Ubiquitous non-Majorana Zero-Bias Conductance Peaks in Nanowire Devices

TL;DR

This study reveals that zero-bias conductance peaks in nanowire devices, previously attributed to Majorana bound states, can also arise from trivial subgap states, emphasizing the need for careful interpretation in Majorana research.

Contribution

The paper introduces a multiband model explaining zero-bias peaks as trivial phenomena, challenging the assumption that all such peaks indicate Majorana states.

Findings

Zero-bias peaks are common and not exclusive to Majorana states.

Trivial subgap states can produce zero-bias conductance peaks.

A multiband model explains the origin of these peaks.

Abstract

We perform tunneling measurements on indium antimonide nanowire/superconductor hybrid devices fabricated for the studies of Majorana bound states. At finite magnetic field, resonances that strongly resemble Majorana bound states, including zero-bias pinning, become common to the point of ubiquity. Since Majorana bound states are predicted in only a limited parameter range in nanowire devices, we seek an alternative explanation for the observed zero-bias peaks. With the help of a self-consistent Poission-Schr\"odinger multiband model developed in parallel, we identify several families of trivial subgap states which overlap and interact, giving rise to a crowded spectrum near zero energy and zero-bias conductance peaks in experiments. These findings advance the search for Majorana bound states through improved understanding of broader phenomena found in superconductor-semiconductor systems.

Figures

Equations

$${\psi }_A=\frac{1}{\sqrt{2}}\left[{\varphi }_{ϵ}+{\varphi }_{-ϵ}\right],$$

$${\psi }_B=\frac{i}{\sqrt{2}}\left[{\varphi }_{ϵ}-{\varphi }_{-ϵ}\right].$$

Full text

Majorana bound states (MBS) are predicted in various intrinsic and engineered topological superconductors Fu and Kane (2008); Alicea (2010); Sau et al. (2010); Lutchyn et al. (2010); Oreg et al. (2010); Alicea (2012); Beenakker (2013). They attract sustained attention primarily thanks to the hypothesized non-Abelian rules for the two-MBS exchange Read and Green (2000). Tunneling experiments reported signatures of MBS by studying zero-bias conductance peaks Mourik et al. (2012); Das et al. (2012); Deng et al. (2012); Finck et al. (2013); Churchill et al. (2013); Nadj-Perge et al. (2014); Albrecht et al. (2016a); Deng et al. (2016); Chen et al. (2017); Gül et al. (2018); Zhang et al. (2018); Suominen et al. (2017); Nichele et al. (2017). The primary challenge for the tunneling evidence is that zero-bias anomalies in transport are widespread in mesoscopic systems. They have many known non-MBS origins such as Kondo effect Lee et al. (2012), weak antilocalization Pikulin et al. (2012), reflectionless tunneling Popinciuc et al. (2012), and supercurrent Zuo et al. (2017). Luckily, most of these phenomena can be ruled out for each particular Majorana experiment through their distinct dependence on the in-situ tunable parameters or through device design.

Yet, zero-bias anomalies of non-topological origin that closely resemble MBS, and cannot be straightforwardly ruled out, have also been identified. Most remarkably, trivial Andreev Bound States (ABS) have been demonstrated to result in zero-bias peaks Lee et al. (2014). This includes peaks that appear at finite magnetic field and exhibit some degree of pinning to zero bias or near-zero oscillations, both being features that MBS and ABS share. Trivial ABS can exist both in the topologically superconducting regime and in the trivial regime, or they can be a result of strong MBS hybridization Kells et al. (2012); Moore et al. (2018); Vuik et al. (2018). Thus ABS can be found in a much wider range of system parameters than MBS. Understanding of the full ABS phenomenology is therefore central to the unambiguous demonstration of MBS.

