Complementary response of static spin-stripe order and superconductivity to non-magnetic impurities in cuprates

TL;DR

This study investigates how non-magnetic Zn impurities affect static spin-stripe order and superconductivity in La214 cuprates, revealing a high sensitivity of both phenomena to impurities and suggesting their intertwined nature.

Contribution

It provides the first detailed analysis of impurity effects on spin-stripe order and superconductivity, highlighting their correlated suppression in cuprates.

Findings

Spin-stripe order temperature decreases linearly with Zn doping.

Both spin-stripe order and superconductivity are similarly suppressed by impurities.

Static spin-stripe order is highly sensitive to impurities within CuO2 planes.

Abstract

We report muon-spin rotation and neutron-scattering experiments on non-magnetic Zn impurity effects on the static spin-stripe order and superconductivity of the La214 cuprates. Remarkably, it was found that, for samples with hole doping x = 1/8, the spin-stripe ordering temperature T_so decreases linearly with Zn doping y and disappears at around y = 4\%, demonstrating a high sensitivity of static spin-stripe order to impurities within a CuO_2 plane. Moreover, T_so is suppressed by Zn in the same manner as is the superconducting transition temperature T_c for samples near optimal hole doping. This surprisingly similar sensitivity suggests that the spin-stripe order is dependent on intertwining with superconducting correlations.

Equations

$$N_{e^{+}\alpha (t)}=N_0{e}^{-\frac{t}{{\tau }_{\mu }}}(1+A_0\vec{P}(t){\hat{n}}_{\alpha })+N_{bgr}.\alpha =F,B$$

$$A(t)=\frac{N_{e^{+}F}(t)-N_{e^{+}B}(t)}{N_{e^{+}F}(t)+N_{e^{+}B}(t)}=A_{0}P(t),$$

$$P(t)=V_{m}\Bigg{[}{\frac{2}{3}e^{-\lambda_{T}t}J_{0}(\gamma_{\mu}B_{\mu}t)}+\frac{1}{3}e^{-\lambda_{L}t}\Bigg{]}+(1-V_{m})e^{-\lambda_{nm}t}.$$

Full text

††preprint: preprint()

One of the most astonishing manifestations of the competing ordered phases occurs in the system La2-xBaxCuO4 (LBCO) Bednorz , where the bulk superconducting (SC) transition temperature $T_{\rm c}$ exhibits a deep minimum at $x$ = 1/8 Moodenbaugh ; Kivelson ; Vojta . At this doping level muon-spin rotation (${\mu}$SR), neutron, and x-ray diffraction experiments revealed two-dimensional static charge and spin-stripe order Tranquada1 ; Tranquada2 ; Abbamonte ; Huecker ; Luke ; GuguchiaPRL ; Fujita2004 . The collected experimental data indicate that the tendency toward uni-directional stripe-like ordering is common to cuprates Kivelson ; Vojta ; Kohsaka ; Julien ; Keimer2015 . However, the relevance of stripe correlations for high-temperature superconductivity remains a subject of controversy. On the theoretical front, the concept of a sinusoidally-modulated pair-density wave (PDW) SC order, intimately intertwined with spatially modulated antiferromagnetism, has been introduced Berg1 ; Fradkin ; Himeda . On the experimental front, quasi-two-dimensional superconducting correlations were observed in La1.875Ba0.125CuO4 (LBCO-1/8) and La1.48Nd0.4Sr0.12CuO4, coexisting with the ordering of static spin-stripes, but with frustrated phase order between the layers Tranquadareview ; Tranquada2008 ; Li ; Valla ; Shen ; Ding2008 . Recently, it was found that in La2-xBaxCuO4 (0.11 ${\leq}$ $x$ ${\leq}$ 0.17) the 2D SC transition temperature $T_{\rm c1}$ and the static spin-stripe order temperature $T_{\rm so}$ have very similar values throughout the phase diagram Guguchiaarxiv ; GuguchiaNJP . Moreover, a similar pressure evolution of $T_{\rm c}$ and $T_{\rm so}$ in the stripe phase of $x$ = 0.155 and 0.17 samples was observed. These findings were discussed in terms of a spatially modulated and intertwined pair wave function Berg1 ; Fradkin ; Himeda ; Xu2014 . There are also a few reports proposing the relevance of a PDW state in sufficiently underdoped La2-xSrxCuO4 Jakobsen2015 and YBa2Cu3O6-x Lee ; Yu . At present it is still unclear to what extent PDW order is a common feature of cuprate systems where stripe order occurs.

