In-plane Penetration Depth Research Articles

In-Plane Magnetic Penetration Depth in Sr_{2}RuO_{4}: Muon-Spin Rotation and Relaxation Study.

We report on measurements of the in-plane magnetic penetration depth (λ_{ab}) in single crystals of Sr_{2}RuO_{4} down to ≃0.015 K by means of muon-spin rotation-relaxation. The linear temperature dependence of λ_{ab}^{-2} for T≲0.7 K suggests the presence of nodes in the superconducting gap. This statement is further substantiated by observation of the Volovik effect, i.e., the reduction of λ_{ab}^{-2} as a function of the applied magnetic field. The experimental zero-field and zero-temperature value of λ_{ab}=124(3) nm agrees with λ_{ab}≃130 nm, calculated based on results of electronic structure measurements reported in A. Tamai et al. [High-resolution photoemission on Sr_{2}RuO_{4} reveals correlation-enhanced effective spin-orbit coupling and dominantly local self-energies, Phys. Rev. X 9, 021048 (2019)PRXHAE2160-330810.1103/PhysRevX.9.021048]. Our analysis reveals that a simple nodal superconducting energy gap, described by the lowest possible harmonic of a gap function, does not capture the dependence of λ_{ab}^{-2} on T, so the higher angular harmonics of the energy gap function need to be introduced.

Oxygen isotope effect on the superfluid density within the [formula omitted]-wave and [formula omitted]-wave pairing channels of YBa[formula omitted]Cu[formula omitted]O[formula omitted

We report on measurements of the oxygen isotope (16O/18O) effect (OIE) on the transition temperature Tc and the zero-temperature in-plane magnetic penetration depth λab(0) in the stoichiometric cuprate superconductor YBa2Cu4O8 by means of muon-spin rotation/relaxation. An analysis of the temperature evolution of λab−2 in terms of coexisting isotropic s-wave and anisotropic d-wave order parameters (s+d-wave) reveals that the OIE on the superfluid density ρs(0)∝λab−2(0) stems predominantly from the d-wave component while the contribution of the s-wave one is almost zero. The OIE on the transition temperature Tc is found to be rather small: δTc/Tc=−0.32(7)%, compared to the total OIE on the superfluid density ρs(0): δρs(0)/ρs(0)=−2.8(1.0)%. The partial OIE’s on the corresponding d-wave and s-wave components of ρs(0) are δρs,d(0)/ρs(0)=−3.0(1.2)%, and δρs,s(0)/ρs(0)=0.2(1.2)%, respectively. Our results demonstrate that polaron formation in the CuO2 planes is the origin of the observed OIE in the d-wave channel. In the much weaker s-wave channel, fermionic quasiparticles are present, which do not contribute to the OIE on ρs(0). Our results support the original idea of K. Alex Müller on the polaronic nature of the supercarriers in high-temperature cuprate superconductors.

Distorted superconducting nodal line on a single Fermi surface in the anisotropic organic superconductor λ−(BETS)2GaCl4

The superconducting (SC) gap structure appearing in systems with single Fermi surface (FS) is generally described by the single gap function with $s$- or $d$-wave symmetry. The organic superconductor $\ensuremath{\lambda}$-(BETS)${}_{2}{\mathrm{GaCl}}_{4}$ [BETS = (${\mathrm{CH}}_{2}{)}_{2}{\mathrm{S}}_{2}{\mathrm{Se}}_{2}{\mathrm{C}}_{6}{\mathrm{Se}}_{2}{\mathrm{S}}_{2}$(${\mathrm{CH}}_{2}{)}_{2}$] endeavors to examine a novel SC gap structure on the distorted single cylindrical FS. Here, we show an example of the formation of the distorted SC nodal line by using the positive muon spin rotation (${\ensuremath{\mu}}^{+}\mathrm{SR}$) spectroscopy on $\ensuremath{\lambda}$-(BETS)${}_{2}{\mathrm{GaCl}}_{4}$. Our analysis method of the ${\ensuremath{\mu}}^{+}\mathrm{SR}$ data reveals that the nodal line has a narrower width than that of the traditional $d$-wave by the steepness factor of 4.6(2.1), and a flat part with the maximum gap exists. We found that the amplitude of the SC gap is $2\mathrm{\ensuremath{\Delta}}/{k}_{\text{B}}{T}_{c}$ = 3.9(2) and the in-plane penetration depth is ${\ensuremath{\lambda}}_{\text{ac}}$(0) = 560(5) nm. Our present study gives insight into the relation of the FS distortion and the unusual Cooper pair formation mediated by the anisotropic spin fluctuations.