In this manuscript, we demonstrate that multiple coexisting and coupled ABS can lead to ubiquitous zero-bias peaks that share spectroscopic signatures with MBS. Our NbTiN/InSb devices have been designed for Majorana experiments, and they yield tunneling resonances that pin near zero source-drain voltage bias at finite external magnetic field, as expected for MBS. However, extended gate voltage sweeps reveal multiple families of states localized near the superconductor. We identify these states as being responsible for the omnipresent zero-bias resonances. The frequency of occurrence of zero-bias features, i.e. their ubiquity, makes it highly unlikely that all of them originate from topologically superconducting segments of the nanowire. A self-consistent multiband model developed in parallel Woods, Benjamin D. and Chen, Jun and Frolov, Sergey M., and Stanescu, Tudor D. (2019) finds a generic presence of overlapping and coupled trivial ABS for the device geometry used in the experiment. The model identifies that trivial ABS can persist near zero bias due to spectral crowding as well as level repulsion.

The basic MBS theories make a number of simple predictions for the tunneling manifestations of MBS in spin-orbit nanowires Lutchyn et al. (2010); Oreg et al. (2010). In long quantum wires MBS should only appear within the topologically superconducting phase described by the boundary equation: $E_{Z}>\sqrt{\Delta^{2}+\mu^{2}}$, where $E_{Z}=g\mu_{B}B/2$ is the Zeeman energy, with $g$ the effective Landé $g$-factor, $\mu_{B}$ the Bohr magneton. $\Delta$ is the induced superconducting gap at $B=0$, and $\mu$ is the chemical potential in the quantum wire. In the limits of zero temperature, hard induced gap and weak tunnel coupling, MBS manifest as a $2e^{2}/h$ peak in conductance at zero tunneling bias. The peak emerges after the bulk superconducting gap in the nanowire closes and re-opens. Beyond the re-opening point the peak is robust - meaning it does not deviate from zero bias until superconductivity is fully suppressed by external field or another subband crosses the Fermi level. MBS come in pairs, and therefore a correlated zero-bias peak should be observed on the opposite end of the nanowire. Spin-orbit anisotropy implies that zero-bias peaks should vanish for a specific magnetic field orientation that is collinear with the effective spin-orbit field.

The demonstration of all of the above basic tunneling predictions in the same nanowire will likely amount to proof of Majorana bound states beyond reasonable doubt. To date, this has not been possible, despite steady progress in growth and fabrication Krogstrup et al. (2015); Shabani et al. (2016); Gazibegovic et al. (2013). A given device can be tuned to display one or more tunneling signatures of MBS but not to simultaneously confirm all of the basic expectations. The discrepancies may still be consistent with MBS but ascribed to experimental limitations such as finite temperature, soft induced gap, disorder, short nanowire length, critical field anisotropy. When experimental limiatations are accounted for, MBS are expected to result in conductance peak oscillations around zero bias, reduced peak height, no gap closing and/or re-opening and distorted topological phase boundary. As noted above, several theories and experiments furthermore point out that these features are shared between imperfect MBS and trivial ABS making the two effects challenging to distinguish. In this manuscript we study the phenomenology of low-bias resonances without assuming MBS, but with a goal of deeper understanding the superconductor-semiconductor hybrid system.

Devices are fabricated using InSb semiconductor nanowires with NbTiN contacts (nominally identical to that used in Chen et al. (2017), though device-to-device variations are common (Fig.1(a), see supplemental materials for additional devices). Prior to the deposition of NbTiN, sulfur passivation is carried out followed by a gentle Ar plasma cleaning in order to obtain a transparent superconductor/semiconductor interface. A normal metal Pd contact is then fabricated to perform tunneling spectroscopy by varying bias voltage $V$ between normal and superconducting contacts. Electrical measurements are performed in a dilution refrigerator at a base temperature of $30$ mK, by a standard low-frequency lock-in technique (See detailed measurement conditions in Supplemental Material).

The electrostatic coupling of gates to the nanowire is enhanced due to half-coverage of the nanowire by the superconductor, as well due to the use of a thin layer of high-$\kappa$ gate dielectric (HfO2, 10 nm). The gate effect is much stronger than in fully-covered nanowires Gül et al. (2018), or where side gates and/or thicker dielectric layers are used Mourik et al. (2012); Albrecht et al. (2016b). Stronger electrostatic coupling allows us to tune the density underneath the superconductor in a wider range, and observe a larger variety of subgap states as shown below. On the flip side, partial coverage may result in weaker induced superconductivity and soft gap Vuik et al. (2016). Following a standard procedure for Majorana experiments Mourik et al. (2012), we create a single tunnel barrier between the normal and superconducting contacts by tuning gate $FG$ (once set $FG$ remains fixed). The gates left of $BG1$ are set to large negative voltages ($-2.5~{}$V) and not changed during the measurements. Those gates have no significant effect on the subgap states studied here.