To further explore the interplay between static stripe order and superconductivity in cuprates we used non-magnetic impurity substitution at the Cu site as an alternative way of tuning the physical properties. Since the discovery of cuprate HTSs much effort was invested in the investigation of the effect of in-plane impurities. It is now well established that in cuprate HTSs nonmagnetic Zn ions suppress $T_{c}$ even more strongly than magnetic ions XiaoG ; Fukuzumi ; Komiya . Such behaviour is in sharp contrast to that of conventional superconductors. This observation led to the formulation of an unconventional pairing mechanism and symmetry of the order parameter for cuprate HTSs. In addition, in several cases a ground state with static antiferromagnetic (AF) correlations is stabilized by Zn-doping Mendels1994 ; Akoshima2000 ; Watanabe2000 ; Adachi2004 ; Adachi2004v2 ; Koike2012 . Up to now much less is known concerning impurity effects on the static stripe phase in cuprates at 1/8 doping. From specific heat and neutron scattering measurements it was inferred that Zn doping leads to stripe destruction AnegawaZn ; Takeda ; Fujita . Such an effect is very interesting and it was not predicted theoretically. However, no systematic impurity effect studies on static stripe order have been carried out up to now. Moreover, specific heat is a very indirect method to characterize the stripe phase in cuprates. Therefore, it is very important to use experimental techniques which can directly probe stripe formation and its evolution with impurity doping.

In this letter, we report on systematic muon-spin rotation ${\mu}$SR, neutron scattering, and magnetization studies of Zn impurity effects on the static spin-stripe order and superconductivity in the La214 cuprates. Remarkably, it was found that in these systems the spin-stripe ordering temperature $T_{\rm so}$ decreases linearly with Zn doping $y$ and disappears at $y{\simeq}4\%$. This means that $T_{\rm so}$ is suppressed in the same manner as the superconducting transition temperature $T_{c}$ by Zn impurities. These results suggest that the stripe and SC orders may have a common physical mechanism and are intertwined.

In a ${\mu}$SR experiment, positive muons implanted into a sample serve as an extremely sensitive local probe to detect small internal magnetic fields and ordered magnetic volume fractions in the bulk of magnetic systems. Thus ${\mu}$SR is a particularly powerful tool to study inhomogeneous magnetism in materials suppl . Neutron diffraction experiments Eiger allow to directly probe the incommensurate spin structure of spin-stripe order and thus provide crucial complementary information to the ${\mu}$SR technique.

Figures 1(a) and (b) show representative zero-field (ZF) ${\mu}$SR time spectra for polycrystalline La1.875Ba0.125Cu1-yZnyO4 samples with $y=0$ and 0.02, respectively, recorded at various temperatures. For $y=0$, damped oscillations due to muon-spin precession in internal magnetic fields are observed below $T_{\rm so}\approx 35$ K, indicating the formation of static spin order in the stripe phase Luke ; Arai ; Nachumi ; GuguchiaNJP ; AndreasSuter ; Crawford . It is seen in Fig. 1(b), that for the $y=0.02$ sample the oscillating signal appears only below $T\approx 10$ K, showing strong suppression of the static spin-stripe order with Zn doping. We have studied this novel effect systematically as a function of Zn doping.

