Emergent vortices at a ferromagnetic superconducting oxide interface

Evidence for inhomogeneous magnetism in our heterostructures is immediately apparent in the temperature-dependent normal-state resistance R_xx(T). Upon cooling from room temperature, R_xx(T) passes through a minimum at low temperature in both samples (figure 2(a)), then rises logarithmically prior to the onset of superconductivity at ∼ 0.3 K. The minima are located at T_m = 10 K and 25 K for samples ANN and NAN, respectively. A logarithmic divergence in the low-temperature resistance is a signature of the Kondo effect, i.e. scattering off dilute magnetic impurities² . Within the Kondo scenario, ${\rm D}{{R}_{xx}}/{\rm D}{\rm log} T$ (figure 2(a) inset) is proportional to the concentration of free spins: for sample ANN, ${\rm D}{{R}_{xx}}/{\rm D}{\rm log} T\sim$ 60 $\Omega /{\rm logK}$, two orders of magnitude higher than the 0.45 $\Omega /{\rm logK}$ found in sample NAN. However, this does not necessarily imply that sample ANN contains a larger number of magnetic scattering centres; any parallel conduction from a non-magnetic electron gas will lead to a reduction in the measured Kondo resistance. Since the total number of carriers in sample NAN is more than an order of magnitude higher than in sample ANN, it is entirely plausible that a large rise in R(T) due to Kondo scattering is being masked by parallel transport through non-magnetic regions of the heterostructure. In agreement with this concept, data from a heterostructure with ${{n}_{2{\rm D}}}\;\gt$ 10¹⁵ cm⁻² display an even smaller Kondo anomaly (figure S2), thus suggesting that the Kondo effect arises within a minority carrier population close to the interface.

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The large difference in carrier density between samples ANN and NAN is also likely to be responsible for the variation in T_m which we observe. T_m is of the order of the Kondo temperature ${{T}_{{\rm K}}}=\frac{1}{2}{{(\Gamma U)}^{1/2}}{\rm exp} [\pi {{\epsilon }_{0}}({{\epsilon }_{0}}+U)/\Gamma U]$, where ₀ is the energy of the magnetic impurity relative to the Fermi level E_F, Γ is the width of this impurity level and U the Coulomb repulsion [28]. Γ should not vary significantly as the carrier density increases, and we expect U to decrease due to enhanced screening. We therefore anticipate that the higher T_K in sample NAN is due to a rise in ₀ from the increased carrier density and the corresponding rise in E_F. Importantly, this suggests that the same localized states are responsible for magnetism in both our heterostructures. Although n_2D in sample ANN lies close to the Lifshitz critical density [29, 30], the Hall resistance varies linearly with magnetic field (figure S1a) and shows no signs of multiband transport. This implies that a clear majority of carriers in sample ANN occupy the lowest-lying Ti 3d_xy band. Since the localized magnetic states must lie below the Fermi level, our data favour a d_xy origin for ferromagnetism.

Both samples ANN and NAN are superconductors, with critical temperatures T_c = 0.28 K and 0.31 K, respectively. The resistive transitions in a range of magnetic fields perpendicular and parallel to the interface are shown in figure 2(b) and (c). As expected, the resistance does not vanish even at 0.035 K in zero-field, due to the series resistance from the LaAlO₃ layer. Assuming that the interface exhibits homogeneous superconductivity, we obtain R(LaAlO₃) = 910 Ω and 3.4 Ω for samples ANN and NAN, thus highlighting the high O²⁻ vacancy density in the LaAlO₃ layer of sample NAN which is responsible for its large n_2D [31]. It should be noted that we cannot exclude inhomogeneities at the superconducting interface from contributing to our residual resistance. In fact, interfacial inhomogeneity should be expected due to cation disorder [32], the tendency of the d_xy electrons to localize and form ferromagnetic zones [24] and dielectric inhomogeneity for ${{V}_{g}}\gt 0$ [33]. For sufficiently high ferromagnetic zone densities close to a shallow superconducting channel, a percolative zero-resistance current path may vanish. Furthermore, in sample B it is plausible that clusters of oxygen vacancies at the AlO₂ surface could locally overdope the interface beyond the maximum carrier density for superconductivity; theory also suggests that such clustering will enhance ferromagnetism [18, 19, 34]. Emergent inhomogeneity is therefore a natural consequence of vacancy-doping the LaAlO₃/SrTiO₃ interface and in no way reflects negatively on our results.

