Julien Barrier

I have recently been asked by a journalist to comment my work on Brown-Zak fermions (see my previous blog here ). As part of the discussion, I was asked to introduce the concept of a quasiparticle.

In condensed matter physics, simple equations are usually used to describe complex behaviours. For example, it is very simple to consider the equations of an electron not interacting with its surroundings, because the number of atoms or any other particle is usually in the order of $N_A \approx 10^{23}$, which would make analytical resolutions impossible to solve. When one tries to make the model a tad more realistic, in semiconductors for example, it is possible to treat the effect of surrounding electrons as “perturbations” on the free electron. By adding a different mass to that free electron, its behaviour will be different, slowed down or accelerated (basic understanding with newton’s law: $\Sigma F = ma$). That way, electrons in semiconductors, for example, can be modelled as a “quasiparticle”, that is, move like a heavy or light free particle.

The idea for first proposed by the Russian physicist Lev Landau in the early 1950s. He was studying the properties of liquid helium, a superfluid that can flow without any friction. He found that the behaviour of liquid helium could be explained by treating it as if it were made up of not He atoms, but of a different kind of particles that were not actually present in the liquids. These particles, which he called quasiparticles, were excitations of the underlying quantum system that could be described by their own equations of motion. This was a major breakthrough in our understanding of condensed matter physics. It has been used to explain the behaviour of a wide variety of materials, including superconductors, semiconductors, insulators. Quasiparticles are now an essential part of the theoretical framework for understanding condensed matter physics.   Let us give important examples of how the idea of quasiparticles has been used to explain the behaviour of materials:

• Superconductors. In a superconductor, the electrons are able to move without any resistance. This is because they are paired up into what are called Cooper pairs. These Cooper pairs can be thought of as quasiparticles that are different from the electrons that make up the normal state of the material.
• Semiconductors and metals. In semiconductors and metals electrons can move through the ideal crystal without any resistivity, and in real crystals all the resistivity appears due to scattering of electrons on defects, and due to the oscillations of the crystal lattice (when the position of atoms is deflected from their equilibrium).

In this picture, electrons just ignore the atoms of the crystal, while one would think that those atoms also should be an obstacle for electrons. This is happening because electrons in solid state conductors are not the same electrons which we usually discuss in vacuum. Electrons in crystals are quasiparticles, which have a very different mass from the mass of the free electron, and in some exotic cases the even have a different charge (e.g. fractional quantum Hall effect).

The idea of quasiparticles is a powerful tool that has helped us to understand a wide variety of phenomena in condensed matter physics. It is likely to continue to play a major role in our understanding of this fascinating field of science.

Since I defended my PhD, I have been applying to multiple research fellowships, with a relatively high success rate. I notably secured a prestigious Marie-Skłodowska Curie Fellowship from the European comission, applying 2 days after my PhD viva. For this reason my friend Alexey Berdyugin (now Assistant Professor at NUS Singapore) asked me to give him advice on his proposal for Singapore’s NRF Fellowship. I spent quit a bit of time thinking about my past strategies and writing down my advice, so I thought I could share this to a broader audience instead.

Here I attached the text I sent to Alexey, althought I edited it to keep his project confidential. He is not proposing to work on non-centrosymmetric materials as written below.

About the readership problem

It is important to understand how your proposal will be judged, and how to convince the panel that it is worth funding. I think the scheme is similar for many funding agencies. Here I describe how it works for all grants from the EPSRC, or other private funders in the UK like Leverhulme, and for the Marie-Curie scheme.

First, there will be a referee. Typically, they are doing research in relation to your grant application, but it is not always the case. It is likely that they have a high level of knowledge about the general field. The referee will spend several hours reading and judging your proposal. They will receive one or two applications to judge. They will not be involved in the decision of whether to fund you or not. They will, however, write a report and a recommendation, which will be sent to the second group of readers: members of the committee.

This is step 2. The job of the committee is to produce an ordered list of rankings, with a score assigned to each application. The grant that is the highest in the ranking list gets funded, then the second one, etc. until the budget runs out. Here you want to get a high mark. You need to convince the referee to convince the committee that your project is what they are looking for.