The temperature dependence of the average internal field $B_{\rm\mu}$, which is proportional to the ordered magnetic moment, is shown in Fig. 2(a) for various Zn dopings $y$. As evident from Fig. 2(a), $B_{\mu}(0)$, the internal magnetic field extrapolated to zero-temperature, does not depend on the Zn content $y$, while $T_{\rm so}$ changes substantially with increasing $y$. Specifically, $T_{\rm so}$ decreases from $T_{\rm so}\simeq 32.5$ K for $y=0$ to $T_{\rm so}\simeq 4$ K for $y=0.04$. Figure 2(b) shows that a very similar behavior is observed for $B_{\rm\mu}$ measured on polycrystalline samples of the related compound La1.48Nd0.4Sr0.12Cu1-yZnyO4. Note that the low-temperature value of $B_{\rm\mu}$ is enhanced by the ordering of the Nd moments. A similar suppression of $T_{\rm so}$ by Zn impurities was also observed in single-crystal samples of La1.48Nd0.4Sr0.12Cu1-yZnyO4 ($y=0$, 0.016). It seems that this effect is a generic feature of cuprates with static stripe order. We note that in all the above mentioned systems, despite the suppression of $T_{\rm so}$ with Zn doping, the magnetic volume fraction $V_{\rm m}$ at the base temperature stays nearly 100% [see Fig. 2(c)]. The bulk LTT structural phase transition temperature also stays nearly uneffected by Zn-doping (see supplementary Note II and supplementary Fig. S1 suppl ).

The observed Zn impurity effects on $T_{\rm so}$ in La1.875Ba0.125Cu1-yZnyO4 and La1.48Nd0.4Sr0.12Cu1-yZnyO4 are summarised in Fig. 3. It is a remarkable finding that $T_{\rm so}$ linearly decreases with increasing Zn content $y$. Such a behaviour is reminiscent of the well known linear suppression of the SC transition temperature $T_{\rm c}$ in cuprates XiaoG ; Fukuzumi ; Komiya . Since the superconducting volume fraction in 1/8 doped samples is tiny and the bulk $T_{\rm c}$ is also very low, it is difficult to follow the SC properties of these systems as a function of Zn content. Alternatively, in Fig. 3 we plot $T_{\rm c}$ values for optimally doped La1.85Sr0.15CuO4 XiaoG ; Fukuzumi ; Komiya and La1.845Ba0.155CuO4 (see below) as a function of Zn content. Strikingly, suppression of $T_{\rm so}$ goes in a very similar manner as the well known impurity-induced $T_{c}$ suppression.

We have confirmed the Zn doping effect on the static spin-stripe order by neutron diffraction experiments on single-crystal samples of La1.48Nd0.4Sr0.12Cu1-yZnyO4 ($y=0$, 0.016) composition . The magnetic ordering wave vectors are ${\bf Q}_{so}=(0.5-\delta,0.5,0)$ and $(0.5,0.5+\delta,0)$, i.e., they are displaced by ${\delta}$ from the position of the magnetic Bragg peak in the AF parent compound La2CuO4 Tranquada1 ; Tranquada2 . In Fig. 4(a) we show $h$ scans through the $(0.5+\delta,0.5,0)$ magnetic superlattice peaks, recorded at $T=1.75$ K for the samples $y=0$ and 0.016. It is clear that the intensity and incommensurability do not change with Zn doping. However, $T_{\rm so}$ is strongly suppressed from $T_{\rm so}\simeq 50$ K Tranquada1 for $y=0$ to $T_{\rm so}\simeq 10$ K for $y=0.016$, as demonstrated in Fig. 4(b), where the peak intensity is shown as a function of temperature.

