Probing valley phenomena with gate-defined valley splitters

Report

Torres Luna et al introduce theoretically valley-splitters defined using patterned gates on bilayer graphene, using KWANT to perform numerical simulations. The manuscript presents successively a geometry to achieve valley splitting, the effect of a magnetic field and of misaligned gates on that geometry, before presenting two applications: resonant tunnelling through quantum dot levels and the influence of inter-valley coherent order in a cavity. I found the paper hard to follow, with the figures not introduced clearly in the text and a lack of connection between items. This kind of architectures are highly appealing in the graphene device community, and this paper could provide novel strategies to induce and probe valley effects in practical devices, but much clarity would be needed.

We thank the referee for the report and for acknowledging the timeliness of our work. In this new version, we included all the suggestions to improve the clarity of our manuscript. We reply to the requests below.

Requested changes

1- The geometry in figure 1 is not described clearly, with the order of panels not matching the discussion in the text. I would suggest to add a schematic of the proposed device as an introduction

We thank the referee for the suggestion. We have included device schematics in Figure 1 and in all subsequent figures in the manuscript. In our reviewed version, we also introduce each of the figure panels in the main text.

2- Panel 2b is not described

We indeed missed a description of panel 2b in our previous version. We now updated the figures, the corresponding captions, and their descriptions in the text.

3- Figure 4 and 5 would benefit from a device schematics, or at least boundaries of the different elements in panel 4a and 5c.

We thank the referee for the suggestion. We followed the same device scheme of Fig. 1 throughout the text.

4- description of the figures and their relation to the overarching story line is missing

We went over the text and improved the figures' descriptions.

5- Bilayer graphene was shown experimentally to present several phases, with different orders (discussed in ref 6 of the present manuscript and references therein), which are used in the cavity simulation. It would be beneficial to discuss how the phase diagram of bilayer graphene influences the simulation.

To the best of our knowledge, none of these ordered phases have been experimentally observed in confined structures. Therefore, we only expect the interacting phase diagram of bilayer to play a role in the cavity simulations. In the cavity, there could indeed be many phases, that change as a function of the displacement field and filling factor. The section 3.2 in our manuscript is devoted to understanding the influence of symmetry-broken phases in the transport behavior of the device depicted in Fig. 5a. Particularly, we argue that the transport through such a device can be used as a probe for valley order. To illustrate the operation of this probe, we include an intervalley coherent order parameter in the cavity and show how to detect it with transport. We reviewed the text to emphasize this aspect of our proposal.

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Report

The manuscript "Probing valley phenomena with gate-defined valley splitters" theoretically investigates the transport properties of a gate-defined valley splitter in bilayer graphene using numerical tight-binding simulations implemented in KWANT.

The manuscript consists of the following parts:

• An Introduction
• The section "device" describes the gate-defined electrostatic landscape considered by the authors and introduces the bilayer graphene tight-binding Hamiltonian they implemented in KWANT. The section also contains Magnetotransport simulations studying the achieved valley polarisation and a study of misalignment effects when the gates are mutually not perfectly aligned.
• A section on applications discussing resonant tunnelling through quantum dot levels and probing the valley-dependent order parameter in transmission through a cavity.
• A conclusion

Gate-defined architectures have attracted some attention in the community over recent years. In this context of recent research activities, the present work may be of some interest to colleagues in the field. However, I have some concerns about the work presented in the manuscript, mainly concerning some technical details and what novelty the results bring compared to earlier works (see detailed below). I can therefore not recommend publication as is.

We thank the referee for the report and for acknowledging the possible interest of the community on our work. We reply to all the concerns on the technical details and novelty of our work listed below.

Requested changes

1. On page 3, the authors report to use a scaling factor of s=16 to reduce computational cost. I assume they refer to scaling in tight binding models as introduced in ref Phys. Rev. Lett. 114, 036601 (2015). A scaling of s=16 seems very large. In Phys. Rev. Lett. 114, 036601 (2015) for monolayer and arXiv:2403.03155 for bilayer graphene (both references are not cited in the manuscript), a scaling of s=4 has been used. Did the authors check systematically as a function of the scaling factor that the tight-binding results, most notably the transport data, do not change for scalings that large?

