SciPost Submission Page
Triple Andreev dot chains in semiconductor nanowires
by Hao Wu, Po Zhang, John P. T. Stenger, Zhaoen Su, Jun Chen, Ghada Badawy, Sasa Gazibegovic, Erik P. A. M. Bakkers, Sergey M. Frolov
|As Contributors:||Sergey Frolov · Po Zhang|
|Arxiv Link:||https://arxiv.org/abs/2105.08636v2 (pdf)|
|Date submitted:||2021-09-02 03:18|
|Submitted by:||Frolov, Sergey|
|Submitted to:||SciPost Physics|
Kitaev chain is a theoretical model of a one-dimensional topological superconductor with Majorana zero modes at the two ends of the chain. With the goal of emulating this model, we build a chain of three quantum dots in a semiconductor nanowire. We observe Andreev bound states in each of the three dots and study their magnetic field and gate voltage dependence. Theory indicates that triple dot states acquire Majorana polarization when Andreev states in all three dots reach zero energy in a narrow range of magnetic field. In our device Andreev states in one of the dots reach zero energy at a lower field than in other two, placing the Majorana regime out of reach. Devices with greater uniformity or with independent control over superconductor-semiconductor coupling should can realize the Kitaev chain with high yield. Due to its overall tunability and design flexibility the quantum dot system remains promising for quantum simulation of interesting models and in particular for modular topological quantum devices.
Submission & Refereeing History
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- Report 2 submitted on 2021-11-14 23:31 by Prof. Pientka
- Report 1 submitted on 2021-10-02 00:02 by Dr Prada
Reports on this Submission
Report 2 by Falko Pientka on 2021-11-14 (Invited Report)
This paper experimentally investigates a setup with three quantum dots, each proximitized by a superconductor. This work is part of the search for Majorana states and aims for a kind of bottom up approach to build a Kitaev chain from individual quantum dots. The authors claim to observe Andreev states in all three dots as well as a "large degree of control over Andreev states in multiple dots".
The paper concludes with a thorough discussion of experimental challenges and limitations of the current device geometry.
I certainly appreciate the endeavor to build superconducting quantum-dot arrays, which may be interesting even beyond the main motivation of this paper - building a Kitaev chain. I also think the discussion of the advantages and drawbacks of this particular geometry, e.g., in comparison to theoretically proposed devices, should be helpful to people in the field, theorists and experimentalists alike.
I am however skeptical of the claimed findings of the paper. While the data is presented well and the plots are well organized, I was confused by the interpretation of the data. While reading the paper, I was constantly asking myself what type of transport mechanism the authors are imagining to occur in their device. One could imagine different transport scenarios for this three dot setup:
a) All tunnel couplings are relatively weak so that states in the dots are sharp.
In a simple noninteracting picture, transport can only occur if the bias voltage is above a certain threshold and if the levels of the different dots are tuned to be in resonance such that electrons can hop from one dot to the next. Of course there could be additional effects such as cotunneling, which could relax this criterion.
b) Tunneling to the leads is weak, but tunneling between the dots is strong. In this case the dot states are delocalized and one is probing effectively only a single dot.
c) One lead is somewhat strongly coupled to the adjacent dot.
In this case, the lead is effectively larger and one is really only probing resonances in one or two dots (depending on the dot-dot tunneling).
The authors never really clarify which regime they believe their device to be in. Only at the very end they make a statement
"We also show in supplementary materials that the tunnel barrier located at the left end near N1 is significantly more open compared to the right tunnel barrier at N2. The inter-dot barriers are also low."
This statement seems extremely important to me, since it facilitates the interpretation of the data. This finding (which is unfortunately only included in the SM, I think it should be part of the main text and should appear there much earlier) suggests to me that option c) is realized in this experiment and one should interpret the data in the following way: dot 1 and 2 are sufficiently well coupled to N1 so that this experiment is effectively two normal leads coupled to a single dot (dot 3). Of course there are resonances in dot 1 and 2 (one can see them in Fig 3a), but they are sufficiently broad, that their exact location in energy does not significantly affect transport.
Of course, I am open to the possibility that my viewpoint is oversimplified and gates 1 and 2 also have an effect. I just think that the evidence presented in the paper does not contradict the picture that all features can be explained in a simple model with a single dot sandwiched between two normal leads.