Going further, we studied the Zn-impurity effects on $T_{\rm so}$ and $T_{\rm c}$ in La1.845Ba0.155CuO4. This compound ($x$ ${\textgreater}$ 1/8) exhibits a well defined bulk SC transition with $T_{\rm c}$ = 30 K and at the same time shows static spin-stripe order $T_{\rm so}$ ${\simeq}$ $T_{\rm c}$ = 30 K Guguchiaarxiv . This enables us to study impurity effects on $T_{\rm so}$ and $T_{\rm c}$ simultaneously in the same sample. Figure 5a shows the temperature dependence of the magnetic volume fraction $V_{m}$ extracted from ZF-${\mu}$SR data for La1.845Ba0.155Cu1-yZnyO4 ($y$ = 0, 0.02, and 0.04). The low temperature value of $V_{m}$ increases with increasing Zn content $y$ and reaches 100 ${\%}$ for the highest Zn content $y$ = 0.04. On the other hand, $T_{{\rm so}}$ decreases with increasing $y$ similar as for 1/8-doping. The values of $T_{{\rm so}}$ and $T_{{\rm c}}$ (see the supplementary Note III and supplementary Figs. S2 and S3 suppl ) as a function of Zn content $y$ are shown in Fig. 5(b). Again, with increasing $y$ both $T_{{\rm c}}$ and $T_{{\rm so}}$ decrease linearly with the same slope, indicating that Zn impurities influence $T_{{\rm c}}$ and $T_{{\rm so}}$ in the same manner.

What is the significance of this surprising correlation? Let us start with the fact that it is unusual to have spin order occur in a hole-doped cuprate at a temperature of $\sim 35$ K. Just a couple of percent of hole doping is generally sufficient to wipe out antiferromagnetic order Matsuda2002 . One common point of view is that antiferromagnetism and superconductivity are competing orders Sachdev2000 . From that perspective, one might take the occurrence of spin-stripe order as evidence that hole-pairing and superconductivity have been suppressed. In that case, we might expect the impact of Zn-doping on $T_{\rm so}$ to be similar to its impact on the Néel temperature in La2CuO4. That assumption leads to a problem, however, as experiment has demonstrated that it takes, not 4%, but $\sim 40\%$ Zn to destroy Néel order Vajk2002 . One could also take account of the fact that the Zn tends to induce static Cu spin order in its immediate neighborhood Mahajan1994 ; Julien2000 , which, with random locations of the Zn sites, could lead, at higher Zn concentrations, to some disorder from neighboring pinned stripe domains being out of phase with one another; however, a shortening of the spin correlation length only becomes apparent with at least 3% Zn doping Fujita ; Birgenau , while the drop in $T_{\rm so}$ is clear at much lower Zn concentrations.

Consider instead that previous experiments provide evidence that spin-stripe order coexists with two-dimensional superconducting correlations in LBCO-1/8 Tranquada2008 ; Li . Here, the superconducting and spin orders are intertwined Fradkin . Superconducting correlations within charge stripes must establish Josephson coupling across the spin stripes, while the spins in neighboring stripes must establish an effective exchange coupling via the fluctuating pairs in the intervening charge stripe. A Zn ion will locally suppress hole motion, thus eliminating local superconducting coherence and weakening the superconductivity Balatsky . Local suppression of hole hopping will also disrupt the effective exchange coupling between spin stripes, leading to a reduction in $T_{\rm so}$.

Previous $\mu$SR studies of Zn doping in LSCO and YBa2Cu3O7 have established the “Swiss-cheese“ model: a fixed carrier density per Zn atom is removed from the superfluid density, as if each Zn removes a fixed areal fraction of the superfluid Nachumi1996 . The linear relationship between $T_{c}$ and the average superfluid density, valid for underdoped through optimally-doped cuprate HTSs, then explains the reduction of $T_{c}$ with increasing Zn concentration Uemura1989 . For the stripe-ordered systems, it is plausible that both the superconducting and spin-stripe orders will respond in a similar fashion.

In conclusion, static spin-stripe order and superconductivity in cuprate systems La2-xBaxCu1-yZnyO4 ($x$ = 0.125, 0.155) and La1.48Nd0.4Sr0.12Cu1-yZnyO4 were studied by means of magnetisation, ${\mu}$SR, and neutron scattering experiments as a function of nonmagnetic Zn impurity concentration. High sensitivity of the static spin-stripe order temperature $T_{\rm so}$ to impurities in the CuO2 plane was demonstrated. Namely, the spin-stripe ordering temperature $T_{\rm so}$ strongly decreases linearly with Zn doping and disappears at about 4% Zn content. More strikingly, $T_{\rm so}$ is suppressed in the same fashion as is the superconducting transition temperature $T_{\rm c}$ by Zn impurities. These results strongly suggest that the existence of the stripe order requires intertwining with the SC pairing correlations, such as occurs in the proposed PDW state. The present findings should help to better understand the complex interplay between stripe order and superconductivity in cuprates. More generally, since charge and spin orders are often observed in other transition-metal oxides, investigation of impurity effects and disorder on stripe formation may become an interesting research avenue in correlated electron systems.