We explore the magnetic field evolution of tunneling conductance in Figs. 1(b)-(d). At zero field, this device exhibits a soft but otherwise featureless superconducting gap characterized by smooth evolution of suppressed conductance within the gap as a function of bias. Such soft gap presents a decoherence pathway for futuristic topological qubits but it does not prevent us from studying the subgap spectroscopy here. In Fig.1(b), the evolution within the magnetic field range $0-300$ mT looks like a closing of the induced gap: the suppressed conductance window around zero bias shrinks and two branches of high conductance move from the apparent induced gap edges ($V=\pm 250\mu V$) toward lower bias reaching zero bias at around 300 mT. Beyond $B=300$ mT, an apparent zero bias resonance is observed over a significant range of magnetic field, up to at least $B=1$ T. This range, expressed in Zeeman energy using a lower bound on InSb g-factor of 30, greatly exceeds the bias width of that resonance—thus we identify it as ‘pinned’ to zero energy (line traces in Fig. S1 of the Supplemental Materials).

Fig. 1(c) shows that with a minor variation in $BG1$ a single zero-bias resonance can be transformed into a pair of low-bias resonances oscillating around zero bias as magnetic field is increased to $1$ T (up to $2$ T in Supplemental Material). Such oscillations are consistent with MBS in a short nanowire Stanescu et al. (2013), and in fact data in Fig. 1 (b) can also be interpreted as similar oscillations of smaller amplitude, less than the resonance width. Fig. 1(d), however, conveys a different picture. After another change in $BG1$ that should not alter the bulk density in any significant way, we can resolve that the apparent oscillations are actually superimposed of two unrelated pairs of resonances moving to zero bias at different magnetic fields, $0.4$ T and $0.7$ T. This demonstrates that the visibility of different branches can be strongly affected by minor changes in gate voltages, and some of the branches may become invisible in color maps, creating the appearance of a sole zero bias resonance or a pair of oscillating resonances, both being important signatures of MBS.

The ubiquity of zero-bias features like those in Fig.1 is demonstrated in Fig.2 (Device B). Because of the extended range of $BG1$ shown and because of the strong electrostatic coupling of $BG1$ to the nanowire, a large number of transient resonances can be seen crisscrossing the subgap region without sticking to zero bias at zero field (Fig.2(a)). These are due to states localized near the tunneling barrier. At finite magnetic field $B=0.3$ T, the transient resonances are still visible, but another set of features tightly confined close to zero bias is now observed throughout the presented range of $BG1$ (Fig.2(b)). Close to 30 distinct ZBP regions are observed. If all of these ZBPs were due to topological superconductivity, we would expect being able to tune through tens of 1D subbands, which is inconsistent with quantum point contact measurements on similar nanowires van Weperen et al. (2013). Data in Fig.2(b) are similar to barrier gate scans in Mourik et al. Mourik et al. (2012), which used the same nanowires and superconductors, though a different gate layout with a weaker $BG1$ coupling.

We zoom in on a representative $BG1$ range in Figs.2(c)-(e). At zero field the inside of the induced gap for $|V|<250\mu V$ is featureless on this scale (Fig.2(c)). In the same gate range at finite field $B=0.3$ T (Fig.2(d)), three oscillations around zero bias and higher bias subgap states are observed. At a higher field $B=0.5$ T (Fig. 2(e)), an extended zero-bias peak is observed. Over a range of $BG$ between $1.61$ V and $1.62$ V the ZBP vanishes, however this is an artifact due to charge jumps, i.e. charge rearrangements near the gate leading to a momentary shift in the electrostatic potential. Such charge jumps are also ubiquitous and appear in many published results Zhang et al. (2018).