For instance, I am skeptical of the argument that the rather narrow features at low bias voltage above 0.2T in Fig 5a are states associated with dots 1 and 2. This seems to be in contradiction with the observation in the supplementary material that transport features are very broad when the current is flowing through dot 1 and 2, which means I would not expect sharp features. Also I think the fact that Fig 5b, with a different current path that does not pass dot 1 and 2, is more broadened does not allow one to conclude that the features are gone. They are relatively weak features in Fig 5a, so one would expect them to be invisible in a broadened Fig 5b, even if they belong to states in dot 3.
In summary, I do not see transport features in the data that can only be explained in terms of Andreev states localized in dots 1 and 2 (except for the very weak modulation in Fig 3a, in which gate 3 is tuned a particularly sensitive point close to a resonance of the third dot). I can see even less support for the claim that those states can be tuned by magnetic field.
From abstract, introduction, and conclusion, however, I would have expected this to be the case. See, e.g., the following quotes1:
"At zero magnetic field, Andreev bound states exist in all three dots. Andreev bound states from each dot are gate-dependent."
"In our device Andreev states in one of the dots reach zero energy at a lower field than in other two"
If my interpretation is correct and all features indeed originate from the third dot, then the findings in this paper would be much less interesting. Nevertheless it may be worth publishing these results in a more specialized journal. In this case an explanation of the transport mechanism needs to be included and the interpretation of the data needs adjustment. Unfortunately, I cannot recommend this article in its present form for publication in SciPost Physics.
If my interpretation can be falsified by additional data, I am willing to reconsider this assessment.
But even then there is a fundamental point I do not understand about this experiment.
If really one of the states in dot 1 or 2 would cross zero energy as a function of magnetic field, why would this show up as a zero bias peak in a three dot setup? If the third dot, the one with the smallest couplings to the leads, was detuned from zero, there should be no peak at zero bias. In other words, any resonance in a three dot setup can be interpreted either as a resonance of the dot with the weakest coupling in case two dots have a sufficiently strong coupling to the leads, or as resonances of multiple dots that need to be aligned, in case that multiple dots have a weak coupling to the leads. Hence, end-to-end transport measurements are no the best way to analyze the individual dots in multi-dot chains.
Here are some additional suggestions:
One possibility to prove that certain states belong to dot 1 and 2 is to use different current paths, e.g., from N1 to S1, S1 to S2, and so on. I understand that the broadening does not permit to gain much information from them, but the broadening itself is pretty interesting as I have discussed above. Why couldn't one study such an alternative setup as a function of any of the barrier gate voltages? The dI/dV in Fig S10a is two orders of magnitude above what is measured in other figures, so there seems to be some room to increase barrier heights without suppressing the current beyond what is measurable.
I don't understand the following statement:
"In Figs. 2(a,b), Andreev bound states appear more flat. We interpret this as D1 and D2 being stronger coupled to leads S1 and S2."
I would rather think that the strong broadening of states in 1 and 2 means, one is effectively only probing the resonance in dot 3.
"In our device, transport is dominated by one of the quantum dots in the chain, D3 ,"
is unclear. Are the authors trying to say, that dot 3 has the weakest couplings to the leads and hence all visible resonances should be interpreted as originating in the third dot?
On grammar: there are a number of indefinite articles missing throughout the paper, please check. One example on p.3:
For quantum dot chain with hard induced gap, zero-bias peak only occurs
-> For a quantum dot chain with a hard induced gap, a zero-bias peak only occurs
What does the statement on p 3:
" If we set Vd3=365mV, we have zero-bias peaks on both nanowire ends."
mean? You only probe end-to-end transport, so all features are associated with both ends.
Report 1 by Elsa Prada on 2021-10-2 (Invited Report)
In this work the authors bring into practice the so-called Kitaev chain model, or at least one version of it. For that they use an InSb semiconducting nanowire and with the help of several back gates, they confine three quantum dots (QDs) in series, attached to source and drain contacts. Moreover, these dots are coupled to superconducting NbTiN contacts (in the shape of "fingers" deposited above the QD regions), bringing in principle these dots into the superconducting state. In this way, they create a chain of three "Andreev" QDs. Subsequently, they perform transport measurements of current and differential conductance. Particularly, they apply a voltage bias through the left contact, and measure current and differential conductance from the right contact while floating the superconducting fingers. By changing parameters such as the back-gate voltages below each dot region and an applied axial magnetic field, they look for signatures of Andreev levels inside the superconducting gap and analyze their behavior. The goal is to find zero bias anomalies compatible with the existence of non-local Majorana states at the outer dots of the device.