We determine the thickness of the superconducting channels at our interfaces using two distinct fitting techniques to our temperature-dependent perpendicular and parallel upper critical fields ${{H}_{c2\bot ,//}}(T)$ (figure 2(d)). Within linearized (GL) theory [35], ${{H}_{c2\bot }}(T)={{\Phi }_{0}}/2\pi {{\xi }^{2}}(T)$ and ${{H}_{c2//}}(T)={{\Phi }_{0}}\sqrt{3}/\pi \xi (T)d$, where Φ₀ is the magnetic flux quantum. This yields GL coherence lengths $\xi (0)=52\pm 2$ nm, $60\pm 2$ nm and superconducting channel widths $d=18\pm 1$ nm, $9\pm 1$ nm for ANN and NAN respectively. A scaling analysis for 2D superconductors [25, 36] provides an alternative means to determine d using $H_{c2//}^{2}(T)$ = $\frac{\pi {{\Phi }_{0}}}{2{{d}^{2}}}$ ${{H}_{c2\bot }}(T)$. This yields superconducting layer thicknesses $d=16\pm 1$ nm and $9\pm 1$ nm for samples ANN and NAN, respectively, which are in excellent agreement with the values from GL theory. The larger d observed in sample ANN may be caused by a partial suppression of superconductivity very close to the interface in sample NAN due to its higher carrier density. Here, it is important to remember that the as-grown vertical charge distribution profile is a key factor in determining the absolute conducting layer thickness and may vary significantly between heterostructures [7, 25, 37]. The crucial point is that $d\ll \xi$ in both ANN and NAN-type heterostructures; our samples are therefore 2D superconductors.

It is of paramount importance to determine whether LaAlO₃/SrTiO₃ interfaces host a conventional superconducting phase (where we define 'conventional' as type-II thin film s-wave superconductivity). The argument for unconventional superconductivity stems from the presence of a strong Rashba spin–orbit coupling (RSOC) from broken inversion symmetry at the interface [38], which may potentially stabilize a helical FFLO state with a maximal T_c for non-zero in-plane fields $\left| {{H}_{//}} \right|\gt 0$ [15]. Since the RSOC is inversely proportional to n_2D [7], we focus our search for exotic superconductivity on sample ANN. Figure 3(a) shows the angle-dependent upper critical field ${{H}_{c2}}(\theta )$, accurately described by the Tinkham model based on fluxoid quantization in conventional superconducting films with $d\ll \xi$ [39]:

$$\begin{eqnarray}&&\left| \frac{{{H}_{c2}}(\theta ){\rm sin} \theta }{{{H}_{c2\bot }}} \right|+{{\left( \frac{{{H}_{c2}}(\theta ){\rm cos} \theta }{{{H}_{c2//}}} \right)}^{2}}=1\end{eqnarray} \tag{ 1 }$$

Even at T = 0.1 K, ${{H}_{c2//}}$ exceeds the Pauli limit ${{H}_{{\rm P}}}\equiv 1.84\;{{T}_{c}}$ = 0.52 T; this is consistent with the strong RSOC expected at the interface, as spin–orbit scattering is known to raise H_P [40, 41]. Since ${{H}_{c2//}}\gt {{H}_{{\rm P}}}$, GL theory only provides an upper limit rather than an absolute value for d. However, the deviations between calculated and true d are small [35] and do not impact our discussion since superconductivity remains confined within 20 nm of the interface. In figure 3(b), we track ${{T}_{c}}({{H}_{//}})$ for sample ANN but find no peak at ${{H}_{//}}\gt 0$: the data closely follow the GL model. To confirm this, we examine the MR ${{R}_{xx}}({{H}_{//}})$ at T = 0.25 K (just below ${{T}_{c}}(0\;{\rm T})$) for both samples in figure 3(c): a maximum T_c at ${{H}_{//}}\gt 0$ will create a point of inflection or local minimum in ${{R}_{xx}}({{H}_{//}})$, as depicted in figure 3(d). No such features are visible, and we therefore find no support for a helical FFLO phase in our heterostructures. It should be noted that a maximum T_c for ${{H}_{//}}\gt 0$ has been reported for LaAlO₃/SrTiO₃ films grown by molecular beam epitaxy (MBE) [42], although it remains unclear whether this result is due to helical superconductivity or some other mechanism (such as quenched phase fluctuations, a field-dependent coherence length or another as-yet unidentified consequence of the strong RSOC). This difference between PLD- and MBE-grown films merits further attention, since it suggests that the 10–20 nm thick superconducting layers in PLD-grown heterostructures may be too broad for the broken symmetry of the interface to stabilize any exotic order parameter. However, ${{R}_{xx}}({{H}_{//}})$ in our samples does reveal one unusual feature of interest: a small non-linearity at low field (leading to a peak in ${\rm D}{{R}_{xx}}/{\rm D}{{H}_{//}}$), which we will demonstrate to be a signature of ferromagnetism.

