Giant orbital magnetoelectric effect and current-induced magnetization switching in twisted bilayer graphene

Recently, quantum anomalous Hall effect with spontaneous ferromagnetism was observed in twisted bilayer graphenes (TBG) near 3/4 filling. Importantly, it was observed that an extremely small current can switch the direction of the magnetization. This offers the prospect of realizing low energy dissipation magnetic memories. However, the mechanism of the current-driven magnetization switching is poorly understood as the charge currents in graphenes are generally believed to be non-magnetic. In this work, we demonstrate that in TBG, the twisting and substrate induced symmetry breaking allow an out of plane orbital magnetization to be generated by a charge current. Moreover, the large Berry curvatures of the flat bands give the Bloch electrons large orbital magnetic moments so that a small current can generate a large orbital magnetization. We further demonstrate how the charge current can switch the magnetization of the ferromagnetic TBG near 3/4 filling as observed in the experiments.


Summary of Reply:
We thank both referees for their careful reading of the manuscript. We are gratified by Referee 2's appreciation of our work, and his/her enthusiastic support for publication in Nature Communications.
Referee 1 made many useful comments, which helped us realize where we had not clearly explained some subtle but important points. Based on this invaluable input, we have now improved our explanations. However, we believe that the key novel findings of our manuscript have been overlooked by Referee 1. Specifically, the giant magnetoelectric effect (current-induced magnetization effect) we studied in twisted bilayer graphene (TBG) is fundamentally different from the large magnetization effect caused by valley imbalance in gapped bilayer graphene.
Moreover, we would like to emphasize that, the current-induced magnetic switching in TBG is a very interesting and important phenomenon. The current needed to switch a magnetic bit of a ferromagnetic TBG is in the order of 10nA. This is six orders of magnitude smaller than the current needed to switch the magnetization direction in state of the art spin-orbit torque devices, which require currents of about 10mA for magnetic switching [Nat. Nanotech. 11 878(2016)]. This phenomenon has great potential applications in current-controlled magnetic memory devices and it deserves a clear theoretical explanation.
In our work, we have not only explained the mechanism of the current-induced switching in TBG, but also predicted that many other moire systems can also have large current-induced magnetization. We believe that our work provides a clear theoretical guidance for the study of the current-induced magnetization effects in moire materials.
In our reply to the referees below, we also highlight the difference between our work and the work by Nandkishore and Levitov mentioned by Referee 1. We hope that our clarification here and in more detail below helps Referee 1 reassess the importance of our work.
Reply to the Comments of the Referees.

Referee 1:
Comment 1: "The main claim of the paper is that that the lattice symmetry reduction in twisted graphene bilayers by itself promotes a large out-of-plane magnetization per valley. When the valley degeneracy is lifted presumably by interaction effects (spontaneous broken symmetry), a large net magnetization can emerge. Those effects are claimed to be further enhanced by strain and sub-lattice symmetry breaking."

Reply 1:
We thank the Referee for the careful reading of the manuscript.
The Referee points out correctly that each valley of twisted bilayer graphene (TBG) can carry a large out-of-plane magnetization (implicitly recognizing that the magnetization is opposite for each valley so that the net magnetization is zero). The Referee then correctly notes that "when the valley degeneracy is lifted presumably by interaction effects (spontaneous broken symmetry), a large net magnetization can emerge".
We differ however with the Referee on the role these observations play in our manuscript: The main intent of our work is not to show that valley polarization in TBG will cause net magnetization. As the Referee points out [Nandkishore et at. PRL 107,097402 (2011)), this connection is already well-known. Nor is the claim that lattice symmetry reduction can enhance the valley polarization central to our narrative.
Instead, the main contribution of our work is to provide an explanation for why an exceedingly small current (6 orders of magnitude smaller current than that needed for switching a magnetic bit using state of the art spin-orbit torque) is capable of switching the magnetization direction of TBG when the TBG is ferromagnetic. This behavior was observed in two recent experiments (by Goldhaber Gordon's group at Stanford [Science 365, 605-608 (2019)] and Young's group at UCSB [Science.aay5533]). These striking experimental findings were not understood nor explained theoretically previously. In the following, we would like to show the importance and broad interest of our work:

i) Explaining current-induced magnetization switching in TBG:
To explain the current-induced magnetization switching observed, there are three important steps laid out in our manuscript (we have revised the manuscript to try to make these steps clearer and more prominent.) First, even without valley polarization and with the TBG being non-ferromagnetic, a small current can induce a large out-of-plane magnetization in TBG (this is very different from the case of conventional gapped bilayer graphene, where the current-induced magnetization is zero).
Second, this large current-induced magnetization can happen in TBG when: a. The three-fold rotational symmetry in TBG is broken (by strain, for example). b. The orbital magnetization of the Bloch electrons is large (due to the large Berry phase of the flat bands of TBG). And the strain further enhances the orbital magnetization. c. The density of states of the flat band is large at the Fermi energy.
We pointed out through our calculations that these three conditions can be satisfied by TBG coupled to boron nitride substrates. It is important to note that this large currentinduced magnetization effect can happen at a general filling factor of the TBG, whether or not the TBG is in a quantum anomalous Hall phase. Moreover, the direction of the magnetization is reversed when the direction of the current is reversed, just as seen in experiment when magnetization can be detected through anomalous Hall measurements.
Third, when there is ferromagnetism in TBG (such as near the ¾ filling), the magnetization induced by current can couple to the magnetization of the TBG and result in current-controlled magnetization switching. Therefore, our theory accounts for the experimental findings of Goldhaber-Gordon and Young regarding current-induced switching.
ii) The novelty and importance of our work.
1. Our work explains why an extremely small current can control the magnetization direction of a ferromagnetic TBG which is an important experimental finding with possible applications in ultra-low dissipation magnetic memory devices.
With our understanding, we also predicted that there is current-induced magnetization effect in other moire materials such as in twisted transition metal dichalcogenides (TMDs), monolayer graphene coupled to boron nitride substrate etc. This work will lead experimentalists and theorists to further study the current-induced magnetization in moire materials.
2. The current-induced magnetization effect discussed in our work is a very special kind of magnetoelectric effect.
As dicussed at the end of the Discussion Section, current-induced magnetization has been studied in non-centrosymmetric materials with spin-orbital coupling three decades ago [1]. This effect is also called the Edelstein effect. The magnetization induced is due to spin magnetic moments. However, the current-induced magnetization is zero when the spin-orbit coupling is zero. In our work, we showed that in TBG, the current-induced magnetization is very large even in the absence of spin-orbit coupling. And the magnetization induced is purely orbital in nature. This kind of orbital magnetization was missed in Edelstein's theory which ignored the effect of Berry curvature of the band. In TBG, the large orbital Edelstein effect is caused by the large Berry curvature and also the large density of states of the flat bands.
3. In the Reply to Comment 3, we will show that the current-induced magnetization in TBG is very different from the magneto-optical Kerr effect studied by Nandkishore and Levitov for gapped bilayer graphene.

Comment 2:
"The model is based on a continuum calculation of twisted graphene bilayers proposed in ref. [3][4][5]. Substrate effects are accounted by breaking the C_2^{\prime} symmetry, which protects the Dirac points of the moire BZ, resulting in the opening of a gap, and also with strain effects, which displace those points and break C_3 symmetry."

Reply 2:
We thank the Referee for a nice summary of our calculations.

Comment 3: "
The understanding that gapped graphene bilayers in general can have a giant magnetoelectric effect when the valley degeneracy is explicitly broken (for instance in the quantum anomalous Hall state or else by selectively exciting valleys with light in dichalcogenides) has been known for a while and I would say is hardly surprising by now (see for instance Nandkishore et at. PRL 107,097402 (2011))."

Reply 3:
We agree with the Referee that gapped graphene bilayers in general can have a large magnetization when the valley degeneracy is explicitly broken. But this is not the giant magnetoelectric effect that we studied in the present work.
The giant magnetoelectric effect in our manuscript refers instead to the large orbital magnetization induced by a small current. In the case of gapped graphene bilayers discussed by Nandkishore and Levitov [PRL 107,097402 (2011)], the current-induced magnetization effect is indeed zero due to the C 3 rotational symmetry. If the C 3 rotational symmetry in a conventional gapped graphene bilayer is broken by strain, the induced magnetization at a given current is three orders of magnitude smaller than the case of strained TBG that we studied (as shown in Fig.R1 below). In the following, we provide a detailed account of why TBG is very different from gapped bilayer graphene.