Let me start with some general comments and then I will get more specific:
I find the subject of this work is of high interest for the community of hybrid semiconducting nanowires, Andreev quantum dots and for the search of Majorana states in this kind of platform. Having analyzed in depth in the literature the "continuous" nanowire model (although this is still an ongoing effort), it is only natural to try to start studying experimentally more sophisticated models such as this Kitaev chain. If I understand correctly, this is one of the first (if not the first) attempts to try to bring this model into experimental reality. For that alone, I find this study is worth considering and publishing.
To approximate a chain just out of three sites (three dots) looks a bit wishful, at least from the point of view of a theorist, but this is definitively a first step. Besides, we all must recognize the tremendous ability and craftsmanship to get a working device of this type must have taken. As the authors note:
"Even though the desired regime is not accessible, this device with new geometry has over- come many nanofabrication challenges...".
"These experiments took several years to complete. Device fabrication is challenging and required significant development to realize multiple narrow superconducting contacts to a nanowire. Measurements are time-consuming due to the large number of experimental parameters (7 gates and 5 contacts). Yes, in the future it would be interesting to use different superconductors or different fabrication methods such as SAG, but this cannot be switched out and tried without a huge investment."
I can only praise the authors for this effort, and again, that itself deserves publication in my opinion.
Another issue is what is going on exactly inside such a complicated device and whether this type of system is better or presents practical advantages to create Majorana bound states than the continuous wire one. With respect to the first question, the authors carry out an extremely simplified model (I would even say naive), but for me that is enough because this is essentially an experimental paper and it is our job as theorists to make sense of the experimental evidence. In any case, I have several observations about this below. With respect to the second question, I honestly think that if it is difficult to get the right conditions for Majorana state formation in the continuous nanowire device, it is only going to be worse in this type of multi-section wires. More on this later.
This said, I still find that this line of research is worth pursuing, since advantages may appear or be possible in the future, and definitively it will bring to the community a better understanding of these wires and Andreev states inside them. I see this work as a first attempt to build an Andreev Kitaev chain. The results are negative, meaning that the necessary conditions for the topological phase are not met, but they are presented with honesty and many interesting conclusions can be driven out of them. For example, the necessity to have rather similar or homogeneous QD's and good proximity effect in all of them. Thus, this work can be seen as a “towards a simulated Kitaev chain” paper, as another Referee put it. This is perfectly fine for me, we cannot pretend that one single study, particularly as difficult as this one, solves all the challenges at the first try. Surely many more studies, both experimental and theoretical, will be necessary to bring this idea to success. This in itself is also good because it can stimulate discussion and bring citations to the journal.
Finally, I think this paper is very well written. I also appreciate the clarity and honesty with which the results, specially the negative ones, are presented, and the effort to bring the non-expert audience to the discussion with sections such as "limitations", "further reading", "background" and "discussion of different models".
Now I proceed with several comments and criticisms:
1) In the Introduction, and as a motivation for their work, the authors (only) cite Refs.  and . However, as the PRL Referee noted and the authors agreed, this work is not trying to emulate those types of chains precisely. The authors themselves added a very useful section "Discussion on different models" where it is clear that the type of device they build is more similar to Refs. [28,29]. I am more familiar with  and, indeed, I find that this work is similar or perhaps identical in spirit to Ref. , and has less to do at the end with  and .
I think this fact should be acknowledged already at the level of the Introduction to avoid misinterpretations or confusion.
Concerning citations, I want to bring another reference to the attention of the authors, because I think it is pretty much related:
"Effects of the electrostatic environment on superlattice Majorana nanowires", Phys. Rev. B 100, 045301 (2019).
In this work there is only one back gate below the semiconductor, but otherwise it is very similar, with several superconducting fingers on top of the wire. Among other things, this superlattice creates and inhomogeneous potential along the wire in the form of potential barriers and wells similar to what it is studied in Ref.  and what must be happening in the author's device. In this work we study carefully all the parameter variations produced in this complicated device when taking into account the 3D geometry and electrostatic environment realistically, and their effect on the phase diagram and Majorana formation. More on this below.