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Figure 4 displays in-plane MR loops for both films. Hysteresis is observed at low fields ($|{{H}_{//}}|$ $\lt \;0.15$ T), with peaks in ${{R}_{xx}}({{H}_{//}})$ at negative/positive ${{H}_{//}}$ after sweeping down/up from positive/negative ${{H}_{//}}$. This pattern is distinct from the hysteresis seen in granular superconductors [43]; we may therefore immediately absolve granularity from responsibility for our data. Although hysteretic MR is generally a signature of ferromagnetism, the collapse of the hysteresis above T_c indicates that a combination of superconductivity and ferromagnetism must be responsible for our MR peaks. It should be noted that this does not imply that superconductivity is somehow responsible for the emergence of ferromagnetism. Instead, the small MR hysteresis generated by ferromagnetism in the normal state ($T\gt {{T}_{c}}$) falls below our experimental resolution, and we are therefore unable to determine the Curie temperature of the ferromagnetic phase.

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Non-zero resistance below T_c for a superconductor in a magnetic field implies a loss of phase coherence, caused by mobile flux vortices. A vortex can be considered as a tube of diameter $\sim \;2\xi$: since our anisotropic MR indicates that $d\ll \xi$, the superconducting channels are too shallow to allow vortex penetration parallel to an in-plane magnetic field. Instead, any vortices must be oriented within $\pm \;{{{\rm sin} }^{-1}}\frac{d}{2\xi }$ of the normal to the interface ($\lt$ 11° in our samples). Interpreting our MR hysteresis therefore requires us to identify a means of generating out-of-plane vortices within the in-plane field configuration used in our experiments.

If we assume that the applied magnetic field ${{H}_{//}}$ lies perfectly in-plane, then the only physically viable out-of-plane flux sources in LaAlO₃/SrTiO₃ are dipole fields from discrete ferromagnetic zones, polarized in-plane (figure 5(a)). To pass through the superconducting channel, the out-of-plane components of these dipole fields must be quantized. Arrays of vortices and antivortices therefore form around each pole; their size and density depend on the shape and total moment of each ferromagnetic zone. This situation is analogous to artificial 2D superconductor/ferromagnet nanodot heterostructures [16, 17].

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In addition to these dipole fields, we must also consider 'stray' perpendicular fields, both from the Earthʼs magnetic field and any misalignment of ${{H}_{//}}$ with respect to the interfacial plane. In a bulk three-dimensional (3D) superconductor, such fields would lie far below the lower critical field ${{H}_{c1}}=\frac{{{\Phi }_{0}}}{4\pi \lambda _{{\rm L}}^{2}}{\rm ln} \frac{{{\lambda }_{{\rm L}}}}{{{\xi }_{0}}}$ (where ${{\lambda }_{{\rm L}}}$ is the London penetration depth) and hence be completely shielded. However, in a superconducting thin film, the screening of a field applied normal to the film is reduced since ${{\lambda }_{{\rm L}}}$ is replaced by the Pearl depth ${{\lambda }_{\bot }}=\lambda _{{\rm L}}^{2}/d\approx {{m}_{e}}/{{\mu }_{0}}{{n}_{s}}{{e}^{2}}d$, where m_e is the electron mass and ${{n}_{s}}\sim {{n}_{2{\rm D}}}/d$ is the superfluid density [44]. ${{\lambda }_{\bot }}\gg {{\lambda }_{{\rm L}}}$, leading to a strong suppression of H_c1. As an example, consider sample ANN: using ${{n}_{2{\rm D}}}\;=$ 2.3 × 10¹³ cm⁻² and $d\;=$ 18 nm we obtain ${{\lambda }_{\bot }}=$ 1.2 × 10⁻⁴ m and ${{H}_{c1}}\;=$ 9.6 × 10⁻⁸ T. This represents a reduction by three orders of magnitude compared with ${{H}_{c1}}\;=$ 3.3 × 10⁻⁴ T for similarly doped bulk SrTiO₃.