The committee will be composed of several people, each of them having read several proposals. They will probably spend 30min-1h max per proposal. This time will be split between reading the report from the referee and your own proposal. They will use both of this to present your project to all other members of the committee, who will not have read the report. The presenting member is usually from a different field. They will probably be physicists, but they might work on polymer elasticity, acoustic waves or whatever. Not condensed matter. They might even be electrical engineer or any other applied science. As a result, they will probably not recommend a mark of 100%. All other members of the committee will have other proposals to present, and they will not have read yours. It is very likely that they work in a very different field from yours, like chemistry or biology. They might skim through it while the presenting member talks about yours, but that’s all. However, they will still vote on your proposal in order to rank it. So you want the plan to be as clear as possible so that the presenting member gives an accurate description and will also recommend a good mark. If it is too complicated, you can be sure that the presenting member will keep your proposal for the end of the session, when there is no longer enough time to present it clearly, and he will not bother recommend funding.

For this Singapore grant, it looks quite similar in principle. In the application guide, they say that it is a 3 stage process:

1. shortlist by host institutions;
2. shortlist by NRF Fellowship evaluation panel (FEP), comprising eminent scientists (this last part is irrelevant)
3. interview.

So you need to convince one or more institutions that your project is worth being selected. As you already have a position at NUS, this is all politics. Identify who is on the shortlisting committee, and go out of your way to talk to them. Convince them that the project is worth being selected. After all, it would be good that they maximise the chances of their own academics. Otherwise, why would they hire you in the first place? For the evaluation panel, it is all on the proposal, based on recommendations from the school. However I don’t really understand if and at which stage it gets reviewed by an academic in your field.

The structure of your proposal.

Once you understand that your proposal will be judged by a lot of people who will not even read it properly, you understand that you should put all the relevant information (a) in the beginning of the proposal and (b) in the headers. Everything that grants you points should be there, because it might be the only thing the committee reads. Indeed, what I would recommend would be the following: your abstract or “executive summary” as they call it here will contain independent sentences that can just be picked by the presenting member of the committee. You want the sentences to be simple enough that they can be directly re-used by the referee and the presenting member of the committee. And these sentences will also be the first sentences of your main paragraphs. Don’t worry too much about this executive summary at the beginning of rewriting your proposal, you should write this at the end.

The Fellowship application guide gives you an example of the structure. In fact, it is not an example, it should be followed if you want to get funded. This is:

a) Title page
b) Table of contents
c) Executive Summary
Main body (should not exceed 8 pages) (this means you should write exactly between 7.5 and 8 pages)
d) Aims and objectives
e) Reasons for wanting to do your research in Singapore
f) Background and significance
g) Research design and methods
h) Milestones and deliverables
Annexes
i) References
j) Publication list
k) Patents and industry link.\

I think it is a very good structure, that can be reused in other proposals (unless they suggest something else!). In different proposals, point e will be changed to reasons to select a specific institution. Here I will only comment on c, d, e, f, g, h.

You should not necessary write them in order. You will be more productive if you write them in the following order: g1, f1, d, h, g2, f2, e and c.

1. Research plan (g1)

The first thing you want to do is to create a number of work packages. Work packages would be something like small projects. They are internally consistent. At that stage, you want to define broadly what you want to do, and define 3 to 4 tasks for each of these work packages. A task is, for example “fabricate the non-centrosymmetric material devices” or “set up the measurement platform including software control”. I find that 3 work packages each having 2 to 4 tasks is ideal for a project that lasts up to 4 years, and you might add a fourth work package for proposals that are for 5 or more years. Don’t go above that otherwise it becomes too complicated. Remember about the poor presenting member of the committee who will have to justify funding you. On top of that, you will add a few work packages for communications and training activities. Usually, I would have WP5: Training and management and WP6: Outreach, communication and dissemination of results, which are associated to their own research objectives (RO), milestones and deliverables. This way, you tell the committee that you have a good intention of doing all the administrative work related to managing the proposal, you won’t just be a scientist in their ivory tower. E.g. you will train the next generation of scientists as part of the funding, you will dedicate time to meet with the patent office of your institution to transform knowledge into innovation, you will deliver yearly reports to the funding agency, etc.