Unconventional superconductivity in CuxBi2Se3 from magnetic susceptibility and electrical transport

Although the Cu doped Bi2Se3 topological insulator was discovered and intensively studied for almost a decade, its electrical and magnetic properties in normal state, and the mechanism of ‘high-Tc’ superconductivity regarding the relatively low-carrier density are still not addressed yet. In this work, we report a systematic investigation of magnetic susceptibility, critical fields, and electrical transport on the nominal Cu0.20Bi2Se3 single crystals with = 4.18 K, the highest so far. The composition analysis yields the Cu stoichiometry of x = 0.09(1). The magnetic susceptibility shows considerable anisotropy and an obvious kink at around 96 K was observed in the magnetic susceptibility for H∥c, which indicates a charge density anomaly. The electrical transport measurements indicate the two-dimensional (2D) Fermi liquid behavior at low temperatures with a high Kadowaki–Woods ratio, A/γ2 = 30.3a0. The lower critical field at 0 K limit was extracted to be 6.0 Oe for H∥ab. In the clean limit, the ratio of energy gap to Tc was determined to be Δ0/kBTc = 2.029 ± 0.124 exceeding the standard BCS value 1.764, suggesting Cu0.09Bi2Se3 is a strong-coupling superconductor. The in-plane penetration depth at 0 K was calculated to be 1541.57 nm, resulting in an unprecedented high ratio of Tc/λ−2(0) ≅ 9.86. Moreover, the ratio of Tc to Fermi temperature is estimated to be = 0.034. Both ratios fall into the region of unconventional superconductivity according to Uemura’s regime, supporting the unconventional superconducting mechanism in CuxBi2Se3. Finally, the enhanced Tc value higher than 4 K is proposed to arise from the increased density of states at Fermi energy and strong electron–phonon interaction induced by the charge density instability.

In-plane magnetic penetration depth in FeSe1-xSx (x = 0, 0.04, 0.09) and FeTe1-xSex (x = 0.40) single crystals

ABSTRACTThe magnetization anomaly of FeSe1–xSx(x = 0, 0.04, 0.09) and FeTe1–xSex(x = 0.40) single crystals are examined through our Modified Phenomenological Ginzburg–Landau (MPGL) theory applicable to two-band superconducting systems. The theory in the London limit calculates the values of λab(0) as 445.695, 372.058 and 432.957 nm of FeSe1–xSx for x = 0, 0.04, 0.09 which are very close to the experimental values reported by Hafiez et al. [Superconducting properties of sulfur-doped iron selenide. Phys Rev B. 2015;91(16):165109–165120] and as 534 nm for FeTe1–xSex(x = 0.40) single crystal, which is close to the experimental value reported by Yadav and Paulose [Upper critical field, lower critical field and critical current density of FeTe0.60Se0.40 single crystals. New J Phys 2009;11:103046]. In addition, the variation of temperature-dependent in-plane magnetic penetration depth λab(T) with the temperature (T) has been presented, which agrees well as reported by Hafiez et al. [Superconducting properties of sulfur-doped iron selenide. Phys Rev B. 2015;91(16):165109–165120] and Yadav and Paulose [Upper critical field, lower critical field and critical current density of FeTe0.60Se0.40 single crystals. New J Phys. 2009;11:103046].