Such an exceptionally low H_c1 enables almost any perpendicular flux source to penetrate the superconducting channel in the form of vortices. The Earthʼs magnetic field strength is ∼ 5 × 10⁻⁵ T, corresponding to an equilibrium vortex spacing of 6.9 μm: since our cryostat does not incorporate any background magnetic shielding, this field alone could generate up to 2600 vortices between the voltage contacts of our Hall bars. Furthermore, we estimate a maximum error of ± 0.1° in the alignment of our $ab$ planes with ${{H}_{//}}$; the resultant out-of-plane flux ${{H}_{\bot }}={{H}_{//}}{\rm sin} 0.1$ exceeds H_c1 at ${{H}_{//}}\gt$ 5.5 × 10⁻⁵ T. At ${{H}_{//}}=$ 0.01 T the 'misalignment' field ${{H}_{\bot }}$ is already capable of creating a further 900 vortices in our Hall bars, and the vortex density will continue to rise linearly with ${{H}_{//}}$. These 'misalignment vortices' will become pinned at the poles of discrete ferromagnetic regions [45], above or below the superconducting channel (figure 5(b)). Similar polar pinning has previously been imaged in Pb/Co nanodot hybrids [46]. As the dipole size increases, a crossover occurs from pinning misalignment vortices to independently creating vortex/antivortex pairs (figure 5(a)). Misalignment vortices are still pinned by larger dipoles, although the pole at which the vortex is pinned oscillates as the dipole moment increases [47].

We therefore have two possible sources of vortices in LaAlO₃/SrTiO₃: (a) misalignment vortices from stray perpendicular fields and (b) the out-of-plane components of in-plane ferromagnetic dipole fields. Recently, a 'dip-hump' MR feature was reported in W nanowires and perforated TiN films [48], which at first glance appears similar to our peaks in ${{R}_{xx}}({{H}_{//}})$. This feature is caused by a giant re-entrant pinning due to vortex confinement in nanoscale device geometries. One might therefore wonder whether a similar re-entrant pinning could occur for misalignment vortices in LaAlO₃/SrTiO₃, thus removing the requirement for ferromagnetic dipoles to be present. However, a comparison of our data with this re-entrance scenario reveals a crucial difference: the MR dip-hump in [48] is permanently present, independently of the field orientation, sweep direction and magnetic history of the sample. In stark contrast, the hysteretic peak in our ${{R}_{xx}}({{H}_{//}})$ data is only observed after reversing the orientation of an in-plane field. Subsequently, it does not re-appear until the field orientation is reversed once more; the absence of any peak from the Up and Down sweep data for ${{H}_{//}}\gt 0$ in figure 4(b) provides an excellent example of this acute sensitivity of ${{R}_{xx}}({{H}_{//}})$ to the magnetic history of the interface. This hysteresis pattern is incompatible with geometric pinning; instead, it is reminiscent of the magnetisation of a hard ferromagnet with large coercive and remanent fields. The MR hysteresis at the LaAlO₃/SrTiO₃ interface therefore corresponds to the reversal of the polarization of an array of ferromagnetic dipoles, which we will discuss in detail later in this paper.

First, we determine the location of these dipoles with respect to the superconducting channel using simple symmetry arguments. If the dipoles whose presence we have deduced are buried inside the channel (figure 5(a), centre panel) then one might expect a vortex/antivortex pair to form at each pole. However, this cannot occur due to the 2D nature of the superconducting channel: $d\ll \xi$, and so the channel is essentially homogeneous normal to the interface. The vortex–antivortex pair therefore self-annihilates, leaving a purely horizontal field within the channel. Similarly, a small in-plane dipole located inside the superconducting channel cannot pin a vortex at either of its poles, since at the south pole the attractive force from the field below the dipole is balanced by a repulsion above it (and vice versa for the north pole). Furthermore, while clearly revealing a competitive interaction, the minor destructive effect of polarized ferromagnetic zones on superconductivity (visible at ${{H}_{//}}=0$ in figures 4(a) and (b) as well as figure 7(b)) does not support a large dipole population inside the channel.

