Superconductivity drives magnetism in delta-doped La2CuO4

TL;DR

This study investigates how superconductivity influences magnetism in delta-doped La2CuO4 superlattices, revealing a complex interaction where superconductivity enhances magnetic order, facilitated by minimal disorder and charge redistribution.

Contribution

It demonstrates the interplay between superconductivity and antiferromagnetism in highly controlled, low-disorder delta-doped La2CuO4 superlattices using molecular beam epitaxy and muSR techniques.

Findings

Superconductivity couples with antiferromagnetic order in a non-trivial way.

Magnetic volume fraction increases below the superconducting transition.

Charge redistribution may enhance magnetic order due to superconducting gap opening.

Abstract

The understanding of the interplay between different orders in a solid is a key challenge in highly correlated electronic systems. In real systems this is even more difficult since disorder can have a strong influence on the subtle balance between these orders and thus can obscure the interpretation of the observed physical properties. Here we present a study on delta-doped La2CuO4 superlattices. By means of molecular beam epitaxy whole LaO-layers were periodically replaced through SrO-layers providing a charge reservoir, yet reducing the level of disorder typically present in doped cuprates to an absolute minimum. The induced superconductivity and its interplay with the antiferromagnetic order is studied by means of low-energy muSR. We find a quasi-2D superconducting state which couples to the antiferromagnetic order in a non-trivial way. Below the superconducting transition temperature, the magnetic volume fraction increases strongly. The reason could be a charge redistribution of the free carriers due to the opening of the superconducting gap which is possible due to the close proximity and low disorder between the different ordered regions.

Tables

Table 1. Table 1: Estimates of the magnetic and superconducting thicknesses. The first and second column gives the magnetic volume fraction at T c subscript 𝑇 𝑐 T_{c} and zero temperature respectively. The superlattice repetition length is d N = ( N + 1 ) ⋅ UC / 2 subscript 𝑑 𝑁 ⋅ 𝑁 1 UC 2 d_{N}=(N+1)\cdot\mathrm{UC}/2 , with UC = 1.32 UC 1.32 \mathrm{UC}=1.32  nm. The magnetic layer thickness is therefore defined as d M ​ ( T ) = f M ​ ( T ) ⋅ d N subscript 𝑑 M 𝑇 ⋅ subscript 𝑓 M 𝑇 subscript 𝑑 𝑁 d_{\rm M}(T)=f_{\rm M}(T)\cdot d_{N} . An upper limit for the superconducting layer thickness is thus d S ​ ( 0 ) = d N − d M ​ ( 0 ) subscript 𝑑 𝑆 0 subscript 𝑑 𝑁 subscript 𝑑 M 0 d_{S}(0)=d_{N}-d_{\rm M}(0) . The last column gives the T c subscript 𝑇 𝑐 T_{c} ’s of the superlattices.

N	$f_M\left(T_c\right)$	$f_M\left(0\right)$	$d_N$	$d_M\left(T_c\right)$	$d_M\left(0\right)$	$d_S\left(0\right)$	$T_c$	${\lambda }_{\mathrm{ab}}^{-2}$
			(nm)	(nm)	(nm)	(nm)	(K)	($\mu m^{-2}$)
3	0	0	2.64	0	0	2.64	28.8	9.5(3)
7	0.13	0.5	5.28	0.69	2.64	2.64	17.8	8.4(3)
8	0.27	0.7	5.94	1.6	4.16	1.78	24.3	11.3(4)
9	0.10	0.4	6.60	0.66	2.64	3.96	26.8	9.4(2)
11	0.32	0.54	7.92	2.53	4.28	3.64	24.6	9.5(3)