Acknowledgments. The ${\mu}$SR experiments were carried out at the ${\pi}$M3 beam line of the Paul Scherrer Institute (Switzerland), using the general purpose instrument (GPS). The neutron scattering experiments were carried out with the three-axis spectrometer EIGER at the Swiss Spallation Neutron Source SINQ at the Paul Scherrer Institut (PSI), Switzerland. We are grateful to S.A. Kivelson for valuable discussions. Z.G. gratefully acknowledges the financial support by the Swiss National Science Foundation (Early Postdoc Mobility SNFfellowship P2ZHP2-161980 and SNFGrant 200021-149486). Z.G. thanks Martin Mansson for useful discussions. A.S. acknowledges support from the SCOPES grant No. SCOPES IZ74Z0-160484. Work at Columbia University is supported by US NSF DMR-1436095 (DMREF) and NSF DMR-1610633 as well as by REIMEI project of Japan Atomic Energy Agency. JMT is supported at Brookhaven by the U.S. Department of Energy, Office of Basic Energy Sciences, under contract No. DE-SC0012704.

I SUPPLEMENTAL MATERIAL

II METHODS

Sample preparation: Polycrystalline samples of La1.875Ba0.125Cu1-yZnyO4 with $y$ = 0, 0.02, 0.04 and La1.48Nd0.4Sr0.12Cu1-yZnyO4 with $y$ = 0, 0.01, 0.02, 0.04 were prepared by the conventional solid-state reaction method using La2O3, Nd2O3, BaCO3, SrCO3, and CuO. The single-phase character of the samples was checked by powder x-ray diffraction. The single crystals of La1.48Nd0.4Sr0.12Cu1-yZnyO4 ($y$ = 0, 0.016) were grown by the traveling solvent floating zone method. All the measurements were performed on samples from the same batch.

Principles of the ${\mu}$SR technique: Static spin-stripe orders in La1.875Ba0.125Cu1-yZnyO4 with $y$ = 0, 0.02, 0.04 and La1.48Nd0.4Sr0.12Cu1-yZnyO4 with $y$ = 0, 0.01, 0.02, 0.04 were studied by means of zero-field (ZF) ${\mu}$SR experiments. The ${\mu}$SR experiments were carried out at the ${\pi}$M3 beam line of the Paul Scherrer Institute (Switzerland), using the general purpose instrument (GPS). In a ${\mu}$SR experiments an intense beam ($p_{\mu}$ = 29 MeV/c) of 100 ${\%}$ spin-polarized muons is stopped in the sample mounted inside of a gas-flow 4He cryostat on a sample holder with a standard veto setup providing essentially a background free ${\mu}$SR signal. The positively charged muons thermalize in the sample at interstitial lattice sites, where they act as magnetic microprobes. In a magnetic material the muons spin precess in the local field $B_{{\rm\mu}}$ at the muon site with the Larmor frequency ${\nu}_{{\rm\mu}}$ = $\gamma_{{\rm\mu}}$/(2{\pi})$$B_{{\rm\mu}} (muon gyromagnetic ratio $\gamma_{{\rm\mu}}$/(2${\pi}$) = 135.5 MHz T*-1*). In a ZF ${\mu}$SR experiment positive muons implanted into a sample serve as an extremely sensitive local probe to detect small internal magnetic fields and ordered magnetic volume fractions in the bulk of magnetic materials. Thus, ${\mu}$SR is a particularly powerful tool to study inhomogeneous magnetism in materials.

