Crossed Andreev Reflection in InSb Flake Josephson Junctions

TL;DR

This paper investigates superconducting interference in InSb flake Josephson junctions, revealing an even-odd pattern linked to crossed Andreev reflection, with implications for topological states and non-local entanglement.

Contribution

It demonstrates the occurrence of crossed Andreev reflection in InSb flakes and links it to specific edge conduction and unique interference patterns, advancing understanding of non-local superconducting processes.

Findings

Observation of an even-odd interference pattern.

Identification of edge-specific conduction related to crystal facets.

Evidence supporting crossed Andreev reflection as the underlying mechanism.

Abstract

We study superconducting quantum interference in InSb flake Josephson junctions. An even-odd effect in the amplitude and periodicity of the superconducting quantum interference pattern is found. Interestingly, the occurrence of this pattern coincides with enhanced conduction at both edges of the flake, as is deduced from measuring a SQUID pattern at reduced gate voltages. We identify the specific crystal facet of the edge with enhanced conduction, and confirm this by measuring multiple devices. Furthermore, we argue the even-odd effect is due to crossed Andreev reflection, a process where a Cooper pair splits up over the two edges and recombines at the opposite contact. An entirely $h/e$ periodic SQUID pattern, as well as the observation of both even-odd and odd-even effects, corroborates this conclusion. Crossed Andreev reflection could be harnessed for creating a topological state of matter or performing experiments on the non-local spin-entanglement of spatially separated Cooper pairs.

Tables

Table 1. Table S1: Geometry parameters for all 7 JJs. The width, W 𝑊 W , and length, L 𝐿 L , are obtained from the SEM images. The areas, A 𝐴 A and A eff subscript 𝐴 eff A_{\mathrm{eff}} , are the relevent areas for the applied perpendicular magnetic field, where the effective area includes flux focusing from the superconducting contacts due to the Meissner effect.

	JJ1	JJ2	JJ3	JJ4	JJ5	JJ6	JJ7
$W$ (nm)	770	1280	855	790	1360	970	1150
$L$ (nm)	240	240	185	425	205	465	670
$L/W$	0.31	0.19	0.22	0.54	0.15	0.48	0.58
$A$ ($\mu$m2)	0.18	0.31	0.16	0.34	0.28	0.45	0.77
$A_{\mathrm{eff}}$ ($\mu$m2)	0.63	0.74	0.95	1.21	1.37	0.81	1.03

Equations

$$I_c(\Phi )=I_{c0}\left|\mathcal{I}(\Phi )+f\right|,$$

$$I_{\text{c}}(\Phi )=I_{\text{c,0}}\left|\frac{\sin (\pi \Phi /{\Phi }_0)}{\pi \Phi /{\Phi }_0}\right|,$$

$$I_{\text{c}}(\Phi )=I_{\text{c,0}}\left|\cos (\pi \Phi /{\Phi }_0)\right|,$$

$$I_{\text{c,ball}}(\Phi )=$$

$$\sum _{k=1}^{\infty }(-1{)}^{k+1}\frac{L}{{\xi }_{\text{T}}\cos {\theta }_{x_1-x_2}}\frac{\sin k\left[\frac{\pi \Phi }{W{\Phi }_0}(x_1+x_2)+\varphi \right]}{\sinh \frac{kL}{{\xi }_{\text{T}}\cos {\theta }_{x_1-x_2}}};$$

$$I_{\text{c,diff}}(\Phi )$$

$$S_l(u)$$

$$f_{\text{diff}}(\nu )=\frac{{\cos }^2\pi \nu /2}{(1-{\nu }^2{)}^2}.$$

$$I_c(\Phi )\propto f=e^{-0.238{\Phi }^2}.$$

$$f\sim {\Gamma }^{-1}\frac{k_{B}T}{\Delta }{e}^{-2\pi (k_{B}T/\Delta )(W/{\xi }_s)}.$$