Equations

$${\left(\frac{{\sigma }_{\mathrm{sc}}(T)}{{\gamma }_{\mu }}\right)}^2=0.00371\frac{{\Phi }_0^2}{\lambda (T{)}^4},$$

$$A(t)$$

Full text

The copper oxide based high-temperature superconductors (cuprates) exhibit rich and complex physics Keimer et al. (2015). Strong electron correlations drive the parent compounds into an insulating, antiferromagnetic ground state. Upon sufficiently high doping of the copper oxide planes by electrons or holes, superconductivity appears. Still, even for doping levels were the highest superconducting transition temperature, $T_{c}$, is reached, short range antiferromagnetic correlations sustain. In some cuprates, the competition between superconducting and magnetic orders causes a tendency towards electronic phase separation, especially on the underdoped side of the phase diagram. The phase coexistence of superconductivity and antiferromagnetic stripe order in the $\mathrm{La_{2-x-y}}M_{\rm y}\mathrm{Sr_{x}CuO_{4}}$ family was observed at finite temperatures by neutron scattering for $M=\mathrm{Nd}$ Tranquada et al. (1995) and $\mu$SR/NMR for $M=\mathrm{Eu}$ Klauss (2004). Subsequent intense theoretical efforts showed (Corboz et al. (2014) and references therein) that within the $t-J$ model, there is close competition between uniform $d$-wave superconductivity and various stripe states and the real ground state is very susceptible to disorder. One source of disorder in the cuprates are the dopant atoms, which is adding another level of complexity Andersen et al. (2010). In this respect, superoxygenated $\mathrm{La_{2}CuO_{4+\delta}}$ Wells (1997); Mohottala et al. (2006) is an interesting family. There the excess oxygen is intercalating in a self-organized manner into the structure of antiferromagnetic and superconducting regions Fratini et al. (2010) quite remarkably so that the magnetism and superconductivity set in at the same temperature, independent of Sr content and characteristic of optimally doped oxygen-stoichiometric $\mathrm{La_{2-x}Sr_{x}CuO_{4+\delta}}$ Udby et al. (2013). Furthermore, the concomitant magnetic propagation vector remains consistent with that of the stripe ordered cuprates.

In this paper we demonstrate a novel approach to dope $\mathrm{La_{2}CuO_{4}}$. Rather than randomly substituting lanthanum by strontium, which leads to micro-scale disorder, we replace single planes of LaO with SrO dopant planes using atomic layer-by-layer molecular beam epitaxy Baiutti et al. (2015); Wang et al. (2016). This allows a much better control over the disorder compared to bulk $\mathrm{La_{2-x}Sr_{x}CuO_{4+\delta}}$ and, at the same time, gives another degree of freedom, namely the separation of the charge reservoirs. In this way a system on the mesoscopic scale can be engineered, allowing to tune the interplay between superconducting and antiferromagnetic ground states. Figure 1 depicts a sketch for a selection of such superlattices which we call $\delta$-doped $\mathrm{La_{2}CuO_{4}}$. The distance between SrO dopant layers can be labeled $N$ which is the number of half-unit-cells separating them, and hence we will abbreviate this family by $\delta$-LCON.

Utilizing low-energy muon spin rotation techniques, we find a non-trivial enhancement of the magnetic volume fraction below the superconducting transition of the $\delta$-LCON superlattices in striking resemblance to bulk superoxygenated $\mathrm{La_{2-x}Sr_{x}CuO_{4+\delta}}$. Furthermore, it is shown that the superfluid density of $\delta$-LCON is in-line with the Uemura relation Uemura et al. (1988), namely that the superfluid density is anomalously small and proportional to $T_{c}$ on the underdoped side.

LE-$\mu$SR allows to study internal magnetic field distributions of any material Morenzoni et al. (2004), and thus is very well suited to investigate systems with a complex interplay between magnetic and superconducting ground states. By tuning the implantation energy of the positive muon, the stopping range can be varied between 5 and 300 nm (see also supplementary S.2). For this study an implantation energy $E_{\rm impl}$ was chosen such that the full muon beam stops in the center of the superlattice. The full stopping distribution can be found in the supplementary material. In order to obtain information about the superconducting state it is possible either to study the vortex state or the Meissner state. From measurements in the vortex state the magnetic field distribution is provided. For a regular vortex lattice, the second moment of the magnetic field distribution is proportional to the muon depolarization rate, $\sigma(T)$, and directly related to the magnetic penetration depth $\lambda(T)$ as (see Ref.Brandt (1988))