In the past, I have found that the work packages would be reorganised during the writing and editing. When I started working on my Marie Curie application I designed 3 work packages, but I noticed later that the third one necessitated different techniques and theoretical background. I decided to remove it and split the remaining two WPs into 4. This is quite a bit for a 2-year project, but it worked.

Keep in mind that you cannot have a work package that is “fabrication of devices” and another which is “measurement of devices”. In fact, you cannot have a work package that is dependent upon the completion of another, because the funder will always think of the worst outcome possible: you do not finish WP1, therefore not WP2. If you continue like this, you imagine no result at the end, therefore no need to fund the project. Whereas if you have 4 independent WPs, that could individually go wrong, but in total, the probability of success will be much higher.

2. Identify open questions (f1)

Your research objectives need to answer a question. For this, you would need to list the open questions in your field. You did it well already. You will need to write them as Q1, Q2, etc. and they will correspond to RO1, RO2, etc. For example Q4 could be “there were theoretical predictions of xxx in non-centrosymmetric materials, but so far no experimental proof of their existence and consequences”. Once you have your list of questions, keep them on the side, you will use them later.

3. Aims and objectives (d)

In this section, you should list the research objectives related to your project. In my opinion, it shouldn’t be a very long section, because it will be related to the open questions that you defined in (f). There will also be some background in (f) so any detail will not be necessary here. I find that it is a section that is more complicated that it seems because it has to find a common ground between two contradictory goals: it needs to be ambitious and show significance, but at the same time prove to be achievable. It is better to be relatively precise here. For example, RO4 could be: “I will study the existence and consequence of xxx in non-centrosymmetric materials”, where xxx is the exact same wording as in Q4.

4. Milestones and deliverables (h)

Now, you have a list of what you want to do (the work packages), open questions and research objectives, you can think about the milestones and deliverables. When will you say that you will have finished a work package? What will you need to do. It is usually not a paper, as the paper might come 1 or 2 years after you get the deliverable. For example you can have “M4.1: existence of xxx in non-centrosymmetric materials proven”, and “M4.2: consequences of xxx in non-centrosymmetric materials elucidated”. The deliverable will be something that the funder expects to get at that time. Usually it will take the form of a report on activities, including research, communication activities. For this, you should get advice from your research support officer, they would know that better, and provide information on what the funder seems to expect in terms of deliverables. Again, this will not be a long paragraph, but it should be precise. You need to prove that you will deliver the deliverables. A paper in Science cannot be a deliverable, because there is only a 10% possible success. However, a dataset placed on an open repository can be a deliverable, because you are practically sure that you will measure something.

5. Research design and methods (g2)

This is (almost) independent from the rest. Here you will list whatever you can do, how you do. You are the expert on this. Show it. I have had the intuition that these sorts of method section are not really here to describe all the details of the methodology, but are more like an opportunity to showcase any competitive advantage you might have. Write down exactly where and how your technique is unique and give details on this, the rest should be written very quickly. For example, in my Marie-Curie fellowship, I wrote a full paragraph on the cryo-SNOM techniques that is unique in the host lab, whereas all transport, cryogenic methods and sample fabrication, represented, in total, the same number of words. For those, I would recommend one or two sentences to describe quickly the technique. This is because there are too many groups that master these techniques. But it can be a technique you have and that is unique in the department or in Singapore: it is understood that you are at an early stage of your career and your portfolio is still expanding. Here again, you can write down which technique will be associated with each WP.

In the NRF fellowship application guidance they also ask you to determine the resources that you will need, like equipments, postdoctoral fellows or PhD students. It is good to estimate wisely. E.g. if you already supervise 1 PhD student and 1 postdoc, ask for 1 more student and one more postdoc. Don’t ask for 3 of each, because there is no proof yet that you will be able to manage something that big. I would write these request for resources as an independent section, placed at the end of the research design and method.

