domain: spi phys math info

text:
Mehdi Othmani-Guibourg

Amal El Fallah-Seghrouchni

Jean-Loup Farges

Decentralized multi-agent patrolling strategies using global idleness estimation

Keywords: Multi-agent systems, Multi-agent patrolling, Artificial Neural Networks

This paper presents preliminary results in the challenge of developing decentralised strategies approaching the performances of centralised ones. Indeed, the latter are better than the former due to centralisation of information. The approach studied here involves the estimation of node idlenesses derived from the paths of all agents, also known as real idlenesses, on the basis of those derived from the path of each agent considered alone, also known as individual idlenesses. This relation between real and individual idlenesses is learnt using traces of execution of a centralised strategy by optimising an error criterion. The strategy thereupon, uses online the learnt relation and is assessed according to certain evaluation criteria. The results indicate that such a relation between perceived and real idlenesses is not a function, leading to large values of the fitting criterion. Finally, the assessment of the strategy shows that performances are good in terms of mean interval but unsatisfactory in terms of quadratic mean interval.

Introduction

The patrol task is well-suited for being shared in space and time by several agents. There are a wide variety of problems that may be reformulated as a particular multi-agent patrol task. As a concrete example, the monitoring of an area by a swarm of drones does face the problem of coordinating them to patrol that area in order to detect certain events. A feature of multi-agent patrolling (MAP) is the difficulty to derive analytic results from its equations. Thereby it appears that the only method enabling to predict its behaviour is to simulate the local interactions of its components. Thus, the quality of a patrolling strategy is evaluated in simulation and it is consensual that a good strategy is one that minimises the time lag between two passages on the same place and for all places.

Different types of strategies were proposed, however, few works concentrate on the problematic of using Artificial Neural Networks (ANN) for the MAP [START_REF] David | Stanley:Multirobot Behavior Synchronization through Direct Neural Network Communication[END_REF] [4] [START_REF] Sales | Multi-agent Autonomous Patrolling System Using ANN and FSM Control[END_REF]. We propose and evaluate new strategies using not only machine learning models, in particular an ANN based on the rectifier linear ReLU, but also a multi-dimensional linear model. In this way, new strategies embedding these models are introduced.

The Section 2 presents the background useful to understand proposed developments: the MAP and ANN types used by the strategies. Then, Section 3 describes new strategies based on idleness estimation by ANN. In Section 4 these strategies are analysed. Finally, Section 5 draws some conclusions and indicates directions for further works.

Background

Multi-agent patrolling

The MAP model consists of a society of agents noted A, able to move in a graph noted G = (V, E) representing a discretisation of the area to patrol. V = {1, .., N } is the set of nodes standing for the places to visit, and E is the set of edges connecting them. At each edge corresponds a transit time representing its travel time. At each node is associated a dynamic variable named idleness, indicating the time elapsed since it has not been visited by any agent [START_REF] Chevaleyre | Theoretical Analysis of the Multi-agent Patrolling Problem[END_REF]. The vector of idlenesses of all nodes at time t is noted I t (v) and the idleness of a node v, I t (v). At the beginning of a patrolling, agents are positioned on nodes and all idlenesses are set to 0. Finally, each time an agent arrives at a node v, it shall decide, among the edges connecting v, the next edge to travel.

A strategy of agent is an information processing method allowing each agent to take a decision each time it arrives at a node. Whatever the strategy considered, each agent intends actions based on its knowledge regarding idlenesses of nodes. Indeed, agents make idleness estimates that can be produced assuming different hypotheses:

individual idleness: each agent considers only its own visits to update its estimated node idleness. It corresponds to the case where communication between agents is not possible. In the case of a mission with only one agent, individual idleness corresponds to real idleness, also called global idleness. shared idleness: all agents consider visits of all agents to reset estimated node idleness. In the case of perfect instantaneous communication between agents, or a mission with only one agent, shared idleness corresponds to real idleness, also called global idleness.

Among the wide family of strategies [START_REF] Almeida | Combining idleness and distance to design heuristic agents for the patrolling task[END_REF], two regarded as representative strategies are relevant here: Conscientious Reactive (CR) and Heuristic Pathfinder Cognitive Coordinated (HPCC). CR selects the next node to visit as the one with the highest individual idleness in its neighbourhood. HPCC is based on a perfect communication between agents: shared idlenesses are estimated by a coordinator on the basis of all paths of agents. Two methods are used: the first one called Heuristic selects the next node to visit, and the second one called Pathfinder chooses the path to go there [START_REF] Almeida | Combining idleness and distance to design heuristic agents for the patrolling task[END_REF] [START_REF] Othmani-Guibourg | Multi-agent patrolling in dynamic environments[END_REF].

Evaluation criteria relevant to establish aggregation measures based on interval between visits for a node are the Mean Interval (MI) and the Quadratic Mean Interval (QMI). In order to better evaluate the contribution of each agent when the population size varies, these criteria are normalised by multiplying values by the number of agents.

Artificial neural networks

ANNs are a special kind of machine learning models. Among the large variety of ANN, single layer and multi-layer perceptrons [START_REF] Werbos | Beyond Regression: New Tools for Prediction and Analysis in the Behavioral Sciences[END_REF] [6] are composed with one or several stacked layers of neurons, each one corresponding to a function that maps the outputs of the previous layer with the output of the current layer. Several kinds of functions are used. For example, the identity, the logistic sigmoid and the linear rectifier (ReLU), f (z) = max(0, z). Each neuron computes a weighted and biased sum of the previous layer's outputs, which is finally passed through its function, making up thereby the neuron's output. When the functions are non-linear, they provide a basis for developing an approximation of the function to be learned.

Networks are generally optimised using gradient-descent-based methods by minimising a cost function representing the difference between the output of the network and its desired value [8] [6]. A quite common cost is the Mean Squared Error (MSE).

Strategies based on idleness estimation

This section presents the design of the three strategies using an idleness estimation called Heuristic Pathfinder Mean Predictor (HPMP), Heuristic Pathfinder Linear Predictor (HPLP) and Heuristic Pathfinder ReLU Predictor (HPRP).

Formal definition

Estimator-based strategies The three strategies use an estimator based on a trained statistical model noted m(., .): the decision-making process is carried out first by computing an estimate of the global idleness from the trained model, then by making the decision regarding the next node to visit with respect to this estimate. In our context, a temporal series representing the successive idlenesses each time an agent stands upon a node, the latter will be called an idleness flow.

Let I a t and Ît being the vectors of individual idlenesses of the agent a and the corresponding estimated global idlenesses, respectively, at the time t. Then, given that ∀t ∈ T, ∀a ∈ A, ∀I a t ∈ R N , Ît is defined such as:

Ît = min( max(m(I a t , θ), 0), I a t ) (1)
where θ is the set of the statistical model's parameters, and min and max are componentwise functions ensuring that whatever the output of the model, the estimation of global idleness is positive and lower than the individual idleness. For any agent, such an estimator will output estimates of the global idlenesses, called estimated idlenesses, from the current individual idlenesses fed as input. All the agents embed the same estimator, i.e. the same trained model. This strategy can be thought of as a reactive strategy using an artefact for estimating missing information regarding the area to patrol, and taking into account the idleness of nodes and thereby implicitly the agents' positions.

For a given scenario, the model learns to predict the global idleness vector corresponding to a current agent's individual idleness vector. Then, agents applies to the estimated global idlenesses the two methods described in the Section 2, namely the Heuristic method to select the next node to visit and the Pathfinder method to choose the path to go there.

However, the relation between individual idlenesses and global idlenesses may not be a function. The following theorem presents conditions under which this relation is not a function.

Theorem 1. Let G = (V, E) be a graph, let A be a society of agents and consider two runs of a given strategy, arbitrarily named first and second run. If:

in initial state, a node u ∈ V is occupied with an agent a 1 ∈ A for the first run and with an agent a 2 ∈ A for the second run, a 1 and a 2 may have the same agent identifier, and a next node v ∈ V is selected by the strategy for the agent a 1 in the first run and for the agent a 2 in the second run and for the first run, it exist w ∈ V, w = v that is occupied by an agent a 3 ∈ A, a 3 = a 1 , a 3 = a 2 or which has already been reached by a 3 when, at time t, a 1 arrives at v and for the second run, no agent has reached w, when at time t, a 2 arrives at v, then the relation between, the individual idleness I a t and the global idleness I t is not a function.

Proof. At time t, the individual idlenesses for a 1 in the first run and for a 2 in the second run are equal: both have I a t (j) = t, ∀j = v and I a t (v) = 0. For the first run I t (w), the global idleness of w, is equal to 0 if a 3 occupies it, or equal to t -τ < t, where τ > 0 is the travel time of agent a 3 from its initial position to w, otherwise. For the second run I t (w) = t. Thus, to the same individual idleness corresponds two different values of the global idleness. Hence, the relation between the individual idleness I a t and the global idleness I t is not a function.

Models As indicated by the Eq. 1, for all the models studied here, the input and output both of dimension N , stand for the vector of individual idlenesses and the vector of estimated idlenesses, respectively.

First, the mean model consists of a model which estimates for each node, the global idleness as being the average of all global idlenesses of this node over all the global idleness flows. With such a model noted M ean, an agent a ∈ A carries out the estimation of the global idlenesses at t ∈ T as following:

∃θ = {B ∈ M N ×1 (R)} : m(I a t , θ) = M ean(I a t , θ) = B (2)
When such a model is used as global idleness estimator, the corresponding strategy is called Heuristic Pathfinder Mean Predictor (HPMP).

