Phase Coherence and Andreev Reflection in Topological Insulator Devices

Topological insulators (TIs) have attracted immense interest because they host helical surface states. Protected by time-reversal symmetry, they are robust to non-magnetic disorder. When superconductivity is induced in these helical states, they are predicted to emulate p-wave pairing symmetry, with Majorana states bound to vortices. Majorana bound states possess non-Abelian exchange statistics which can be probed through interferometry. Here, we take a significant step towards Majorana interferometry by observing pronounced Fabry-Perot oscillations in a TI sandwiched between a superconducting and normal lead. For energies below the superconducting gap, we observe a doubling in the frequency of the oscillations, arising from the additional phase accumulated from Andreev reflection. When a magnetic field is applied perpendicular to the TI surface, a number of very sharp and gate-tunable conductance peaks appear at or near zero energy, which has consequences for interpreting spectroscopic probes of Majorana fermions. Our results demonstrate that TIs are a promising platform for exploring phase-coherent transport in a solid-state system.


2
When electrons travel far distances without scattering off of impurities, their wave-like nature becomes apparent through signatures of interference. A prominent example is ballistic transmission through a barrier between two semi-infinite leads. Imperfect transmission through the interfaces causes partial reflection of impinging electrons. Within the barrier, constructive interference of reflected waves leads to periodic modulation of total transmission through the barrier, resulting in resonant transmission whenever the electron wavelength is an integer multiple of twice the barrier length. This is equivalent to k F L = πn, where k F is the Fermi wave vector, L is the barrier length, and n is a nonzero integer.
This quantum phenomenon, known as Fabry-Perot interference due to its similarity to the eponymous classical optical effect, has been observed in a variety of mesoscopic systems, including nanotubes [16], nanowires [17], and graphene [18]. In topological systems with protected edge states, interferometry takes on a new significance as particles traveling through the edge states can acquire an additional phase if their paths enclose exotic quasiparticles known as anyons [3]. A class of anyons known as Majorana bound states (MBSs) can be realized in a topological insulator with induced superconductivity [2]. MBSs can encode quantum information non-locally, realizing topologically protected qubits whose states can be read out through interferometry [4][5][6].
We study Andreev reflection in thin (∼ 10 nm) flakes of the 3D topological insulator Bi 2 Se 3 connected to both superconducting and normal metal leads separated by L = 100 − 250 nm. An example of a device with L = 230 nm is pictured in Fig. 1a. Either top or back gates are used to tune the chemical potential of the surface states, depicted in Here we focus on one sample, with qualitatively similar results obtained from others (see Supplementary Materials). At low temperature, we observe an enhancement in conductance at source-drain voltages (V ) below the niobium gap, as expected for Andreev reflection (Fig. 1c) [19][20][21]. Near V = 0, however, there is a downturn in the conductance. This is consistent with the re-entrant resistance effect [22,23], in which Andreev reflection is cancelled out by the proximity effect in the limit of zero energy and zero temperature [23,24]. Maximum conductance occurs at either a characteristic temperature or V that is governed by the Thouless energy E T =hD/L 2 , where D is the diffusion constant and L is the separation between the superconducting and normal metal leads on the bismuth selenide. Magnetotransport and ARPES measurements [25] allow us to calculate E T through where v F is the Fermi velocity, l is the mean free path, µ is the mobility, N 2D is the 2D carrier density. This results in E T = 30 µV for L = 230 nm, consistent with our transport measurements (T max = 900 mK and V max = 85 µV). At high temperature (Fig. 1d), the conductance dip disappears and only a broad peak from Andreev reflection remains. We note in passing a remarkable similarity between Fig. 1 and the predictions of Ref. [26], which posited that anomalous Andreev bound states on the surface of a topological superconductor would generate an asymmetric zero bias tunneling conductance peak [27] that splits at low temperature. However, we feel that our data is To demonstrate phase coherent transport, we turn to gate-tuned measurements. In such oscillations can be observed without special care for cleanliness, as achieved in other materials through current annealing or isolating the material from dielectric impurities by suspension from the substrate. This is likely due to the topological protection of the helical surface states, although we do not rule out the contribution from trivial states. The peakto-peak amplitude of the oscillations are ≈ 0.2e 2 /h, independent of normal state resistance, which varies from 600 Ω to 21 kΩ in our three devices. This suggests that while bulk carriers might contribute to the zero bias conductance, the interference originates from a finite number of surface states.