Full text

Induced superconductivity in semiconductors with strong spin-orbit interaction (SOI) attracted much interest for its potential applications in topological quantum computation Alicea (2012). A semiconducting Josephson junction (JJ) offers a platform to study the induced superconductivity by means of superconducting quantum interference (SQI) Tinkham (1996). Recently, induced superconductivity in edge channels in the quantum Hall regime Amet et al. (2016) and in a predicted two-dimensional topological insulator Hart et al. (2014); Pribiag et al. (2015), interesting for topological zero modes such as parafermions or Majoranas ,are investigated using SQI. Additionally, an oscillation with both $h/e$ and $h/2e$ periodic components, before connected to topological edge states Pribiag et al. (2015), is observed in a trivial InAs quantum well and attributed to crossed Andreev reflection (CAR) in the JJ Baxevanis et al. (2015); de Vries et al. (2018).

Crossed Andreev reflection is a process where the quasiparticles that form a Cooper pair, are spatially separated but still entangled. The entanglement of these quasiparticles holds promise in harnessing electrons in a solid-state environment to for example test the Einstein-Podolsky-Rosen paradox Einstein et al. (1935) – of fundamental importance to both quantum communication and computation. Additionally, coupling two one-dimensional (1D) structures (i.e. nanowires or edge states) via CAR is interesting for engineering a topological state of matter hosting parafermions Klinovaja and Loss (2014) or MajoranasGaidamauskas et al. (2014). To observe pronounced CAR in a device, normal or direct Andreev reflection needs to be suppressed. In this regard, quantum dots Recher et al. (2001); Hofstetter et al. (2009) or Luttinger liquids Recher and Loss (2002) can be utilized. Two-dimensional (2D) systems, such as a 2D electron gas Russo et al. (2005) connected to a superconductor, offer a scalable and flexible platform for more complex device geometries. Therefore, exploiting coupled 1D edge channels in a 2D material for Cooper pair splitting, combines the large CAR amplitude and flexibility in device design Sato et al. (2012); Baxevanis et al. (2015).

Here, we obtain measurements of CAR in a JJ made of an InSb flake – a 2D nanostructure. We observe both even-odd and odd-even Fraunhofer patterns, and an entirely $h/e$ periodic SQUID pattern in JJs with enhanced conduction at both edges. We argue that these $h/e$ effects are caused by a flux independent supercurrent due to CAR, where the quasiparticles are spatially separated over the two edges.

InSb is known for its large g-factor de la Mata et al. (2016) and strong SOI Nilsson et al. (2009), and earlier works referred to the flakes as nanosails de la Mata et al. (2016) or nanosheets Pan et al. (2016); Zhi et al. (2019). The InSb flakes are grown with the vapor-liquid-solid technique Gazibegovic et al. (2017); SM . The crystal facet on their edges is (110) when stemming from a wire surface SM . This (110) facet is known for having electron accumulation at its surface Kreutz et al. (1983); Ritz and Lüth (1985), because the lack of Sb atoms results in band bending Whitman et al. (1990). Considering the geometry, we expect strong band bending at the edges of the flake with (110) facets as sketched in Fig. 1(a,b). To fabricate devices, we use a micro-manipulator to transfer the flakes to a Si/SiOx substrate that serves as a global bottom gate. Then, two NbTiN contacts are deposited after treating the surface with a sulfur solution to remove the native oxides S_e ; Gül et al. (2017). The geometrical parameters, such as contact separation, $L$, and width, $W$, of all JJs are presented in the Supplemental Material SM . The JJs are measured in a quasi four-terminal current bias setup at a temperature of 300 mK, unless stated otherwise. Characterization of the superconductivity provides us an estimate of the superconducting gap, $\Delta$, of 1.4 meV, consistent with values found earlier for NbTiN Gül et al. (2017), and the induced superconducting coherence length, $\xi_{s}$, of 1.2 µm at $V_{\mathrm{BG}}=15$ V (see Supplemental Material for details SM ).