When the estimator corresponds to a linear model noted Lin or Linear, the strategy is termed Heuristic Pathfinder Linear Predictor HPLP. With such a model an agent a ∈ A carries out the estimation of the global idlenesses at t ∈ T as follows:

∃θ = {W ∈ M N (R)} : m(I a t , θ) = Lin(I a t , θ) = W • I a t ( 3
)
with W being the model's weight matrix. Training such a model corresponds to figure out the W minimising a certain distance between m(I a t , θ) and I t . Finally, an MLP composed with H hidden ReLU layers as described in the Section 2, is termed ReLU model, while its corresponding strategy is called Heuristic Pathfinder ReLU Predictor (HPLP). With such a model noted M LP H ReLU , an agent a ∈ A carries out the estimation of the global idlenesses at t ∈ T as following: Three different graphs were selected to evaluate the strategies: the maps Islands, Grid and A, as shown in the Figure 1 [START_REF] Othmani-Guibourg | Multi-agent patrolling in dynamic environments[END_REF]. For each map we tested the strategies CR, HPCC and the idleness-predictor-based strategies were trained from HPCC's simulation and tested. The tests were performed over population sizes of 5, 10, 15 and 25 agents and for each size we selected 100 random starts, also called runs. For each start, each strategy was tested over 3000 periods and, in average, an agent visits 650 nodes during one execution of 3000 periods. In doing so, the sequences used to train the models have approximately a length of 650 idleness vectors. For each scenario we trained 8 statistical models by minimising the MSE: a mean-based model, a linear model, three Multi-layer Perceptron (MLP) with sigmoid units and three different artificial neural networks with rectifier linear units (ReLU): an architecture with only one ReLU layer simply termed ReLU, another one with one hidden ReLU layer and the output layer being also a ReLU layer, termed ReLU Output (ReLUO), and finally an MLP with ReLU activation termed ReLU MLP. The data base was divided into a training base and a validation base with

∃θ = {W h , W out ∈M N (R) : h ∈ [|1, H|]} : m(I a t , θ) = M LP H ReLU (I a t , θ) = W out • ReLU ( W H • ReLU ( W H-1 • ReLU (. . . W 2 • ReLU ( W 1 • I a t ) ) ) . . .) ) ) (

Experiments and results

Scenarios and training

Performance results

To evaluate their performances, the studied strategies were tested and compared with CR and HPCC using normalised MI and QMI as evaluation criteria.

Fig. 2 show all the results for the topologies Islands, A and Grid for the normalised MI. Not surprisingly, HPCC always outperformed all the others strategies on all the maps and for all the population sizes of agents. First, except for the map Grid, HPLP overwhelmingly outperforms the reactive strategy CR, while on the map Grid it is slightly better than CR. Unlike the others maps, this little difference can be explained in considering that, Grid being a topology where the nodes are uniformly distributed, the strategy CR is well adapted. However, HPLP remains better than CR on this map, except for the population size of 15 agents where they are approximately equal. Then, except for the map Islands, HPLP has always better performances over this criterion than HPMP. On that map, the performances of HPLP are approximately equal to the ones of HPMP for 5 and 10 agents. However, for 15 and 25 agents the former is worse than the latter. Results on Islands for 15 and 25 agents, on A for 5 and 15 agents and Grid for all the population sizes, seem to show that agents do not benefit from the presence of each other. Indeed, unlike HPCC which has a decreasing or stable normalised MI, it increases for HPLP. For the map Islands, HPRP is the best idleness predictor strategy over the normalised MI, except for 10 agents where it is approximately equal to HPLP and HPMP. On that map, it is also slightly better than HPCC for 5 agents. For the map A, HPRP is by far the best strategy. In average it is better than HPCC of 74 periods. Finally, as previously stated while comparing HPLP and HPRP to it, HPMP is most of the time the worst idleness predictor strategy and for the map Grid it is even worse than CR of 39 periods in average. The two models trained and used as a part of the two strategies HPLP and HPRP are thereby better than HPMP.

The Fig. 3 shows the results for the normalised QMI. Unlike the results of MI, on the three topologies the idleness predictor-based strategies are worst than HPCC and CR, the coordinated and the decentralised ones, respectively. For Islands, the HPRP is the worst strategy, while for the maps A and Grid, it is HPMP, which is the worst. For the latter maps, HPLP is incomparably better than the other two idleness predictor strategies with a difference in average with HPRP of 574 for A and 1161 periods for Grid. However, HPLP has worse performances than CR of 1522, 310 and 211 periods on the Islands, A and Grid, respectively. Finally the idleness predictor-based strategies show bad performances over the criterion QMI. QMI as quadratic mean takes better into account the difference of time intervals between the nodes and thereby measures the tendency of nodes to be equitably visited. A node with a long interval will have a little impact on MI, while it will have a large one on the QMI due to its quadratic growth.

Thereupon, the results show that good performances in average i.e. over MI are balanced by the ones of QMI. These results show the tendency of idleness predictor agents to visit a particular inferred set of nodes at the expense of the other ones.

Conclusion and perspectives

We proposed and evaluated new strategies for the MAP. Those strategies are based on learning the relation between individual and global idlenesses. The assessment of selected strategies based on estimations of global idlenesses using learned model indicates good results in terms of MI, but also unsatisfactory results in terms of QMI.

Theorem 1 indicates that there may be no significant expectation for approximation improvement. Data analysis methods should be applied in order to check the presence of conditions implying that the relation between individual and global idlenesses is not a function. Other future research will aim at modifying the strategies in order to improve their performance in terms of QMI. One track is to consider some randomisation process when exploiting the estimation of global idlenesses by the model. For example, defining a probability distribution with Ît as a mean and the strategy could sample in this distribution, idlenesses' estimate. The global idleness approximation problem could be further investigated with other structures of ANN and better learning algorithms. Finally, it should be noted that there is a large set of possibilities for using ANN to try to learn some information from centralised strategies that is useful for decentralised strategies. For example, nodes sequences of centralised strategies could be learned using Long Short-Term Memory ANN architectures and directly used in a decentralised strategy. This kind of approach could be compared to the approach proposed here.

  [START_REF] Guo | 9 Collaborative Robots for Infrastructure Security Applications[END_REF] with ReLU being the element-wise ReLU activation, and ∀h ∈ [|0, H|], W h the weight matrix of the layer h.

Fig. 1 :

 1 Fig. 1: Graphs used during assessment.

Fig. 2 :

 2 Fig. 2: Normalised MI of the evaluated strategies in ordinate for the three maps w.r.t. the population sizes of agents in abscissa.

Fig. 3 :

 3 Fig. 3: Normalised QMI of the evaluated strategies in ordinate for the three maps w.r.t. the population sizes of agents in abscissa.


Bruno Grenet

Ilya Volkovich

One (more) line on the most Ancient Algorithm in History

We give a new simple and short ("one-line") analysis for the runtime of the well-known Euclidean Algorithm. While very short simple, the obtained upper bound in near-optimal.

Introduction

Perhaps the most ancient algorithm recorded in history is the Euclidean Algorithm. Published in 300 BC, the algorithm computes the greatest common divisor (gcd) of two integer numbers. (See Algorithm 1 for the description of the algorithm.) The following is a quote from [START_REF] Knuth | The art of computer programming[END_REF]: "[The Euclidean algorithm] is the granddaddy of all algorithms, because it is the oldest nontrivial algorithm that has survived to the present day."

The Euclidean Algorithm serves as a subroutine for many tasks such as: finding Bézout's coefficients, computing multiplicative inverses and the RSA algorithm, Chinese Remainder Theorem and others. For more details on these and other applications, see [START_REF] Knuth | The art of computer programming, Volume I: Fundamental Algorithms[END_REF][START_REF] Knuth | The art of computer programming[END_REF]. Yet, despite its simplicity, the actual runtime complexity cannot be easily inferred from its description due to the recursive nature of the algorithm.

As the operations inside each iteration are "elementary", the actual runtime complexity is dominated by the number of (recursive) iterations of the algorithm. Indeed, various upper bounds for this number were given [START_REF] Lamé | Note sur la limite du nombre des divisions dans la recherche du plus grand commun diviseur entre deux nombres entiers[END_REF][START_REF] Sedgewick | Algorithms[END_REF][START_REF] Knuth | The art of computer programming, Volume I: Fundamental Algorithms[END_REF][START_REF] Knuth | The art of computer programming[END_REF][START_REF] Cormen | Introduction to Algorithms[END_REF]. The first one, presented in [START_REF] Lamé | Note sur la limite du nombre des divisions dans la recherche du plus grand commun diviseur entre deux nombres entiers[END_REF], ties the number of iteration with the so-called Fibonacci numbers. Another kind of analysis shows that the larger argument shrinks by a factor of at least 2 every two iterations (see e.g. [START_REF] Knuth | The art of computer programming, Volume I: Fundamental Algorithms[END_REF][START_REF] Cormen | Introduction to Algorithms[END_REF]). Yet, the proof of this claim requires a somewhat non-trivial case-analysis. Note that in the case of univariate polynomials, the analysis is much simpler: the degree of the highest-degree polynomial decreases at each step. A similar easy argument was lacking in the integer case since the same phenomenon does not occur: it does not hold that the bit-size of the largest integer decreases at each step.

Our new analysis provides a clean, "one-line" upper bound which turns out to be optimal. The main idea is based on the so-called Potential Method (see e.g. [START_REF] Cormen | Introduction to Algorithms[END_REF]). More specifically, we define appropriate potential functions and show that these functions lose a constant fraction of their mass with every recursive iteration of the algorithm. At the same time, the functions are bounded away from zero. Formally, we give a simple proof to the following theorems: Theorem 1. For any x, y ∈ N, the Euclidean Algorithm performs at most log 1.5 (x+y)+1 iterations.

Using a slightly more sophisticated potential function yields an even tighter result.

Theorem 2. For any x, y ∈ N, the Euclidean Algorithm performs at most log φ (φx + y) iterations.

Here φ = 1+ √ 5 2 ≅ 1.61803398875 denotes the golden ratio.

Our analysis should be contrasted with the analysis of [START_REF] Lamé | Note sur la limite du nombre des divisions dans la recherche du plus grand commun diviseur entre deux nombres entiers[END_REF] which states that if the Euclidean algorithm requires m iterations, then it must be the case that x ≥ F m+2 and y ≥ F m+1 , where F i -s are the Fibonacci numbers: 1, 1, 2, 3, 5, . . .. And, indeed, selecting x and y as consecutive Fibonacci numbers demonstrates the tightness of this analysis. Now for a general pair x and y we have:

φx + y ≥ φF m+2 + F m+1 ≥ F m+3 φ m+1 .
Therefore m log φ (φx + y), implying the optimality of our analysis. * LIRMM, Université de Montpellier, CNRS, Montpellier, France. Email: bruno.grenet@lirmm.fr.

