell penetration depth was measured with the external \textit{dc} fields up to 9T for two different sample orientations with respect to the direction of applied magnetic field. The absolute value of the London …
The absolute values of the conductivity in the normal state sigma_n and of the low temperature penetration depths lambda(0) were measured for a number of different samples of the YBaCuO family. We found a striking correlation between sigma_n and 1/lambda^2, regardless of doping, oxygen reduction or defects, thus providing a simple method to predict the superconducting penetration depth and to have an estimate of the sample quality by measuring the normal-state conductivity.
Measurements of the magnetic field penetration depth λin superconductor Mo_3Sb_7 (T_c~2.1 K) were carried out by means of muon-spin-rotation. The absolute values of λ, the Ginzburg-Landau parameter κ, the first H_{c1} and the second H_{c2} critical fields at T=0 are λ(0)=720(100)nm, κ(0)=55(9), μ_0H_{c1}(0)=1.8(3)mT, and μ_0H_{c2}(0)=1.9(2)T. The zero temperature value of the superconducting energy gap Δ(0) was found to be 0.35(1)meV corresponding to the ratio 2Δ(0)/k_BT_c=3.83(10). At low temperatures λ^{-2}(T) saturates and becomes constant below T~0.3T_c, in agreement with what is expected …
We have measured the temperature dependence of the absolute value of the magnetic penetration depth $λ(T)$ in a Ca$_{10}$(Pt$_{3}$As$_{8}$)[(Fe$_{1-x}$Pt$_{x}$)$_{2}$As$_{2}$]$_{5}$ (x=0.097) single crystal using a low-temperature magnetic force microscope (MFM). We obtain $λ_{ab}$(0)$\approx$1000 nm via extrapolating the data to $T = 0$. This large $λ$ and pronounced anisotropy in this system are responsible for large thermal fluctuations and the presence of a liquid vortex phase in this low-temperature superconductor with critical temperature of 11 K, consistent with the interpretation of th…
Measurements of the magnetic field penetration depth $λ$ in the ternary boride superconductor Li$_2$Pd$_3$B ($T_c\simeq7.3$ K) have been carried out by means of muon-spin rotation ($μ$SR). The absolute values of $λ$, the Ginzburg-Landau parameter $κ$, and the first $H_{c1}$ and the second $H_{c2}$ critical fields at T=0 obtained from $μ$SR were found to be $λ(0)=252(2)$ nm, $κ(0)=27(1)$, $μ_0H_{c1}(0)=9.5(1)$ mT, and $μ_0H_{c2}(0)=3.66(8)$ T, respectively. The zero-temperature value of the superconducting gap $Δ_0=$1.31(3) meV was found, corresponding to the ratio $2Δ_0/k_BT_c=4.0(1)$. At low …
The presence of magnetic fields and/or transport currents can cause penetration of vortices in superconductors. Their motion leads to dissipation and resistive state arises, which in turn strongly affects the performance of superconducting devices such as single-photon and single-electron detectors. Therefore, an understanding of the dissipation mechanisms in mesoscopic superconductors is not only of fundamental value but also very important for further technological advances. In the present work, we analyzed the contributions and interplay of the dissipative mechanisms due to the locally indu…
Measurement of the penetration depth as a function of temperature using millimeter wave transmission in the range 130-500GHz are reported for three YBCO thin films. The experiment provides the absolute value for the penetration depth at low temperature: the derivation from the transmission data and the experimental uncertainty are discussed. We find a zero temperature penetration depth of 199+/-20nm, 218+/-20nm and 218+/-20nm, for YBCO-50nm/LaAlO3 (pristine), YBCO-130nm/MgO and YBCO-50nm/LaAlO3 (irradiated) respectively. The penetration depth exhibits a different behavior for the three films. …
We have measured the temperature dependence of the microwave surface impedance Z_s = R_s + iΩμ_0 of a MgB2 film at a frequency Ω/2Πof 18 GHz employing a dielectric resonator technique. We found that the temperature dependence of the magnetic field penetration depth \Lamda can be fitted by \Lamda(T)= \Lamda(0) [1-(T/Tc)^2] ^(-1/2) with \Lamda(0)=(260 +/- 20)nm. The absolute value of \Lamda(0) was confirmed by direct measurements employing submillimeter wave transmission spectroscopy at 430 GHz. The analysis of the \Lamda(T) data below Tc/2 revealed significant deviations from the quadratic temp…
A study of magnetic flux penetration in a superconducting film patterned with arrays of micron sized antidots (microholes) is reported. Magneto-optical imaging (MOI) of a YBCO film shaped as a long strip with perpendicular antidot arrays revealed both strong guidance of flux, and at the same time large perturbations of the overall flux penetration and flow of current. These results are compared with a numerical flux creep simulation of a thin superconductor with the same antidot pattern. To perform calculations on such a complex geometry, an efficient numerical scheme for handling the boundary…
