Breakdown of the Hebel-Slichter effect in superconducting graphene due to the emergence of Yu-Shiba-Rusinov states at magnetic resonant scatterers

TL;DR

This paper studies how magnetic impurities and Yu-Shiba-Rusinov states affect spin relaxation in superconducting graphene, revealing a breakdown of the Hebel-Slichter effect at resonances due to spectral weight redistribution.

Contribution

It demonstrates that Yu-Shiba-Rusinov states cause the Hebel-Slichter effect to fail at resonances, providing new insights into quasiparticle spin dynamics in superconducting graphene.

Findings

Hebel-Slichter effect observed off resonance

Resonant impurities lead to Yu-Shiba-Rusinov states

Spin relaxation decreases with temperature at resonance

Abstract

Employing analytical methods and quantum transport simulations we investigate the relaxation of quasiparticle spins in graphene proximitized by an $s$-wave superconductor in the presence of resonant magnetic and spin-orbit active impurities. Off resonance, the relaxation increases with decreasing temperature when electrons scatter off magnetic impurities---the Hebel-Slichter effect---and decreases when impurities have spin-orbit coupling. This distinct temperature~dependence (not present in the normal state) uniquely discriminates between the two scattering mechanisms. However, we show that the Hebel-Slichter picture breaks down at resonances. The emergence of Yu-Shiba-Rusinov bound states within the superconducting gap redistributes the spectral weight away from magnetic resonances. The result is opposite to the Hebel-Slichter expectation: the spin relaxation decreases with decreasing temperature. Our findings hold for generic $s$-wave superconductors with resonant magnetic impurities, but also, as we show, for resonant magnetic Josephson junctions.

Equations

$$\begin{gathered} H_g=-\sum _{m,n,\sigma }(t{\delta }_{⟨mn⟩}+\mu {\delta }_{mn})c_{m\sigma }^{†}{c}_{n\sigma }^{} \\ +\Delta \sum _m{c}_{m\uparrow }^{†}{c}_{m\downarrow }^{†}+h.c.; \end{gathered}$$

$$V_o=\sum _{\sigma }[(\epsilon -\mu )d_{\sigma }^{†}{d}_{\sigma }^{}+\omega d_{\sigma }^{†}{c}_{0\sigma }^{}]+\Delta d_{\uparrow }^{†}{d}_{\downarrow }^{†}+h.c.;$$

Full text

superconductivity, graphene, spin relaxation, resonance, Yu-Shiba-Rusinov states, Hebel-Slichter effect, magnetic Josephson junctions

Introduction.

Superconducting spintronics investigates the interplay between the electron spin phenomena Žutić et al. (2004) and macroscopic quantum coherence of superconducting structures Eschrig (2011, 2015); Linder and Robinson (2015). A versatile platform for superconducting spintronics is offered by 2D layered materials. Indeed, there is a growing family of 2D superconductors—twisted bilayer graphene Cao et al. (2018); Yankowitz et al. (2019), 2D topological insulators Sajadi et al. (2018); Fatemi et al. (2018), or transition-metal dichalcogenides Shi et al. (2015); Jo et al. (2015); Xi et al. (2016); Navarro-Moratalla et al. (2016); Costanzo et al. (2016)—which could serve as a source of Cooper pairs. At the same time there are high-mobility 2D (semi)metals and semiconductors whose spin properties, in particular spin relaxation (SR), can profoundly change when proximitized by superconductors.

Measurements of SR in graphene have not yielded a unique mechanism for electron spin flips Tombros et al. (2007); Ohishi et al. (2007); Pi et al. (2010); Yang et al. (2011); Avsar et al. (2011); Lundeberg et al. (2013); Guimarães et al. (2014); Raes et al. (2016); Drögeler et al. (2016); Dushenko et al. (2016). Ensuing intense scientific discussions Seneor et al. (2012); Han et al. (2014); Roche and Valenzuela (2014); Roche et al. (2015); Feng et al. (2017); Garcia et al. (2018); Rybkin et al. (2018) have focused on spin-orbit and exchange impurities as possible culprits. The principal difficulty in setting one against the other lies, unlike in conventional materials Žutić et al. (2004), in the absence of a systematic temperature behavior of the measured spin relaxation. However, the absence of SR anisotropy Raes et al. (2016) points towards magnetic resonant impurities Kochan et al. (2014a, 2015) as the main source of spin-flip scattering in graphene.