Superconducting quantum interference measurements are performed by measuring the switching current of the JJs while varying the flux through them with an out-of-plane magnetic field. Interestingly, the SQI pattern of JJ1 in Fig. 1(c) does not show the regular Fraunhofer pattern Tinkham (1996) as observed in Fig. 1(d) for JJ2. The pattern instead displays an even-odd effect, which means the amplitude of the side lobes is not monotonically decaying but alternating. The first side lobe has a smaller amplitude than the second, and the amplitude of the third side lobe is zero. We describe this even-odd effect with a positive, magnetic field independent supercurrent offset $f$, added to the expected interference pattern $\mathcal{I}(\Phi)$:

[TABLE]

where $I_{c0}$ is the critical current at zero magnetic field, it increases (decreases) the amplitude of the lobes with the same (opposite) sign. Examples of Fraunhofer and SQUID patterns with different positive offsets are presented in Fig. 2(d-e). A SQUID pattern with such an offset is reported before Pribiag et al. (2015); de Vries et al. (2018), however the even-odd Fraunhofer pattern is not experimentally studied to our knowledge.

The SQI pattern in our InSb flake JJs cannot be described by the standard Fraunhofer pattern Tinkham (1996), because our JJs does not satisfy the limit of $W\gg L$ (i.e. for JJ1 $W=1280$ nm and $L=240$ nm). Therefore, we use a theoretical model for rectangular JJs as proposed by Barzykin et al. in Ref. 27, where the supercurrent is calculated over all possible quasiparticle trajectories for either a ballistic or diffusive junction. The expressions for the resulting SQI patterns for both cases are provided in the Supplemental Material for convenience SM . We use the ballistic model in Fig. 1 (dashed gray lines), since we estimate the mean free path, $l_{\text{MFP}}$, to be around 250 nm, similar to $L$ [Fig. 3(b)]. The SQI pattern of JJ2 is well resembled by the calculated SQI pattern [Fig. 1(d)], showing that the larger periodicity of $1.5\Phi_{0}$ is due to the rectangular geometry of the JJ Barzykin and Zagoskin (1999). Note that the SQI patterns are compensated already for flux focusing due to the Meissner effect SM . In Fig. 1(c), the dashed gray line is calculated using the ballistic model Barzykin and Zagoskin (1999) with an offset of $f=0.09$ incorporated with equation 1. The even-odd behavior of the SQI pattern is qualitatively reproduced, supporting our assumption of a flux independent supercurrent.

In the calculations we have implicitly taken into account that the supercurrent is homogeneously distributed throughout the JJ. To check this, we reconstruct the current density distribution with the method described by Dynes and Fulton Dynes and Fulton (1971), which we are allowed to use, since the JJs are in the short junction limit, $\xi_{s}>L$ Hui et al. (2014). The current density distribution for JJ2, plotted in Fig. 1(f), is homogeneously distributed, whereas for JJ1 [Fig. 1(e)] it reveals a large peak at the center of the JJ. Apart from having a supercurrent through the center of the JJ, the peak could also be due to a magnetic field independent supercurrent. Because the Dynes-Fulton method is based on a Fourier transform, constant (or zero frequency) components end up at $x\,=\,0$ – the center of the distribution. Such a supercurrent offset cannot be due to a partial short in the JJ, since we confirmed the supercurrent can be pinched off by the global bottom gate SM . Because of its insensitivity to magnetic field, the offset cannot stem from mechanisms that occur at a certain magnetic field either Yokoyama et al. (2014); Meier et al. (2016). An effect that however could cause a magnetic field independent supercurrent is CAR Recher et al. (2001). CAR describes an Andreev pair of which one quasiparticle encircles the junction area [Fig. 2(a)] and therefore acquires a phase proportional to the flux through the junction area. That extra phase can either directly van Ostaay et al. (2011) or by interference of two different Andreev pairs, result in a flux independent supercurrent Baxevanis et al. (2015); Jacquet et al. (2015); Liu et al. (2017).

