Application of a tunnel-diode resonator (TDR) technique for studies of the vortex response in magnetic superconductors is described. Operating at very small excitation fields and sufficiently high frequency, TDR was used to probe small-amplitude linear AC response in several types of single crystals where long-range magnetic order coexists with bulk superconductivity. Full local - moment ferromagnetism destroys superconductivity and can coexist with it only in a narrow temperature range ($\sim 0.3$ K). In contrast, weak ferromagnetic as well as antiferromagnetic orders can coexist with bulk superconductivity and may even lead to enhancements of vortex pinning. By analyzing the Campbell penetration depth we find sharp increase of the true critical current in the vicinity of the magnetic phase transitions. We conclude that critical magnetic fluctuations are responsible for this enhancement.
The penetration of an $ac$ magnetic signal into a type II superconductor residing in the Shubnikov phase depends on the pinning properties of Abrikosov vortices. Within a phenomenological theory, the so-called Campbell penetration depth $λ_{\rm \scriptscriptstyle C}$ is determined by the curvature $α$ at the bottom of the effective pinning potential. Preparing the sample into a Bean critical state, this curvature vanishes and the Campbell length formally diverges. We make use of the microscopic expression for the pinning force density derived within strong pinning theory and show how flux pene…
The magnetic penetration depth $λ(T,H,j)$ was measured in the presence of a slowly relaxing supercurrent, $j$. In single crystal $\mathrm{Bi_2Sr_2CaCu_2O_8}$ below approximately 25 K, $λ(T,H,j)$ is strongly hysteretic. We propose that the irreversibility arises from a shift of the vortex position within its pinning well as $j$ changes. The Campbell length depends upon the ratio $j/j_{c}$ where $j_{c}$ is the critical current defined through the Labusch parameter. Similar effects were observed in other cuprates and in an organic superconductor.
Type-II superconductors owe their magnetic and transport properties to vortex pinning, the immobilization of flux quanta through material inhomogeneities or defects. Characterizing the potential energy landscape for vortices, the pinning landscape (or short, pinscape), is of great technological importance. Besides measurement of the critical current density $j_c$ and of creep rates $S$, the $ac$ magnetic response provides valuable information on the pinscape which is different from that obtained through $j_c$ or $S$, with the Campbell penetration depth $λ_{\rm \scriptscriptstyle C}$ defining a…
Campbell penetration depth in a single crystal of heavy fermion superconductor CeCoIn$_5$
Ames National Laboratory（艾姆斯国家实验室）；Iowa State University（爱荷华州立大学）；Missouri University of Science and Technology（密苏里科技大学）；Brookhaven National Laboratory（布鲁克海文国家实验室）；Shanghai Advanced Research in Physical Sciences (SHARPS)（上海物理科学研究院）；Vinca Institute of Nuclear Sciences, University of Belgrade（贝尔格莱德大学温卡核科学研究所）
Magnetic penetration depth, $λ_{m}$, was measured as a function of temperature and magnetic field in single crystals of low carrier density superconductor YPtBi by using a tunnel-diode oscillator technique. Measurements in zero DC magnetic field yield London penetration depth, $λ_{L}\left(T\right)$, but in the applied field the signal includes the Campbell penetration depth, $λ_{C}\left(T\right)$, which is the characteristic length of the attenuation of small excitation field, $H_{ac}$, into the Abrikosov vortex lattice due to its elasticity. Whereas the magnetic field dependent $λ_C$ exhibit …
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 …
A true critical current density, $j_{c}$, as opposite to commonly measured relaxed persistent (Bean) current, $j_{B}$, was extracted from the Campbell penetration depth, $λ_{C}(T,H)$ measured in single crystals of LiFeAs. The effective pinning potential is non-parabolic, which follows from the magnetic field - dependent Labusch parameter $α$. At the equilibrium (upon field - cooling), $α(H)$ is non-monotonic, but it is monotonic at a finite gradient of the vortex density. This behavior leads to a faster magnetic relaxation at the lower fields and provides a natural \emph{dynamic} explanation f…