Here we show that in (proximitized) superconducting graphene (SCG) the two types of impurities yield distinct temperature characteristics due to coherence effects. Particularly striking is the prediction that resonant magnetic scatterers cause SR whose temperature dependence is opposite to that predicted by the perturbative Hebel-Slichter effect Hebel and Slichter (1957, 1959); Hebel (1959). Since the non-perturbative analytic and quantum transport simulation methods we use are not specific to graphene, this prediction applies to resonant scattering in all s-wave superconductors. Furthermore, we demonstrate that it also applies to superconducting resonant Josephson junctions with magnetic tunnel barriers.

Although measurements of SR in SCG have not yet been performed (which makes theoretical predictions particularly motivating), they are within the current experimental reach. Indeed, proximity induced superconductivity in graphene has been experimentally demonstrated in lateral Josephson junctions Heersche et al. (2007); Komatsu et al. (2012); Calado et al. (2015); Indolese et al. (2018) and vertical stack geometries Tonnoir et al. (2013); Di Bernardo et al. (2017), as well as in alkaline-intercalated graphite Li et al. (2013); Ludbrook et al. (2015); Chapman et al. (2016). The induced superconducting gap ranges from tens of $\upmu\mathrm{eV}$ Heersche et al. (2007) to 1 meV Rickhaus et al. (2012) ($T_{\rm c}\simeq 6.593$ K). Both $s$-wave Heersche et al. (2007) and $p$-wave Di Bernardo et al. (2017) pairings were convincingly demonstrated; see the comprehensive review Lee and Lee (2018) for more details.

Rationale.

Yafet showed  Yafet (1983), using the first order perturbation theory, that the SR rate in superconductors is modified from the normal state as $1/\tau_{s}^{\rm SC}\sim\langle(u_{\mathbf{k}}u_{\mathbf{q}}\pm v_{\mathbf{k}}v_{\mathbf{q}})^{2}/\tau_{s}^{\rm N}\rangle$, where $u$ and $v$ are standard BCS coherence factors and $\langle\cdots\rangle$ denotes thermal broadening. The plus sign applies to interactions that are odd under time reversal, such as exchange. As a result, $1/\tau_{s}^{\rm SC}>1/\tau_{s}^{\rm N}$, which is the Hebel-Slichter effect Hebel and Slichter (1957, 1959); Hebel (1959). On the other hand, the minus sign is for time-reversal symmetric interactions such as spin-orbit coupling, in which case $1/\tau_{s}^{\rm SC}<1/\tau_{s}^{\rm N}$, as demonstrated experimentally for aluminum Yang et al. (2010); Quay et al. (2015) 111For more details about the charge and spin accumulation of QPs in superconductors, their non-equilibrium separation, and relaxation, consult Refs. Zhao and Hershfield (1995); Yamashita et al. (2002); Quay et al. (2013).. We will see that in SCG $1/\tau_{s}^{\rm N}$ and $1/\tau_{s}^{\rm SC}$ can differ by orders of magnitude due to the coherence effects, which allows for an unprecedented experimental feasibility to disentangle the dominant SR mechanism. Our methodology overcomes two shortcomings of the standard theory of Yafet, Hebel, and Slichter. First, we calculate the SR rate to all orders of perturbation theory allowing us to consider resonant scattering and, second, we also include sub-gap Yu-Shiba-Rusinov (YSR) states Yu (1965); Shiba (1968); Rusinov (1968); *Rusinov1968alt; Costa et al. (2018) which have no normal-state counterpart and which take away considerable spectral weight from the scattering states.