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If the moments are instead distributed symmetrically above and below the channel (figure 5(a), left panel), then annihilation still tends to occur due to the attractive force between vortices and antivortices, leaving little quantized flux in the channel. Only an asymmetric ferromagnetic zone distribution across the channel creates stable vortex/antivortex pairs (right panel); therefore, ferromagnetism must be confined to either the top or the base of the channel.

To differentiate between these two scenarios, i.e. to determine whether the dipole moments are located above or below the superconducting channel, we directly examine the electron gas beneath this channel for any evidence of magnetism. In sample NAN, SdH oscillations develop for ${{H}_{//}}\;\gt$ 2.5 T (figures 4(b) and 5(c)). Since (i) $d\;\leqslant$ 20 nm, (ii) our field-effect doping proves that the SrTiO₃ is a good dielectric and (iii) no oscillations are visible for ${{H}_{\bot }}\lt 4$ T, there is no bulk-like 3D electron gas extending throughout the SrTiO₃ substrate (such as that seen in LaAlO₃/SrTiO₃ films grown at very low O₂ pressure [49]). Instead, our observed SdH effect is characteristic of a low carrier-density conducting 'tail' lying below the superconducting channel [50], penetrating the SrTiO₃ to a depth of at least twice the cyclotron radius ${{r}_{g}}=\hbar {{k}_{F}}/eH$ (k_F is the Fermi wave vector). At ${{n}_{2{\rm D}}}=6.9\times {{10}^{14}}$ cm⁻², the SdH frequency F = 25 T; assuming Fermi surface sphericity for simplicity, ${{k}_{F}}=\sqrt{2\pi F/{{\Phi }_{0}}}$ (from the Onsager relation) and we obtain 2r_g = 88 nm. This is much larger than the 9 nm superconducting channel at ${{V}_{g}}\;=$ 0 in sample NAN; therefore, the majority of the 'tail' generating the SdH oscillations must lie below the superconducting channel.

SdH oscillations display clear signatures of both the scattering and magnetism within the electron gas responsible for their formation. In ferromagnetic materials, the SdH effect exhibits peak amplitude 'beating' and field-reversal asymmetry [51], while in dirty SrTiO₃ showing Kondo behaviour, scattering from localized electrons has been predicted to suppress the observation of SdH oscillations until $H\;\gtrsim$ 20–30 T [52]. From figure 5(c), our oscillations are field-reversal symmetric, 'beating'-free and emerge at merely 2.5 T. This implies that the electron gas below the superconducting channel is clean, non-magnetic and homogeneous: it cannot host ferromagnetic dipoles. To support this conclusion, we also consider the magnetic field distribution from any dipoles lying below the superconducting channel: the vertical component of the dipole field will be quantized as it passes through the channel, strongly distorting the flux profile when viewed from above. No evidence for any such distortion has been seen by scanning SQUID microscopy [9, 53, 54]. We deduce that our SrTiO₃ substrates are non-magnetic: the ferromagnetic dipoles must therefore be confined to the interface, above the superconducting channel. Here, they create and/or pin vortices as depicted in figure 5(a) (right panel) and 5(b).

