Skew Andreev reflection in ferromagnet/superconductor junctions

TL;DR

This paper theoretically investigates how a semiconducting interface with spin-orbit coupling causes asymmetric Andreev reflection in ferromagnet/superconductor junctions, leading to charge imbalances and tunable supercurrents.

Contribution

It introduces the concept of skew Andreev reflection caused by spin-orbit effects at the interface, revealing a giant, tunable transverse supercurrent response.

Findings

Skew AR causes significant charge imbalances and Hall currents.

The effect is greatly enhanced compared to normal AR.

Transverse supercurrents can be widely tuned via interface properties.

Abstract

Andreev reflection (AR) in ferromagnet/superconductor junctions is an indispensable spectroscopic tool for measuring spin polarization. We study theoretically how the presence of a thin semiconducting interface in such junctions, inducing Rashba and Dresselhaus spin-orbit coupling, modifies AR processes. The interface gives rise to an effective momentum- and spin-dependent scattering potential, making the probability of AR strongly asymmetric with respect to the sign of the incident electrons' transverse momenta. This skew AR creates spatial charge carrier imbalances and transverse Hall currents flow in the ferromagnet. We show that the effect is giant, as compared to the normal regime. We provide a quantitative analysis and a qualitative picture of this phenomenon, and finally show that skew AR also leads to a widely tunable transverse supercurrent response in the superconductor.

Figures

Equations

$${\hat{\mathcal{H}}}_{\mathrm{BdG}}=\left[\begin{array}{cc}{\hat{\mathcal{H}}}_{\mathrm{e}}& {\hat{\Delta }}_{\mathrm{S}}(z)\\ {\hat{\Delta }}_{\mathrm{S}}^{†}(z)& {\hat{\mathcal{H}}}_{\mathrm{h}}\end{array}\right],$$

$$V_{\mathrm{eff}}=V_{\mathrm{SC}}{d}_{\mathrm{SC}}-\sigma \alpha k_x,$$

$$\begin{gathered} G_{\eta ,z}=-\frac{G_{0}A}{8{\pi }^2}\sum _{\sigma =\pm 1}\int {\mathrm{d}}^2{\mathbf{k}}_{\parallel }\frac{k_{\eta }}{k_z^{\sigma }}\{\left[{\left|r_{\mathrm{e}}^{\sigma ,\sigma }(eV)\right|}^2+{\left|r_{\mathrm{e}}^{\sigma ,-\sigma }(eV)\right|}^2\right] \\ +\left[{\left|r_{\mathrm{h}}^{\sigma ,-\sigma }(-eV)\right|}^2+{\left|r_{\mathrm{h}}^{\sigma ,\sigma }(-eV)\right|}^2\right]\}, \end{gathered}$$

Full text

Due to the extraordinary properties occurring at their interfaces, ferromagnet/superconductor (F/S) heterostructures attract considerable interest Eschrig (2011); Linder and Robinson (2015); Gingrich et al. (2016). Such junctions might not only offer novel tools for controlling and measuring charge and spin currents, but might also bring new functionalities into spintronics devices.

While early efforts focused on detecting spin-polarized quasiparticles in superconductors via spin transport experiments Tedrow and Meservey (1971, 1973); Meservey and Tedrow (1994), current progress in the rapidly growing field of superconducting spintronics Linder and Robinson (2015) opened several promising perspectives, ranging from the observation of long spin lifetimes and giant magnetoresistance effects Yang et al. (2010) to the generation and successful manipulation of superconducting spin currents Wakamura et al. (2015); Beckmann (2016); Bergeret and Tokatly (2016); Espedal et al. (2017); Linder et al. (2017); Ouassou et al. (2017); Jeon et al. (2018); Montiel and Eschrig (2018). But the interplay of magnetism and superconductivity gets even more interesting when spin-orbit coupling (SOC) of the Rashba Bychkov and Rashba (1984) and/or Dresselhaus Dresselhaus (1955) type is present Žutić et al. (2004); Fabian et al. (2007). Prominent examples are spin-triplet pairing mechanisms Bergeret et al. (2001); Volkov et al. (2003); Keizer et al. (2006); Halterman et al. (2007); Eschrig and Löfwander (2008); Eschrig (2011); Sun and Shah (2015), leading to long-range superconducting proximity effects Duckheim and Brouwer (2011); Bergeret and Tokatly (2013, 2014); Jacobsen and Linder (2015), and Majorana states Nilsson et al. (2008); Duckheim and Brouwer (2011); Lee et al. (2012); Nadj-Perge et al. (2014); Dumitrescu et al. (2015); Pawlak et al. (2016); Ruby et al. (2017); Livanas et al. (2019), which are expected to form in superconducting proximity regions in the presence of SOC.