The muons $\mu^{+}$ implanted into the sample will decay after a mean life time of ${\tau}$μ = 2.2 ${\mu}$s, emitting a fast positron $e^{+}$ preferentially along their spin direction. Various detectors placed around the sample track the incoming $\mu^{+}$ and the outgoing $e^{+}$. When the muon detector records the arrival of a ${\mu}$ in the specimen, the electronic clock starts. The clock is stopped when the decay positron $e^{+}$ is registered in one of the $e^{+}$ detectors, and the measured time interval is stored in a histogramming memory. In this way a positron-count versus time histogram is formed. A muon decay event requires that within a certain time interval after a $\mu^{+}$ has stopped in the sample a $e^{+}$ is detected. This time interval extends usually over several muon lifetimes ($e.g.$ 10${\mu}$s). After a bunch of muons stopped in the sample, one obtains a histogram for the forward ($N_{\rm e^{+}F}$) and the backward ($N_{\rm e^{+}B}$) detectors, which in the ideal case has the following form:

[TABLE]

Here, the exponential factor accounts for the radioactive muon decay. $\vec{P}$(t) is the muon-spin polarization function with the unit vector ${\hat{n}_{\alpha}}$ (${\alpha}$ = F,B) with respect to the incoming muon spin polarization. $N_{\rm 0}$ is number of positrons at the initial time $t$=0. $N_{\rm bgr}$ is a background contribution due to uncorrelated starts and stops. $A_{0}$ is the initial asymmetry, depending on different experimental factors, such as the detector solid angle, efficiency, absorption, and scattering of positrons in the material. Typical values of $A_{0}$ are between 0.2 and 0.3.

Since the positrons are emitted predominantly in the direction of the muon spin which precesses with ${\omega_{\mu}}$, the forward and backward detectors will detect a signal oscillating with the same frequency. In order to remove the exponential decay due to the finite life time of the muon, the so-called asymmetry signal $A$(t) is calculated:

[TABLE]

where, $N_{e^{+}F}$(t) and $N_{e^{+}B}$(t) are the number of positrons detected in the forward and backward detectors, respectively. The quantities $A(t)$ and $P(t)$ depend sensitively on the spatial distribution and dynamical fluctuations of the magnetic environment of the muons. Hence, these functions allow to study interesting physics of the investigated system.

Analysis of ZF-${\mu}$SR data:

The ${\mu}$SR signals in the whole temperature range were analyzed by decomposing the signal into a magnetic and a nonmagnetic contribution:

[TABLE]

Here, $V_{\rm m}$ denotes the relative volume of the magnetic fraction, and $B_{\mu}$ is the average internal magnetic field at the muon site. ${\lambda_{T}}$ and ${\lambda_{L}}$ are the depolarization rates representing the transversal and the longitudinal relaxing components of the magnetic parts of the sample. $J_{0}$ is the zeroth-order Bessel function of the first kind. This is characteristic for an incommensurate spin density wave and has been observed in cuprates with static spin stripe order Nachumi . ${\lambda_{nm}}$ is the relaxation rate of the nonmagnetic part of the sample, where spin-stripe order is absent. The ${\mu}$SR time spectra were analyzed using the free software package MUSRFIT AndreasSuter .

III Studies of transport, thermodynamic and structural properties

Figure 6a shows the temperature dependence of the $c$-axis resistance for the single crystal of La1.48Nd0.4Sr0.12CuO4. A clear jump in resistance ($R$) appears at $T_{\rm LTT}$ ${\simeq}$ 71 K, which is indicated by a arrow and ascribed to the structural phase transition from low-temperature orthorhombic (LTO) to low-temperature tetragonal (LTT) phase. Upon lowering the temperature below $T_{\rm LTT}$ ${\simeq}$ 71 K, $R$ monotoneously increases down to 12 K, below which it decreases and reaches zero resistance state towards the base temperature. This indicates that while the onset of SC transition is at $T_{\rm SC}$ in La1.48Nd0.4Sr0.12CuO4 is at ${\simeq}$ 12 K, bulk superconductivity is only reached at the base temperature $T_{\rm c}$ ${\simeq}$ 2 K.