We briefly consider the other possible sources of vortices in LaAlO₃/SrTiO₃ which have previously been discussed in the literature. Spatial inhomogeneities in the RSOC or in-plane polarization can in principle create vortices in helical superconductors [38], but—apart from the fact that we find no evidence for helical superconductivity—these only emerge above a critical field and are hence incompatible with our low-field hysteresis. It has been suggested that Bloch domain walls in a continuous ferromagnetic layer may generate vortices, which then propagate when the magnetic field is swept [55]. However, this scenario is unlikely for several reasons. Firstly, shape anisotropy is a key factor in domain formation for thin ferromagnetic layers, and the walls should be of Néel rather than Bloch-type, i.e. their magnetization points in-plane and the out-of-plane flux is greatly reduced. A recent magnetic force microscopy study of carrier-depleted LaAlO₃/SrTiO₃ has indeed confirmed the presence of Néel-type walls [56]. Secondly, there is no evidence in the literature for a continuous ferromagnetic layer coexisting with superconductivity; in contrast, SQUID microscopy [9] and theoretical considerations [34] both indicate inhomogeneous ferromagnetism. Thirdly, we acquire R_xx(H) data in stable fields with our superconducting magnet in persistent mode, i.e. ${\rm D}H/{\rm D}t=0$, so there is no driving force for domain-wall propagation (and our ac current negates spin–torque transfer effects on the domains). Finally, equilibrium domain formation in a ferromagnet is driven by the requirement to minimize the total free energy by balancing the exchange, anisotropy, magnetostatic and Zeeman energies. The domain structure therefore evolves with the applied field, which should generate mobile vortices (and hence a rise in the MR) regardless of the field sweep direction. This pattern is not present in our data: no peak or anomaly in ${{R}_{xx}}({{H}_{//}})$ is observed upon sweeping large in-plane fields down to zero. We therefore find no evidence for mobile domain walls in our LaAlO₃/SrTiO₃ heterostructures.

Having confirmed that a series of discrete ferromagnetic dipoles above the superconducting channel is responsible for vortex formation in LaAlO₃/SrTiO₃, we now consider the physical mechanism driving the hysteretic MR in detail. Peak formation in ${{R}_{xx}}({{H}_{//}})$ arises due to ferromagnetic polarity reversal, a process illustrated in figure 6(a). This mainly occurs via dipole rotation rather than domain wall propagation due to the limited zone size (⪅ 10 μm from SQUID data), the presence of hysteresis up to $\left| H \right|$ = 0.15 T (implying a large coercive field H_coerc) and flux conservation within the channel. During repolarization, a torque ${\bf m}\times {\bf H}$ acts on ferromagnetic zones, where ${\bf m}$ is the dipole moment. The vortex/antivortex arrays around the poles are rotated through 180° in-plane: while mobile, the vortices develop an electric field in their cores and dissipate energy, thus creating a hysteretic peak in ${{R}_{xx}}({{H}_{//}})$. Dipoles which are too small to independently generate vortex/antivortex pairs contribute to the observed hysteresis by pinning misalignment vortices at their poles; these vortices are also rotated around their respective dipoles upon ferromagnetic polarity reversal. However, all vortex motion is impeded by pinning due to impurities within the underlying superconducting channel, which broadens the MR peak and increases the effective coercive field of the dipoles. Within the framework of the Anderson flux creep model [57], we may write:

$$\begin{eqnarray}&&{{\nu }_{{\rm depin}}}\;\propto \;{{{\rm e}}^{-(U-|m||H|+{\bf m}\cdot {\bf H})/{{k}_{B}}T}},\end{eqnarray} \tag{ 2 }$$

where ν_depin is the depinning frequency, U is the pinning potential, $|H|\gt {{H}_{{\rm coerc}}}$, and we disregard depinning by the 'shaking' effects of our ac current. ν_depin scales linearly with the induced electric field and hence ${{R}_{xx}}({{H}_{//}})$. Assuming an infinite vortex supply, constant U and ferromagnetic polarization antiparallel to ${{H}_{//}}$, an increase $\Delta H$ will generate $\Delta {{R}_{xx}}({{H}_{//}})\;\propto \;{{{\rm e}}^{2mH}}\Delta H$. Such assumptions are unrealistic, since our samples contain finite numbers of vortex-inducing ferromagnetic dipoles which may be pinned at arbitrary angles to ${{H}_{//}}$ during repolarization; we also expect local variation in the superconducting channel pinning potential from inhomogeneous defect distributions. Nevertheless, this relation permits a qualitative understanding of the temporal evolution of our hysteretic peaks (figure 6(b)). Compared with a reference data-set (sweep 1) acquired at field increment ${{\delta }_{H}}\;=$ 0.005 T, a long pause in the data acquisition at ${{H}_{//}}$ = 0.005 T (sweep 2) has little effect on ${{R}_{xx}}({{H}_{//}})$, since ${{H}_{{\rm coerc}}}\;\gt$ 0.005 T for most of the dipoles and depinning is limited. In contrast, measuring with no pause but a smaller increment ${{\delta }_{H}}/3$ (sweep 3) yields a larger drop in ${{R}_{xx}}({{H}_{//}})$, since fewer vortices are depinned at each data point. This dependence of ${{R}_{xx}}({{H}_{//}})$ on the measurement duration cannot be explained within a geometric pinning scenario [48]. Above ${{H}_{//}}\;\sim$ 0.15 T, ferromagnetic repolarization is complete: ${{R}_{xx}}({{H}_{//}})$ therefore falls to its background level.