While SOC in bulk materials plays the key role for intrinsic anomalous Hall effects Hall (1881); Wölfle and Muttalib (2006); Nagaosa (2006); Sinitsyn (2008); Nagaosa et al. (2010), recent theoretical studies Vedyayev et al. (2013a, b); Matos-Abiague and Fabian (2015); Dang et al. (2015, 2018); Zhuravlev et al. (2018) predicted that interfacial SOC in F/normal metal (N) tunnel junctions can give rise to extrinsic tunneling anomalous Hall effects (TAHEs) in the N, owing to spin-polarized skew tunneling of electrons through the interface. The unique scaling of the associated TAHE conductances could make the effect a fundamental tool for identifying and characterizing interfacial SOC, thus providing the input for tailoring systems that could, e.g., host Majoranas. Although first experiments on granular junctions Rylkov et al. (2017) confirmed the predictions, the extremely small TAHE conductances remain one of the main obstacles. Sizable TAHE conductances require either interfacial barriers with large SOC, such as ferroelectric semiconductors (SCs) Zhuravlev et al. (2018), or different junction compositions.

In this Rapid Communication, we consider F/SC/S junctions, in which the N electrode is replaced by a S. We demonstrate that, analogously to the tunneling picture in the normal-conducting case, skew reflection 111Conventional skew scattering actually refers to momentum- and spin-dependent scattering of spin-polarized charge carriers on magnetic impurities. To clearly differentiate between that and our reflection-based mechanism (which does not require the presence of impurities at all), we rely on the term skew reflection. of spin-polarized carriers at the barrier leads to TAHEs in the F. Due to the presence of a S electrode, we distinguish two skew reflection processes: skew specular reflection (SR) and skew Andreev reflection (AR). By formulating a qualitative physical picture including both processes, we assert that skew SR and skew AR can act together and significantly enhance the TAHE compared to all previously studied (normal) systems. Special attention must be paid to skew AR, which transfers Cooper pairs across the barrier into the S. The electrons forming one Cooper pair are thereby also subject to the proposed skew reflection mechanism. We discuss that the result is a transverse supercurrent response, initially deduced from a phenomenological Ginzburg-Landau treatment Mironov and Buzdin (2017), with widely tunable characteristics. Both findings, relatively giant TAHE conductances in the F and transverse supercurrents in the S, are distinct fingerprints to experimentally detect skew AR and characterize the junctions’ interfacial SOC.

We consider a biased ballistic F/SC/S junction grown along the $\hat{z}$-direction, in which the two semi-infinite F and S regions are separated by an ultrathin SC barrier [see Fig. 1(a)].

The barrier may be composed of a thin layer of zincblende materials (e.g., GaAs or InAs) and introduces potential scattering, as well as strong interfacial Rashba Bychkov and Rashba (1984) and Dresselhaus Dresselhaus (1955) SOC Žutić et al. (2004); Fabian et al. (2007).