The most unusual feature of the Fabry-Perot oscillations in Fig. 2 is a doubling of their frequency when V is less than the superconducting gap of niobium, as shown in the lower graph in Fig. 2b. Coulomb repulsion can split otherwise degenerate levels [29], but the high degree of coupling between the leads and the TI should minimize charging effects.
Furthermore, the peaks in conductance at low bias are evenly separated in V BG , unlike the uneven spacing expected for split degeneracies. Instead, this behavior can be understood as a modification of phase coherent transport by Andreev reflection. Low energy electrons passing from the cavity to the superconducting lead will be retro-reflected as a hole with high probability in the absence of significant barrier, depicted in the lower cartoon in Fig. 2b.
As first pointed out by de Gennes and Saint-James, the reflected hole traverses the cavity and reflects off the interface with the normal metal lead. The holes do not interfere with the incident electrons; instead, the hole returns to the superconducting lead to undergo Andreev reflection again, leading to an emitted electron. This reflected electron can traverse the cavity and subsequently interfere with incident electrons. Because in this process the reflected particles travel four times the cavity length rather than twice the cavity length, the corresponding geometric resonances occur with twice the frequency. Although this interplay between Andreev reflection and geometric resonances was predicted over 50 years ago [9,10], we believe this to be the first observation of a clear cross-over from de Gennes-Saint-James resonances to conventional Fabry-Perot behavior.
In Fig. 2c, we explore the temperature dependence of the Fabry-Perot oscillations. The amplitude of the oscillations is rapidly suppressed by thermal fluctuations, decreasing by roughly a factor of two between 20 mK and 1.6 K. This is consistent with the temperature dependence of the phase-coherence length l φ from weak anti-localization measurements in Hall bar devices, helping to confirm that the oscillations are an interference phenomenon.
We next consider transport in magnetic fields applied perpendicular to the TI surface.
While in-plane fields have no influence up to at least 100 mT, with out-of-plane fields we observe a steady evolution of the background signal from universal conductance fluctuations and suppression of superconductivity, as shown in Fig. 2d. While the amplitude of the Fabry-Perot oscillations are similarly reduced, their phase does not change appreciably, suggesting that the fields we use are too small to observe the phase shift expected from Klein backscattering [30]. Small shifts can be observed in the features from Fabry-Perot oscillations, but it is difficult to reliably disentangle them from the evolving background signal due to universal conductance fluctuations.
Strikingly, in two of our samples we observe a number of narrow, low energy conductance peaks emerge at large magnetic fields. An example is shown in Fig. 3a, with data from the second device shown in the Supplementary Materials. The location of the peaks can be tuned readily with V BG , causing them to even form a zero bias conductance peak for certain gate ranges with height ≈ e 2 /h, indicated in Fig. 3b. The peaks split and reform in a quasiperiodic fashion that does not seem to be directly related to the Fabry-Perot oscillation period. In Fig. 4a, we detail the field evolution of one of these zero bias anomalies. We find that the zero energy resonance forms at finite field and can persist for at least up to 120 mT. At elevated temperatures (Fig. 4b), we find these anomalies broaden and merge with the background.
These peaks are notable because no such features are seen in these two samples at zero magnetic field. They appear only when large fields are applied perpendicular to the TI surface and are accompanied by a collapse of the minigap governed by the Thouless energy (depicted in Fig. 4a). They share many of the properties of zero bias anomalies observed in spectroscopic studies of nanowires coupled to superconducting leads [11][12][13][14][15]. While such studies were interpreted in terms of Majorana fermions, here we are more cautious. For example, weak anti-localization can also generate low energy resonances due to coherence between electron-hole trajectories that repeatedly impinge upon the superconductor-semiconductor interface [31]. Although originally proposed to describe diffusive semiconductors, such trajectories can exist in ballistic cavities due to multiple reflections off the boundary opposite of the superconductor-semiconductor interface [32]. In this respect, this phenomenon is similar to the de Gennes-Saint-James resonances but is restricted to near zero energy.
It has been pointed out that weak anti-localization in superconductor-semiconductor devices could survive in the presence of broken time-reversal symmetry, leading to low energy anomalies that are nearly identical in appearance to those attributed to Majorana fermions [33]. The persistence of the de Gennes-Saint-James resonances in our devices with magnetic field strongly suggests that the analogous trajectories for weak anti-localization can also remain phase coherent. This is in contrast to the previous assumption in the nanowire experiments that the large magnetic field would suppress weak anti-localization [11][12][13][14][15]. We note that while Refs. [34] and [35]