Our lattice scaling model is indeed inspired by Phys. Rev. Lett. 114, 036601 (2015), and therefore we now add a proper reference. It is however different from the bilayer graphene model in arXiv:2403.03155 since our tight-binding model is built in a honeycomb crystal structure and includes trigonal warping effects coming from higher-order hoppings. We developed this scaling independently and now included Appendix A to appropriately present it. The validity of the lattice scaling depends on the typical length scales of the physical system. The works pointed out by the referee study graphene at high magnetic fields. In this regime, lattice scaling is only valid if $l_B \gg a$, where $l_B = \sqrt{\frac{\hbar}{eB}}$ is the magnetic length and $a$ the lattice constant. The largest magnetic field in our simulations is $\sim 1T$, resulting in $l_B\approx 25nm$. The lattice constant in graphene is $a\approx 1.4 \text{\AA}$, thus the scaled lattice has a lattice constant of $\tilde{a} = 16 \times a \approx 2.3nm$. Therefore, in all calculations, the lattice constant is an order of magnitude smaller than the magnetic length. Besides the magnetic field, the other requirement for our calculations is that $\lambda_F \gg a$, which is fulfilled in all our simulations since they are performed at low electron density. We added to Appendix A the scaling dependence on the band structure of narrow channels and in bulk bilayer graphene. In these calculations, we show the band structure within the parameters for which we perform transport calculations. From our length scale analysis and bandstructure plots, we do not expect significant changes in transport calculations.

1. I wonder about the role of edge states in the simulations, both along the edges of the gates and at the physical borders of their lattice in the simulations. Can the authors show plots of the local density of states to show where the wave functions are localised? If there is electronic density at the borders of the sample, how do the authors exclude their contribution to transport in the simulations?

Indeed the zigzag edge states play a role in transport if not properly handled. We remove edge states from our simulations by adding a staggered potential at the edges. As a result, the edge states are pushed to higher energies (note that no edge states are visible in the bandstructure plots of the device). Thus, these edge states do not contribute to transport in the regimes our calculations were performed. We indeed forgot to mention this additional step in the construction of our model and added it to the main text.

1. The work discusses the effects of misalignment when the gates are not perfectly aligned. I am also wondering about the role of alignment of the gates wrt the BLG graphene lattice. Does it matter how the gates are aligned wrt to the lattice, i.e., e.g. along the armchair or the zigzag direction? If yes, I see significant challenges for any experimental realisation, as the orientation of the lattice is notoriously hard to determine in any experiment. How would the authors suggest to mitigate such experimental issues? If not, what is the reason for the device's design choice, as shown in Fig 6? Why are the gates at precisely this angle? Could one choose any shape of gates (curved, straight, bent,…), and the states would follow that shape at the D>0/D<0 interface?

The helical transmission is topologically protected. Thus, it is independent of the lattice orientation. The angle choice was arbitrary and any other choice would lead to the same behavior. To demonstrate this robustness we added Appendix B where we change the injection angle over a wide range. We observe no significant change in the transmission of the helical channels changing the angle between the arms.

1. On page 5, the authors discuss "a cavity with valley order". It is not clear to me what type of valley order the authors have in mind and what would be its origin.

In the manuscript we state that the type of valley order considered in our simulations is the so-called intervalley coherence (IVC). This is an standard name in the literature that refers to states at the equator of the (valley) Bloch sphere (see for example Phys. Rev. B 107, L201119 (2023) and Nature Physics 20, 1413–1420 (2024)). This order is predicted to appear in Bernal bilayer graphene due to electronic interactions. We choose to illustrate this order parameter since typical transport experiments (quantum oscillations) cannot distinguish intervalley coherent (states in the equator of the Bloch sphere) to valley polarized phases (states at the poles of the Bloch sphere). Thus, probing intervalley coherent often requires local probes such as scanning tunneling microscopy which is impossible in the double-gated samples as the one in the cited experiments. We now expand on the meaning of intervalley coherence and the motivation to study it in the text.

1. There exist a multitude of similar works on gate-defined valley polarisers in bilayer graphene, e.g. https://doi.org/10.1038/s41467-020-15117-y, DOI: 10.1126/science.aao5989, https://doi.org/10.1103/PhysRevApplied.11.044033 (references not cited in the manuscript). Therefore, I fail to see what novelty or advantage the current work brings compared to these earlier works.

The goal of our manuscript is three-fold: (i) present valley splitters as a transport probe for valley polarization that does not require external magnetic fields or polarized light, (ii) perform atomistic tight-binding simulations of gate-defined valley splitters, and (ii) propose applications for these devices. Additionally, we introduce a scaling model for bilayer graphene that preserves the lattice symmetry and includes trigonal warping effects at low density. Our device proposal consists of two valley filters with opposite polarization, thus distinguishing our work from the three references pointed out by the referee as none of them show spatial splitting of valley states. Indeed, among the three references, we only cite one of them. Since valley filters are a building block of our device, we added the corresponding references and also more recent works addressing the misalignment issues mentioned in our work. Our proposal differs from these works for two reasons: (i) the device we consider is a valley splitter, not a valley filter; (ii) the second part of our manuscript covers applications of the device, which to the best of our knowledge were not proposed before.

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