We observe that the near-zero bias states often merge continuously into the transient resonances above the induced gap. This implies a relation between the two types of features. This behavior is expected in quantum dots strongly coupled to superconductors, where transport resonances due to ABS split from and merge into the induced gap as the dot occupation changes from even to odd Eichler et al. (2007); Lee et al. (2014). In this framework, the regime in Fig.2 is consistent with several coupled quantum dots formed near the superconductor. Note that the absence of Coulomb blockade suggests open quantum dots and transparent contact to the superconductor. The open dots may be connected both in series and in parallel.

To model our devices we perform 3D Schrödinger-Poisson calculations that incorporate geometric and electrostatic details of the experimental device Woods et al. (2018). The calculations naturally capture the multi-band nature of the system and its highly inhomogeneous electrostatic potential, which turn out to be the crucial elements responsible for the ubiquitous zero bias peaks. The inhomogeneity arises due to device geometry, while disorder is not included in the model. A detailed description of the model can be found in the conjoint theoretical paper Woods, Benjamin D. and Chen, Jun and Frolov, Sergey M., and Stanescu, Tudor D. (2019).

First, we demonstrate that the model generates ubiquitous zero bias peaks, as seen in the experiment, by calculating the differential conductance Blonder et al. (1982) as a function of the $BG1$ voltage. The results are shown in Fig. 3 (compare with Fig.2). At zero magnetic field (Fig. 3(a)), the differential conductance is characterized by multiple sub-gap resonances that approach or cross zero bias without sticking. At finite field (Fig. 3(b)), one notices features that are confined near zero energy. Examples of differential conductance maps as a function of Zeeman energy and bias are shown in Figs. 3(c)-(d). In Fig. 3(c), we notice an in-gap mode that collapses to zero energy at $E_{Z}\approx 0.7~{}$meV, then splits at higher $E_{Z}$. A slight change in $BG1$ generates a low-energy mode that remains near zero bias over a large range of $E_{z}$ (Fig. 3(d)).

Next, we address the key question regarding the nature of the low-energy states by studying the band and real-space structure of the corresponding wave functions (also see the Supplemental Material). We find that the ubiquitous low-energy states are not MBS emerging in a segment of the wire, or partially-separated MBS induced by soft confinement, but rather ABS pinned near zero energy by level repulsion. As detailed in the conjoint theory paper Woods, Benjamin D. and Chen, Jun and Frolov, Sergey M., and Stanescu, Tudor D. (2019), inter-band coupling can give rise to ABS that stick near zero energy due to anti-crossings between multiple modes approaching zero energy at different magnetic fields. For example, in Fig. 3(d) one can distinguish two low-energy modes that cross zero energy at $E_{Z}\approx 0.7~{}$meV and $E_{Z}\approx 1.1~{}$meV, respectively, displaying an anti-crossing behavior (near $E_{Z}\approx 0.9~{}$meV). Evidence of similar level repulsion behavior can be found in the experimental results shown in Fig. 1(b-c). We note that the inter-band coupling arises from the evolution along the length of the wire of the transverse profiles of the various bands due to the electrostatic potential nonuniformity. As explained in the theory paper Woods, Benjamin D. and Chen, Jun and Frolov, Sergey M., and Stanescu, Tudor D. (2019), single-subband models cannot capture this zero-bias pinning behavior for short nanowire segments of 200 nm (the width of $BG1$) without assuming overlapping MBS.