6. Write the background and significance section (f)

You already identified the open questions in the field. Now you need to write this so that it fits with the rest. It says you should split it in two subsections: 1. Background and 2. Significance. For the background subsection, add 1 or 2 sentence of context before your open question. This will be one paragraph. For the significance section, it will be a bit more complicated. You will have to pin down what consequence the research will have on other people lives, on the physics community. In summary, what are the reasons to fund your research? In my experience, you can write 3 paragraphs, and each will address the following question: what is the expected scientific impact, what is the expected social impact of the research (i.e. are there policies in Singapore that go into this direction? E.g. what is the government doing for quantum computation, how much investments, etc? how does your proposal match with the political strategy), and what is the economical impact (e.g. startups, patents, etc.). You will need precise examples and numbers.

7. Reasons for wanting to do research in Singapore (c)

Reasons to do your research in Singapore isn’t really a reason for you to choose as you are already submitting a proposal. In fact, I take this as a disguised question to justify what is the reason that the office of the NRF choses your project vs any other, on their own criteria. What I did for example in the Leverhulme application, at the stage of internal review, is asking the head of School and head of Department what the research priorities were, both within the School and in the Department. Then I picked 3 among all of them and justified that my research was aligned with these priorities. The consequence is that the University will hire many more people in this field in the future, and will have a shared interest in your success. There were other reasons that I justify Manchester for the project, but these have less weight, I could have written this in any other university that had cleanroom + helium regeneration, and there are many!

8. Finally, write the executive summary (e)

For this, the application guidelines are very explicit, they say:

i) Articulate the big problem your research wants to solve.

• The problem should be significant
• The problem and related ideas need to be understood by non-scientists^a
• The problem should be important
• People should care when you solve it successfully
• You should justify the significant difference between the proposed research beyond your postdoctoral work.

ii) Explain why you are qualified to do this research

• Give examples on how you have led a team to solve a scientific problem
• Give examples of previous successful proposals
• Showcase your technical skills necessary to solve the research problem
• Detail how you will use the funds

Each of these points will be one sentence (and no more!) of the executive summary. The details of this will be explained in all the other sections. Except point marked with (a) which is irrelevant at this stage. In addition, you will write sentences that will relate the open questions (Q), the research objectives (RO) and the work packages (WP). You can write the impact there. The construction will be very simple: “We need to understand XXX (Q1) in order to solve big problem 1(impact).” You write the 4 of them corresponding to your 4 research objectives. After that, you add “I will measure YYY (WP1) in order to understand XXX (Q1).” It is not a problem if “XXX” is repeated at several sentences of interval: it brings clarity to the presenting committee member. The details for the referee will be in section g: research design and method. In addition, the NRF guidance reads: “Your first sentence might start: ‘The broad objective of the research proposal is…’”. Here, ‘might’ should not be understood as ‘can’ but ‘must’. Below I gave an example of the abstract (called executive summary here) for my Marie-Curie proposal, with highlighted key points that appear clearly.

Now that you have written your executive summary, take every sentence from it and use them as the first sentence of every section or subsection. That way, the presenting committee member will be in a position where they can identify very quickly where to find more detailed information when discussing the proposal. Don’t rephrase and use synonyms for technical concepts, because they will not be an expert and it will just confuse them.

At the very end, check the formatting requirements. The application guide states you should have the following:

Page format	Font type	Font size	Line spacing	Alignment
A4	Arial	12	Single	Justify

You will get eliminated if this is not what you do. It means that all titles should be the same size. The trick here is to use combinations of bold, italic, underline and colours. However, they don’t give any detail on the margin, so it can virtually be 0 to fit more text. In practice they might not like it, so do between 0.5 and 1cm.

Example abstract of my Marie Curie fellowship.