Model and Methodology.

To describe SCG we use the tight-binding Hamiltonian Uchoa and Castro Neto (2007),

[TABLE]

$t=2.6$ eV stands for the conventional nearest neighbor hopping, $\mu$ for the chemical potential (doping level) taking the normal-phase’s charge neutrality point as a reference, and $\Delta$ models the $T$-dependent global on-site $s$-wave pairing. We assume the BCS-like temperature dependence $\Delta(T)=\Delta_{0}\tanh{[1.74\sqrt{T_{\rm c}/T-1}]}$ with realistic proximity values of $\Delta_{0}=1$ meV and $T_{\mathrm{c}}\simeq 6.593$ K. The operator $c_{m\sigma}^{(\dagger)}$ annihilates (creates) an electron with spin $\sigma$ at the graphene lattice site $m$, $\delta_{mn}$ represents the usual Kronecker symbol, and $\delta_{\langle mn\rangle}$ its nearest-neighbor analog (unity for the direct graphene nearest-neighbor sites and zero otherwise). Orbital interactions with an adatom—annihilation and creation operators $d_{\sigma}$ and $d_{\sigma}^{\dagger}$—are governed by the impurity-site hybridization $\omega$, on-site energy $\varepsilon$, and the proximity pairing $\Delta$, combining into Ratto and Blandin (1967) (see the inset of Fig. 1)

[TABLE]

This orbital perturbation is complemented by a local spin-dependent term $V_{s}$ comprising (1) exchange interaction, $V_{s}^{(1)}=-J\,\mathbf{s}\cdot\mathbf{S}$, between the itinerant electron spin $\mathbf{s}$ and the impurity $1/2$-spin $\mathbf{S}$ Kochan et al. (2014a), and (2) local SOC in the vicinity of an adatom Gmitra et al. (2013); Irmer et al. (2015); Zollner et al. (2016); Frank et al. (2017); Kochan et al. (2017); SM  To be specific, we use hydrogen and fluorine adatoms, both of which induce sizable SOC enhancements Gmitra et al. (2013); Irmer et al. (2015) and can also carry magnetic moments Yazyev (2010); Xie et al. (2011); Hong et al. (2012); Nair et al. (2012); Gonzalez-Herrero et al. (2016); Szałowski (2016); Wellnhofer et al. (2019); Sousa et al. . We also assume low concentrations $\eta$ (per carbon atom) of dilute spin-active impurities Zuckermann (1965) for the independent scatter picture to be valid.

Our methodology employs the full analytical approach, calculating the spin-flip rates from the T-matrix, as well as numerical Kwant simulations of spin-flip scattering probabilities, providing together coherent and consistent qualitative and quantitative pictures. Detailed methodology is in Supplementary Material SM .

Results.

Adatoms on graphene give rise to resonances Skrypnyk and Loktev (2007); Wehling et al. (2010); Ferreira and Mucciolo (2015); Irmer et al. (2018), which strongly modify graphen’s transport properties Skrypnyk and Loktev (2010); Ferreira et al. (2014); Stabile et al. (2015); Garcia and Rappoport (2016); Katoch et al. (2018); Irmer et al. (2018); Lee et al. (2019), particularly when they lie close to the Dirac point. Figure 1 demonstrates how normal-state resonances affect the population of quasiparticle states in SCG at different chemical potentials. Panel 1(a) shows the DOS of graphene covered by 100 ppm of resonant nonmagnetic impurities, while panel 1(b) displays the corresponding quasiparticle DOS (QPDOS). We present resonant and off-resonant doping limits and conclude that QPDOS gets strongly modified near the coherence peaks as $\mu$ approaches the normal-state resonance. This makes sense since BCS theory gives  QPDOS$(E)=[E/\sqrt{E^{2}-\Delta^{2}}]$ DOS$(\mu)$.