We investigate the spatial characteristics of the low-energy states at zero magnetic field by mapping the zero bias conductance as a function of $BG1$ and $FG$. The experimental results are shown in Fig. 4(a), while the numerical results are given in Fig. 4(b). The remarkable common feature is the presence of three types of resonances characterized by different slopes, which we attribute to distinct families of low-energy states having different couplings to the gate potentials. The nearly-vertical resonances in Fig.4(a) are generated by states coupled primarily to the $FG$ gate (white dashed lines). These resonances are also seen in the calculation (Fig.4(b)), although their amplitude is reduced (see also the Supplemental Materials). The wavefunction profile of a typical state associated with this type of resonance (Fig.4(c)) reveals that most of its weight is located in the $FG$ region (see inset). The two additional sets of resonances are generated by states electrostatically coupled to both $FG$ and $BG1$ (green and yellow). As revealed by the wave function profiles shown in Figs.4(d)-(e), these states have significant weight in both $BG1$ and $FG$ regions. However, the transverse profiles (see insets) show that the state in Fig.4(e) is located closer to $BG1$ and farther away from $FG$ as compared to the state in Fig.4(d), which explains the different slopes of the corresponding resonances. The existence of these distinct families of states demonstrates that the low-energy physics is controlled by modes localized in different adjacent regions (i.e. the $FG$ region and the covered and uncovered $BG1$ regions) that are coupled to one another. At finite magnetic field, this generically produces low-energy ABS resonances pinned near zero energy through the inter-band coupling mechanism discussed in detail in Ref. Woods, Benjamin D. and Chen, Jun and Frolov, Sergey M., and Stanescu, Tudor D. (2019). We note a discrepancy in gate voltage and conductance scales between Fig.4(a) and Fig.4(b), likely as a result of device-dependent gate screening variations and high sensitivity of conductance to tunneling rates.

In conclusion, we have demonstrated that many of the commonly discussed features of MBS in nanowires, such as gap closing, zero-bias pinning in magnetic field or gate, and peak oscillations around zero bias, are ubiquitous and easily observed when ensembles of trivial ABS are present. Evidence of MBS in tunneling experiments should therefore be accompanied by detailed studies of subgap resonances in the extended gate voltage range. For example, an earlier study of a similar device has revealed zero-bias peaks occupying a large continuous region of field-gate space with a boundary similar to the basic topological condition Chen et al. (2017).

Nevertheless, since tunneling measurements have so far not yielded a definite MBS proof, it is intuitively attractive to explore more sophisticated techniques, e.g. the fractional Josephson effect Rokhinson et al. (2012), Majorana fusion or even braiding Aasen et al. (2016). However, the added measurement complexity will not help resolve the experimental limitations of the tunneling experiments, since the limitations remain rooted in the growth and fabrication. It is also unclear whether advanced techniques can reveal signatures unique to MBS, and whether they are better at distinguishing MBS from ABS Houzet et al. (2013). At the same time, tunneling remains powerful in surveying the subgap spectra in proximitized nanowires, thereby guiding device design and fabrication towards a more ideal regime in which MBS can be demonstrated unambiguously.

We thank V. Mourik and K. Zuo for comments on the manuscript. T.D.S. acknowledges NSF DMR-1414683. S.M.F. acknowledges NSF DMR-1743972, NSF PIRE-1743717, ONR and ARO.

I Supplemental Material for ’Ubiquitous non-Majorana Zero-Bias Conductance Peaks in Nanowire Devices’

I.1 Experimental measurements conditions

Measurements are performed in a dilution refrigerator at a base temperature of $30$ mK, by standard low-frequency lock-in technique ($77.77$ Hz, $5~{}\mu$V, time constant $100$ ms, setting time $300$ ms). Multiple stages of filtering are used to enhance signal-to-noise ratio(a $\pi$-filter at room temperature, a copper-powder filter and a RC filter at base temperature). Estimated electronic temperature is around $80$ mK, which is extracted from the minimal observed subgap peaks FWHM. DC bias is scanned in steps of $5-10~{}\mu$V. The voltage drop on the devices is calibrated (voltage drop on filters is subtracted). For all the measurements, bias voltage is applied to the normal contact and the superconducting contact is grounded. All the measurement data are plotted using Spyview with an image processing of lowpass $(1,0)$, which smoothen the data in the bias voltage but not in gate or magnetic field.

I.2 Fig. S1 Line traces of conductance maps in the main text and Devices configuration

I.3 Fig. S2 Zero bias resonances at magnetic field up to 2T

I.4 Fig. S3 Persistent ZBP in Device C

I.5 Fig. S4 Study of ABS and their apparent g-factors in Device A

In Fig. S4 we examine ABS in Device A. The apparent g-factor extracted from the dispersion of ABS in magnetic field can vary strongly with only minor variations in gate voltage. This is due to quantum dot singlet-doublet physics: in the even-parity regime the resonance is pinned to the gap and moves above the gap at higher field, leaving the gap edge to be the lowest energy resonance. In the odd-parity regime the resonance exhibits opposite behavior: it moves towards zero bias in magnetic field. At the transition between the even- and odd- parity regimes the competition of two opposite field behaviors causes the apparent g-factor to be strongly reduced.