† Department of EECS, CSE Division, University of Michigan, Ann Arbor, MI. Email: ilyavol@umich.edu.

Improved Analysis

In this section we show that using a slightly more sophisticated potential function yields an even tighter result. As before, for the sake of analysis, let us fix x > y and let m denote the number of the (recursive) iterations of algorithm. In addition, let φ = 1+ √ 5 2 ≅ 1.61803398875 denote the golden ratio.

Definition 2.7. For 1 ≤ i ≤ m, we define s i

∆ = x i + 1 φ • y i . As before, for 1 ≤ i ≤ m: s i ≥ 1. The following lemma mirrors Lemma 2.5. Lemma 2.8. For 1 ≤ i ≤ m: s i ≤ 1 φ • s i-1 . Proof. Recall that s i-1 = x i-1 + 1 φ • y i-1 .
Let us divide x i-1 by y i-1 with a reminder. In particular,

x i-1 = q i-1 • y i-1 + r i-1 , where 0 ≤ r i-1 ≤ y i-1 -1.
Observe that s i = y i-1 + 1 φ • r i-1 and in addition, q i-1 ≥ 1, as x i-1 > y i-1 . We obtain the following:

s i-1 = q i-1 + 1 φ y i-1 +r i-1 ≥ 1 + 1 φ y i-1 +r i-1 = φ•y i-1 +r i-1 = φ y i-1 + 1 φ • r i-1 = φ•s i .
Recall that 1 + 1 φ = φ. The proof of Theorem 2 follows by repeating the previous argument. Actually, a slightly better bound can be obtained: Since the algorithm stops as soon as y m ≤ 1, y m-1 ≥ 2, and moreover x m-1 > y m-1 . Thus s m ≥ 3 + 2 φ = φ 3 and since s m ≤ s 1 φ -m+2 , m ≤ log φ (x + 1 φ y) -1. This bound is tight, attained for consecutive Fibonacci numbers.

Conclusion & Open Questions

In this short note we add one (more) line of analysis to the well-known Euclidean Algorithm. The new analysis does not require any background and can be taught even in an introductory-level undergraduate class. It would be nice to see if we could replace other kind of analyses that rely on Fibonacci numbers by a potential argument. One such example is the analysis of the height of an AVL-Tree [START_REF] Adelson-Velsky | An algorithm for the organization of information[END_REF][START_REF] Sedgewick | Algorithms[END_REF].

The Algorithm and the Runtime Analysis

In this section we review the algorithm and give the new, simplified runtime analysis. The description of the algorithm is given below in Algorithm 1. The algorithm is given non-negative integers, x and y as an input. We assume w.l.o.g that x > y.

Input: Two non-negative integers, x > y ∈ N Output: gcd(x, y)

1 if y = 0 then return x; 2 if y = 1 then return 1; 3 return gcd(y, x mod y)

Algorithm 1: Euclidean Algorithm

For the sake of analysis, let us fix x > y and let m denote the number of the (recursive) iterations of algorithm.

Definition 2.1. For 1 ≤ i ≤ m, let x i and y i denote the values of x and y in the i-th iteration of the algorithm, respectively. In particular,

Proof. Follows from the fact that y i = x i-1 mod y i-1 = x i-1 mod x i , which attains values between 0 and x i -1.

Next, we will show that s i is a potential function for this algorithm. In particular, the function loses a constant fraction of its mass in every step, yet it is bounded away from zero. The following is immediate from the definition, given the above observation.

Corollary 2.4. For 1 ≤ i ≤ m: s i ≥ 1.

The following lemma is the heart of the argument.

Proof. Recall that s i-1 = x i-1 + y i-1 . Let us divide x i-1 by y i-1 with a reminder. In particular,

Observe that s i = y i-1 + r i-1 and in addition, q i-1 ≥ 1, as x i-1 > y i-1 . We obtain the following:

The second inequality holds since r i-1 < y i-1 .

By applying the lemma repeatedly, we obtain:

Theorem 1 follows immediately from the above.

Proof of Theorem 1. By Corollaries 2.4 and 2.6:

Therefore, m ≤ log 1.5 (x + y) + 1.


Daniel Rhodes

Sang Hoon Chae

Rebeca Ribeiro-Palau

James Hone

Disorder in van der Waals heterostructures of 2D materials

Realizing the full potential of any materials system requires understanding and controlling sources of disorder that can obscure intrinsic properties and hinder device performance.

Here we examine progress in understanding and reducing both intrinsic and extrinsic disorder in two-dimensional (2D) graphene and transition metal dichalcogenides (TMDs) incorporated into layered van der Waals heterostructures. In the case of mechanically exfoliated graphene, the dominant disorder is extrinsic and can be reduced by encapsulation within crystalline hexagonal boron nitride (hBN). Continuing improvement in encapsulation and device fabrication techniques has led to dramatic performance improvement and enabled the study of a wide range of novel phenomena. In the case of TMDs, both extrinsic and intrinsic disorder are relevant. The electronic and optical performance of semiconducting TMDs improves with the reduction of external disorder, but commonly used materials show high defect density which must be reduced to enable similar progress as demonstrated in graphene. These atomic defects can also be harnessed to provide useful electronic, optical,

chemical, and magnetic properties.

Atomically thin 2D materials have generated wide interest for electronic and optoelectronic applications, and as hosts for novel physical phenomena such as topologically protected states, valley effects, moiré physics, and exciton superfluidity. Toward these goals, minimizing disorder is crucial in order to reveal intrinsic properties and improve device performance and repeatability.

A particular challenge is that 2D systems are by definition entirely surface, and therefore highly susceptible to external sources of disorder. Thus the challenge of controlling disorder in 2D materials includes both intrinsic disorder arising from crystalline imperfections, and extrinsic disorder arising from the environment. Below we review progress toward this goal in two specific systems: mechanically exfoliated graphene, in which intrinsic disorder is minimal and dramatic improvement is seen with encapsulation by hBN in a van der Waals heterostructure; and semiconducting TMDs for the intrinsic crystal quality must also be addressed in order to achieve similar progress.

Figure 1 summarizes the main sources of intrinsic and extrinsic disorder in 2D materials.

Sources of extrinsic disorder, depicted in Fig. 1c, include: strain, adsorbates, surface roughness, charged impurities in the substrate, and oxidation. These sources of disorder can affect multiple properties by introducing electrostatic potential variation, scattering of charge carriers [START_REF] Ma | Charge Scattering and Mobility in Atomically Thin Semiconductors[END_REF] , locally modifying the electronic band structure [START_REF] Sachs | Midgap states and band gap modification in defective graphene/h-BN heterostructures[END_REF] , and acting as recombination centers for excitons [START_REF] Wang | Ultrafast Dynamics of Defect-Assisted ElectronHole Recombination in Monolayer MoS 2[END_REF] .

Sources of intrinsic disorder (depicted in Fig. 1a for a hexagonal lattice but applicable to all 2D crystals), include atomic defects such as vacancies, anti-sites, substitutions, edges, and grain boundaries. Such grain boundaries are common in large area, polycrystalline monolayer films (see Fig. 1b, lower panel), and can dominate the materials' properties. In graphene, transport across a (1) has collapsed [START_REF] Favron | Photooxidation and quantum confinement effects in exfoliated black phosphorus[END_REF] . Figure reproduced with permission: adapted from ref. [START_REF] Huang | Grains and grain boundaries in single-layer graphene atomic patchwork quilts[END_REF] , Nature Publishing Group (b); ref. [START_REF] Shin | Indirect Bandgap Puddles in Monolayer MoS 2 by Substrate-Induced Local Strain[END_REF] , Wiley (d); ref. [START_REF] Favron | Photooxidation and quantum confinement effects in exfoliated black phosphorus[END_REF] , Nature Publishing Group (e).

single grain boundary can decrease conductivity by a factor of three [START_REF] Tsen | Tailoring Electrical Transport Across Grain Boundaries in Polycrystalline Graphene[END_REF] . In TMDs, grain boundaries can produce local strain that can either reduce or increase the band gap depending on the tilt angle between each grain [START_REF] Wang | Bandgap broadening at grain boundaries in single-layer MoS 2[END_REF] . The role of defects and grain boundaries in 2D materials has been reviewed in detail [START_REF] Hus | Spatially-resolved studies on the role of defects and boundaries in electronic behavior of 2d materials[END_REF] . In order to understand how electronic properties depend on other types of disorder, it is important to investigate single crystalline material. Below, we will continue to discuss intrinsic disorder in the context of single crystal monolayers produced through mechanical exfoliation.

Mechanically exfoliated graphene provides a case study for a material in which virtually most of the disorder is extrinsic. Figure 1b shows an annular dark-field scanning transmission electron microscopy (ADF-STEM) image of the graphene lattice showing the absence of any vacancies or Stone-Wales defects [START_REF] Huang | Grains and grain boundaries in single-layer graphene atomic patchwork quilts[END_REF] . In fact, extensive investigation by transmission electron microscopy (TEM) and scanning tunnelling microscopy (STM) have shown that graphene flakes obtained by mechanical exfoliation from bulk graphite crystals are remarkably free of defects [START_REF] Huang | Grains and grain boundaries in single-layer graphene atomic patchwork quilts[END_REF] . In spite of the high quality of mechanically exfoliated graphene, early devices fabricated on SiO 2 substrates showed evidence of substantial disorder in electrical transport studies [START_REF] Novoselov | Two-dimensional gas of massless Dirac fermions in graphene[END_REF] . This disorder can be directly visualized by mapping the local potential variation using a scanning single electron transistor [START_REF] Martin | Observation of electronhole puddles in graphene using a scanning singleelectron transistor[END_REF] or by scanning tunnelling spectroscopy (STS) [START_REF] Xue | Scanning tunnelling microscopy and spectroscopy of ultra-flat graphene on hexagonal boron nitride[END_REF][START_REF] Decker | Local Electronic Properties of Graphene on a BN Substrate via Scanning Tunneling Microscopy[END_REF] . For charge-neutral graphene on SiO 2 , chemical potential variations of ∼100 meV (corresponding to carrier density variations of δn ∼ 10 12 cm -2 ) on the length scale of tens of nanometers can be readily observed. These electron-hole 'puddles' have been shown to arise from trapped charge impurities in the SiO 2 substrate 14 and surface adsorbates.