To find out whether a central current path or a magnetic field independent supercurrent due to CAR causes the even-odd effect, we continue by studying the gate dependence of the SQI patterns. The even-odd SQI pattern from JJ1 changes drastically as a function of gate voltage [Fig. 2(b)]. The amplitude of the first side lobe decreases as $V_{\text{BG}}$ is reduced (highlighted by the red arrows), and becomes zero at $V_{\text{BG}}=3$ V. Then, for the bottom two traces of Fig. 2(b), the SQI pattern takes a cosinusoidal shape, known as a SQUID pattern Tinkham (1996). When the amplitude of the second lobe drops below the offset of the SQI pattern, the periodicity of the SQI pattern changes, see also the curve for $f=1$ in Fig. 2(e). It doubles from 1.3$\Phi_{0}$ at $V_{\text{BG}}=15$ V to 2.7$\Phi_{0}$ at $V_{\text{BG}}=3$ V, and the SQI pattern becomes entirely $h/e$ periodic for $V_{\text{BG}}\leq 1.2$ V. This is different from the observed $h/e$ SQUID in Ref. 5 and Ref. 7, where the amplitude was larger than the offset and therefore an $h/2e$ oscillation is observed simultaneously. Our observation confirms that the $h/e$ periodicity is not a unique signature of a topological JJ de Vries et al. (2018).

The changes in the SQI pattern are reflected in the extracted current density distributions in Fig. 2(c). Note that $J$ is spanning half of the width for $V_{\text{BG}}\leq 3$ V compared to $V_{\text{BG}}\geq 6$ V, because we used the same area and flux periodicity for the calculation of all traces. The center peak in the current density disappears at $V_{\text{BG}}=3$ V. Such a local effect in $J$ is not likely to be caused by changing the global gate. Additionally, the offset in the SQI patterns persists, even though there is no center peak anymore. Therefore, we disregard a current path at the center of the JJ as an explanation for the even-odd effect. Interestingly, between $V_{\text{BG}}=3$ V and 0 V, the SQUID pattern translates to a current density distribution with edge conduction only [bottom trace of Fig. 2(c)], in agreement with JJ1 having electron accumulation at both edges [Fig. 1(a)]. This enhanced conduction at both edges of the JJ, in combination with the $h/e$ periodicity is consistent with the occurrence of CAR Baxevanis et al. (2015); de Vries et al. (2018). To substantiate this, we consider this mechanism in detail, and study additional JJs.

The CAR trajectories encircling the JJ area consist of the edges of the flake, and two paths along the contacts, as sketched in Fig. 2(a). The latter could arise of combining doping from and a finite barrier to the superconducting contact de Vries et al. (2018). The estimated induced superconducting coherence length of 1.2 µm is close to the typical junction circumference of 2 µm SM . Due to the difference between the circumference and $L$, the CAR is expected to be suppressed with temperature before direct Andreev reflection (and supercurrent) diminishes. A temperature dependence is however ambiguous, since the side lobes (and with that the even-odd effect) disappear before the switching current is suppressed SM . The magnitude of the offset reaches a value of 1 in Fig. 2(b), which is in range of what one can expect for a combination of a small coupling to the contact, while maintaining the Fermi velocity in the edges Baxevanis et al. (2015); SM . Additionally, electron-electron interaction in the 1D edges could also reduces the direct Andreev reflection and lead to a large f Sato et al. (2019). The the large offset $f$ and $h/e$ periodicity mean that the CAR amplitude exceeds the direct Andreev reflection, an interesting topic and regime for future experiments.

