Zero-bias conductance peaks at zero applied magnetic field due to stray fields from integrated micromagnets in hybrid nanowire quantum dots

We thank Dr. Ostrovsky. We provide discussion and address requested changes.

1. In the introductory part of the paper (page 2) there is the following statement about Majorana bound states: "The quantization is rapidly lost with departure from ideal conditions." This statement is at least misleading. In fact, the major scientific interest in the Majorana bound states is due to their topological origin, which provides a certain degree of robustness with respect to external conditions. It is exactly this robustness, that is required to realize quantum computations with the Majorana states. The authors should rephrase their statement in a way that would not lead to misunderstanding.

We agree that the notion of conductance quantization is misleading, because really it is only expected in an infinitely long nanowire with zero disorder and at zero temperature. In fact, the paper that popularized the idea to search for quantized conductance has a dramatic calculation in its Figure 5 that shows how quickly any quantization is erased when realistic terms are added to the Hamiltonian.

We modify the sentence the following way: “The quantization is rapidly lost with departure from ideal conditions (see Figure 5 in ref. [22]).”

It is also necessary to explain in more detail the distinction between topological Majorana and usual Andreev bound states and, if possible, formulate some criteria to distinguish them. More on this in the next item.

We added a more background to discuss ABS on page 2. It is an open problem as to what are the specific criteria for identifying MZM and distinguishing them from ABS. This question will have more practical value when real candidate MZM are identified and the data cannot be ruled out as trivial ABS by simply considering a slightly larger set of data than what is shown in a paper.

1. The authors claim that Majorana bound states exhibit a zero bias peak of at most e2/he2/h while usual Andreev states can have higher tunneling conductance. On page 2, it is written that the zero-bias peak due to usual Andreev bound states can get larger than 2e2/h2e2/h. Factor of two here is probably a misprint. A single bound state cannot provide such a high conductance. On the other hand, if two Majorana bound states located e.g. at opposite ends of an SNS junction have a small but for some reason non-negligible overlap (normally the overlap should be exponentially small with the distance between them), they will hybridize into a single normal Andreev bound state with a small but nonzero energy. What would be than the magnitude of the conductance peak? Will it abruptly grow from e2/he2/h to 2e2/h2e2/h once the small overlap is taken into account (given that all other conditions are ideal)? What will happen if the width of this peak (e.g. due to temperature) is larger than the energy of the Andreev state itself and the peak visually occurs at zero energy? This point needs some more explanation.

Zero bias peaks due to MZM in nanowires measured by tunneling can reach conductance values of 2e2/h or they can be shorter. But they cannot be taller than 2e2/h when a basic Majorana model is used (See Figure 1 in Reference 22). The factor 2 is due to resonant Andreev reflection and particle-hole symmetry. Zero-bias peaks due to ABS can be both taller and shorter than 2e2/h. One way this can happen is, indeed, when multiple ABS coalesce near zero bias. In Figure 3 we subtracted a particular series resistance to plot the plateau-like zero-bias peak near e^2/h, which is an allowed value for Majorana, however it is below the quantized Majorana conductance value. This is done to illustrate that plateaus can appear at different values of conductance and by themselves do not indicate the presence of MZM. In this paper we do study SNS junctions but we only study supercurrents in them. We study zero bias peaks in what nominally are NS junctions, so we do not expect coupled MZM across the junction. When two MZM are hybridized across the junction there will be no tunneling barrier to detect the zero-bias peak if a two-terminal geometry is used. If another lead is added for tunneling near the junction, then in the simplest Majorana model the detected peak would transition from a quantized value of 2e^2/h to a new value determined by MZM coupling, tunnel coupling and temperature.

1. According to the statement on page 4, the device 1 is fabricated with two junctions but the results are analyzed as if there was only one junction. Why does it work like this?

We do not have enough control knobs in this particular experiment to answer the question ‘why’. For instance, we only have one single gate electrode so we cannot selective address the two junctions.

In the context of this specific study this question is not central because the point here is to show examples of data that challenge exotic narratives, and we do have enough information to rule out such narratives, e.g. that the hysteretic critical current is not because of the exchange interaction induced in the semiconductor by the ferromagnet.

By way of speculation, one possibility is that the left junction is too close to the gold lead, and it is poisoned by the normal metal so that the critical current of that junction is both suppressed and does not manifest as a sharp switch. Since these are two-terminal measurements the normal state resistance of that junction is simply folded into the series resistance present in the circuit and subtracted out.

We thank Prof. Bouchiat. In response to your comment, we provide an expanded Duration and Volume of Study section:

Duration and Volume of Study
This article is written based on more than 5500 datasets from 5 separate dilution refrigerator cooldowns. For each cooldown, we measure several devices. We measured 19 SN devices, and 20 SNS devices (7 of 20 devices have CoFe strips). Among them, 4 SN devices exhibit clear ABSs; 2 SNS devices show ABSs; and 4 SNS devices have measurable supercurrent. The yield is limited by fabrication issues, device quality and occasional static discharge damage to devices. Within each device, the behaviors in the main text require fine-tuning. The more general behaviors are presented in the supplementary materials.