The $AC$ magnetic penetration depth $λ(T,H,j)$ was measured in presence of a macroscopic $DC$ (Bean) supercurrent, $j$. In single crystal BSCCO below approximately 28 K, $λ(T,H,j)$ exhibits thermal hysteresis. The irreversibility arises from a shift of the vortex position within its pinning well as $j$ changes. It is demonstrated that below a new irreversibility temperature, the nonequilibrium Campbell length depends upon the ratio $j/j_c$. $λ(T,H,j)$ {\it increases} with $j/j_c$ as expected for a non-parabolic potential well whose curvature {\it decreases} with the displacement. Qualitatively…
Measuring the $ac$ magnetic response of a type II superconductor provides valuable information on the pinning landscape (pinscape) of the material. We use strong pinning theory to derive a microscopic expression for the Campbell length $λ_{\rm \scriptscriptstyle C}$, the penetration depth of the $ac$ signal. We show that $λ_{\rm \scriptscriptstyle C}$ is determined by the jump in the pinning force, in contrast to the critical current $j_c$ which involves the jump in pinning energy. We demonstrate that the Campbell lengths generically differ for zero-field-cooled and field-cooled samples and pr…
In type-II superconductors, the macroscopic response of vortex matter to an external perturbation depends on the local interaction of flux lines with the pinning landscape (pinscape). The (Campbell) penetration depth $λ_{\mathrm{\scriptscriptstyle C}}$ of an $ac$ field perturbation is often associated with a phenomenological pinning curvature. However, this basic approach is unable to capture thermal hysteresis effects observed in a variety of superconductors. The recently developed framework of strong-pinning theory has established a quantitative relationship between the microscopic pinscape …
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 in-plane magnetic penetration depth, $λ_m(T)$, was measured in single crystals of SrPd$_2$Ge$_2$ superconductor in a dilution refrigerator down to T=60 mK and in magnetic fields up to $H_{dc} = 1$ T by using a tunnel diode resonator. The London penetration depth, $λ$, saturates exponentially approaching $T\rightarrow 0$ indicating fully gapped superconductivity. The thermodynamic Rutgers formula was used to estimate $λ(0) = 426$ nm which was used to calculate the superfluid density, $ρ_s(T)=λ^2(0)/λ^2(T)$. Analysis of $ρ_s(T)$ in the full temperature range shows that it is best described b…
New modes of the magnetic flux penetration in SQUIDs with ferromagnetic junctions are predicted theoretically. Characteristic multinode anharmonicity of the current-phase relation implies the coexistence of stable and metastable 0 and $π$ states for a large interval of strength of the ferromagnetic barrier influence at low temperature. As a consequence, the coexistence of integer and half-integer fluxoid configurations appears in both rf and dc SQUIDs, and generates two flux jumps per one external flux quantum. In dc SQUIDs, new flux jumps are manifested as two dips in the critical current dep…
Applying the strong pinning formalism to the mixed state of a type II superconductor, we study the effect of thermal fluctuations (or creep) on the penetration of an ac magnetic field as quantified by the so-called Campbell length $λ_\textrm{C}$. Within strong pinning theory, vortices get pinned by individual defects, with the jumps in the pinning energy ($Δe_\mathrm{pin}$) and force ($Δf_\mathrm{pin}$) between bistable pinned and free states quantifying the pinning process. We find that the evolution of the Campbell length $λ_{\rm C}(t)$ as a function of time $t$ is the result of two competin…
Superconducting characteristics such as the Meissner-Ochsenfeld state, screening supercurrents and hysteresis loops of type-II superconductors were observed from the temperature and magnetic field dependences of the magnetic moment, m(T, H), in graphite powders reacted with sulfur for temperatures below 9.0 K. The temperature dependence of the lower critical field Hc1(T) was determined and the zero-temperature penetration depth, lambda(0), was estimated (lambda (0) = 227 nm). The superconductivity was observed to be highly anisotropic and to coexist with a ferromagnetic state that has a Curie …
We have investigated the doping dependence of the penetration depth vs. temperature in electron doped Pr$_{2-x}$Ce$_x$CuO$_{4-δ}$ using a model which assumes the uniform coexistence of (mean-field) antiferromagnetism and superconductivity. Despite the presence of a $d_{x^2-y^2}$ pairing gap in the underlying spectrum, we find nodeless behavior of the low-$T$ penetration depth in underdoped case, in accord with experimental results. As doping increases, a linear-in-$T$ behavior of the penetration depth, characteristic of d-wave pairing, emerges as the lower magnetic band crosses the Fermi level…