Multigap Superconductivity in RbCa2Fe4As4F2 Investigated Using μSR Measurements

The superconducting properties of the recently discovered double Fe$_2$As$_2$ layered high-$T_c$ superconductor RbCa$_2$Fe$_4$As$_4$F$_2$ with $T_c\approx$ 30~K have been investigated using magnetization, heat capacity, transverse-field (TF) and zero-field (ZF) muon-spin rotation/relaxation ($\mu$SR) measurements. Our low field magnetization measurements and heat capacity (C$_p$) reveal an onset of bulk superconductivity with $T_{\bf c}\sim$ 30.0(4) K. Furthermore, the heat capacity exhibits a jump at $T_{\bf c}$ of $\Delta$C$_p$/$T_{\bf c}$=94.6 (mJ/mole-K$^2$) and no clear effect of applied magnetic fields was observed on C$_p$(T) up to 9 T between 2 K and 5 K. Our analysis of the TF-$\mu$SR results shows that the temperature dependence of the magnetic penetration depth is better described by a two-gap model, either isotropic $s$+$s$-wave or $s$+$d$-wave than a single gap isotropic $s$-wave or $d$-wave model for the superconducting gap. The presence of two superconducting gaps in RbCa$_2$Fe$_4$As$_4$F$_2$ suggests a multiband nature of the superconductivity, which is consistent with the multigap superconductivity observed in other Fe-based superconductors, including ACa$_2$Fe$_4$As$_4$F$_2$ (A=K and Cs). Furthermore, from our TF-$\mu$SR study we have estimated an in-plane penetration depth $\lambda_{\mathrm{ab}}$$(0)$ =231.5(3) nm, superconducting carrier density $n_s = 7.45 \times 10^{26}~ $m$^{-3}$, and carrier's effective-mass $m^*$ = 2.45\textit{m}$_{e}$. Our ZF $\mu$SR measurements do not reveal a clear sign of time reversal symmetry breaking at $T_{\bf c}$, but the temperature dependent relaxation between 150 K and 1.2 K might indicate the presence of spin-fluctuations. The results of our present study have been compared with those reported for other Fe pnictide superconductors.

Dependence of the absolute value of the penetration depth in (Ba1−xKx)Fe2As2 on doping

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 ($\lambda_{ab}$) in underdoped, optimally-doped, and overdoped samples. The results suggest an abrupt increase of $\lambda_{ab}$ 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 $x=0.19$ and $0.58$ is compatible with previously observed structural symmetries at those doping levels.

Nodal superconductivity coexists with low-moment static magnetism in single-crystalline tetragonal FeS: A muon spin relaxation and rotation study

We report muon spin relaxation and rotation ($\mu$SR) measurements on hydrothermally-grown single crystals of the tetragonal superconductor~FeS, which help to clarify the controversial magnetic state and superconducting gap symmetry of this compound. $\mu$SR time spectra were obtained from 280~K down to 0.025~K in zero field (ZF) and applied fields up to 20 mT. In ZF the observed loss of initial asymmetry (signal amplitude) and increase of depolarization rate~$\Lambda_\mathrm{ZF}$ below 10~K indicate the onset of static magnetism, which coexists with superconductivity below $T_c$. Transverse-field $\mu$SR yields a muon depolarization rate $\sigma_\mathrm{sc} \propto \lambda_{ab}^{-2}$ that clearly shows a linear dependence at low temperature, consistent with nodal superconductivity. The $s{+}d$-wave model gives the best fit to the observed temperature and field dependencies. The normalized superfluid densities versus normalized temperature for different fields collapse onto the same curve, indicating the superconducting gap structure is independent of field. The $T=0$ in-plane penetration depth $\lambda_{ab}$(0) = 198(3) nm.