To shine light on the correlation between the even-odd effect and having enhanced conduction at both edges, we study the gate dependence of JJ2 as well. The periodicity of the SQI patterns in Fig. 3(a) grows slightly from $1.3\Phi_{0}$ to $1.5\Phi_{0}$ as $V_{\text{BG}}$ is lowered from 30 V to 1.2 V. Meanwhile the mobility $\mu$ and $l_{\text{MFP}}$ decrease [Fig. 3(b)], and for $V_{\text{BG}}<3$ V the length $L$ of the JJ is larger than $l_{\text{MFP}}$ and the JJ changes from ballistic to diffusive. In Fig. 3(c), calculated SQI traces for ballistic and diffusive transport are plotted Barzykin and Zagoskin (1999), consistent with the transition observed in our data. Furthermore, the strong increase in periodicity and suppression of the side lobe amplitudes for gate voltages below $V_{\mathrm{BG}}=1.2$ V highlight a transition from a 2D to a 1D diffusive regime. The measured SQI pattern at $V_{\text{BG}}=-0.3$ V is well reproduced by the theoritical curves for a (quasi) 1D JJ [Fig. 3(c)], as described Barzykin and Zagoskin (1999); Chiodi et al. (2012) and observed before Chiodi et al. (2012); Amado et al. (2013). Entering the 1D regime is in line with having a single edge with (110) facet and enhanced edge conduction in JJ2 [Fig. 1(b)]. Continuing the argument, not finding an even-odd effect in JJ2, strengthens the connection between the even-odd SQI pattern and having enhanced conduction at both edges.

Two other devices (JJ3 and JJ4) also reveal an even-odd SQI pattern, and show enhanced conduction in both their edges with a (110) facet SM . Furthermore, JJ5-JJ7, having a single (110) edge, do not show an even-odd SQI pattern SM . Interestingly, the SQI pattern from JJ3 [Fig. 4(b)] shows a negative offset, or odd-even effect. The third lobe has a similar amplitude to the second, thus is not smaller as expected for a standard Fraunhofer pattern. By adding a negative offset to the calculated SQI pattern from Ref. 27 [dashed gray line in Fig.4(b)], we indeed find good agreement with the data. For comparison we plotted the standard Fraunhofer patterns with negative offsets in Fig. 4(c). Having either a positive or negative offset to the switching current due to CAR depends on microscopic details regarding the spin mixing in the JJ Baxevanis et al. (2015). To be more precise, spin mixing with predominantly spin conserving or spin-flip processes, refer to a positive or negative offset, respectively. In our InSb flakes, the spin mixing is probably caused by strong spin-orbit interaction in the InSb Nilsson et al. (2009). The observation of a negative offset is, to our knowledge, unique to CAR Baxevanis et al. (2015), and therefore strongly supports that CAR is causing the observed $h/e$ periodic SQI patterns.

In conclusion, we observe $h/e$ effects due to crossed Andreev reflection in the SQI patterns of InSb flake Josephson junctions with enhanced conduction at both edges. We identified crystal facet (110) to have enhanced edge conduction, and can thus in the future choose to either study or circumvent them. The observed $h/e$ SQUID pattern reveals that the CAR amplitude can exceed the direct Andreev reflection in a 2D semiconducting Josephson junction. The InSb flakes therefore provide a promising platform to use CAR for creating topological zero modes Klinovaja and Loss (2014); Gaidamauskas et al. (2014) or for applications in Cooper pair splitting Einstein et al. (1935); Sato et al. (2012).

Acknowledgments

This work has been supported by funding from the Netherlands Organization for Scientific Research (NWO) and Microsoft.

Author contributions

S.C.B developed the substrate preparation, S.G. grew the flakes, D.C. and E.P.A.M.B. contributed to the growth and R.L.M.o.h.V developed the flake deposition procedure. J.S., F.K.d.V. and M.S. fabricated the devices, performed the measurements and analyzed the data. F.K.d.V. and J.S. wrote the manuscript with comments from all authors. J.S. and L.P.K. supervised the project.