I.6 Fig. S5 Two independent ABS in Device A

I.7 Fig. S6 Gate-tunable ABS in Device D

I.8 Device setup in 3D simulations and additional simulation data

A comparison of the experimental and theoretical zero bias differential conductance was shown in Fig. 4 of the main text. While both conductance maps show peaks with three different slopes, the vertical (white) lines in the simulation are not as steep as those seen in the experiment. Indeed the vertical (white) lines in the experiments are nearly completely independent of the $BG1$ voltage, suggesting the $FG$ and $BG1$ regions are not as well coupled as our model predicts. Moreover, the features associated with the experimental vertical lines in Fig. 4(a) are broader than those seen in the simulation.

To study how reducing the coupling between the $FG$ and $BG1$ regions changes the conductance maps we reduce the hopping between the last and first layers of the $BG1$ and $FG$ regions, and recalculate the zero bias differential conductance. The result is shown in Fig. S8 for $t^{{}^{\prime}}=0.7t$, where $t^{{}^{\prime}}$ is the hopping parameter between the last and first layers of the $BG1$ and $FG$ regions, respectively, and $t$ is the hopping parameter between all other layers. The conductance map has a broader and more vertical feature starting in the lower right corner that shows a closer resemblance to the vertical features in Fig. 4(a).

A possible explanation for the reduction in coupling involves our neglection of self-consistency in the charge distribution along the length of the wire. As mentioned in the main text and explained completely in Ref. Woods et al. (2018), our numerical method in general involves self-consistency both when constructing the effective 1D multi-orbital model and when solving the 1D problem. However, here we chose to ignore the self-consistency when solving the 1D problem due to the large parameter space that needed to be explored. This amounts to ignoring the charge redistribution along the length of the wire, which can produce potential barriers between gate regions Woods et al. (2018). This extra potential barrier between the $FG$ and $BG1$ regions would then explain the reduced coupling and the broader vertical features seen in the experiment.

I.9 Majorana Representation of trivial low energy states

We claim in the main text that zero bias peaks in the differential conductance scans arise from topological trivial states rather than topological MBSs. To quantify this claim, we make use of the Majorana representation of the zero energy states that are responsible for the zero bias peaks. Recall that any Bogoliubov-de Gennes (BdG) Hamiltonian can be written as a linear combination of two Majorana modes. Let $\phi_{\epsilon}$ and $\phi_{-\epsilon}$ be eigenstates of the BdG Hamiltonian that are related to each other by particle-hole symmetry with energies $\epsilon$ and $-\epsilon$, respectively. From these particle-hole symmetric states we can construct

[TABLE]

By construction these states satisfy the Majorana condition of being equal parts particle and hole, however they are not eigenstates of the BdG Hamiltonian unless $\epsilon=0$. Note that $\phi_{\pm\epsilon}=\frac{1}{\sqrt{2}}\left[\psi_{A}\pm i\psi_{B}\right]$. This is therefore named the Majorana representation of an eigenstate of the BdG Hamiltonian. For topological Majorana states, the wavefunctions $\psi_{A}$ and $\psi_{B}$ should be localized on opposite ends of the wire. In contrast, a trivial Andreev Bound state will have highly overlapping $\psi_{A}$ and $\psi_{B}$. Fig. S9(a) shows the spectrum of the setup without the lead attached. The parameters are the same as the differential conductance scan shown in Fig. 3(d). Note that the absence/presence of the lead alters the position and shapes of the features when comparing the spectrum to the differential conductance. We plot the Majorana wavefunctions in Figs. S9(b)-(d) coming from the corresponding zero energy states in Fig. S9(a). The Majorana wavefunctions are significantly overlapped, indicating that the zero energy states are not MBSs.