Theoretical and experimental analysis indicates that scattering from this electrical potential disorder provides the dominant scattering mechanism for charge carriers in graphene at low tempera-ture, providing the connection between charge disorder and electronic mobility [START_REF] Chen | Charged-impurity scattering in graphene[END_REF][START_REF] Adam | A self-consistent theory for graphene transport[END_REF] . Roughness of SiO 2 can introduce strain variation in the 2D system at similar length scales which cause charge fluctuations [START_REF] Xue | Scanning tunnelling microscopy and spectroscopy of ultra-flat graphene on hexagonal boron nitride[END_REF] . In TMDs, bending strain induced by surface roughness, which strongly modifies the local bandstructure, is likely the dominant source of disorder on SiO 2 substrates. Figure 1d shows a color map for the bandgap of monolayer MoS 2 on SiO 2 , characterized by scanning tunneling spectroscopy (STS) [START_REF] Shin | Indirect Bandgap Puddles in Monolayer MoS 2 by Substrate-Induced Local Strain[END_REF] . Significant modulations in the bandgap, 1.23-2.65 eV, are observed with ∼80% of the MoS 2 under sufficient enough strain to cause a direct-to-indirect bandgap transition.

Oxidation is another source of extrinsic disorder in many 2D materials that, unlike graphene, are air-sensitive [START_REF] Geim | Van der Waals heterostructures[END_REF] . A prime example is semiconducting black phosphorus (BP): small bumps of adsorbed water are seen by atomic force microscopy (AFM) immediately after exfoliation in ambient conditions (top panel of Fig. 1e 3 ), and TEM imaging shows that few-layer BP completely collapses after 20 s of exposure to air (bottom panel of Fig. 1e). Recent work has shown that a photoassisted oxidation reaction by oxygen dissolved in adsorbed water is the dominant mechanism for degradation [START_REF] Favron | Photooxidation and quantum confinement effects in exfoliated black phosphorus[END_REF] .

A straightforward and effective way to reduce disorder arising from the substrate is to suspend the 2D layer so that it is not in contact with the SiO 2 . Indeed, a study of suspended graphene [START_REF] Du | Approaching ballistic transport in suspended graphene[END_REF] has shown dramatically increased carrier mobility and greatly reduced charge inhomogeneity, reaching a residual carrier density δn ∼ 2 × 10 9 cm -2 . This lower disorder permits the observation of fractional quantum Hall (QH) states [START_REF] Bolotin | Observation of the fractional quantum Hall effect in graphene[END_REF][START_REF] Feldman | Unconventional Sequence of Fractional Quantum Hall States in Suspended Graphene[END_REF] , interaction-driven ground states [START_REF] Jr | Transport spectroscopy of symmetry-broken insulating states in bilayer graphene[END_REF] , and hydrodynamic phenomena [START_REF] Nam | Electronhole collision limited transport in charge-neutral bilayer graphene[END_REF] . However, suspended devices are limited in important ways: they are limited to low carrier density and weak gate capacitance, complex geometries are extremely challenging, and multilayer structures are impossible. Therefore, a major goal in the field is to create substratesupported devices with charge disorder (i.e. δn) approaching the cleanest suspended graphene. Toward this end, the most successful approach to date utilizes hexagonal boron nitride (hBN), a layered wide-bandgap (∼6 eV) insulator [START_REF] Taniguchi | Synthesis of high-purity boron nitride single crystals under high pressure by using BaBN solvent[END_REF] with a crystalline structure similar to graphene, as a replacement for conventional dielectrics such as SiO 2 . Below we examine the effects of hBN encapsulation on disorder and materials performance, first for graphene and then for TMDs.

Reducing extrinsic disorder in graphene-hBN heterostructures 60 nm × 60 nm STM topography (Fig. 2a) and charge density (Fig. 2b) maps of monolayer graphene on hBN and SiO 2 substrates show the clear advantage of hBN [START_REF] Decker | Local Electronic Properties of Graphene on a BN Substrate via Scanning Tunneling Microscopy[END_REF] . Roughness in the graphene/hBN charge density for the surface (2.3 × 10 10 cm -2 rms) is much lower than what is observed for graphene/SiO 2 (8.2 × 10 10 cm -2 rms). Both the roughness and charge density inhomogeneities are dramatically reduced on hBN substrates, making hBN an ideal candidate for supporting 2D materials [START_REF] Xue | Scanning tunnelling microscopy and spectroscopy of ultra-flat graphene on hexagonal boron nitride[END_REF][START_REF] Decker | Local Electronic Properties of Graphene on a BN Substrate via Scanning Tunneling Microscopy[END_REF][START_REF] Dean | Boron nitride substrates for high-quality graphene electronics[END_REF] .

Figure 2c depicts the evolution of graphene devices as continued improvements to fabrication techniques have reduced extrinsic disorder. These are divided into four approximate 'generations', beginning with device fabricated on SiO 2 10 (Generation 1). The first graphene-hBN devices [START_REF] Dean | Boron nitride substrates for high-quality graphene electronics[END_REF] (Generation 2) were assembled by using polymers to pick up graphene flakes and place them onto hBN. Although these devices showed lower disorder than those on SiO 2 12, 13 , they remained affected by polymer residue from the transfer and lithographic processing. In addition, this residue can impede assembly of multilayer structures by aggregating into 'bubbles' between layers. Improved performance was seen upon the removal of residue (by annealing in oxygen or [START_REF] Decker | Local Electronic Properties of Graphene on a BN Substrate via Scanning Tunneling Microscopy[END_REF] , ACS Publications (a,b); ref. [START_REF] Novoselov | Two-dimensional gas of massless Dirac fermions in graphene[END_REF] , Nature Publishing Group (e); ref. [START_REF] Dean | Boron nitride substrates for high-quality graphene electronics[END_REF] , Nature Publishing Group (f).

by 'sweeping' with an AFM tip [START_REF] Garcia | Effective Cleaning of Hexagonal Boron Nitride for Graphene Devices[END_REF] ) and in micron-scale hBN-encapsulated graphene [START_REF] Ponomarenko | Tunable metalinsulator transition in double-layer graphene heterostructures[END_REF] . Large-area hBN-encapsulated devices (Generation 3) are assembled by using the top layer in the heterostructure to 'pick up' subsequent layers, using only van der Waals forces, so that the interior surfaces are never in contact with the polymers [START_REF] Wang | One-Dimensional Electrical Contact to a Two-Dimensional Material[END_REF] (Generation 3). The graphene-hBN interfaces produced by this van der Waals assembly technique show no evidence of contamination in cross-sectional imaging by scanning transmission electron microscopy (STEM), and bubble-free areas tens of µm in size can be achieved; electrical contact to the encapsulated graphene is achieved by metalization of the graphene edge exposed after etching the assembled heterostructure [START_REF] Wang | One-Dimensional Electrical Contact to a Two-Dimensional Material[END_REF] . These structures reduce, but do not eliminate, the effects of charge disorder in the underlying SiO 2 substrate, and intervening metal gates used to eliminate these effects can introduce variations in the electrostatic potential as well [START_REF] Osvald | On barrier height inhomogeneities at polycrystalline metal-semiconductor contacts[END_REF] . The lowest disorder is instead achieved by utilizing graphite gates [START_REF] Zibrov | Tunable interacting composite fermion phases in a half-filled bilayergraphene Landau level[END_REF] (Generation 4). Local potential variation around metal contacts is also minimized by using graphite for electrical contacts, in essence creating devices entirely from van der Waals materials.

A simple yet accurate technique to estimate charge inhomogeneity in graphene devices is to measure the width of the peak in resistance as a function of gate-induced carrier density across the charge neutrality point (CNP) [START_REF] Adam | A self-consistent theory for graphene transport[END_REF][START_REF] Adam | Transport in suspended graphene[END_REF] . A more rigorous method is to directly measure carrier density by Hall effect, but the two approaches give very similar results [START_REF] Du | Fractional quantum Hall effect and insulating phase of Dirac electrons in graphene[END_REF] . The CNP resistance peaks for four generations of graphene devices are shown in Fig. 2d, showing a dramatic narrowing from δn ∼ 10 12 cm -2 (Gen 1) to 6 × 10 9 cm -2 (Gen 4)within a factor of three of the lowest reported for suspended graphene [START_REF] Feldman | Unconventional Sequence of Fractional Quantum Hall States in Suspended Graphene[END_REF] . Similar estimates of charge disorder are obtained by STM mapping [START_REF] Xue | Scanning tunnelling microscopy and spectroscopy of ultra-flat graphene on hexagonal boron nitride[END_REF][START_REF] Decker | Local Electronic Properties of Graphene on a BN Substrate via Scanning Tunneling Microscopy[END_REF] .

At this point the origin of the remaining disorder is unclear; small amounts of environmental impu-rities such as water or hydrocarbons may still be present at the graphene-hBN interface, and charge disorder may arise from oxygen or carbon impurities in the hBN, which occur at concentrations similar to the charge fluctuations seen in STM [START_REF] Taniguchi | Synthesis of high-purity boron nitride single crystals under high pressure by using BaBN solvent[END_REF] . Beyond measurement of disorder by CNP peak width, electron transport measurements can also be used to extract the carrier mobility, which increases as δn decreases. Gen 3 and 4 devices show ballistic conduction, with an electronic mean free path limited by device size (typically <20 µm) rather than impurity scattering, which provides a lower limit of ∼10 6 cm 2 V -1 s -1 (at n = 10 12 cm -2 ) on the intrinsic carrier mobility [START_REF] Wang | One-Dimensional Electrical Contact to a Two-Dimensional Material[END_REF][START_REF] Banszerus | Ballistic Transport Exceeding 28 µm in CVD Grown Graphene[END_REF] . A more accurate determination of the intrinsic mobility would require assembly of devices on the scale of 100 µm, which is currently difficult given the size of mechanically exfoliated hBN flakes.