Figure 2 illustrates various characteristics of spin-flip scattering off magnetic impurities in normal and superconducting graphene for two representative impurities: hydrogen—panels 2(a),(c),(e)—and fluorine—panels 2(b),(d),(f). Particularly, Figs. 2(a) and (b) display the SR rates in SCG for different temperatures in the presence of 280 ppm of magnetic impurities, as a function of the chemical potential $\mu$ and the superconducting gap with temperature $T$. Solid lines corresponds to analytical T-matrix calculation, and the symbols with the same color to the corresponding Kwant simulation, for details see SM . Hydrogen Kochan et al. (2014a) with a nonzero magnetic moment—$\omega=7.5$ eV, $\varepsilon=0.16$ eV, and $J=-0.4$ eV—gives rise to a narrow normal-state resonant region near the Dirac point; see the corresponding magnetic DOS in Fig 2(c) (concentration $\eta=0.1\%$ is exaggerated for resolution purposes). In contrast, fluorine—$\omega=5.5$ eV, $\varepsilon=-2.2$ eV, and $J=0.5$ eV—develops Irmer et al. (2015); Wellnhofer et al. (2019) a wide resonance region spreading below the Dirac point; see the magnetic DOS in Fig 2(d) (again with the elevated concentration $\eta=1\%)$.

The striking impact of resonances on quasiparticle SR rates is seen from Figs. 2(a) and (b). The shaded regions show the SR rate in the normal phase ($T=T_{\mathrm{c}}$). Lowering $T$ below superconducting $T_{c}$ we observe an intriguing behavior: for off-resonant doping regions, $1/\tau_{s}^{\rm SC}>1/\tau_{s}^{\rm N}$ in accordance with the Hebel-Slichter scenario Hebel and Slichter (1957, 1959); Hebel (1959), whereas quasiparticle SR sharply drops,  $1/\tau_{s}^{\rm SC}\ll 1/\tau_{s}^{\rm N}$, at resonances. To further quantify those effects, the insets of Figs. 2(a) and (b) represent the corresponding Hebel-Slichter ratios, (1/\tau_{s}^{\rm SC})\bigl{/}(1/\tau_{s}^{\rm N}), taken at two representative $\mu$’s as functions of $T/T_{\rm c}$. For the off-resonant value of $\mu=500$ meV, both impurity cases lead to a notable enhancement of the superconducting SR rates by almost a factor of four (graphs with red symbols), while we see a strong decrease of the SR rates by almost three orders of magnitude (graphs with black symbols in logarithmic scale) inside the resonant regions—we use $\mu=-80$ meV for hydrogen and $\mu=-300$ meV for fluorine. This can serve as a powerful experimental evidence—observing strongly depleted SR rates in the SCP when lowering $T$ signifies the presence of resonant magnetic impurities.