Magnetization and muon spin relaxation or rotation (muSR) measurements have been performed to study the superconducting and magnetic properties of Sr3Ir4Sn13. From magnetization measurements the lower and upper critical fields of Sr3Ir4Sn13 are found to be 81(1) Oe and 14.4(2) kOe, respectively. Zero-field muSR data show no sign of any magnetic ordering or weak magnetism in Sr3Ir4Sn13. Transverse-field muSR measurements in the vortex state provided the temperature dependence of the magnetic penetration depth. The dependence of penetration depth with temperature is consistent with the existence…
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…
Measurements of the in-plane magnetic field penetration depth λ_{ab} in Fe-based superconductors with the nominal composition SmFeAsO_0.85 (T_c\simeq52K) and NdFeAsO_0.85 (T_c\simeq51K) were carried out by means of muon-spin-rotation. The absolute values of λ_{ab} at T=0 were found to be 189(5)nm and 195(5)nm for Sm and Nd substituted samples, respectively. The analysis of the magnetic penetration depth data within the Uemura classification scheme, which considers the correlation between the superconducting transition temperature T_c and the effective Fermi temperature T_F, reveal that both fa…
The renormalization of the magnetic field penetration depth $λ$ owing to the electron-electron correlation is discussed with its application to high-$T_{\rm c}$ cuprates. The formula for the current carried by quasiparticle with the Umklapp scattering is derived, on the basis of which we investigate how the value of $λ^{-2}$ deviates from that of $n/m^*$ where $n$ and $m^*$ are the carrier density and the effective mass respectively. Although this deviation is small in the case of weak momentum dependence of the vertex, this is large and negative owing to the non-negligible value of the backfl…
We report combined experimental and theoretical analysis of superconductivity in CaK(Fe$_{1-x}$Ni$_x$)$_4$As$_4$ (CaK1144) for $x=$0, 0.017 and 0.034. To obtain the superfluid density, $ρ=\left(1+Δλ_L(T)/λ_L(0) \right)^{-2}$, the temperature dependence of the London penetration depth, $Δλ_L (T)$, was measured by using tunnel-diode resonator (TDR) and the results agreed with the microwave coplanar resonator (MWR) with the small differences accounted for by considering a three orders of magnitude higher frequency of MWR. The absolute value of $λ_L (T \ll T_c) \approx λ_L(0)$ was measured by usin…
We have measured the penetration depth of overdoped Y0.9Ca0.1Ba2Cu3O7-delta (Ca-YBCO) thin films using two different methods. The change of the penetration depth as a function of temperature has been measured using the parallel plate resonator (PPR), while its absolute value was obtained from a quasi-optical transmission measurements. Both sets of measurements are compatible with an order parameter of the form: Delta*dx2-y2+i*delta*dxy, with Delta=14.5 +- 1.5 meV and delta=1.8 meV, indicating a finite gap at low temperature. Below 15 K the drop of the scattering rate of uncondensed carriers be…
Images of flux vortices in superconductors acquired by transmission electron microscopy should allow a quantitative determination of their magnetic structure but so far, only visual comparisons have been made between experimental images and simulations. Here, we make a quantitative comparison between Fresnel images and simulations based on the modified London equation to investigate the magnetic structure of flux vortices in MgB2. This technique gives an absolute, low-field (~30 Oe) measurement of the penetration depth from images of single vortices. We found that these simulations gave a good…
We present a detailed analysis of two chip-based superconducting trap architectures capable of levitating micrometer-sized superconducting particles in the Meissner state. These architectures are suitable for performing novel quantum experiments with more massive particles or for force and acceleration sensors of unprecedented sensitivity. We focus in our work on a chip-based anti-Helmholtz coil-type trap (AHC) and a planar double-loop (DLP) trap. We demonstrate their fabrication from superconducting Nb films and the fabrication of superconducting particles from Nb or Pb. We apply finite eleme…
Temperature-dependent London penetration depth, $λ(T)$, was measured in optimally - doped, $x=$0.35, as-grown ($T_c \approx$25 K, RRR=$ρ(300K)/ρ(T_c)$=4.5) and annealed ($T_c \approx$35 K, RRR=6.4) single crystals of SrFe$_2$(As$_{1-x}$P$_{x}$)$_2$ iron - based superconductor. Annealing increases the RRR and decreases the absolute value of the London penetration depth from $λ(0) = 300 \pm 10$ nm in as-grown sample to $λ(0) = 275 \pm 10$ nm. At low temperatures, $λ(T) \sim T$ indicating superconducting gap with line nodes. Analysis of the full - temperature range superfluid density is consisten…