[TABLE]

where $\gamma_{\mu}$ is the muon gyromagnetic ratio, $\sigma_{\rm sc}=\sqrt{\sigma(T)^{2}-\sigma(T>T_{c})^{2}}$, $\Phi_{0}=2.067\cdot 10^{-15}(\mathrm{Tm}^{2})$ is the flux quantum. Figure 2 (a) shows the temperature dependence of $\sigma$ in the vortex state, and (b) presents the magnetic field probability distribution ($z$-components) of the vortex state given by the Fourier transform of the muon spin polarization function (Supplementary S.2.1.2). The marked high field shoulder is typical for a regular vortex lattice. Since the film thickness, $d\simeq 40$ nm, is small compared to the London penetration depth, $\lambda_{\rm L}$, $\lambda(T)$ in Eq.(1) represents an effective magnetic penetration depth Clem (1991). The relation between them is approximately given by $\lambda_{\rm L}^{2}(T)=c_{0}\lambda(T)\cdot d$, with $c_{0}=1/2$. Measurements on optimally doped $\mathrm{La_{2-x}Sr_{x}CuO_{4}}$ with $d=40$ nm were scaled such that we obtained the bulk data, resulting in a $c_{0}=4.3$. We chose this factor to estimate $\lambda_{\rm L}$ for the superlattices. The Uemura plot in (d) shows, that the $\delta$-LCON superlattices are in line with the hole doped cuprates. Measurements in the Meissner state (zero field cooled, $H_{\rm ext}<H_{c1}$) should show a corresponding magnetic field shift as depicted by the dash-dotted line in Fig.2 (c). The absence of the Meissner state demonstrates that superconductivity is layered in nature and likely localized around the charged SrO-layers.

In metal-insulator superlattices of the form $R\times[3\times\mathrm{La_{1.55}Sr_{0.45}CuO_{4}}+N\times\mathrm{La_{2}CuO_{4}}]$ the charge transfer effects throughout the superlattices was modeled quantitatively Loktev and Pogorelov (2008); Suter et al. (2012). This is possible since the chemical potential as function of Sr doping in $\mathrm{La_{2-x}Sr_{x}CuO_{4}}$ has been experimentally determined Ino et al. (1997). The result shows that superconducting layers along the interfaces form with an extend of about 1 UC. As for the $\delta$-LCON superlattices, the Josephson coupling in the vortex state breaks down (field geometry as in Fig.2 (a)), and the Meissner state is suppressed (as in Fig.2 (c)). These findings are further supported by the temperature dependence of $\sigma(T)$ which does not follow the expected behavior $\sigma(T)\propto[1-(T/T_{c})^{r}]$, $r\simeq 2\ldots 6$.

The situation is very reminiscent to the case of intercalated Bi2212 and Bi2202 Baker et al. (2009) where the interlayer spacing between adjacent CuO2-layers was tuned by intercalating guest molecules. Above a critical separation the Josephson coupling between adjacent layers is getting too weak and only the dipole-dipole interaction remains to align the pancake vortices. The $\sigma$ versus $T$ behavior found there is essentially identical to what is shown in Fig.2 (a). The superconducting state of the $\delta$-LCON superlattices can be summarized such that superconducting layers are forming rather localized at around the SrO-layers. The distance between these quasi-2D superconducting layers ranges from $\sim 2.6$ nm for $\delta$-LCO3 up to $\sim 7.9$ nm for $\delta$-LCO11, thus the Josephson coupling between layers is essentially suppressed and only dipolar interaction between vortices can stabilize the vortex lattice. Therefore the superconducting ground state is extremely anisotropic. A very recent infrared spectroscopy study of charge dynamics in $\delta$-LCON confirms that the superconducting state in this system is essentially two-dimensional Boris and et al. .