The system can be modeled by means of the stationary Bogoljubov–de Gennes (BdG) Hamiltonian De Gennes (1989),

[TABLE]

where $\hat{\mathcal{H}}_{\mathrm{e}}=[-\hbar^{2}/(2m)\,\bm{\nabla}^{2}-\mu]\,\hat{\sigma}_{0}-(\Delta_{\mathrm{XC}}/2)\,\Theta(-z)\,(\hat{\mathbf{m}}\cdot\hat{\bm{\sigma}})+V_{\mathrm{SC}}\,d_{\mathrm{SC}}\,\hat{\sigma}_{0}\,\delta(z)+\hat{\mathcal{H}}_{\mathrm{SC}}^{\mathrm{SOC}}\,\delta(z)$ represents the single-electron Hamiltonian and $\hat{\mathcal{H}}_{\mathrm{h}}=-\hat{\sigma}_{y}\,\hat{\mathcal{H}}_{\mathrm{e}}^{*}\,\hat{\sigma}_{y}$ its holelike counterpart ($\hat{\sigma}_{0}$ and $\hat{\sigma}_{i}$ indicate the $2\times 2$ identity and the $i$th Pauli matrix; $\hat{\bm{\sigma}}=[\hat{\sigma}_{x},\,\hat{\sigma}_{y},\,\hat{\sigma}_{z}]^{\top}$ is the vector of Pauli matrices). The F is described within the Stoner model with exchange energy $\Delta_{\mathrm{XC}}$ and magnetization direction $\hat{\mathbf{m}}=[\cos\Phi,\,\sin\Phi,\,0]^{\top}$, where $\Phi$ is measured with respect to the $\hat{x}$-axis. Following earlier studies de Jong and Beenakker (1995); Žutić and Valls (1999, 2000); Costa et al. (2017, 2018), the ultrathin SC layer is included into our model as a $\delta$-like barrier with height $V_{\mathrm{SC}}$ and width $d_{\mathrm{SC}}$; its SOC enters the Hamiltonian Žutić et al. (2004); Fabian et al. (2007) $\hat{\mathcal{H}}_{\mathrm{SC}}^{\mathrm{SOC}}=\alpha\,(k_{y}\,\hat{\sigma}_{x}-k_{x}\,\hat{\sigma}_{y})-\beta\,(k_{y}\,\hat{\sigma}_{x}+k_{x}\,\hat{\sigma}_{y})$, where the first part accounts for SOC of the Rashba type and the second part resembles linearized Dresselhaus SOC 222Higher-order SOC terms can be neglected since the main current contributions come from states with small $k_{x}$ and $k_{y}$ Matos-Abiague and Fabian (2009), both with the effective strengths $\alpha$ and $\beta$, respectively. Inside the S electrode, the S pairing potential, $\hat{\Delta}_{\mathrm{S}}(z)=|\Delta_{\mathrm{S}}|\,\Theta(z)$ ($|\Delta_{\mathrm{S}}|$ is the isotropic energy gap of the S), couples the electron and hole blocks of the BdG Hamiltonian. Note that although writing $\hat{\Delta}_{\mathrm{S}}$ in that way is a rigid approximation, neglecting proximity effects, this approach still yields reliable results for transport calculations Likharev (1979); Beenakker (1997). For the sake of simplicity, we further assume the same Fermi levels, $\mu$, and equal effective carrier masses, $m$, in the F and S.

Assuming translational invariance parallel to the barrier, the solutions of the BdG equation, $\hat{\mathcal{H}}_{\mathrm{BdG}}\,\Psi^{\sigma}(\mathbf{r})=E\,\Psi^{\sigma}(\mathbf{r})$, can be factorized according to $\Psi^{\sigma}(\mathbf{r})=\psi^{\sigma}(z)\,\mathrm{e}^{\mathrm{i}\,(\mathbf{k_{\parallel}}\cdot\mathbf{r_{\parallel}})},$ where $\mathbf{k_{\parallel}}=[k_{x},\,k_{y},\,0]^{\top}$ ($\mathbf{r_{\parallel}}=[x,\,y,\,0]^{\top}$) denotes the in-plane momentum (position) vector and $\psi^{\sigma}(z)$ are the BdG equation’s individual solutions for the reduced one-dimensional scattering problem along $\hat{z}$. The latter account for the different involved scattering processes at the SC interface: incoming electrons with spin $\sigma$ [$\sigma=+(-)1$ for spin up (down), which effectively indicates a spin parallel (antiparallel) to $\hat{\mathbf{m}}$] may either undergo AR or SR, or may be transmitted as quasiparticles into the S.