In-plane magnetic penetration depth of superconducting CaKFe4As4

The temperature dependence of the in-plane magnetic penetration depth (${\ensuremath{\lambda}}_{ab}$) in an extensively characterized sample of superconducting ${\mathrm{CaKFe}}_{4}{\mathrm{As}}_{4}\phantom{\rule{0.28em}{0ex}}({T}_{\mathrm{c}}\ensuremath{\simeq}35\phantom{\rule{0.28em}{0ex}}\mathrm{K})$ was investigated using muon-spin rotation ($\ensuremath{\mu}\mathrm{SR}$). A comparison of ${\ensuremath{\lambda}}_{ab}^{\ensuremath{-}2}(T)$ measured by $\ensuremath{\mu}\mathrm{SR}$ with the one inferred from angle-resolved photoemission spectroscopy (ARPES) data confirms the presence of multiple gaps at the Fermi level. An agreement between $\ensuremath{\mu}\mathrm{SR}$ and ARPES requires the presence of additional bands, which are not resolved by ARPES experiments. These bands are characterized by small superconducting gaps with an average zero-temperature value of ${\mathrm{\ensuremath{\Delta}}}_{0}=2.4(2)\phantom{\rule{0.28em}{0ex}}\mathrm{meV}$. Our data suggest that in ${\mathrm{CaKFe}}_{4}{\mathrm{As}}_{4}$ the ${s}^{\ifmmode\pm\else\textpm\fi{}}$ order parameter symmetry acquires a more sophisticated form by allowing a sign change not only between electron and hole pockets, but also within pockets of similar type.

Two-gap superconductivity with line nodes in CsCa2Fe4As4F2

We report the results of a muon-spin rotation ($\mu$SR) experiment to determine the superconducting ground state of the iron-based superconductor CsCa$_2$Fe$_4$As$_4$F$_2$ with $T_{\rm c} \approx 28.3\,$K. This compound is related to the fully-gapped superconductor CaCsFe$_4$As$_4$, but here the Ca-containing spacer layer is replaced with one containing Ca$_2$F$_2$. The temperature evolution of the penetration depth strongly suggests the presence of line nodes and is best modelled by a system consisting of both an $s$- and a $d$-wave gap. We also find a potentially magnetic phase which appears below $\approx 10\,$K but does not appear to compete with the superconductivity. This compound contains the largest alkali atom in this family of superconductors and our results yield a value for the in-plane penetration depth of $\lambda_{ab}(T=0)=423(5)\,$nm.

Doping effect on the physical properties of Ca10Pt3As8(Fe2As2)5 single crystals

Ca10Pt3As8(Fe2As2)5 is a unique parent compound for superconductivity, which consists of both semiconducting Pt3As8 and metallic FeAs layers. We report the observation of superconductivity induced via chemical doping in either Ca site using rare-earth (RE) elements (RE = La, Gd) or Fe site using Pt. The interlayer distance and the normal-state physical properties of the doped system change correspondingly. The coupled changes include (1) superconducting transition temperature Tc increases with increasing both doping concentration and interlayer distance, (2) our Tc value is higher than previously reported maximum value for Pt doping in the Fe site, (3) both the normal-state in-plane resistivity and out-of-plane resistivity change from non-metallic to metallic behavior with increasing doping concentration and Tc, and (4) the transverse in-plane magnetoresistance (MRab) changes from linear-field dependence to quadratic behavior upon increasing Tc. For La-doped compound with the highest Tc (~35 K), upper critical fields (, ), coherence lengths (ξab, ξc), and in-plane penetration depth (λab) are estimated. We discuss the relationship between chemical doping, interlayer distance, and physical properties in this system.

Penetration depth of Ba1−xKxFe2As2 single crystals explained within a multiband Eliashberg s± approach

We investigate the penetration depth of high-quality ${\mathrm{Ba}}_{1\ensuremath{-}x}{\mathrm{K}}_{x}{\mathrm{Fe}}_{2}{\mathrm{As}}_{2}$ single crystals by a planar waveguide resonator technique, in a cavity perturbation approach. The experimental ${\ensuremath{\lambda}}_{L}$ is compared to calculations based on the three-band Eliashberg equations within the $s\ifmmode\pm\else\textpm\fi{}$ wave model. To this end, the anisotropy of the penetration depth is taken into account. In fact, the agreement between theory and experiment is remarkable. The low-temperature value of the in-plane penetration depth, ${\ensuremath{\lambda}}_{L,ab}(5\phantom{\rule{4.pt}{0ex}}\text{K})=220\phantom{\rule{4pt}{0ex}}\mathrm{nm}$, and the total plasma frequency, ${\ensuremath{\omega}}_{p}=1.0\phantom{\rule{4pt}{0ex}}\mathrm{eV}$, are also consistent with earlier results. This overall consistency validates the model itself, thus allowing us to estimate parameters that are missing in literature, such as the plasma frequency for each band: it turns out that ${\ensuremath{\omega}}_{p,1}={\ensuremath{\omega}}_{p,3}=0.32\phantom{\rule{4pt}{0ex}}\mathrm{eV}$ and ${\ensuremath{\omega}}_{p,2}=0.89\phantom{\rule{4pt}{0ex}}\mathrm{eV}$, with the subscripts 1 and 2 denoting the two hole bands and 3 the equivalent electron band.