The reduction of disorder in successive generations of graphene devices is better reflected in more sensitive measurements such as the quantum Hall effect (QHE). Gen 1 devices (Fig. 2e)

show four-fold degenerate integer states, with the sequence famously reflecting the Berry phase [START_REF] Novoselov | Two-dimensional gas of massless Dirac fermions in graphene[END_REF] .

Gen 2 devices (Fig. 2f), under the same experimental conditions, show breaking of the four-fold symmetry [START_REF] Dean | Boron nitride substrates for high-quality graphene electronics[END_REF] , reflecting the improved quality. In fully-encapsulated structures (Gen 3), at high magnetic fields [START_REF] Wang | Evidence for a fractional fractal quantum Hall effect in graphene superlattices[END_REF] further details of the fractional quantum Hall effect (FQHE) can be observed. In Gen 4 devices (Fig. 2g) well developed FQHE can be observed from magnetic fields as low as 6 T [START_REF] Chen | Competing Fractional Quantum Hall and Electron Solid Phases in Graphene[END_REF] .

In particular, Gen 4 devices that probe bulk insulating states capacitively [START_REF] Zibrov | Even-denominator fractional quantum hall states at an isospin transition in monolayer graphene[END_REF] or through transport measurements in a Corbino geometry [START_REF] Zeng | Ultra-high quality magnetotransport in graphene using the edge-free Corbino geometry[END_REF] show well developed integer QHE states emerging at magnetic fields as small as B ∼ 50 mT, symmetry-broken integer states at around B = 1 T, and easily resolvable FQHE states by B ∼ 5 T. When comparing to Gen 4 devices in the Hall bar geometry these results show that edges remain as a source of intrinsic disorder still to be overcome.

Quantum transport measurements allow a more direct benchmark of 4th-generation graphene devices to GaAs/AlGaAs quantum wells with mobilities exceeding 3 × 10 7 cm 2 V -1 s -1 , which represent the 'gold standard' for low-disorder 2D electron gas (2DEG) devices [START_REF] Schreiber | Onset of quantum criticality in the topological-to-nematic transition in a two-dimensional electron gas at filling factor ν = 5/2[END_REF] . In the low-field regime, Shubnikov-de Haas (SdH) oscillations indicate a quantum scattering time τ q of 0.2-0.3 ps in graphene, whereas GaAs 2DEGs show values above 20 ps [START_REF] Shi | Microwave photoresistance in an ultra-high-quality GaAs quantum well[END_REF] . Likewise, the Landau level broadening measured by measuring the activation gap state as a function of magnetic field yields Γ ∼ 10 K, for edge [START_REF] Chen | Competing Fractional Quantum Hall and Electron Solid Phases in Graphene[END_REF] and edgeless [START_REF] Zeng | Ultra-high quality magnetotransport in graphene using the edge-free Corbino geometry[END_REF] graphene devices and 0.1-1 K for GaAs [START_REF] Pan | Fractional quantum Hall effect at Landau level filling ν = 4/11[END_REF] . Thus the cleanest graphene devices still show roughly two orders of magnitude higher bulk disorder than GaAs.

However, graphene stringer interactions strength and the ability to tune its carrier density enables the investigation and control of QH states. As a result, nearly every QH state originally observed in GaAs has now been seen in graphene (including recently-reported ordered electron solids [START_REF] Chen | Competing Fractional Quantum Hall and Electron Solid Phases in Graphene[END_REF] ), with the notable exception of the 'stripe' phases [START_REF] Koulakov | Charge Density Wave in Two-Dimensional Electron Liquid in Weak Magnetic Field[END_REF] . Beyond QH physics, new regimes such as hydrodynamic transport are accessible in low-disorder graphene due to strong carrier-carrier interactions [START_REF] Nam | Electronhole collision limited transport in charge-neutral bilayer graphene[END_REF] .

Moreover, graphene-hBN heterostructures offer greater flexibility in architecture versus conventional heterostructures. For instance, graphene layers can be placed extremely close to external patterned gates allowing strong bandstructure modification [START_REF] Forsythe | Band structure engineering of 2d materials using patterned dielectric superlattices[END_REF] , or to each other to study interaction effects in the strong coupling limit [START_REF] Li | Negative Coulomb Drag in Double Bilayer Graphene[END_REF] . Likewise, the weak interlayer van der Waals interaction allows for the emergence of moiré superlattices that give rise to 'Hofstader Butterfly' physics in graphene-hBN devices [START_REF] Dean | Hofstadters butterfly and the fractal quantum Hall effect in moir superlattices[END_REF][START_REF] Yu | Hierarchy of Hofstadter states and replica quantum Hall ferromagnetism in graphene superlattices[END_REF] , and a strongly modified bandstructure in twisted bilayer graphene that can host both insulating and superconducting phases [START_REF] Cao | Unconventional superconductivity in magic-angle graphene superlattices[END_REF] . Therefore graphene-hBN heterostructures can be seen as a complementary materials system with quality still below that of GaAs, but with the benefit of providing access to a wide range of new physical phenomena.

Extrinsic disorder in semiconducting transition metal dichalcogenides

TMDs are compounds of the family MX 2 , where M is a transition metal (Nb, Mo, W, etc.) and X is a chalcogen (S, Se, Te). For this discussion, we will focus on monolayer semiconducting Mo-and W-based TMDs with a trigonal prismatic, 2H, P6 3 /mmc structure, akin to the honeycomb structure of graphene [START_REF] Wang | Electronics and optoelectronics of two-dimensional transition metal dichalcogenides[END_REF] . As with graphene, SiO 2 substrates introduce substantial extrinsic disorder to monolayer TMDs. Two main approaches have been utilized to reduce the effects of this disorder: i) Deposition of high-κ dielectric (i.e. HfO 2 or Al 2 O 3 ) [START_REF] Yu | Realization of Room-Temperature Phonon-Limited Carrier Transport in Monolayer MoS 2 by Dielectric and Carrier Screening[END_REF] , which can both screen charged impurities and reduce charged impurity density by surface passivation; and ii) encapsulation within hBN [START_REF] Cui | Multi-terminal transport measurements of MoS 2 using a van der Waals heterostructure device platform[END_REF] , which moves the conducting channel away from the SiO 2 surface. Charge disorder from the SiO 2 surface can also be screened by placing a conducting backgate under the hBN [START_REF] Pisoni | Interactions and magnetotransport through spin-valley coupled Landau levels in monolayer MoS 2[END_REF] . Reducing this extrinsic disorder improves electronic transport properties, as seen in measurements of carrier mobility and quantum transport measurements, as well as optical properties, as seen in photoluminescence spectra.

The electronic mobility of monolayer TMDs reflects the separate contributions of temperaturedependent scattering from phonons (both within the material and in the substrate) and static scattering from charge disorderboth are modulated by the dielectric environment and carrier density.

Figure 3a shows theoretical calculations of the mobility for monolayer MoS 2 at fixed electron density n = 1 × 10 13 cm -2 on SiO 2 with HfO 2 encapsulation 4 . At low-temperature, the mobility is entirely dominated by charged impurity scattering, varying inversely with impurity density N i . At higher temperatures, scattering from optical phonons dominates the electronic transport behavior Superconductivity is only observed in encapsulated samples. f, PL spectrum at 4 K for the monolayer TMDs on SiO 2 , top, versus encapsulated by hBN, bottom [START_REF] Cadiz | Excitonic Linewidth Approaching the Homogeneous Limit in MoS 2 -Based van der Waals Heterostructures[END_REF] . Adapted from ref. [START_REF] Ma | Charge Scattering and Mobility in Atomically Thin Semiconductors[END_REF] , APS (a); ref. [START_REF] Pisoni | Interactions and magnetotransport through spin-valley coupled Landau levels in monolayer MoS 2[END_REF] , APS (c); ref. [START_REF] Fallahazad | Shubnikov-de Haas Oscillations of High-Mobility Holes in Monolayer and Bilayer WSe 2 : Landau Level Degeneracy, Effective Mass, and Negative Compressibility[END_REF] , APS (d); ref. [START_REF] Cadiz | Excitonic Linewidth Approaching the Homogeneous Limit in MoS 2 -Based van der Waals Heterostructures[END_REF] , APS (f). Figure 3b shows the measured temperature-dependent mobility of monolayer MoS 2 in three different environments: on SiO 2 without 46 (red) and with [START_REF] Radisavljevic | Mobility engineering and a metalinsulator transition in monolayer MoS 2[END_REF] (blue) HfO 2 encapsulation, and encapsulated with hBN 48 (black). On SiO 2 , the room-T mobility of MoS 2 ∼10 cm 2 V -1 s -1 , well below the predicted phonon-limited value shown above. Such devices also show a large hysteresis with gate voltage, indicating the presence of mobile charges at the surface [START_REF] Late | Hysteresis in Single-Layer MoS 2 Field Effect Transistors[END_REF] . HfO 2 encapulation reduces this hysteresisconsistent with passivation of mobile chargesand increases mobility [START_REF] Yu | Realization of Room-Temperature Phonon-Limited Carrier Transport in Monolayer MoS 2 by Dielectric and Carrier Screening[END_REF] , as does encapsulation with hBN [START_REF] Cui | Multi-terminal transport measurements of MoS 2 using a van der Waals heterostructure device platform[END_REF] . In both HfO 2 -and hBN-encapsulated devices, a power-law temperature dependence is observed upon cooling, consistent with theory (Fig. 3a), indicating that the mobility is in the intrinsic phonon-limited regime. We note that hBN-encapsulation (vs.