I InSb flakes growth

II Devices

III Induced superconductivity

IV Superconducting quantum interference models

IV.1 Fraunhofer and SQUID SQI patterns

The standard Fraunhofer and SQUID SQI patterns are given by Tinkham (1996):

[TABLE]

[TABLE]

where $I_{\text{c,0}}$, is the critical current at zero flux, $\Phi$ is the flux and $\Phi_{0}=h/2e$ the superconducting flux quantum.

IV.2 Ballistic and difussive SQI

The superconducting quantum interference in a JJ with a length, $L$, and finite width, $W$, is described analytically in Ref. Barzykin and Zagoskin (1999). For convenience we included their main results here. The final expressions for the critical current as a function of flux in a ballistic JJ ($I_{\text{c,ball}}$) and diffusive JJ ($I_{\text{c,diff}}$) are Barzykin and Zagoskin (1999):

[TABLE]

[TABLE]

where, $\phi$ is the superconducting phase difference, $v_{\text{F}}$ and $\lambda_{\text{F}}$ are the Fermi velocity and wavelength, $\xi_{\text{T}}$ is the normal metal coherence length ($\hbar v_{\text{F}}/2\pi k_{\text{B}}T$), $\tan\theta_{x_{1}-x_{2}}=(x_{2}-x_{1})/L$ is the angle of the Andreev reflection with respect to the vector perpendicular to the contact. $K_{1/2}$ is a modified Bessel equation and $q_{\text{T}}=1/\tilde{\xi}_{\text{T}}^{2}$, where $\tilde{\xi}_{\text{T}}$ is the diffusive coherence length. The SQI pattern in the limit of $L\ll W$ for a diffusive JJ reads Barzykin and Zagoskin (1999):

[TABLE]

The geometrical parameters used for the different devices, can be found in table S1. Furthermore, we use an effective mass of 0.02 $m_{e}$ Qu et al. (2016) and a temperature of 300 mK (or 50 mK) to extract parameters from the Hall bar measurement (Fig. S6). For the calculated SQI patterns in Fig. 1 and Fig. 4, at $V_{\text{BG}}=15$ V, we find $v_{\text{F}}=2.5\cdot 10^{6}$  m/s, $\lambda_{\text{F}}$= 2.3 nm, and $\xi_{\text{T}}$=10 $\mu$m; and for Fig. 3, at $V_{\text{BG}}=3$ V, $v_{\text{F}}=1.5\cdot 10^{6}$  m/s, $\lambda_{\text{F}}$= 4.0 nm, and $\xi_{\text{T}}$=5.9 $\mu$m. Because the devices are in the crossover regime from ballistic to diffusive transport, we use the ballistic coherence length in both cases, so we set $\tilde{\xi_{\text{T}}}=\xi_{\text{T}}$. With a transmission value of $T\leq 0.6$ we expect to have a sinusoidal current phase relation Haberkorn et al. (1978), and therefore only use $k=1$ for eq. 4. For eq. 5 we sum up to $l=100$. For both we confirmed that the periodicity does not change for summing over a larger $k$ or $l$, respectively. In all figures in the main text we plot the normalized critical current, either $I_{\text{c}}(\Phi)/I_{\text{c}}(0)$ or $f_{\text{diff}}(\Phi)/f_{\text{diff}}(0)$.

IV.3 1D diffusive model

For a one-dimensional system in the diffusive limit, the critical current decays monotonically, following Ref. 7:

[TABLE]

IV.4 CAR offset $f$

The offset in the SQI pattern caused by CAR is expressed as Baxevanis et al. (2015):

[TABLE]

In example for JJ1: $T=300$ mK, $W\,=\,770$ nm, $\Delta=1.4$ meV, $\xi_{s}=1.2$ µm; resulting in $f=\Gamma^{-1}\cdot 0.02$. To reach $f=1$, $\Gamma=0.02$, which means there should be a relatively strong barrier between the contacts and the edge states.

V Temperature dependence SQI pattern JJ1

VI Hall bar measurement

VII Extra devices: JJ3-JJ7