$\mu$SR is a well-established method to study magnetic systems Reotier and Yaouanc (1997). Reasons are that the ground state can be studied in zero applied magnetic field, and a sensitivity of about $10^{-3}\mu_{\rm B}$ per unit cell is reached. The top panels of Fig.3 (a-c) show the time evolution of the muon spin asymmetry, $A(t)=A_{0}\,P(t)/P(0)$. $A_{0}$ is the instrumental asymmetry and $P(t)$ the muon decay asymmetry (see also supplementary material S.2). For the $\delta$-LCO3 superlattice $A(t)$ shows a Gaussian like time evolution typical for a paramagnetic state where the loss of the polarization is solely governed by the dephasing of the muon spin ensemble due to the quasi static nuclear magnetic dipole fields Yaouanc and Dalmas de Reotier (2010). The very weak temperature dependence of $A(t)$ is an indication of the gradual slowing down of high frequency short range magnetic correlation still present in the system. In Fig.3 (e) the temperature dependence of the initial asymmetry, $A(t=0)$, is presented which stays constant in the whole observed temperature range. These zero field results show that a SrO-layer separation of $\sim 2.6$ nm is close enough to fully suppress the AFM ground state of the $\mathrm{La_{2}CuO_{4}}$ layers due to charge transfer. Essentially, $\delta$-LCO3 is behaving as a metal with short range AFM correlations.

$\delta$-LCO11 shows a drastically different behavior. The full time spectra shown in Fig.3 (b) change from an initially Gaussian like behavior at high temperature, towards an exponential one at low temperature. At short times and low enough temperature, spontaneous zero field precession is found (see Fig.3 (c)). This shows that $\delta$-LCO11, differently to $\delta$-LCO3, undergoes an antiferromagnetic transition. To be able to quantify the changes in the asymmetry spectra, the following zero field fit model was assumed:

[TABLE]

Since the muon stopping distribution is covering the whole superlattice (see the supplementary material), the asymmetry spectrum, $A(t)$, will be a superposition of muons experiencing a paramagnetic surrounding (close to the SrO doping layers) and muons stopping in an antiferromagnetic surrounding (far from the SrO layers). The first term describes the paramagnetic response of the sample. $\Delta$ is the width of the magnetic field distribution due to nuclear dipoles and $\lambda$ is describing the slowing down of high frequency short range magnetic correlations. The second term describes the regions which are antiferromagnetically ordered. The zero field precession signal is well described by a zero-order spherical Bessel function. The last term, $A_{\rm bkg}$, describes a background signal due to muons not stopping in the sample. For a more detailed discussion of Eq.(2) see the supplementary material. The value of the internal magnetic field $B_{\rm int}$ is a very sensitive measure of the doping level in $\mathrm{La_{2-x}Sr_{x}CuO_{4}}$ Borsa et al. (1995); Savici et al. (2002). We find $B_{\rm int}(T\to 0)=40(2)$ mT which allows to estimate an upper doping level in the antiferromagnetic regions of $x<0.01$. Furthermore, this value shows that the full electronic Cu moment of about $0.64\,\mu_{\rm B}$ is present in the antiferromagnetic state. The zero field time spectra and temperature dependencies of the asymmetries of the $\delta$-LCON, $N=7,8,9$ are found in the supplementary materials. The loss of the temperature dependent paramagnetic asymmetry $1-A_{1}(T)/A_{0}$ reflects the growth of the magnetic volume fraction. Its behavior is rather surprising as can be seen in Fig.3 (f). At about $T=150$ K, $A_{1}(T)$ starts to gradually decrease towards lower temperature. At $T_{c}$ a clear trend change can be observed, with a substantially faster increase of the magnetic volume fraction.