In this work we study thermodynamic manifestations of the quantum criticality in multiband unconventional superconductors. As a guiding example we consider the scenario of magnetic quantum critical point in the model that captures superconductivity coexistence with the spin-density wave. We show that in situations when the superconducting order parameter has incidental nodes at isolated points, quantum magnetic fluctuations lead to the renormalization of the relative $T$-linear slope of the London penetration depth. This leads to the nonmonotonic dependence of the penetration depth as a functi…
NiBi3 polycrystals were synthesized via a solid state method. X-ray diffraction analysis shows that the main phase present in the sample corresponds to NiBi3 in a weight fraction of 96.82 % according to the refinement of the crystalline structure. SEM - EDS and XPS analysis reveal a homogeneous composition of NiBi3, without Ni traces. The powder superconducting samples were studied by performing magnetic measurements. The superconducting transition temperature and critical magnetic fields were determinated as Tc = 4.05 K, Hc1 = 110 Oe and Hc2 = 3,620 Oe. The superconducting parameters were GL …
Muon-spin rotation experiments were performed on the pnictide high temperature superconductor SmFeAsO$_{1-x}$F$_{x}$ with x=0.18 and 0.3. We observed an unusual enhancement of slow spin fluctuations in the vicinity of the superconducting which suggests that the spin fluctuations contribute to the formation of an unconventional superconducting state. An estimate of the in-plane penetration depth $λ_{ab}(0)=190(5)$\thinspace nm was obtained, which confirms that the pnictide superconductors obey an Uemura-style relationship between $T_{\mathrm{c}}$ and $λ_{ab}(0)^{-2}$.
Superconductors usually display either type-I or type-II superconductivity and the coexistence of these two types in the same material, for example at different temperatures is rare in nature. We the employed muon spin rotation (muSR) technique to unveil the superconducting phase diagram of the dodecaboride ZrB12 and obtained clear evidence of both type-I and type-II characteristics. Most importantly, we found a region showing unusual behavior where the usually mutually exclusive muSR signatures of type-I and type-II superconductivity coexist. We reproduced that behavior in theoretical modelin…
We present a detailed investigation of the magnetic and superconducting properties of Ca1-xNaxFe2As2 single crystals with x = 0.00, 0.35, 0.50, and 0.67 by means of the local probe techniques Moessbauer spectroscopy and muon spin relaxation experiments. With increasing Na substitution level, the magnetic order parameter as well as the magneto-structural phase transition are suppressed. For x = 0.50 we find a microscopic coexistence of magnetic and superconducting phases accompanied by a reduction of the magnetic order parameter below the superconducting transition temperature Tc. A systematic …
We report a clear anomaly in the rf response of a tunnel-diode resonator measured in borocarbide crystals in dc fields along the $c-$axis. We associate the anomaly in the dynamic magnetic susceptibility, $χ$, with a structural transition in the vortex lattice. $χ$ is sensitive to the Campbell penetration depth $λ_{C}$ of the rf perturbation into the mixed state. $λ_{C}$ depends on the vortex lattice elastic moduli, which in turn depend on the vortex lattice structure. The high-field transition line is clearly mapped and shown to behave in agreement with theory.
This paper has been withdrawn by the authors. We performed additional zero-field muon spin relaxation measurements in the superconducting state of CeIrIn$_5$ and found that the spontaneous fields reported previously below $T_c$ are not present. Thus, there is no evidence for a time-reversal-symmetry-violating superconducting order parameter. These new zero-field measurements, as well as new measurements of the penetration depth in this system, will be reported elsewhere. Our zero-field measurements in CeIr$_{0.5}$Rh$_{0.5}$In$_5$, reporting coexistence of superconductivity and magnetic order, …
The coexistence of charge density wave (CDW) and superconductivity in tantalum disulfide (2H-TaS$_2$) at ambient pressure, is boosted by applying hydrostatic pressures up to 30GPa, thereby inducing a typical dome-shaped superconducting phase. The ambient pressure CDW ground state which begins at TCDW = 76 K, with critically small Fermi surfaces, was found to be fully suppressed at Pc = 8.7GPa. Around Pc, we observe a superconducting dome with a maximum superconducting transition temperature Tc = 9.1 K. First-principles calculations of the electronic structure predict that, under ambient condit…
