Atomic-scale tailoring of spin susceptibility via non-magnetic spin-orbit impurities

Following the discovery of topological insulators, there has been a renewed interest in superconducting systems that have strong spin-orbit (SO) coupling. Here we address the fundamental question of how the spin properties of a otherwise spin-singlet superconducting ground state evolve with increasing SO impurity density. We have mapped out the Zeeman critical field phase diagram of superconducting Al films that were deposited over random Pb cluster arrays of varying density. These phase diagrams give a direct measure of the Fermi liquid spin renormalization, as well as the spin orbit scattering rate. We find that the spin renormalization is a linear function of the average Pb cluster-to-cluster separation and that this dependency can be used to tune the spin susceptibility of the Al over a surprisingly wide range from 0.8$\chi_0$ to 4.0$\chi_0$, where $\chi_0$ is the non-interacting Pauli susceptibility.

F or much of the long history of superconductivity spin-orbit effects were never at the forefront of the larger phenomenological framework. This was certainly true of development of BCS theory. Spin-orbit (SO) scattering does not break time reversal symmetry, nor does it disrupt the pairing amplitude 1 . However, it can dramatically alter the spin states of the system by destroying the spin-singlet symmetry of the ideal BCS ground state 2 . Although this was well understood by the late 1960's, the effects of spin mixing in strong SO scattering systems proved to be somewhat subtle and difficult to measure. One of its earliest reported manifestations was the Knight shift in Hg 3 . In contrast to these inauspicious beginnings, SO coupling is now believed to be a necessary component of several classes of non-conventional superconductors. These include correlated systems having noncentrosymmetric crystal structures such as CePt 3 Si 4,5 and BiPd 6,7 , as well as possible topological superconductors such as Cu x-Bi 2 Se 3 8 . The interplay between SO coupling and superconductivity is also crucial for the possible realization of Majorana fermions in proximitized nanowires 9 .
Notwithstanding the resurgent interest in the SO underpinnings of non-centrosymmetric and topological superconductivity, details of how a otherwise low SO superconductor accommodates a spin-orbit impurity remains unclear 10 . This is particularly true in the case of an interacting system for which Fermi-liquid (FL) renormalizations of basic electronic properties such as the effective mass and spin susceptibility must be included. In this report, we present Zeeman-limited critical field studies of ultra-thin superconducting Al films that were grown over well-separated Pb clusters. We show that the Pb clusters not only serve as spin-orbit impurities but also have a profound effect on the e−e interaction renormalization of the spin susceptibility as described in FL theory [11][12][13] .