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For completeness, we also consider the MR in a field perpendicular to the interface. Small fields ${{H}_{\bot }}\lt {{H}_{c2\bot }}$ will create a high density of vortices pinned at the poles of the ferromagnetic dipoles; however, reversing the direction of ${{H}_{\bot }}$ does not change the in-plane orientation of the dipoles. Increasing the field, a combination of large magnetic anisotropy and strong vortex pinning implies that the dipoles cannot be rotated out-of-plane for ${{H}_{\bot }}\;\lt \;{{H}_{c2\bot }}\;\sim$ 0.1 T. Therefore, no dipole-induced vortex motion can occur in perpendicular fields, and hence no large hysteretic peaks of the type seen in ${{R}_{xx}}({{H}_{//}})$ are expected for ${{H}_{\bot }}$. In agreement with this prediction, no hysteretic peaks are indeed visible in our measured ${{R}_{xx}}({{H}_{\bot }})$ (figure S1b), although a very small hysteresis develops at fields close to zero, comparable to the typical remanence in superconducting magnets. The absence of any reversible (i.e. independent of the field sweep direction) dip-hump feature in our ${{R}_{xx}}({{H}_{\bot }})$ data confirms that geometric pinning does not play a significant role in the vortex dynamics of our heterostructures [48].³ Furthermore, the lack of any peaks in ${{R}_{xx}}(|{{H}_{\bot }}|\;\gt \;0)$ is entirely consistent with our conclusion that the hysteretic peaks in ${{R}_{xx}}({{H}_{//}})$ below T_c are primarily a consequence of the interaction between vortices and in-plane ferromagnetic dipoles.

Applying a back-gate voltage ${{V}_{g}}\gt 0$ to a LaAlO₃/SrTiO₃ heterostructure (see figure 1(a)) should increase the superconducting channel thickness d, since electrons are simultaneously injected at the interface and 'decompressed' into the SrTiO₃ substrate. We confirm this quantitatively in figure 7(a), where we plot ${{H}_{c2//,\bot }}$ for sample NAN at ${{V}_{g}}\;=$ 350 V. G-L and scaling analyses both indicate that d is approximately doubled from its ${{V}_{g}}\;=$ 0 V value of 9 nm.

SQUID microscopy studies have indicated that gating the LaAlO₃/SrTiO₃ interface to modulate its carrier density does not have an effect on the density of ferromagnetic inclusions [53, 54]. This suggests that the total number of vortices within the superconducting channel is unlikely to change with n_2D. However, the Pearl magnetic penetration depth ${{\lambda }_{\bot }}$ will fall as n_2D increases, leading to a decreased pinning efficiency. Our time-dependent data (figure 6(b)) show that depinning controls the hysteresis in ${{R}_{xx}}({{H}_{//}})$, so we anticipate some variation in the shape of the hysteretic peaks upon changing V_g. This is indeed observed: the peaks at ${{V}_{g}}\;=$ 350 V are broader and begin to form at smaller $\left| {{H}_{//}} \right|$ than at 0 V (figure 7(b)).

We attribute the onset of hysteresis at lower fields to weaker pinning due to a reduction in ${{\lambda }_{\bot }}$, since n_2D has risen from 6.9 × 10¹⁴ to 2.4 × 10¹⁵ cm⁻² (figure S1a). The fall in ${{\lambda }_{\bot }}$ is partially offset by the rise in d, which increases the number of available pinning centres in the superconducting channel. In turn, this augments the probability of pinning any given vortex during the rotation of its respective ferromagnetic dipole, which will broaden the hysteretic peaks. However, the strongest pinning is likely to originate from isolated structural defects close to the interface and is thus independent of ${{\lambda }_{\bot }}$ and d. The maximum field at which hysteresis is observed should therefore remain similar regardless of the applied V_g, which is in agreement with our data.