i) The difference our work reveals between TBG and the conventional gapped bilayer graphene mentioned by the Referee.
First, in the case of conventional gapped bilayer graphene without net valley polarization, no matter how large the orbital magnetizations of the Bloch electrons are, a charge current cannot induce an out-of-plane magnetization. The C 3 symmetry will enforce the out-of-plane magnetization to be exactly zero. This is also the case for conventional TMDs.
Second, in the case of conventional gapped bilayer graphene with valley polarization, the current can carry magnetization due to the valley polarization. However, the magnetization direction of the current will always be the same as the magnetization of the sample, so the charged current cannot switch the magnetization direction of the sample.
Third, when strain is applied to conventional gapped bilayer graphene to break the C 3 rotational symmetry, symmetry allows finite current-induced magnetization as in the case of TBG (to the best of our knowledge, this problem has not been studied previously). However, as we show in the following figure (Fig.R1), the current-induced magnetization in this case is three orders of magnitude smaller than the twisted bilayer graphene case. This is due to the small density of states of gapped bilayer graphene. Though the orbital magnetic moments of the Bloch electrons near the band bottom of gapped bilayer graphene can be as large as 15 Bohr magneton near K and -K points (Fig.R1b), these moments decrease quickly away from these high-symmetry points. Near the K points, the only places where the moments are large, the density of states is very small. Therefore, the current-induced magnetization is very small. In Fig.R1c, we show that with 0.1% strain, the current-induced magnetization is three orders of magnitude smaller than the case of TBG. Fig.R1 a) The band structure of gapped bilayer graphene. The gap size is set to be 200 meV as shown in a recent experiment [2,3]. b) The orbital magnetic moment of the Bloch electrons of gapped bilayer graphene. As pointed out by the referee, the orbital magnetic moments can be as large as 15 Bohr magneton near the conduction band bottom. However, the magnetic moment decreases quickly away from the K poiints. c) The magnetoelectric response of a gapped bilayer graphene. When the strain is zero (red line), the current-induced magnetization is always zero. Even when a strain is applied (blue curve), the current-induced magnetization is three orders of magnitude smaller than the case of TBG: The tensor element a, which characterizes the strength of the current-induced magnetization as defined in Eq.7 of the main text, can be as large as 2000 in strained TBG ( Fig.3a of the main text) as compared to about 2.5 in gapped bilayer graphene. In the calculation for c), we considered a current direction which maximizes the magnetization. One can also show that a is not sensitive to the gap size which is set to be 200meV in the calculations.
ii) The difference between our work and the work by Nandkishore and Levitov.
Since the Referee specifically referenced the work by Nandkishore and Levitov as anticipating our present work, we would like to highlight the strong differences: Nandkishore et al. pointed out that: 1. If a bilayer graphene is in the quantum anomalous Hall state, that bilayer graphene can induce finite magneto optical Kerr rotation.
2. In the nematic phase of a bilayer graphene, due to the breaking of the C 3 rotational symmetry, the intensity of the reflected light breaks the C 3 symmetry as well.
The results of the elegant work by Nandkishore et al. are useful for optical detections of spontaneous symmetry breaking phases of suspended bilayer graphene, for which transport measurements can be difficult to perform. Their results are related to the photon-induced optical transition between the bands of the gapped bilayer graphene. In our work, the current-induced magnetization is related instead to the properties of the Fermi surface. The work by Nandkishore et al. is not directly relevant to the current induced magnetization in gapped bilayer graphene. Their results cannot be used to predict the current-induced magnetization and magnetization switching in TBG.
After reviewing the manuscript and the response of the authors, I am persuaded that the work is novel and important enough to merit publication in Nature Communications.
The key finding of the paper is that a broken C3 symmetry in twisted graphene bilayers (due to strain) leads to a net orbital magnetization when a current is injected. The effect does not depend on time reversal symmetry being broken, and can happen at arbitrary filing factors, including in the metallic phase, and hence is different of the standard orbital magnetoelectric effect in Chern insulators and systems with anomalous Hall effects in general. This effect is greatly enhanced by the large density of states in the flat bands.
These results appear to explain the current switching effect observed recently by different groups and is of clear technological importance. The changes in the abstract and in the introduction have significantly improved clarity of the paper. Overall, this is a very interesting result in a very timely problem. For those reasons, I recommend publication in Nature Communications.

Reviewer #3 (Remarks to the Author):
This paper theoretically studies the giant orbital magnetic momentum induced by the small current due to the enhanced Berry curvature in twisted bilayer graphene with reduced symmetry due to strain and substrate. I have examined previous reviewer's comments and responses from authors, and find that author's rebuttal is reasonable. The giant orbital magnetization and consequent magnetization switching is a novel finding and offers a reasonable scenario for the experiment. Considering that the magnetization switching by current in terms of spin-orbit coupling usually requires huge current density, this mechanism without spin-orbit interaction is very promising also for the applications. Thus, I do recommend its publication in Nature Communications.