HfO 2 -encapsulation) provides somewhat larger room-T mobility due to its higher optical phonon energy. [START_REF] Ma | Charge Scattering and Mobility in Atomically Thin Semiconductors[END_REF] At low-T, where phonons are inactive, mobility in SiO 2 -supported devices is limited to ∼170 cm 2 V -1 s -1 . Moreover, at low carrier density such devices show a crossover to a low-T insulating behavior, indicative of disorder-induced localization [START_REF] Radisavljevic | Mobility engineering and a metalinsulator transition in monolayer MoS 2[END_REF] . HfO 2 encapsulation eliminates the low-T insulating behavior seen at low density, but does not improve the low-T mobility at high carrier density. In contrast, encapsulation within hBN improves the low-T mobility by roughly one order of magnitude in monolayer MoS 2 (see Fig. 3b). Following the theory of Fig. 3a, this increased mobility corresponds to a reduced impurity density of N i ∼ 1 × 10 12 cm -2 . We note, however, that the reduction in impurity density with hBN encapsulation is still far less than what has been seen

for graphene, indicating that either: (1) interfaces in hBN-TMD heterostructures are not as clean as for graphene; or (2) TMD crystals possess substantial intrinsic disorder that dominates low-T mobility when extrinsic disorder is reduced. TEM imaging does not find interfacial impurities in MoS 2 -hBN interfaces [START_REF] Cui | Multi-terminal transport measurements of MoS 2 using a van der Waals heterostructure device platform[END_REF] , arguing against the first scenario. This observation motivates careful examination of intrinsic material quality as discussed below.

The improved performance of hBN-encapsulated MoS 2 enables quantum transport measurements in high magnetic fields. In these measurements, the onset of SdH resistance oscillations decreases as the inverse carrier mobility (quantum scattering time). The first such studies (with µ ∼ 1,000 cm 2 V -1 s -1 ) showed the onset of quantum oscillations near 10 T [START_REF] Cui | Multi-terminal transport measurements of MoS 2 using a van der Waals heterostructure device platform[END_REF] . More recent data with graphite gate is shown in Figure 3c 49 . The onset of SdH oscillations at ∼2 T suggests mobility above 4,000 cm 2 V -1 s -1 at ∼100 mK. These oscillations show even-odd behavior, clear evidence of valley Zeeman splitting. Likewise, recent measurements 50 of hole-doped WSe 2 with µ ∼ 2,000 cm 2 V -1 s -1 show strong SdH oscillations with an onset above 5 T, as shown in Fig. 3d. However, fully formed QH stateswith zero longitudinal resistance and clear plateaus in Hall resistancehave yet to be observed in monolayer TMDs, highlighting the need for a further reduction in disorder.

Oxidation in air is another source of extrinsic disorder that strongly affects electronic transport in 2D materials. Encapsulating 2D monolayers with hBN in an inert atmosphere provides excellent protection from degradation, allowing air-sensitive samples to be handled easily. This has been widely used to study air-sensitive TMD superconductors such as NbSe 2 . As shown in Fig. 3e, few-layered NbSe 2 directly exfoliated on SiO 2 does not show superconductivity, and bilayer/trilayer samples show insulating behavior at low-T due to localization from strong disorder [START_REF] Staley | Electric field effect on superconductivity in atomically thin flakes of NbSe 2[END_REF] .

In contrast, hBN-encapsulated NbSe 2 is superconducting even in the monolayer limit [START_REF] Xi | Ising pairing in superconducting NbSe 2 atomic layers[END_REF] . This method of using hBN as a means of isolating samples from oxidation has been expanded to many other layered materials for exploring ferromagnetism (CrI 3 ) [START_REF] Huang | Layer-dependent ferromagnetism in a van der Waals crystal down to the monolayer limit[END_REF] , or QHE (BP) [START_REF] Li | Quantum Hall effect in black phosphorus two-dimensional electron system[END_REF] .

As direct-bandgap semiconductors, monolayer TMDs show efficient photoluminescence (PL), with spectra dominated by strong excitonic resonances. As shown for monolayer TMDs in Fig. 3f, when extrinsic disorder is reduced in monolayer TMDs through hBN encapsulation and sufficient screening of charge disorder in the the underlying SiO 2 substrate, a substantial narrowing of the linewidth is observed [START_REF] Cadiz | Excitonic Linewidth Approaching the Homogeneous Limit in MoS 2 -Based van der Waals Heterostructures[END_REF] . The low-T (∼4 K) linewidth of hBN encapsulated samples is less than 2 meV, significantly reduced from the linewidth of MoSe 2 on bare SiO 2 51, 59 . This reduced linewidth allows for the observation of additional excitonic states at energies close to the highest intensity peaks. In the case of WSe 2 this has allowed for the observation and control, via in-plane magnetic fields, of dark excitonic states [START_REF] Zhang | Magnetic brightening and control of dark excitons in monolayer WSe 2[END_REF] . These linewidths are even sufficiently narrow enough to allow for the observation of optical transitions between Landau levels with an energy spacing of ∼2 meV in electrostatically gated monolayer WSe 2 [START_REF] Movva | Density-Dependent Quantum Hall States and Zeeman Splitting in Monolayer and Bilayer WSe 2[END_REF] . Reduced extrinsic disorder has also facilitated the observation of strong interaction-driven effects such as cyclotron mass renormalization due to electronic-induced doping [START_REF] Wang | Valley-and spin-polarized Landau levels in monolayer WSe 2[END_REF] , biexcitons [START_REF] You | Observation of biexcitons in monolayer WSe 2[END_REF] , and additional many-body states.

Intrinsic disorder in TMDs

As discussed above, reduction in extrinsic disorder through hBN encapsulation improves the properties of TMDs, but not to the same degree as observed for graphene, indicating that TMD an obscured Landau Fan diagram for CVT as compared to flux [START_REF] Gustafsson | Ambipolar Landau levels and strong band-selective carrier interactions in monolayer WSe 2[END_REF] . Adapted from ref. [START_REF] Vancs | The intrinsic defect structure of exfoliated MoS 2 single layers revealed by Scanning Tunneling Microscopy[END_REF] , Nature Publishing Group (a); ref. [START_REF] Zhou | Intrinsic Structural Defects in Monolayer Molybdenum Disulfide[END_REF] , ACS Publications (b); ref. [START_REF] Hong | Exploring atomic defects in molybdenum disulphide monolayers[END_REF] , Nature Publishing Group (c); ref. [START_REF] Lin | Realizing Large-Scale, Electronic-Grade Two-Dimensional Semiconductors[END_REF] , ACS Publications (d);

ref. [START_REF] Gustafsson | Ambipolar Landau levels and strong band-selective carrier interactions in monolayer WSe 2[END_REF] , Nature Publishing Group (f).

crystals do not share the near-perfection of graphene. This is not surprising given that the formation energy for defects in TMDs is much lower than for similar defects in graphene (e.g. 7-8 eV for a graphene vacancy vs. 2.1 eV for a sulfur vacancy in MoS 2 ) [START_REF] Zhou | Intrinsic Structural Defects in Monolayer Molybdenum Disulfide[END_REF][START_REF] Hong | Exploring atomic defects in molybdenum disulphide monolayers[END_REF] .

Direct inspection of TMDs by STM [START_REF] Vancs | The intrinsic defect structure of exfoliated MoS 2 single layers revealed by Scanning Tunneling Microscopy[END_REF] (Fig. 4a) or ADF-STEM [START_REF] Zhou | Intrinsic Structural Defects in Monolayer Molybdenum Disulfide[END_REF] (Fig. 4b) finds a variety of defect types, including chalcogen vacancies, transition metal vacancies, and antisite defects, in which transition metals occupy chalcogen sites. As one example, Fig. 4c shows the measured density of the major defect types [START_REF] Vancs | The intrinsic defect structure of exfoliated MoS 2 single layers revealed by Scanning Tunneling Microscopy[END_REF][START_REF] Hong | Exploring atomic defects in molybdenum disulphide monolayers[END_REF] in MoS 2 single crystals and films grown by the chemical vapor deposition (CVD) technique. In all of these materials, the defect density (ignoring the presence of impurities) is very high, ranging from mid 10 12 to above 10 13 cm -2 . Moreover, the dominant type of point defect varies depending on the material source: in mined 'natural" MoS 2 and CVD-grown films, sulfur vacancies can dominate [START_REF] Vancs | The intrinsic defect structure of exfoliated MoS 2 single layers revealed by Scanning Tunneling Microscopy[END_REF] , whereas in synthetic crystals grown by chemical vapor transport (CVT), metal vacancies and antisites dominate [START_REF] Edelberg | Hundredfold Enhancement of Light Emission via Defect Control in Monolayer Transition-Metal Dichalcogenides[END_REF] . The relative densities of different types of defects determines the crystal doping: for instance, when metal vacancies are dominant, hole (p-type) doping is observed [START_REF] Zhang | Defect Structure of Localized Excitons in a WSe 2 Monolayer[END_REF] .

Elemental impurities are a second defect class that has to be considered for TMDs. In fact, mined MoS 2 can have extremely large impurity concentration of up to 10 14 cm -271 . Artificially grown TMDs from high-purity starting powders show much lower impurity density of ∼8×10 10 cm -2 , in agreement with the purity of the starting materials [START_REF] Addou | Surface Analysis of WSe 2 Crystals: Spatial and Electronic Variability[END_REF] . Thus in synthetic TMDs with vacancy density of ∼10 13 cm -2 , elemental impurities can be neglected as a source of disorder.

However, in materials with substantially lower vacancy density (see below) it will be important to consider both classes of defect.