Results
Parallel critical field measurements. The temperature dependence of the parallel (to the film surface) critical magnetic field was measured in 15 monolayer-thick superconducting Al films having varying densities of embedded Pb clusters. The clusters were typically well defined, each consisting of only a few Pb atoms. Their average separation d was measured directly from an in situ high resolution scanning tunneling microscope. The thickness of the Al films used in this study was much less than superconducting coherence length, ξ~300 Å. In this limit the orbital response to an applied parallel magnetic field is suppressed and the critical field transition is mediated by the Zeeman splitting of the conduction electrons 2 . The Zeeman-limited phase diagram gives one a direct probe of the spin properties of the superconducting condensate. If the SO scattering rate is low, as it is in pristine Al films, the low temperature first-order critical field transition is expected to be near the Clogston-Chandrasekhar 14,15 T c is the zero temperature gap, and μ B is the Bohr magneton 16 .
The spin properties of the BCS condensate are primarily influenced by: (1) Landau FL renormalization the spin susceptibility 13 and (2) spin-orbit scattering which inhibits spin polarization. The Zeeman critical field, itself, is also influenced by these mechanisms, as well as by the reduced film thickness t/ξ, where ξ is the Pippard coherence length 17 . The quasi-classical theory of weak-coupling superconductivity 18,19 (QCTS), as applied to the Zeeman-limited superconductivity [20][21][22] , captures these mechanisms via the corresponding dimensionless parameters 23 : the antisymmetric FL G 0 , the spin-orbit b = ℏ/(3τ so Δ 0 ), where τ so is the spin-orbit scattering time, and the orbital pair breaking c ∝ Dt 2 , where D is the electron diffusivity and t is the film thickness. G 0 is a measure of the renormalization of the spin susceptibility of an interacting Fermi gas, χ = χ 0 /(1 + G 0 ), where χ 0 is the spin susceptibility of a free Fermi gas of effective mass m * .
Numerous studies of the Zeeman critical field transition in ultra-thin Al and Be films have shown that these two light elements have a very low intrinsic spin-orbit scattering rate 24,25 and are true spin-singlet superconductors. Consequently, they make ideal candidates for systematic studies of the effects of SO scattering with a controllable SO impurity density. Early Zeeman critical field studies of Be and Al films showed that one could introduce SO scattering by simply coating them with submonolayer coverages of heavy metals (Z = Au, Pt, or Pb). These studies showed two primary effects on the critical transition. First, SO increases the Zeeman critical field well beyond the Clogston-Chandrashekar limit, due to the fact that SO scattering inhibits the polarization of the spins. Second, the presence of even modest SO scattering drives the transition from first-order to secondorder 26 .
The parallel critical field transitions of 15 monolayer (ML) Al films with varying cluster densities is shown in Fig. 1. The meanfree-path and coherence length of the films were determined by perpendicular critical field measurements as described in ref. 16 . The low temperature sheet resistances of films~10 Ω were insensitive to the cluster density for the range of coverages used in this study. The vertical dashed line represents the Clogston-Chandrasekhar (H CC ) critical field for the ideal case of b = G 0 = c = 0. The critical field of the pure Al film is slightly higher than H CC due to the fact that the SO and FL parameters are not exactly zero in Al. Note that the critical field increases substantially with decreasing cluster separation. We also include the critical field curve of a 15 ML Al that was deposited on 1 ML of Pb. It's critical field was H c~8 T, thus only the tail of critical field trace appears in the plot. In the analysis that follows we define the critical field by the midpoint of the transition.  Numerical analysis of the phase diagrams. In Fig. 2 we plot the temperature dependence of the Zeeman critical field H c of an Al film with a Pb-cluster separation of 3.8 nm. These data represent the Zeeman-mediated phase diagram of the film. The solid line is a best least-squares fit to QCTS where only the SO parameter b and the FL parameter G 0 were varied. The thickness parameter was previously determined from a pure 15 ML Al film. The details of the fitting procedure and its underlying assumptions has been published elsewhere 16 . Here we mention that as long as the phase transition remain second-order, as is the case for the data in Fig. 2, the critical field is obtained as solution to the equation 20 Note that this fitting procedure captures the salient features of the phase diagram. In contrast, if we fix the FL parameter to its pure Al film value G 0 ≃ 0.18 and only vary b, then the fit is much worse, as indicated by the dashed line in Fig. 2. This was also recognized in the early work of Tedrow and Meservey 26 who attempted to fit the phase diagram of Pt-coated Al films, where Pt was used to induce SO scattering. They found that for relatively large values of b, the measured critical fields were in poor agreement with theory, however they did not include FL corrections in their analysis.
Shown as triangle symbols in Fig. 3 are the values of the SO parameter b obtained from samples of varying Pb-cluster density as a function of the cluster coverage on the Si substrate. For point-like impurities with uncorrelated positions the scattering rate, and hence b, is expected to be proportional to the impurity density. Therefore, b should scale as the Pb coverage, which is itself proportional to 1/d 2 . The solid line represents an power-law fit to the data and gives an exponent of 1.1. We can compare our SO scattering rates with those obtained via weak localization measurements on thin Mg films dusted with sub-monolayer coverages of Au 27 . Of course, Mg films do not superconduct but we can nevertheless extract an effective b for the Mg/Au data by simply multiplying the reported SO scattering rates by ℏ/(3Δ 0 ), where Δ 0 is the average gap energy of our Al films. These data are depicted by the circle symbols in Fig. 3. The Mg/Au exhibits an exponent of 1 indicating that the SO scattering rate is simply proportional to the coverage. In our case, the Pb clusters are not point-like and their positions, while random, display some correlations over the length scale d. Such correlations can play an important role in the mobility in doped semiconductors and graphene and could perhaps contribute to the slightly super-linear dependence of b on Pb coverage 28 . Nevertheless, the overall agreement between these two very different experimental probes of heavy element SO scattering is reassuring. Perhaps, the most surprising finding in these analyses is that the antisymmetric FL parameter G 0 is also dramatically affected by the Pb clusters. In fact, in order to fit phase diagrams like that in Fig. 2, we must treat G 0 as an effective free parameter, which we denote with G 0 eff to distinguish it from that of the zero SO FL theory 13 . We should point out that increasing G 0 increases the theoretical critical field, which is true of b as well. However their influences have somewhat different temperature dependencies 16,22 . Consequently, their relative contributions can only be de-convolved by fitting across the entire phase diagram. In Fig. 4 we plot G 0 eff as a function of d. The relative magnitude of the change in G 0 eff with decreasing d is non-perturbative. Indeed, our effective approach is likely not applicable at small cluster separations, since one would expect a ferromagnetic instability at G 0~− 1. However, the analysis is sound in the perturbative limit jG 0 eff j ( 1 and our data suggests that G 0 eff changes sign at an average separation of d~7.5 nm, corresponding to a Pb coverage of 4 × 10 −3 ML. Specifically, the spin correlations change from antiferromagnetic-like to ferromagnetic-like at this critical separation.