Figure 6b shows the specific heat $C_{\rm p}$ as a function of temperature for the single crystals of La1.48Nd0.4Sr0.12Cu1-yZnyO4 ($y$ = 0, 0.016). Clear peak Takeda ; Crawford in $C_{\rm p}$ at $T_{\rm LTT}$ ${\simeq}$ 71 K is observed for $y$ = 0 sample, which is in perfect agreement with the resistance data. In Zn-doped sample $y$ = 0.016 the peak appears at slightly lower temperature $T_{\rm LTT}$ ${\simeq}$ 67 K, indicating very tiny impact of the Zn-doping on the bulk LTT structural phase transition temperature.

IV Superconducting properties of La1.845Ba0.155Cu1-yZnyO4, studied by means of magnetization and muon-spin rotation experiments

Figure 7 shows the temperature dependence of the zero-field-cooled (ZFC) diamagnetic moment $m_{\rm ZFC}$ for La1.845Ba0.155Cu1-yZnyO4 ($y$ = 0, 0.02, 0.04) sample, recorded in a magnetic field of {\mu}_{\rm 0}$$H = 0.5 mT. The diamagnetic moment as well as $T_{c}$ are strongly suppressed by Zn-substitution.

Figures 8a and b show the temperature dependence of the internal magnetic field $B_{\mu}$ and the magnetic volume fraction $V_{m}$ extracted from the ZF-${\mu}$SR data for La1.845Ba0.155Cu1-yZnyO4 ($y$ = 0, 0.02 and 0.04). The low temperature value of $V_{m}$ increases with increasing Zn-content $y$. Note that $V_{m}$ = 100 ${\%}$ for $y$ = 0.04. On the other hand, the static spin-stripe order temperature $T_{{\rm so}}$ decreases with increasing $y$ as it is the case for $T_{{\rm c}}$.

The SC response of the samples was also measured by using TF-${\mu}$SR. For all the samples a diamagnetic shift of {\mu}_{{\rm 0}}$$H_{{\rm int}} experienced by the muons is observed below $T_{{\rm c}}$. This is evident in Fig. 8c where we plot the temperature dependence of the diamagnetic shift {\Delta}$$B_{{\rm dia}} = ${\mu}_{{\rm 0}}$[$H_{{\rm int,SC}}$-$H_{{\rm int,NS}}$] for La1.845Ba0.155Cu1-yZnyO4 ($y$ = 0, 0.02 and 0.04), where {\mu}_{{\rm 0}}$$H_{{\rm int,SC}} denotes the internal field measured in the SC state and {\mu}_{{\rm 0}}$$H_{{\rm int,NS}} is the internal field measured in the normal state. Note that {\mu}_{{\rm 0}}$$H_{{\rm int,NS}} is temperature independent. The SC transition temperature $T_{{\rm c}}$ is determined from the intercept of the linearly extrapolated {\Delta}$$B_{{\rm dia}} curve and its zero line. The diamagnetic shift {\Delta}$$B_{{\rm dia}} decreases strongly with increasing Zn-content $y$, which is consistent with the reduction of the diamagnetic moment as a result of Zn-substitution. In Fig. 8b of the main text, the values of $T_{{\rm c}}$ and $T_{{\rm so}}$ are plotted as a function of Zn-content $y$. Remarkably, both $T_{{\rm c}}$ and $T_{{\rm so}}$ decrease linearly with increasing $y$, indicating that Zn impurities influence $T_{{\rm c}}$ and $T_{{\rm so}}$ in the same way. These experiments give strong support to our high pressure data, showing the simultaneous occurrence of static magnetism and superconductivity in the LBCO-0.155 system.

Figure 9 shows the Zn-doping evolution of the base temperature values of the magnetic volume fraction, the superconducting volume fraction (estimated from the susceptibility data) and the diamagnetic shift, imposed by the SC state (extracted from ${\mu}$SR). It is clear that while the magnetic fraction is enhanced by Zn-doping, the SC fraction is reduced substantially. Since already for 2 ${\%}$-doped sample the SC fraction is small, it is impossible to extract the reliable information about the superfluid density. Only parameters we can get is the critical temperature and the strength of the diamagnetism.