To explain this peculiar resonant depletion of QP SRs in SCG, which is at odds with the well-understood resonant enhancement Kochan et al. (2014a, 2015); Soriano et al. (2015); Wilhelm et al. (2015); Miranda et al. (2019); Thomsen et al. (2015) in the normal phase, we calculate the energies (poles of T-matrix, Loktev and Pogorelov (2015)) of the corresponding YSR states Yu (1965); Shiba (1968); Rusinov (1968); *Rusinov1968alt which develop around magnetic impurities Wehling et al. (2008); Lado and Fernández-Rossier (2016), see Figs. 2(e) and (f). We deduce that the YSR states lie deep inside the superconducting gap at doping levels that correspond to resonances in the normal-phase of graphene. This offers an explanation why the SR rates of quasiparticles sharply decrease. Resonant scattering of QPs counts many contributions, especially those from multiple scatterings and those from virtual states’ tunnelings. Schematically, the scattering matrix element can be written as $V_{\alpha\alpha}+V_{\alpha\mathrm{I}}\tfrac{|\mathrm{I}\rangle\,\langle\mathrm{I}|}{E_{\alpha}-E_{\mathrm{I}}+i0_{+}}V_{\mathrm{I}\alpha}+\ldots$, where $E_{\mathrm{I}}$ represents the energy of any intermediate state (extended or subgap), and $E_{\alpha}$ stands for the energy of an incident QP state. The dominant elements $V_{\alpha\mathrm{I}}$ for spin-flip processes arise from a strong overlap of quasiparticle states $\alpha$ with the magnetic impurity level $\mathrm{I}$=YSR since only the latter gives rise to QP spin flips. While the matrix-elements $V_{\alpha\mathrm{YSR}}$ are roughly the same for all extended QP states $\alpha$ at the coherence peaks, independently of doping (impurity in our model acts like $\delta$-function), this is not true for the energy differences $E_{\alpha}-E_{\mathrm{YSR}}$ in the denominator. Those are, in fact, small in the off-resonant region since $E_{\mathrm{YSR}}$ are aligned with the superconducting gap edges and become large at the resonances, where $E_{\mathrm{YSR}}$ are deep inside the gap. That this must cause the reduced SR is clear from the $T$-dependence of the SR rates; for higher $T$, $\Delta$ gets smaller and hence also the difference $E_{\alpha}-E_{\mathrm{YSR}}$. The opposite conclusion would follow from Yafet’s formula, which does not account for the formation of subgap bound states.

At very low temperatures—data for $T=100$ mK displayed by dashed lines in Figs. 2(a) and (b)—, the SR rates drop down at low dopings since the related QPDOS, $\frac{E}{\sqrt{E^{2}-\Delta^{2}}}\,\text{DOS}(\mu)(-\tfrac{\partial g}{\partial E})$, becomes substantially suppressed upon thermal smearing. However, at larger dopings this suppression is counteracted by the elevated $\text{DOS}(\mu)$.

Finally, we turn to SR due to spin-orbit active impurities. Figure 3 displays the calculated SR rates in SCG as functions of doping at different temperatures for fluorine impurities, see SM for the hydrogen which shows the same qualitative features. We find that the SR rates in SCG decrease (as predicted by Yafet Yafet (1983)) by one order of magnitude with lowering $T$, giving sizable experimental signals. At resonances SR rates get enhanced, similarly to what was predicted for the normal state Bundesmann et al. (2015). This is because a quasiparticle in a resonant state has sufficient time to experience SOC and flip its spin, what gives rise to the increased SR rates, although somewhat less than in the normal state due to lower group velocities of quasiparticles. Thus, observing a global decrease of the SR rate with lowered $T$ spanning a wide doping region would be an unprecedented experimental signal for SOC-dominated SR.

Although motivated by superconducting spintronics we used primarily SCG to perform detailed simulations, our findings are also qualitative and thus expected to be valid for resonant scattering off magnetic impurities in generic s-wave superconductors. In fact, we have analyzed a simple 1D model of a resonant Josephson junction with magnetic tunnel barriers SM , which explicitly demonstrates the reduction of spin-flip probabilities of quasiparticles at resonances due to the appearance of deep immersed YSR states.

Conclusions.

We have shown that the spin relaxation of quasiparticles due to resonant magnetic impurities has the opposite temperature dependence from what is predicted as the Hebel-Slichter effect. The reason is emergence of sub-gap YSR states which redistribute the spectral weight away from the resonances. The anomalous decrease of resonant SR can span three-to-four orders of magnitude, making it a robust and verifiable experimental tool.

Acknowledgements.

D.K. thanks Drs. Ferenc Simon, Yuriy Pogorelov, Lucia Komendová, and Benedikt Scharf for useful discussions, and Dr. Jeongsu Lee for helpful tips regarding numerical implementation. The authors acknowledge support from Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—Project-ID 314695032—SFB 1277 (Subprojects A07, A09, B07), the EU Seventh Framework Programme under Grant Agreement No. 604391 (Graphene Flagship), and the International Doctorate Program Topological Insulators of the Elite Network of Bavaria.