Discussion
The origin of the shift in G 0 eff toward ferromagnetic spin correlations is unknown 29 . It is interesting that G 0 eff is a linear function of cluster separation and not, in contrast to b, a function of cluster density. The linear dependence may represent a proximity effect in which the local FL environment of Pb clusters influences the average G 0 eff of the surrounding Al in a manner that is similar to proximity-induced exchange fields in superconductingferromagnet bilayers 30,31 . Unfortunately, in contrast to Al, bulk Pb is diamagnetic. Consequently there is no straightforward way to independently probe the spin susceptibility and corresponding FL environment of the Pb islands.
Another possibility is that the FL spin renormalization in the Al films is transformed from the single channel value of pristine Al to a more complex effective value in the presence of SO scattering. In a low SO FL the renormalization of the spin susceptibility by e-e interactions only depends upon a single parameter, G 0 . However, in the presence of generic spin-orbit couplings a more complicated relationship between spin susceptibility and the strength of the various spin-dependent interaction channels emerges 32 . It may be possible to calculate G 0 eff using ab-initio methods, similarly to e.g., the treatment of Ni clusters magnetism in Ag 33 . Alternatively, one may be able to extract G 0 eff from a Kondo lattice-like model. It is known that the impurity spin susceptibility in these models can be affected by the Ruderman-Kittel-Kasuya-Yosid (RKKY) interaction 34 .
In summary, we have exploited the Zeeman critical field of ultra-thin superconducting Al films to investigate the evolution of their spin susceptibility as a function of imbedded Pb-island separation. This technique provides a powerful and direct probe of a spin-singlet superconductor's accommodation of local nonpair breaking SO perturbations. By varying the Pb-cluster separation the antisymmetric FL parameter G 0 can be tuned over a wide range G 0 eff $ 0:18 ! À0:75 with a corresponding multifold effect on the spin susceptibility. From a practical standpoint, this allows one to adjust the spin susceptibility to a specific value for the purposes of spintronics applications. For instance, our data suggests that at a separation of~7.5 nm G 0 eff ¼ 0. At this impurity density the spin characteristics of the Al film are transformed into that of a non-interacting Fermi gas with modest SO scattering rate, b ≈ 0.4.

Methods
Transport measurements. The magnetotransport properties of the films were measured on a Quantum Design Physical Properties Measurement System equipped with a 3 He probe. The base temperature of system was 400 mK. Electrical contact was made to the films using a standard 4-probe geometry and phase sensitive detection of the film resistivity. The films were carefully aligned to parallel field orientation using a custom designed mechanically actuated rotating platform fitted to the probe sample mount. After alignment, the parallel critical field was measured as a function of temperature. The cluster separation of our samples varied between d = ∞ for the pristine Al films and d = 0.5 nm for the highest Pb coverages used. The Pb clusters did not appreciably affect the transition temperature of the Al films, nor did they appreciably affect their conductivity. However, as the cluster density was increased, the SO scattering rate also increased. As expected, this produced significant higher Zeeman critical fields than is typical of pristine Al films.
Film synthesis. The Al-Pb-cluster samples used in this study were depositing onto carefully prepared n-doped (n~10 −15 cm −3 ) Si(111) substrates. The substrates were cleaned by flashing them 5 times (via Joule heating) to 1200°C, followed by an anneal at~550°C for 10 min. The Pb clusters were formed by first depositing a small amount of Pb at room temperature (≪1 ML) at a chamber pressure of~8 × 10 −11 Torr and subsequently annealing the sample at~200°C for 10 min. Scanning tunneling microscope (STM) topographs were then used to determine the cluster distribution characteristics. A cluster image corresponding to 0.02 ML of Pb is shown in the inset of Fig. 2. Note that the clusters are only a few atoms in size and thus have a lateral dimension that is much smaller than either the coherence length or the film thickness. Finally, 15 ML of Al was deposited on the cluster matrix at 100 K, followed by room temperature annealing for 12 h. The upper three layers of the resulting composite film was oxidized in order to produce a protective cap. Thus the metallic thickness of the Al films used in critical field studies was approximately 12 ML (~3.2 nm) 16 . We note that because of the clustering tendencies of the Pb atoms, one needs direct STM imaging of the cluster array in order to determine the average separation. If one assumes that the Pb atoms are simply randomly distributed on the Si surface, then the average separation will be substantially underestimated.

Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.  Fig. 4 Effective antisymmetric Fermi liquid parameter as a function of Pbcluster separation. Note that d $ ðPb coverageÞ À 1 2 . The solid line is a linear least-squares fit to the data. The horizontal dashed line represents G 0 for a pure Al film. The arrow represents the average cluster separation at which there is no spin renormalization. Error bars where estimated from standard deviations produced by the least-squares fitting algorithm