In order to quantify this effect, weak transverse field measurements (wTF) were carried out which allow to measure the magnetic volume fraction, $f_{\rm M}$, in an efficient and precise manner. The long-lived oscillation amplitude in the wTF asymmetry represents muons in a non- or paramagnetic environment. Fig.4 (a) shows typical wTF measurements in an applied field of $\mu_{0}H_{\rm ext}=5$ mT. The data were fitted to

[TABLE]

The magnetic volume fraction is given by $f_{\rm M}=1-A_{\rm T}/A_{0}$. For all para- and diamagnetic states $A_{\rm L}\equiv 0$. Therefore, the finite value of $A_{\rm L}$ found below $T_{\rm M}$, the $T=5$ K value is depicted in Fig.4 (a), clearly demonstrates the presence of a magnetic ground state. The low-temperature magnetic volume fraction allows to estimate the superconducting layer thickness. Assuming that the superlattices are laterally homogeneous, with no stripe-like electron patterns within the superconducting layer, a magnetic and superconducting layer thickness can be estimated, as presented in Tab.1. It shows that the upper limit for the superconducting layer thickness $d_{S}(0)\simeq\,$2-4 nm, as sketched in Fig.1. This value is consistent with the dopant profile in $\delta$-LCON as measured by high-resolution and analytical scanning transmission electron microscopy Wang et al. (2016), and hence it is not too surprising that $\delta$-LCO3 shows only marginal signs of magnetism, since $d_{N}\lesssim d_{S}(0)$.

A closer look a the temperature dependence of $f_{\rm M}(T)$ reveals a rather unusual behavior. Typically, $f_{\rm M}(T)$ shows a sharp upturn at $T_{\rm M}$ as found in various copper- and iron-based superconductors Savici et al. (2002); Udby et al. (2013); Bernhard et al. (2012); Guguchia et al. (2014). In contrast, for all $\delta$-LCON, $f_{\rm M}(T)$ increases very gradually, almost linearly, when lowering the temperature. However, at exactly $T_{c}$ there is a clear trend change, $df_{\rm M}/dT$ is strongly increasing. As shown in the supplementary material S.4, this behavior is also present when applying the external magnetic field, $H_{\rm ext}$, parallel to the superlattice layers, thus ruling out that the observed effect is related to the formation of a vortex lattice in the superconducting state. This observation suggests that the magnetic and superconducting ground states are coupled.

In Ref.Kivelson et al. (2001) the authors discuss, in the context of stripe formation, the coupling between incommensurate antiferromagnetic and superconducting order in terms of the thermodynamics of fluid mixtures. They confirm that $f_{\rm M}(T)$ may grow in the superconducting state, albeit not giving a microscopic explanation of the simultaneous onset of magnetism and superconductivity, $T_{\rm M}\approx T_{\rm c}$. Further experimental and theoretical development is necessary in order to gain a comprehensive understanding of the superconductivity-induced long range magnetic order in the $\mathrm{La_{2}CuO_{4}}$-based superconductors. A possible explanation of the trend change of $f_{\rm M}(T)$ at $T_{\rm c}$ could be related to charge redistribution between different phases caused by a lowering of the chemical potential upon the opening of the superconducting gap in the superconducting phase, a similar mechanism as discussed for the superconductivity-induced charge redistributions between different planes in the cuprates Khomskii and Kusmartsev (1992) or between different electronic bands in the multi-gap Fe-based superconductors Charnukha et al. . In $\delta$-LCON, as soon as the regions around the SrO-layers turn superconducting, for holes residing in the antiferromagnetic regions, it would energetically be favorable to migrate into the “active” superconducting layers below $T_{\rm c}$ and thus “cleaning up” the antiferromagnetic layers and leading to a stronger increase of $f_{\rm M}$. This could be possible in these systems due to the mesoscopic proximity. Whatever the explanation will prove correctly, the advantage of systems as the presented $\delta$-LCON over the homogeneously doped bulk cuprates is the much higher level of control over the spatial parameters in these systems. Further high-resolution transmission electron microscopy and resonant X-ray experiments are necessary to verify the correlation of the out-of-plane charge distribution and associated structural distortion Wang et al. (2016) with the onset of the superconductivity in $\delta$-LCON, in order to shed light on the intriguing interplay between superconductivity and long range antiferromagnetic order in the $\mathrm{La_{2}CuO_{4}}$-based superconductors.