A key ingredient governing the behaviour of electron dynamics in condensed matter systems is the valley degree of freedom. This fellowship aims at inducing topological (or valley-polarised) electronic phases in moiré systems and explore their tunability. Moiré quantum materials present a high degree of tunability as the ratio between interaction strength and electronic bandwidth can be optimised, resulting in flat bands and correlated electronic states. Control of such phases with circularly polarised light would allow to understand the valley-polarisation, topological protection, and the mechanisms underlying the emergence of flat-band superconductivity and correlated insulators. The project will use an interdisciplinary methodology combining electronic transport techniques with infrared far-field irradiation and near field imaging, to create, manipulate and explore novel phases with non-trivial topology (valley polarisation) in moiré materials. The researcher will conduct transport measurement with circularly polarised light to selectively induce inter-valley scattering in 1D channels and explore superconducting interferences in this regime. The project will also quantify the influence of circularly polarised light (CPL) on the superconducting phases and correlated insulators in flat band moiré systems, in order to comprehend the interplay of strongly correlated electronic states with CPL and understand their origin. The fellowship will control and explore novel phases with non-trivial topology in moiré materials, providing new understanding of valley-polarised phases. This objective is a steppingstone towards the creation of new optoelectronic devices such as photonic diodes, optical transistors and logic circuits, and could lead to new concepts as topological nanophotonics for optical information processing.

This abstract is composed of the following parts (colour-coded)

1. a quick introduction to situate the context. Note that this is not necessary. I also included a reminder of the goal of the project here to emphasise on this from the very beginning.
2. a summary of the background and significance section.
3. a summary of the research design and methods section.
4. a precise project description.. This is a key sentence that can be repeated as much as possible throughout the project, with the exact same wording.
5. a summary for the content of one work package
6. the aims and objectives corresponding to the described work package.
7. Finally, a sentence to explain the impact of the research and potential next steps. It will be related to the significance section.

Sentences 5 and 6 were written successively for each work package. Once these sentences are written,I repeated them as the header of each section. That way, I think that the committee doesn’t have the impression of being lost in the project. Using the same wording will ensure that even if they don’t understand the jargon, they understand the relation between the background, the aim, the work packages and the expected results.

I’m regularly invited to the Journal club in the group of Roman Gorbachev at the University of Manchester. Each week, two of us present a paper each. It is a good occasion to catch up with research we wouldn’t be confronted with otherwise. This week I talked about a paper published in Nature last Thursday about long lived Andreev-Floquet states in graphene Josephson junctions. It is a good reason for a blog about it.

The paper is: Steady Floquet-Andreev states in graphene Josephson junctions, from the group of Gil-Ho Lee at Postech, South Korea. Before starting with this paper, let me discuss Floquet states.

Take a free electron. It would be described by the Schrödinger equation and have a parabollic energy-momentum dispersion. If now you create a periodic lattice in space, the Bloch theorem tells you that you should get a band structure that is periodic in space. The underlying reason is that position and momentum are conjugate variables.

The Floquet states are analogues of Bloch states in the time domain. Since time and energy are conjugate variables, having a time periodicity in the system would result in a periodicity on the energy axis. In condensed matter systems, Floquet states were actually found using ARPES by the group of Nuh Gedik at MIT almost 10 years ago. The way they were observed was by irradiating a topological insulator with a short laser pulse and simultaneously observing the momentum-energy photoemission image. This shows a clear duplication of the band structure in energy: the Floquet states, but with two main drawbacks: a) the Floquet state exists for the time of the impulsion, a few femtoseconds; b) the laser pulse results in considerable heating of the sample.

Let’s now go back to the recent paper in graphene Josephson junctions. These Floquet states are based on Andreev bound states, that can be observed with tunneling spectroscopy as peaks in the dI/dV after the superconducting gap. By irradiating the graphene sample with a microwave excitation, in the same way that one would apply microwaves to observe Shapiro steps, the group of Gil-Ho Lee observed duplicate peaks at energies above and below the Andreev bound states. They characterised these states and provide a definitive proof of the Floquet nature through minute theoretical analysis. Interestingly, the separation between the Floquet states increases with the RF frequency and their number with the power. They also provide a Bessel function analysis (fig 2e) for these, but to me this is not convincing.

The strength of that paper lies in the fact that Floquet states could be found without heating the sample considerably (low frequency allow to use lower electric fields, therefore less heating) and graphene devices present good electron-phonon coupling, allowing continuous cooling in a dilution refrigerator. Additionally, these Floquet states are not volatile. In principle, nothing prevents them from disappearing. The authors claim that the states do not vary for 25h. In principle, this would allow further tuning of the band structure from microwave irradiations.

As I have done previously, I’m writing some words about statistical analysis that I think is useful for undergrads, based on my teaching experience. There is a particularly interesting example posted on arXiv on Tuesday that allows me to illustrate this theory.