Direct measurement of the temperature dependence of the in-plane magnetic penetration depth in optimally doped BaFe2(As[formula omitted]Px)2 single crystals

Measurements of the temperature dependence of the in-plane magnetic penetration depth, Δλab(T)=λab(T)−λab(0), are presented in the isovalently substituted iron pnictide BaFe2(As1−xPx)2 near optimal doping (Tc ∼ 28K). The data were obtained directly from the shielding magnetic susceptibility of thin single crystals (∼ 20 µm thickness along the crystals c-axis) under magnetic fields parallel to the ab layers. Complications associated to flux penetration were avoided by using fields in the 10−4 T range, which ensured that the samples were well inside the Meissner region. At low temperatures (T < 0.25Tc) our data confirm the linear temperature dependence of λab(T) observed by using other procedures, and that is consistent with a nodal superconducting order parameter. By using values for λab(0) in the literature we obtained the temperature dependence up to Tc of the superfluid density, ns(T)∝1/λab2(T). It is found that samples with slightly different Tc values present a significantly different qualitative behavior of ns(T).

Coexistence of low-moment magnetism and superconductivity in tetragonal FeS and suppression ofTcunder pressure

We report local probe ($\mu$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 ($10^{-2}-10^{-3}\mu_\mathrm{B})$ 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 $\mu$SR we obtain an in-plane penetration depth $\lambda_\mathrm{ab}(0)=223(2)$ nm for FeS. The temperature dependence of the corresponding superfluid density $\lambda_\mathrm{ab}^{-2}(T)$ indicates a fully gapped superconducting state and is consistent with a two gap s-wave model. Additionally, we find that the superconducting $T_\mathrm{c}$ of FeS continuously decreases for hydrostatic pressures up to 2.2 GPa.

Single-vortex pinning and penetration depth in superconductingNdFeAsO1−xFx

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 superconducting in-plane penetration depth in NdFeAsO$_{1-x}$F$_x$, $\lambda_{ab}=320 \pm 60$ nm, which is larger than previous bulk measurements.

Direct measurement of the temperature dependence of the magnetic penetration depth in Ba(Fe1−xNix)2As2 superconductors

The temperature dependence of the in-plane magnetic penetration depth λab of Ba(Fe1−xNix)2As2 single crystals is determined directly from the shielding magnetic susceptibility, measured in the Meissner region with the field parallel to the ab layers. The doping levels studied cover the underdoped, optimally doped and overdoped regimes. At temperatures below 0.5Tc a well-defined power-law behavior λab(T) − λab(0) = ATn (with n ≈ 2.5) is observed. At lower temperatures (T < 0.3Tc) the data are still consistent with n = 2 and , as predicted by the strong pair-breaking scenario proposed by Gordon et al (2010 Phys. Rev. B 81, 180501(R)). The temperature dependence of the superfluid density presents a marked positive curvature just below Tc, which is a sign of two-gap superconductivity. The analysis of ρs(T) in terms of a two-gap model allowed estimation of parameters like the in-band and inter-band couplings, the relative weight of each band, and their dependence on the doping level. A comparison with ρs(T) data obtained by using other techniques in compounds with a similar composition is also presented.