Figures 4d and4e show recent progress in reducing point defect density in TMDs. Figure 4d shows STM images of WSe 2 epitaxially grown on graphene by metal-organic chemical vapor deposition (MOCVD) [START_REF] Lin | Realizing Large-Scale, Electronic-Grade Two-Dimensional Semiconductors[END_REF] , where the dominant defect types are Se vacancies (black circles) and W vacancies substituted by Se (grey circles). Annealing in H 2 Se at 800 °C reduces the density of both defect types by roughly an order of magnitude (to below 10 12 cm -2 ), and largely eliminates nanoparticles on the surface (white dots). Figure 4e shows STM images of the surface of cleaved increases by 2 orders of magnitude [START_REF] Edelberg | Hundredfold Enhancement of Light Emission via Defect Control in Monolayer Transition-Metal Dichalcogenides[END_REF] ; and capacitance measurements of the Landau level spectrum of WSe 2 show much more detail when flux-grown crystals are used [START_REF] Gustafsson | Ambipolar Landau levels and strong band-selective carrier interactions in monolayer WSe 2[END_REF] (Fig. 4f). Finally, we note that an alternative approach to reducing defect density is 'repair' of chalcogen vacancies using thiol chemistry [START_REF] Yu | Towards intrinsic charge transport in monolayer molybdenum disulfide by defect and interface engineering[END_REF] . This approach has been used to achieve large increases in photoluminescence intensity [START_REF] Amani | Near-unity photoluminescence quantum yield in MoS 2[END_REF] , but does not achieve high low-T mobility consistent with low disorder.

Applications of defects and disorder

In the sections above, we have focused on the challenge of minimizing disorder in 2D materials. However, disorder in particular crystal defects can also be used to achieve control over electrical, optical, vibrational, magnetic, and chemical properties toward new functionality. Here we briefly discuss some of the most prominent uses of defects in 2D materials. First, substitutional doping can be used for work function control and interface engineering. Layered assembly d, Diamagnetism and ferromagnetism have been observed in of MoS 2 [START_REF] Guguchia | Magnetism in semiconducting molybdenum dichalcogenides[END_REF] . e, Schematics of restacked nonintercalated and intercalated MoS 2 by alkali ions (Li + , Na + , K + ) and TEA +80 . Adapted from ref. [START_REF] Chuang | Low-Resistance 2d/2d Ohmic Contacts: A Universal Approach to High-Performance WSe 2 , MoS 2 , and MoSe 2 Transistors[END_REF] , ACS Publications (a); ref. [START_REF] He | Single quantum emitters in monolayer semiconductors[END_REF] , Nature Publishing Group (b); ref. [START_REF] Li | All The Catalytic Active Sites of MoS 2 for Hydrogen Evolution[END_REF] , ACS Publications (c); ref. 79 (d); ref. [START_REF] Acerce | Metallic 1T phase MoS 2 nanosheets as supercapacitor electrode materials[END_REF] Nature Publishing Group (e).

of doped 2D semiconductors circumvents problems such as work function mismatch and Fermi level pinning that create large Schottky barriers in metal-TMD contacts [START_REF] Das | High Performance Multilayer MoS 2 Transistors with Scandium Contacts[END_REF] . Figure 5a shows that Nb-doped WSe 2 functions as a near-ideal p-type contact to intrinsic WSe 2 [START_REF] Chuang | Low-Resistance 2d/2d Ohmic Contacts: A Universal Approach to High-Performance WSe 2 , MoS 2 , and MoSe 2 Transistors[END_REF] , achieving contact resistance of ∼0.3 kΩ µm and steep subthreshold slope. Doping of 2D materials has been extensively reviewed [START_REF] Zhao | Doping, Contact and Interface Engineering of Two-Dimensional Layered Transition Metal Dichalcogenides Transistors[END_REF] .

Individual defects in 2D materials can also act as optically active sites. Figure 5b shows the single-photon emission from localized defects in WSe 2 monolayers [START_REF] He | Single quantum emitters in monolayer semiconductors[END_REF] . The measured autocorrelation function g 2 (τ ) of the emission shows a dip below 0.5 at zero delay time (τ = 0), indicating that each defect acts as a single-photon emitter. Tunable single photon emission from semiconducting TMDs has been reported [START_REF] Chakraborty | Voltage-controlled quantum light from an atomically thin semiconductor[END_REF] , and hBN-based quantum emitters are optically active at room temperature, an important attribute for applications in quantum information [START_REF] Tran | Quantum emission from hexagonal boron nitride monolayers[END_REF] . Although the exact mechanism is still uncertain, this emission is likely due to defect sites that are activated in the presence of strain. Greater control over optically active defects, coupled with precise charge injection in van der Waals heterostructures, will allow TMDs to be used for electrically controlled quantum emitters for quantum information processing.

Defects in TMDs also hold promise as catalytic sites that can replace expensive noble metal catalysts in the hydrogen evolution reaction (HER) and other processes. MoS 2 has been demonstrated as an effective catalyst for HER at a moderate overpotential of 0.1-0.2 V [START_REF] Hinnemann | Biomimetic Hydrogen Evolution: MoS 2 Nanoparticles as Catalyst for Hydrogen Evolution[END_REF] . Catalysis occurs at defects including edges, sulfur vacancies, and grain boundaries [START_REF] Li | All The Catalytic Active Sites of MoS 2 for Hydrogen Evolution[END_REF] (Fig. 5c). Changing the sulfur vacancy density simultaneously tunes the density of catalytic sites and doping in the sheet, allowing for optimization of both catalytic activity and charge injection to the TMD layer [START_REF] Li | All The Catalytic Active Sites of MoS 2 for Hydrogen Evolution[END_REF] .

Defects in TMDs can give rise to diamagnetism, ferromagnetism, and spin-valley coupling applications. Recent experimental work has shown that point defects in Mo-based TMDs can be magnetic, as confirmed by calculation of the spin-polarized density of states for an antisite defect (Mo at the chalcogen site), shown in Fig. 5d for MoTe 2 . These spins couple antiferromagnetically to nearby Mo atoms and can show long-range ordering [START_REF] Guguchia | Magnetism in semiconducting molybdenum dichalcogenides[END_REF] . Spin polarized first-principles calculations suggest that one-dimensional edges of two-dimensional MoS 2 or WS 2 carry non-zero net magnetic moments associated with unpaired electrons [START_REF] Zhang | Magnetic Molybdenum Disulfide Nanosheet Films[END_REF] . Atomic defects also give unconventional selection rules for intervalley quasiparticle scattering in 2D TMDs valleytronics [START_REF] Kaasbjerg | Symmetry-forbidden intervalley scattering by atomic defects in monolayer transition-metal dichalcogenides[END_REF] .

Interface chemistry approaches in 2D materials seek to manipulate the electrochemical behaviour of layered systems, and open new pathways to control the charge density in electronic and optoelectronic devices. For instance, it has been shown that intercalation by organolithium chemistry in 2H-TMDs stabilizes a metallic 1T phase via strong electron doping (see schematic image, Fig. 5e 80 ). This method can be further applied to phase-engineer contacts, reducing contact resistance to 200-300 Ω µm in few-layered 2H-MoS 2 , comparable to the best contacts produced through more intricate fabrication methods [START_REF] Kappera | Phase-engineered low-resistance contacts for ultrathin MoS 2 transistors[END_REF] .

Outlook

Reducing disorder in 2D materials has enabled stunning scientific advances and progress toward technological applications, providing strong motivation to continue these efforts. Toward this end, each materials system faces distinct challenges. In graphene devices, the remaining sources of extrinsic disorder must be identified and further reduced. One possible source is environmental contaminants such as water vapour and hydrocarbons which are not completely eliminated from interfaces during heterostructure fabrication. A second possible source is defects in hBN. There has been little work beyond hBN synthesis [START_REF] Taniguchi | Synthesis of high-purity boron nitride single crystals under high pressure by using BaBN solvent[END_REF] in reducing defects and increasing grain size in hBN single crystals. More work is needed in this area, as well as in development of methods to quickly assess defect density in hBN. In hBN-encapsulated 2D semiconductors, on the other hand, intrinsic defects of 2D semiconductors are the dominant source of disorder. The need for careful characterization of materials qualitywhich can vary widely with with synthesis technique and from batch to batchshould be emphasized in ongoing studies of novel phenomena in TMDs. Further progress in the field will require reduction in defect density in both single crystals and large-area films of every material under study. In order to achieve rapid progress toward this goal, it will be important to develop facile techniques for in situ characterization of materials quality. However, this effort is still impeded by a lack of basic understanding of how atomic defects affect easilymeasured bulk properties.

Realizing the potential of 2D materials in applications will require development of materials synthesis and manufacturing techniques that can achieve low disorder on a large scale. A specific challenge is to achieve low extrinsic disorder in large-area films. Toward this end, significant progress has been made in the synthesis of graphene and TMD films at the wafer scale, as well as in the development of clean, wafer-scale transfers of monolayer films [START_REF] Kang | High-mobility three-atom-thick semiconducting films with wafer-scale homogeneity[END_REF][START_REF] Kang | Layer-by-layer assembly of two-dimensional materials into wafer-scale heterostructures[END_REF][START_REF] Shim | Controlled crack propagation for atomic precision handling of wafer-scale two-dimensional materials[END_REF] . Scaling up encapsulation approaches used to reduce disorder in exfoliated flakes will require synthesis of multilayer hBN films with high uniformity and large grain size, which has proven to be extremely challenging to date [START_REF] Jana | Progress in CVD synthesis of layered hexagonal boron nitride with tunable properties and their applications[END_REF] . Alternatively, other dielectric materials may provide sufficient performance: for example, many layered metal oxides to (i.e. MoO 3 , V 2 O 5 , etc.) with bandgaps in excess of 3.5 eV have yet to be extensively explored as as dielectrics for graphene and 2D semiconductors [START_REF] Kalantar-Zadeh | Two dimensional and layered transition metal oxides[END_REF] . The important role of defects and dopants in 2D materials highlights another gap in current capabilities: in conventional semiconductors dopants can be patterned using ion implantation, allowing precise control over dopant density required for high-performance electronic and optoelectronic devices. While techniques such as plasma treatment and wet chemical treatment have demonstrated local doping of 2D semiconductors, no current technique provides precise local control over doping combined with low disorder and good stability. Finally, we note that improving the intrinsic quality of 2D materials will improve fundamental understanding of the behavior of defects by permitting careful study of well-characterized defects embedded in high-quality crystals.