Gaussian distribution & Central Limit Theorem

Let’s a set of $n$ independent variables $x_i$ taken at random from a set with mean $\mu$ and variance $\sigma^2$. Now you can calculate the mean $\bar x$ of these $n$ values. The Central Limit Theorem says that the the distribution of $\bar x$ tends to a Gaussian of mean $\mu$ and variance $\sigma^2 / n$. The important feature of the theorem is that whatever the distribution of the $x_i$, a linear combination of a few variables (almost) always approximates to a Gaussian distribution.

What matters here is that acquiring a large number of data with random error leads to a distribution of values that tends to a Gaussian distribution.

In the treatment of errors

A consequence of the Central Limit Theorem is that in most of the situations, repeating an experiment would produce a spread of values whose distribution is a Gaussian. This approximation becomes better if individual errors contributing to the final value are small.

When this is the case, the fractional area in the tail of a Gaussian (filled area in panel (a) of the figure below is always non-zero, for all values $r$ in which it is taken. It is thus meaningless to speak of a maximum possible error of a given experiment as this would be infinite. Although it would be easy to calculate the error on any experiment, that would not distinguish a precise result from an inaccurate one.

Therefore, it is common to quote $\sigma$, the standard deviation of the Gaussian function, as the accuracy for a measurement. As it is not the maximum possible error, results of measurements can fall more than $\sigma$ away form the expected value, without questioning the validity of the assumptions. In fact, $1\sigma$ includes 68% of all the data points, $2\sigma$ includes 95% of all data points, $3\sigma$, 99.7% of all data points, etc.

When we carry out $n$ measurements, the standard deviation $\sigma_m$ of the mean would be calculated as $\sigma_m = \frac{\sigma}{\sqrt{n}}$, that is the standard error in the mean of $n$ observations is $1/\sqrt{n}$ times the standard error in a single observation.

The question of importance now is: when would a particular data point be considered significant? When one data point falls beyond $2\sigma$ away, the probability that it happens is only 5%. When it falls beyond $3\sigma$, the fractional area is only 0.3%, se we would expect a deviation to occur that frequently. As a result, when a data point falls a few sigma away from a model being tested, there is an evidence that the data point is not consistent with the model. Indeed, we should expect a data point to be $3\sigma$ away, about 3 times if the experiment is repeated 1000 times.

Example: LHCb experiment

This preprint from the LHCb collaboration about possible violation of the standard model, that had a good press coverage.

They measure the decay of beauty quarks (B-mesons) into muons and electrons, probed through the value $R_K$ that is the ratio between these two decays and find $R_K$ = 0.846 $\pm$ 0.044 (that includes a statistical error evaluated at +0.042/-0.039 and a systematic error evaluated at +0.013/-0.012). The standard model would predict a value of 1.00$\pm$0.01. The question is: to what extent are these numbers in agreement?

To answer this, let’s look at panel b in the above figure. This shows the fractional area under the gaussian curve ($|f|>r$ with $f = \frac{x-\mu}{\sigma}$), that is the area of the tail highlighted in panel a. In the LHCb example, $\mu$ = 1, $\sigma$ = 0.044 and $x$ = 0.846. With our formula, this gives $f=3.5$, for which the corresponding probability read on the graph is 0.046%. In reality, they evaluate this more precisely than my rough analysis and get $f=3.1$ with a corresponding probability of 0.1%.

As a result, if 1000 experiments are performed, all of which with the same precision and if the theory is correct and if the experiment is bias-free, then we expect about 1 of them to differ from this value at least as much as this experiment does. The scientists of the LHCb collaboration expect to repeat update the experimental setup to go as far as $5\sigma$, that would give a probability of 6·10⁻⁵%. What will matter in this case is whether they regard the theory (and the experiment) as satisfactory or not, but they will have numbers to base that judgement on.

What this example means, is that if the Standard Model is valid (the Null hypothesis), there is 1 in 10,000 chance the LHCb data is at least as extreme as what they observed. In this field, this is still not enough to claim they found something outside of the Standard Model.