2) In the Introduction the authors rightly mention: "Theory suggests a robust topological superconducting phase in chains of Andreev quantum dots [18, 19]." This is a positive message worth noting. However, while this might be true for the proposals of [18,19], I want to mention that the phase diagram of a superlattice of superconducting fingers is in general more complicated and less robust that that of a continuous wire model. As can be seen in Phys. Rev. B 100, 045301 (2019), there are several types of phenomena that break the topological phase and that are summarized in Fig. 10 (the formation of localized sates, longitudinal subband gaps and longitudinal subband overlaps.). These or similar weaknesses (with a different language) are also found in Ref. . Since the device the authors study is actually similar to these works, this could be at least mentioned.
3) The authors use a many-body description for their Andreev states (singlet ground state, doublet excited states, many-body spectra, etc.). However, they themselves acknowledge that electron interactions are negligible in their experiments: "Dots have a quenched charging energy and exhibit no Coulomb blockade.", "The charging energy is quenched because the dots are covered by huge superconducting leads. It can be assumed to be zero. There are no Coulomb diamonds present in the system." Thus, for me, it doesn't make much sense to use such a many-body interpretation, it is confusing. In my opinion, the authors measure single-particle resonances, Andreev levels, that split under magnetic field.
4) Concerning the previous point, the authors perform a theoretical (simplified) study in the supplementary material. There they introduce interactions, U. If they believe there are no interactions in their device, it is perhaps a bit strange to study precisely the effect of U. Perhaps they could motivate such study a little. Actually, I find it interesting since with this simplified model one can already learn that interactions are detrimental for the topological phase. By the way, the authors should state how they compute the effect of interactions. I assume they use a self-consistent Hartree approximation.
5) I must say that the theoretical model section is rather short and contains very few explanations. I had to make a considerable effort to understand what the authors were trying to prove and the conclusions they were extracting from their simulations (these were only mentioned, not explained, in the figure captions). For example, they could explain a bit more why they calculate the Majorana probability and what's its meaning. Moreover, with just a couple of figures of differential conductance and Majorana probability amplitudes for some parameters, the authors imply that they have their device figured out. See for instance statements such as "While we are motivated by early theoretical proposals, we do not aim to replicate them and instead develop a theory tailored to our devices (see supplementary information for discussion).", "... For such a short chain, within a narrow parameter window, the probability distribution of Majorana wavefunctions indicates a partial separation of two Majorana zero modes localized at two end sites (see supplementary materials for simulation results)."
As I said at the beginning, I think this is an experimental paper and as such, I consider it is more than enough. I even appreciate the effort the authors do to analyze a simple toy model to get an idea of the physics behind the experiments. However, I don't really think the authors perform any kind of serious study of the topological phase or the real behavior of such a sophisticated device, and this should be acknowledged somehow. The authors should refer there to more serious works such as Ref.  or Phys. Rev. B 100, 045301 (2019).
6) Related to the previous point, at the end of Section "Discussion of different models" the authors say: "Models in Refs.  and  are relevant to our study since the wire segments are of a length similar to our devices. However, these models are too optimistic to assume partially separated Majorana already exist in each wire segment, which needs to be first experimentally established." I don't think this is correct. These works (as well as Phys. Rev. B 100, 045301 (2019)) do not "assume" partially separated Majoranas. This partial separation happens immediately for wire segments of the order of the QD regions of their device simply becase these wires have spin-orbit interactions and magnetic fields. The wavefunction in their device is surely not a combination of three differentiated almost point-like wavefunctions in each of the QD regions weakly coupled through tunneling barriers. As the authors themselves acknowledge, the barriers between QD regions are low (or even absent), and thus the wave function is basically spread along the whole wire. When they observe zero bias anomalies, most probably they have quantum well regions between the covered regions by the superconductors, see for instance Fig. 1(c) in Phys. Rev. B 100, 045301 (2019), where the realistic electrostatic problem has been solved. Thus, most probably the shape of the wave function, in the best case scenario, is similar to Fig. 12(h) in Phys. Rev. B 100, 045301 (2019) (but with fewer superconducting fingers) or to Fig. 9 in Ref. .
7) Finally, let me mention that the Majorana localization length for a periodic structure is larger than that for a continuous model wire, see Phys. Rev. B 100, 045301 (2019). Thus, for a similar nanowire length, the Majorana wavefunction overlap is typically larger and the topological mini gap smaller.