We investigated the complex conductivity spectrum of a Co-doped BaFe$_2$As$_2$ epitaxial thin film in the THz region. In the normal state, the complex conductivity shows a Drude-type frequency dependence, while in the superconducting state, the frequency dependence of the complex conductivity changes to that of a typical superconducting materials. We estimated the magnetic penetration depth at absolute zero to be 710 nm and the superconducting gap energy to be 2.8 meV, which is considered to be the superconducting gap opened at the electron-type Fermi surface near the M point. We succeeded in …
The absolute values and temperature, T, dependence of the in-plane magnetic penetration depth, $λ_{ab}$, of La_{2-x}Sr_xCuO_4 and HgBa_2CuO_{4+δ} have been measured as a function of carrier concentration. We find that the superfluid density, $ρ_s$, changes substantially and systematically with doping. The values of $ρ_{s}(0)$ are closely linked to the available low energy spectral weight as determined by the electronic entropy just above T_c and the initial slope of $ρ_{s}(T)/ρ_{s}(0)$ increases rapidly with carrier concentration. The results are discussed in the context of a possible relation…
Recent experiments showed that thinning gallium, iron selenide and 2H tantalum disulfide to single/several monoatomic layer(s) enhances their superconducting critical temperatures. Here, we characterize these superconductors by extracting the absolute values of the London penetration depth, the superconducting energy gap, and the relative jump in specific heat at the transition temperature from their self-field critical currents. Our central finding is that the enhancement in transition temperature for these materials arises from the opening of an additional superconducting gap, while retainin…
In this short review, we attempt to give a comprehensive discussion of studies performed to date by muon-spin spectroscopy (more precisely the relaxation and rotation technique, also know as \muSR technique) on the recently discovered layered iron-based superconductors. On one side, \muSR has been used to characterized the magnetic state of different families of layered iron-based systems. Similarly the subtle interplay of the magnetic state and the structural transition present in some families has been investigated. We will also discuss the information provided by this technique on the inter…
We have carried out non-linear microwave (8 GHz) surface impedance measurements of three YBaCuO thin films in dc magnetic fields $H_{dc}$ (parallel to c axis) up to 12 mT using a coplanar resonator technique. In zero dc field the three films, deposited by the same method, show a spread of low-power residual surface resistance, $R_{res}$ and penetration depth, $λ$ (T=15 K) within a factor of 1.9. However, they exhibit dramatically different microwave field, $H_{rf}$ dependences of the surface resistance, $R_s$, but universal $X_s(H_{rf})$ dependence. Application of a dc field was found to affec…
We report a $μ$SR investigation of a non-centrosymmetric superconductor (LaNiC$_2$) in single crystal form. Compared to previous $μ$SR studies of non-centrosymmetric superconducting polycrystalline and powder samples, the unambiguous orientation of single crystals enables a simultaneous determination of the absolute value of the magnetic penetration depth and the vortex core size from measurements that probe the magnetic field distribution in the vortex state. The magnetic field dependence of these quantities unambiguously demonstrates the presence of two nodeless superconducting energy gaps. …
We provide a detailed account for a new method to measure superconducting stiffness $ρ_{s}$, critical current density $j_c$, and coherence length $ξ$, in one apparatus, without subjecting the sample to magnetic field or attaching leads. The method is based on the London equation $\mathbf{j}=-ρ_{s}\mathbf{A}$, where ${\bf j}$ is the current density and ${\bf A}$ is the vector potential. Using a rotor free $\bf{A}$ and a measurement of $\bf{j}$ via the magnetic moment of a superconducting ring, we determine $ρ_{s}$. By increasing $\mathbf{A}$ until the London equation fails we determine $j_c$ an…
We have investigated the behaviour of the depairing current J_{dp} in ferromagnet/superconductor/ferromagnet (F/S/F) trilayers as function of the thickness d_s of the superconducting layers. Theoretically, J_{dp} depends on the superconducting order parameter or the pair density function, which is not homogeneous across the film due to the proximity effect. We use a proximity effect model with two parameters (proximity strength and interface transparency), which can also describe the dependence of the superconducting transition temperature T_c on d_s. We compare the computations with the exper…