Due to the presence of interfacial SOC, electrons incident on the ultrathin SC are exposed to an effective scattering potential that incorporates besides the usual barrier strength (determined by the barrier’s height and width) also the in-plane momentum- and spin-dependent contribution of the SOC. To extract valuable qualitative trends from our model, we first focus on the simple situation in which only Rashba SOC is present ($\alpha>0$, $\beta=0$), the F’s magnetization is aligned along $\hat{y}$ ($\Phi=\pi/2$), and $k_{y}=0$. In this case, the effective scattering potential reads

[TABLE]

where the first part represents the usual barrier strength and the second the SOC-dependent part. Assuming that SOC is weak and spin-flip scattering becomes negligible, only spin-conserving AR and SR are allowed inside the F, each with certain probabilities. The latter, extracted from an extended Blonder–Tinkham–Klapwijk (BTK) model Blonder et al. (1982) by substituting the effective scattering potential in Eq. (2) [see the Supplemental Material (SM) 333See the attached Supplemental Material, including Refs. Bychkov and Rashba (1984); Dresselhaus (1955); Žutić et al. (2004); Fabian et al. (2007); de Jong and Beenakker (1995); Žutić and Valls (1999, 2000); Costa et al. (2017, 2018); De Gennes (1989); Blonder et al. (1982); Högl et al. (2015a); *Hogl2015; Matos-Abiague and Fabian (2015); Rylkov et al. (2017); Furusaki and Tsukada (1991); McMillan (1968); Moser et al. (2007); Matos-Abiague and Fabian (2009); Nitta et al. (1997); Koga et al. (2002); Chen et al. (2018); Giancoli (1995); Wang et al. (2006); *Wang2007; Zhuravlev et al. (2018), for more details. for details], are shown for incoming spin up electrons as a function of $V_{\mathrm{eff}}$ in Figs. 1(b) and (c), once for the normal state and once for the superconducting junction.

In the first case, AR is completely forbidden, while the probability that the incident electron gets specularly reflected continuously increases with increasing effective scattering potential; note that there is also a finite transmission probability into the right normal-state electrode (not shown). For a constant moderate barrier height and width (black dashed line) and nonzero Rashba SOC, Eq. (2) suggests that incoming spin up electrons with positive $k_{x}$ experience a significantly lower barrier (violet dashed line) and thus undergo skew SR with a lower probability than those with negative $k_{x}$ (orange dashed line). The generated spatial charge imbalance in the F must be compensated by a transverse Hall current flow, $J_{x}$, along $\hat{x}$. Strictly speaking, the situation gets reversed for incident spin down electrons. Nevertheless, since there are more occupied spin up states, both channels cannot completely cancel and a finite Hall current remains.

If the junction becomes superconducting, AR comes into play. Although the AR probability generally decreases with increasing $V_{\mathrm{eff}}$, the crucial point is that AR involves holes. Consequently, skew AR produces simultaneously an electron excess also at negative $k_{x}$, and both skew AR and SR act together to noticeably increase the transverse Hall current.

Another important observation relies on the reflection probabilities at large $V_{\mathrm{eff}}$. In both junction scenarios, the SR probabilities approach unity at $V_{\mathrm{eff}}\gg(\hbar^{2}k_{\mathrm{F}})/(2m)$; this happens much faster in the superconducting case than in the normal state. The scattering potential is then mostly determined by the usual barrier height and width, and the spin-dependent contribution only barely impacts $V_{\mathrm{eff}}$. Therefore, both skew reflection and the Hall currents are expected to be strongly damped in the presence of strong barriers, in superconducting even more than in normal-conducting junctions.