Irradiation with 2.5 MeV electrons at doses up to 5.2 $\times$10$^{19}$ electrons/cm$^2$ was used to introduce point-like defects in single crystals of Ba$_{1-x}$K$_x$Fe$_2$As$_2$ with $x=$ 0.19 ($T_c=$ 14 K), $x=$ 0.26 ($T_c=$ 32 K) and 0.34 ($T_c=$ 39 K) to study the superconducting gap structure by probing the effect of non-magnetic scattering on electrical resistivity, $ρ(T)$, and London penetration depth, $ λ(T)$. For all compositions, the irradiation suppressed the superconducting transition temperature, $T_c$ and increased resistivity. The low - temperature behavior of $λ(T)$ is best de…
The coexistence of weak ferromagnetism and superconductivity in ErNi$_{2}$B$% _{2}$C suggests the possibility of a spontaneous vortex phase (SVP) in which vortices appear in the absence of an external field. We report evidence for the long-sought SVP from the in-plane magnetic penetration depth $Δλ(T)$ of high-quality single crystals of ErNi$_{2}$B$_{2}$C. In addition to expected features at the Néel temperature $T_{N}$ = 6.0 K and weak ferromagnetic onset at $T_{WFM}=2.3 $K, $Δλ(T)$ rises to a maximum at $T_{m}=0.45$ K before dropping sharply down to $\sim $0.1 K. We assign the 0.45 K-maximum…
We explore the dynamic response of vortex lines in a Pb thin film with a periodic array of antidots by means of ac-susceptibility measurements. For low drive field amplitudes, within the Campbell regime, vortex motion is of intra-valley type and the penetration depth is related to the curvature of the pinning potential well, $α$. For dc-fields below the first matching field $H_1$, $α$ reaches its highest value associated with a Mott Insulator-like phase where vortex lines are strongly localized at the pinning sites. For $H_1<H_{dc}<H_2$, the response is mainly due to the interstitial vortices …
We report local probe ($μ$SR) measurements on the recently discovered tetragonal FeS superconductor which has been predicted to be electronically very similar to superconducting FeSe. Most remarkably, we find that low moment ($10^{-2}-10^{-3}μ_\mathrm{B})$ disordered magnetism with a transition temperature of $T_\mathrm{N}\approx 20$ K microscopically coexists with bulk superconductivity below $T_\mathrm{c}=4.3(1)$ K. From transverse field $μ$SR we obtain an in-plane penetration depth $λ_\mathrm{ab}(0)=223(2)$ nm for FeS. The temperature dependence of the corresponding superfluid density $λ_\m…
The influence of the geometry of a thin superconducting sample on the penetration of the magnetic field lines and the arrangement of vortices are investigated theoretically. We compare superconducting disks, squares and triangles with the same surface area having nonzero thickness. The coupled nonlinear Ginzburg-Landau equations are solved self-consistently and the important demagnetization effects are taken into account. We calculate and compare quantities like the free energy, the magnetization, the Cooper-pair density, the magnetic field distribution and the superconducting current density …
Abrikosov vortices in altermagnetic superconductors
Centro de Física de Materiales (CFM-MPC) Centro Mixto CSIC-UPV/EHU（材料物理中心（CFM-MPC）CSIC-巴斯克国立大学混合中心）；Donostia International Physics Center (DIPC)（圣塞巴斯蒂安国际物理中心）
The characteristic features of Ca$_{10}$(Pt$_3$As$_8$)[(Fe$_{1-x}$Pt$_x$)$_2$As$_2$]$_5$ ("10-3-8") superconductor are relatively high anisotropy and a clear separation of superconductivity and structural/magnetic transitions, which allows studying the superconducting gap without complications due to the coexisting order parameters. The London penetration depth, measured in underdoped single crystals of 10-3-8 ($x =$ 0.028, 0.041, 0.042, and 0.097), shows behavior remarkably similar to other Fe-based superconductors, exhibiting robust power-law, $Δλ(T) = A T^n$. The exponent $n$ decreases from…
We have synthesized the polycrystalline samples of Eu1-xKxFe2As2 (x = 0-1) and carried out systematic characterization using x-ray diffraction, ac & dc magnetic susceptibility, and electrical resistivity measurements. We have seen a clear signature of the coexistence of superconducting transition (Tc = 5.5 K) with SDW ordering in our under doped sample viz. x = 0.15. The spin density wave transition observed in EuFe2As2 get completely suppressed at x = 0.3 and superconductivity arises below 20 K. Superconducting transition temperature Tc increases with increase in K content and a maximum Tc = …