A week ago we published a paper in Nature physics showing the formation of van Hove singularities in twisted monolayer-bilayer graphene systems that can be tuned with the application of an electric field or by changing the twist angle. Notably, we show that these can cause strong correlations effects under various conditions. This is an interesting system, reproducing strongly correlated states seen in magic-angle graphene, but for some reason much easier to process.

Role of symmetry

Twisted van der Waals heterostructures usually offer different tuning parameters to create exotic state. In the case of twisted monolayer-bilayer graphene, there are more tuning knobs, compared to twisted bilayer graphene, because the band structure of multilayer graphene (here, bi-layer) is more tuneable than the one of the monolayer. Different stacking orders can exist for trilayer graphene: Bernal (ABA) or rhombohedral (ABC). The former presents a mirror symmetry and is semi-metallic, whereas the latter presents an inversion symmetry and is semi-conducting, with a gap that can be varied with the displacement field.

In twisted monolayer-bilayer graphene, the symmetry is even more reduced with the twist induced between the bilayer and monolayer graphene, that results in ABA-, ABB- and ABC-domains (cf schematic below).

The phase diagram can be tuned via field-effect

In all the studies, twisted monolayer-bilayer graphene (tMBG) devices comprise a top- and back-gate, that allow to control the doping ($n$) or to apply a perpendicular electric field in the material ($D$).

Particularly, Chen et al have shown that a variety of correlated metallic, insulating and topological magnetic state appear, owing to the low symmetry. Particularly, the application of a displacement field endows this system with either the phase diagram of twisted bilayer graphene (at positive displacements) or twisted double bilayer graphene (at negative displacements).

We (Xu et al) have studied tMBG devices with a twist between 1.2° and 1.6° and studied the correlated states that appear asymetrically with the external electric field applied between the layer, and are highly tuneable by varying the twist angle. As shown on the following figure, two resistive peaks emerge at full filling $\nu = \pm 4$, and a gap opens at charge neutrality point (CNP) with increased $D$. The single particle model cannot explain the resistive and metallic states at half and quarter fillings. We have shown that the resistivity is asymmetric for electrons and holes in these states.

Correlated states at quarter filling

Polshyn et al have observed the correlated states corresponding to filling of 1 and 3 electrons per unit cell. In these states, they observe a quantised anomalous hall effect, with a transverse resistance close to $h/2e^2$. This could indicate a spontaneous polarisation of the system into a single-valley projected band. In the state with 3 electrons per unit cell, they show, as well, that the sign of the quantum anomalous hall effect can be reversed via field-effect control of the chemical potential. This is a major breakthrough.

Chen et al observe these correlated states undergoing an orbitally-driven insulating transition above a critical perpendicular magnetic field, for electric fields applied from the monolayer to the bilayer ($D>0$). Particularly, at $\nu = 1$ (one electron per moiré unit cell), they probe the emergence of an electrically tuneable ferromagnetism, in the conduction band, that is accompanied by an anomalous Hall effect.

More recently, He et al reported that orbital magnetism is abundant within the correlated phase diagram, leading to anomalous Hall states within the correlated metallic states around odd integer fillings of the flat conduction band. They also observe correlated Chern insulator that are stabilised within a magnetic field. Importantly, they show that this behaviour at zero electric field is inconsistent with simple spin and valley polarisation and propose an intervalley coherent state with an order parameter, breaking time-reversal symmetry.

All these study show that both electric and magnetic field can tune competition between correlated and topological phase, owing to the rich interplay between competing correlated states in twisted monolayer-bilayer graphene systems. So far, most of the recent study focused in the conduction band at $D>0$, where the system has a phase diagram comparable to twisted double bilayer graphene.

References:

• H. Polshyn et al 2020, Electrical switching of magnetic order in an orbital Chern insulator, Nature;
• S. Chen et al 2020, Electrically tunable correlated and topological states in twisted monolayer-bilayer graphene, Nature Physics;
• S. Xu, et al 2021, Tunable van Hove singularities and correlated states in twisted monolayer-bilayer graphene, Nature Physics;
• M. He et al 2021, Competing correlated states and abundant orbital magnetism in twisted monolayer-bilayer graphene.