As a clear fingerprint to experimentally detect skew AR, our qualitative picture suggests a significant enhancement of the superconducting junctions’ TAHE conductance, compared to the normal-state regime. To evaluate the TAHE conductances along the transverse $\hat{\eta}$-direction ($\eta\in\{x;y\}$), we follow a generalized BTK approach Blonder et al. (1982), yielding the zero-temperature TAHE conductances

[TABLE]

where $G_{0}=(2e^{2})/h$ abbreviates the conductance quantum, $A$ stands for the cross-section area, $k_{z}^{\sigma}=\sqrt{k_{\mathrm{F}}^{2}\,(1+\sigma P)-\mathbf{k}^{2}_{\mathbf{\parallel}}}$ represents the $\hat{z}$-component of the particles’ wave vector in the F with spin polarization $P=(\Delta_{\mathrm{XC}}/2)/\mu$, and $k_{\mathrm{F}}=\sqrt{2m\mu}/\hbar$ is the Fermi wave vector. The reflection coefficients $r_{\mathrm{e}}^{\sigma,\sigma}$ ($r_{\mathrm{e}}^{\sigma,-\sigma}$) correspond to SR (spin-flip SR), while $r_{\mathrm{h}}^{\sigma,-\sigma}$ ($r_{\mathrm{h}}^{\sigma,\sigma}$) indicate AR (spin-flip AR). Unlike for the (longitudinal) tunneling conductance Blonder et al. (1982), SR and AR contribute to the Hall conductances with the same sign since the specularly reflected electron and the Andreev reflected hole move into opposite transverse directions; the different sign in the transverse velocities gets then compensated by the opposite charge of electrons and holes. Therefore, the charge imbalances created by skew AR and SR can indeed give rise to individual Hall currents that flow along the same direction, and finally lead to sizable Hall responses in superconducting junctions.

To elaborate on the TAHE conductances’ main features, we evaluate Eq. (3) for Fe/GaAs/V like model junctions. The spin polarization in Fe is $P=0.7$ (Fermi wave vector $k_{\mathrm{F}}\approx 8\times 10^{7}\,\mathrm{cm}^{-1}$ Martínez et al. (2018)), while $|\Delta_{\mathrm{S}}|\sim 1.6\,\mathrm{meV}$ refers to V’s gap Martínez et al. (2018). The (material-specific) Dresselhaus SOC strength of GaAs can be approximated Fabian et al. (2007); Matos-Abiague and Fabian (2009) as $\beta\approx Zk_{\mathrm{F}}\gamma$, with $\gamma\approx 24\,\mathrm{eV}\,\text{\AA}^{3}$ being the cubic Dresselhaus parameter for GaAs Fabian et al. (2007). The GaAs barrier’s height and width are captured by the dimensionless BTK-like barrier measure $Z=(2mV_{\mathrm{SC}}d_{\mathrm{SC}})/(\hbar^{2}k_{\mathrm{F}})$ (typically, $V_{\mathrm{SC}}\sim 0.75\,\mathrm{eV}$ Matos-Abiague and Fabian (2009) so that $Z=1$ represents a barrier with thickness $d_{\mathrm{SC}}\sim 0.40\,\mathrm{nm}$). Figure 2 shows the dependence of the normalized zero-bias 444Experimental measurements of the TAHE response in the F simultaneously detect a contribution stemming from conventional anomalous Hall effects. To separate both parts, one could exploit the TAHE contribution’s unique voltage dependence Note (3). TAHE conductances, $G_{x,z}$ and $G_{y,z}$, on the F’s magnetization orientation for various barrier strengths, $Z$, and the Rashba SOC parameter $\alpha\approx 42.3\,\mathrm{eV}\,\text{\AA}^{2}$, which lies well within the experimentally accessible values Moser et al. (2007); Matos-Abiague and Fabian (2009). To quantitatively compare the conductance amplitudes, the insets show analogous calculations in the normal-conducting state.