Anisotropic superconductivity in noncentrosymmetric BiPd

We report measurements of London penetration depth $\lambda(T)$ for the noncentrosymmetric superconductor BiPd by using a tunnel diode oscillator. Pronounced anisotropic behavior is observed in the low-temperature penetration depth; the in-plane penetration depth $\lambda_{ac}(T)$ follows an exponential decrease, but the interplane penetration depth $\lambda_b(T)$ shows power-law-type behavior. The superfluid density $\rho_s(T)$, converted from the penetration depth $\lambda(T)$, is best fitted by an anisotropic two-band BCS model. We argue that such a complex order parameter is attributed to the admixture of spin-singlet and spin-triplet pairing states as a result of antisymmetric spin-orbit coupling in BiPd.

BEC MODEL OF HIGH-TcSUPERCONDUCTIVITY IN LAYERED CUPRATES

High-Tcsuperconductivity in layered cuprates is described in a BCS-BEC formalism with linearly-dispersive s- and d-wave Cooper pairs moving in quasi-2D finite-width layers around the CuO2planes. This yields a closed formula for Tcinvolving the layer width, the Debye frequency, the pairing energy and the in-plane penetration depth. The new formula has no free parameters and reasonably reproduces empirical values of superconducting Tcs for 11 different layered superconductors over a wide doping regime including YBCO itself as well as other compounds like LSCO, BSCCO and TBCCO. In agreement with the London formalism, the formula also yields a fair description of the Tcdependence of the lower critical magnetic field in highly underdoped YBCO.

Probing the anisotropic vortex lattice in the Fe-based superconductor KFe2As2using small-angle neutron scattering

Using small angle neutron scattering, the anisotropy of the magnetic vortex lattice (VL), in the heavily hole-doped pnictide superconductor ${\text{KFe}}_{2}{\text{As}}_{2}$, was studied. Well-ordered VL scattering patterns were measured with fields applied in directions between $\mathbf{B}\ensuremath{\parallel}\mathbf{c}$ and the basal plane, rotating either towards $[100]$ or $[110]$. Slightly distorted hexagonal patterns were observed when $\mathbf{B}\ensuremath{\parallel}\mathbf{c}$. However, the scattering pattern distorted more strongly as the field was rotated away from the c axis. At low field, the arrangement of vortices is affected by the anisotropy of penetration depth in the plane perpendicular to the field. By fitting the distortion with the anisotropic London model, we obtain an estimate of $\ensuremath{\sim}$$3.4$ for the anisotropy factor $\ensuremath{\gamma}$ between the in-plane and c-axis penetration depths at the lowest temperature studied. The results further reveal VL phase transitions as a function of field direction. We discuss these transitions using the ``hairy ball'' theorem.

Superconducting and normal-state properties ofAPd2As2(A = Ca, Sr, Ba) single crystals

The synthesis and crystallography, magnetic susceptibility chi, magnetization M, specific heat Cp, in-plane electrical resistivity rho and in-plane magnetic penetration depth measurements are reported for single crystals of APd2As2 (A = Ca, Sr, Ba) versus temperature T and magnetic field H. The crystals were grown using PdAs self-flux. CaPd2As2 and SrPd2As2 crystallize in a collapsed body-centered tetragonal ThCr2Si2-type structure (I4/mmm), whereas BaPd2As2 crystallizes in the primitive tetragonal CeMg2Si2-type structure (P4/mmm), in agreement with literature data. The rho(T) data exhibit metallic behavior for all three compounds. Bulk superconductivity is reported for CaPd2As2 and SrPd2As2 below Tc = 1.27 and 0.92 K, respectively, whereas only a trace of superconductivity is found in BaPd2As2. No other phase transitions were observed. The chi(T) and M(H) data reveal anisotropic diamagnetism in the normal state, with chi_c > chi_ab for CaPd2As2 and BaPd2As2, and chi_c < chi_ab for SrPd2As2. The normal and superconducting state data indicate that CaPd2As2 and SrPd2As2 are conventional type-II nodeless s-wave electron-phonon superconductors. The electronic superconducting-state heat capacity data for CaPd2As2, which has an extremely sharp heat capacity jump at Tc, are analyzed using our recent elaboration of the alpha-model of the BCS theory of superconductivity, which indicates that the s-wave gap in this compound is anisotropic in momentum space.