COMPETING FINANCIAL INTERESTS
The authors report no competing financial interests.

I. ADDITONAL EVIDENCE FOR RE-ENTRANT RESISTANCE EFFECT
Sample T3-V consisted of a 15 nm thick flake of Bi 2 Se 3 with one gold lead and one niobium lead, separated by L = 135 nm. The device was relatively highly conducting (normal state G ≈ 270 e 2 /h), possibly due to larger bulk conductance. As shown in Fig. S1, this device also displayed signs of the re-entrant resistance effect. Because the lead separation is smaller in this device than in the sample described in the main text, the Thouless energy E T =hD/L 2 is correspondingly larger and the suppression in conductance occurs over a wider range in source-drain bias V . This sample possessed a top gate, but the normal state resistance was relatively affected by a top gate bias. Similar behavior occurred in other highly conducting samples, none of which displayed any clear signs of Fabry-Perot oscillations. This supports the assertion of greater bulk contribution to the conductance in these samples as well as demonstrates that the gate-tuned Fabry-Perot oscillations originate from surface states.

II. ADDITIONAL FABRY-PEROT OSCILLATIONS
Sample T3-VII consisted of a 7 nm thick flake of Bi 2 Se 3 with one gold lead and one niobium lead, separated by L = 95 nm. The chemical potential is tuned by a top gate separated from the sample by 30 nm of Al 2 O 3 . This type of gating is susceptible to charge noise. Nonetheless, we were able to resolve both de Gennes-Saint-James resonances at low energies and conventional Fabry-Perot oscillations at energies beyond the niobium gap.
In Fig. S2a, we show an example of a stability plot for this device, with the location of the de Gennes-Saint-James resonances indicated by dashed lines. Note that the geometric resonances can lead to a zero bias peak in conductance. Similar behavior was observed at zero field, but the resonances are more apparent in the 50 mT data set due to less charge noise. In From the slopes of the features in the stability plot (Fig. S2a), we observe that a 1 V change in top gate bias shifts the Fermi energy by 5.7 mV. The Fabry-Perot oscillations at high V possess a periodicity of ∆V T G = 0.4 V; this is equivalent to a periodicity in Fermi energy of ∆E = 2.3 mV. This is consistent with the prediction that ∆E for this device should be larger than that of the device in the main text by the ratio of (235 nm) / (95 nm) = 2.47, based on the formula ∆E = hv F /L.

III. ADDITIONAL EVIDENCE FOR LOW ENERGY RESONANCES
Sample N5-VI consisted of a 17 nm thick and 250 nm wide flake of Bi 2 Se 3 with a narrow (100 nm) gold lead in between two niobium leads. The gold lead was separated by L ≈ 100 nm from either niobium lead. The two niobium leads are connected by a superconducting loop, enabling a phase-bias to be applied to the two superconducting leads. However, the conductance of the device showed no signs of phase-sensitivity. The device responded to large magnetic fields in ways similar to devices with only one superconducting lead.
At zero magnetic field, Sample N5-VI displayed signs of the re-entrant resistance effect.