Our simulations reveal all the TAHE conductances’ important properties. First, we observe the sinelike (cosinelike) variation of $G_{x,z}$ ($G_{y,z}$) with respect to the F’s magnetization angle. Those dependencies follow from symmetry considerations Note (3) and unambiguously reflect the junction’s magnetoanisotropic transport characteristics Matos-Abiague and Fabian (2015). Second, we find that skew AR and SR can indeed act together in superconducting junctions, leading to sizable TAHE conductances (and voltages Note (3)). Specifically, $G_{x,z}$ can be increased by more than one order of magnitude and $G_{y,z}$ still roughly by a factor of 4, compared to normal junctions. However, the full physical mechanism is more complicated than our simple picture in Fig. 1, where we considered one particular combination of in-plane momenta. To obtain the total TAHE conductances, we need to average over all possible configurations [see Eq. (3)], which can—mostly depending on the barrier and Rashba SOC strengths—also reverse the Hall current’s direction, observed, for example, in $G_{y,z}$ by increasing $Z$ from $Z=1$ to $Z=4$ Note (3). Finally, we can confirm the stated connection between the skew reflection mechanism and the TAHE conductances for strong tunneling barriers. As $Z$ increases, $V_{\mathrm{eff}}$ is mostly determined by the bare barrier strength itself [see Eq. (2)], and the momentum- and spin-dependent SOC asymmetry, responsible for the Hall current generation, gets remarkably suppressed (especially in the superconducting regime). As a result, strong barriers significantly decrease the TAHE conductances.

To resolve AR and SR, Fig. 3 shows their spin-resolved conductance contributions. The spin-flip AR part is not separately shown as its amplitudes are up to two orders of magnitude smaller than those of (spin-conserving) AR. Interestingly, the total TAHE conductance is nearly fully dominated by (spin-conserving) AR and SR; both contributions are comparable in magnitude and have the same signs so that they indeed add up, resulting in sizable TAHE conductances. Since spin-flip SR involves electrons with opposite spin, the effective barrier picture in Fig. 1 gets reversed and the related TAHE conductance contribution changes sign. Nevertheless, this contribution is much smaller than those attributed to spin-conserving skew reflections so that it cannot modify the TAHE conductances’ qualitative features.

AR is the crucial scattering process at metal/S interfaces; it transfers Cooper pairs, converting normal into supercurrents, plays an important role in experimentally quantifying Fs’ spin polarization Soulen et al. (1999), and is also essential for the sizable TAHE conductances in the F of our system. Particularly interesting are the transferred Cooper pairs, which are also exposed to the effective scattering potential and may thus trigger a response in terms of a transverse supercurrent in the S Mironov and Buzdin (2017). Within our model, we evaluate the zero-bias supercurrent components, $I_{\eta}$, starting from a generalized Furusaki–Tsukada technique Furusaki and Tsukada (1991) (see the SM Note (3)). For the considered parameters, we stated that the main skew AR contribution to the TAHE conductance comes from the spin-conserving process. As the latter involves spin-singlet Cooper pairs, composed of electrons with opposite transverse momenta and spins, one could generalize our skew reflection picture to a combined one for the two individual Cooper pair electrons. As a consequence, the induced supercurrents’ qualitative features follow the same trends as those of the TAHE conductances in Fig. 2. Figure 4, presenting $I_{\eta}$ as a function of the magnetization angle, $\Phi$, confirms this expectation: the supercurrent components’ dependence on $\Phi$ and their orientations (signs) reflect one-to-one the properties of the TAHE conductances in the F. Even the sign change we explored in $G_{y,z}$ when changing $Z$ from $Z=1$ to $Z=4$ is (qualitatively) transferred into the supercurrent response $I_{y}$. Nevertheless, there is one important difference from the TAHE conductance, concerning the currents’ magnitudes. The supercurrent response always results from two single electrons that tunnel into the S, forming a Cooper pair. In order to generate sizable supercurrents, both electrons must simultaneously skew tunnel into the S (mediated by skew AR), which is less likely to happen at strong barriers than skew tunneling of unpaired electrons. Therefore, the maximal supercurrent amplitudes—several milliamperes for optimal configurations—occur at smaller $Z$ than the maximal TAHE conductance amplitudes in the F.

To conclude, we investigated the intriguing consequences of skew AR and SR at SC interfaces of superconducting tunnel junctions. We predict that the interplay of both skew reflection processes can constructively amplify their effects. Furthermore, also the Cooper pairs transferred into the S via AR cycles are subject to interfacial skew reflections. As a result, both sizable TAHE conductances in the F and characteristically modulating transverse supercurrents in the S are generated, opening new venues for experimental and theoretical studies.