Figure 1 |

 1 Figure 1 | Types of disorder in 2D materials. a, Intrinsic disorder: vacancy, anti-site, substitution, edges

Figure 2 |

 2 Figure 2 | Evolution of graphene device design and resulting performance improvements. Compar-

Figure 3 |

 3 Figure 3 | Reducing extrinsic disorder in TMDs through hBN encapsulation. a, Theoretical calcula-

  and a crossover to a power-law temperature dependence is observed. This model agrees well with experimental values for SiO 2 -HfO 2 encapsulated MoS 2 (Fig.3a, open squares), with N i = 6 × 10 12 cm -2 .

Figure 4 |

 4 Figure 4 | Intrinsic disorder in TMDs and defect control through growth. a, Topographic STM image

WSe 2

 2 crystals grown by chemical vapor transport (CVT)the most commonly utilized technique for TMD crystal growth[START_REF] Qiu | Hopping transport through defect-induced localized states in molybdenum disulphide[END_REF] and grown within a liquid chalcogen flux. The flux technique achieves a large reduction in defect density to below 10 11 cm -2 . Initial studies using flux-derived monolayers show dramatic improvements in performance. For instance, the low-T PL intensity in MoSe 2

Figure 5 |

 5 Figure 5 | Applications of defects and disorder in 2D materials. a, Low-resistance ohmic contacts using

Acknowledgements We would like to acknowledge Matthew Yankowitz and J.I.A. Li for many discussions involving graphene and graphene devices. This work was supported the NSF MRSEC program through Columbia in the Center for Precision Assembly of Superstratic and Superatomic Solids (DMR-1420634). S.H.C. was supported by the Postdoctoral Research Program of Sungkyunkwan University (2016).

Competing Interests

The authors declare that they have no competing financial interests.

Correspondence Correspondence and requests for materials should be addressed to J.H. (email: jh2228@columbia.edu).


Rebeca Ribeiro-Palau

Changjian Zhang

Kenji Watanabe

Takashi Taniguchi

James Hone

Cory R Dean

Twistable electronics with dynamically rotatable heterostructures

come

The electronic properties of two-dimensional materials and their heterostructures can be dramatically altered by varying the relative angle between the layers. This makes it theoretically possible to realize a new class of twistable electronics in which device properties can be manipulated on-demand by simply rotating the structure. Here, we demonstrate a new device architecture in which a layered heterostructure can be dynamically twisted, in situ. We study graphene encapsulated by boron nitride where at small rotation angles the device characteristics are dominated by coupling to a large wavelength Moiré superlattice. The ability to investigate arbitrary rotation angle in a single device reveals new features in the optical, mechanical and electronic response in this system. Our results establish the capability to fabricate twistable electronic devices with dynamically tunable properties.

The weak van der Walls forces between the atomic planes in 2D materials makes it possible to fabricate devices with arbitrary rotational order. This provides a new opportunity in device design where electronic properties are controlled by varying the relative twist angle between layers [1]. Indeed several studies have established that in heterostructures assembled from 2D crystals, electron tunneling between layers varies strongly with rotation [2][3][4][5][6][7][8]. In twisted bilayer graphene (two monolayers in direct contact but with an angle mismatch between the layers) several novel phenomenon have been predicted and observed, including topological valley transport [9][10][11][12][13], and superconductivity [14], as a consequence of angledependent interlayer coupling. Likewise, the formation of interlayer excitons in transition metal dichalcogenide heterostructures is highly sensitive to angle [15][16][17].

The effect of rotational alignment between conducting and insulating 2D layers can be equally significant. A remarkable example is provided by graphene coupled to hexagonal boron nitride (BN). Owing to the closely matched lattice constants a large Moiré superlattice develops near zero angle mismatch [18,19]. This substantially alters the graphene band structure opening an energy gap at the charge neutrality point (CNP) and creating replica Dirac points at higher energies [20][21][22].

Several techniques have been developed to fabricate layered heterostructures with controlled rotation between the layers, including optical alignment of crystal edges [18,[20][21][22], rotational alignment [23] during assembly, and self-alignment through thermal annealing [24,25]). However, in each case a priori understanding of the crystallographic orientation of each layer is required before assembly; motion between the layers during assembly makes it difficult to achieve precise angle control; and most significantly, once assembled the angle can not be further modified. Here, we present a new experimental technique that provides on-demand control of the orientation between layers in a van der Waals heterostructure. We study a BN/graphene/BN structure where we demonstrate in situ control over the length of the Moiré potential and consequently achieve dynamic tunability of the optical, mechanical and electrical properties of the system.

Figure 1a shows a cartoon schematic of our device design. Using the mechanical assembly technique [26], graphene is placed on top of a large flake of BN, and then etched into a Hall bar shape using oxygen plasma. The graphene layer is intentionally misaligned to this BN, producing a short-wavelength Moiré potential that does not significantly alter the intrinsic graphene band structure [27]. Next, a pre-shaped BN structure is transferred on top of the graphene. Finally, electrical contacts are patterned onto the exposed leads of the graphene (see supplementary information). Due to the low mechanical friction between graphene and BN, we are able to freely rotate and translate this top BN layer using an atomic force microscope (AFM). Pushing on one of the arms of the uppermost BN structure rotates this layer (Fig. 1b-d) changing its crystallographic orientation with respect to the graphene layer. The Moiré superlattice wavelength, λ, generated between these crystals is given by:

λ = (1 + δ)a 2(1 + δ)(1 -cos(θ)) + δ 2 (1)
where δ = 0.017 is the lattice mismatch between graphene and BN, a is the lattice constant of graphene and θ is the rotational mismatch between the layers. By rotating the top BN layer, the wavelength of the resulting Moiré superlattice can therefore be dynamically varied.

Imaging with the same AFM provides a real-time measurement of the orientation of the rotatable BN layer relative to the reference frame of the AFM, which we label by the absolute angle (θ A ) (see for example Fig. 1b-d). More significant is the relative angle, θ, between the rotatable BN and encapsulated graphene crystal lattices. To determine this we identify and measure angledependent features in the optical Raman spectrum of the heterostructure.

Figure 1f shows a series of Raman spectra measured from θ A = 34-64 degrees. The most striking variation in the Raman spectra is an increase in the full width at half maximum of the 2D peak (FWHM 2D ) [19,28]. Plotting this as a function of absolute angle (Fig. 1g) we see a well defined maximum, occurring for this device at θ A ∼ 34.2 degrees. Based on previous work [28], we interpret the peak position as corresponding to zero angle alignment between the BN and graphene crystals. This therefore provides a reference which we can use to determine θ for any orientation of the BN layer.

Our result highlights the robustness of Raman spectroscopy as a tool to characterize the rotational order in these van der Waals heterostructures. Moreover, we em-phasize that previous efforts to study effects of twist angle in this system required numerous samples with different fixed angles whereas here we demonstrate a mapping of the angular dependence with better than 0.2 degree precision, in a single tunable device.

One notable disagreement with previous results [28] is an overall narrower 2D line-width in our devices. In previous work this reduced linewidth was proposed to be a consequence of a reduction of the in-plane strain in fully-encapsulated structures, such that areas of full commensuration to the aligned BN disappear [19]. However, our observation of the same linear trend and a reduced linewidth for the misaligned position -for which no commensurate regions are expected -suggests a different origin. We speculate that the real origin of this linewidth reduction is the change of dielectric environment of the graphene by the presence of the second BN, as proposed in ref. [29]. Therefore, our results re-open the question of the existence of commensurate states in encapsulated graphene devices. We additionally have identified a shift The resulting voltage difference is proportional to the tip cant angle (referred to here as the tip deflection) and serves as a measure of the torque force acting at the end of the tip. b, Tip deflection versus time in a translational push of the upper BN structure. Different regimes of the measurement are identifiable: (i) as the tip drags along the surface, tip-substrate friction results in a steady state tip deflection (ii) when the tip encounters the BN structure it initially resists translation and the tip deflection increases. We refer to this as the static friction regime (iii) Once the BN is in motion the tip deflection relaxes slightly providing a measure of the dynamic friction at the BN-graphene interface. c, tip deflection versus absolute angle measured during a continuous rotation of the BN. Two peaks are observed, spaced 60 degrees apart.

in the position of the 2D and G peaks with relative angle (see supplementary information), a full discussion of which is beyond the scope of this report.

Mechanical resistance while pushing the BN imparts a torque to the AFM tip, causing it to cant away from a vertical position and produce a voltage difference in the AFM's photodetector. This can be used to identify variations in frictional forces (Fig. 2a.) When sliding the top BN layer with the AFM tip, in a translational motion far from alignment, we identify three regimes (see Fig. 2b): i) sliding friction between the tip and the substrate before the tip encounters the BN; ii) static friction when the tip encounters the BN but it resists translation; and iii) dynamic friction once the BN begins to move.

Figure 2c shows a plot of the change in the tip deflection under continuous rotation (dynamic friction), where the background due to piezoelectric drift and residual friction has been subtracted. Two prominent peaks appear, separated by 60 degrees. This closely resembles previous measurement of friction between two graphitic structures in which a transition from superlubricity (where the structures are in an incommensurate position and the atomic shear forces are negligible) to a dissipative state was observed at commensurate angles of the 3-fold symmetric hexagonal lattices [30][31][32]. However, since there is a lattice mismatch between graphene and BN there is not true lattice commensurability at any angle and therefore the increase of the friction should have a different origin. Recent numerical simulations suggest that contributions of the Moiré superlattice to the frictional force cannot be neglected and are expected to be maximal for aligned layers [33], which could explain our experimental result. A more detailed study of the interlayer frictional forces will be necessary to fully understand this behavior. While this is beyond the present scope, we note that this new device structure allow us to study mechanical properties, such as frictional force, in atomically flat materials without rugosity contributions. These results also highlight the possibility to use the friction response as an in situ method to monitor and control layer alignment in heterostructures.

Our device design allows us to measure electron transport in the active layer (in this case graphene) while changing the relative orientation of the over-layer. Fig- ure 3a shows a plot of the four-terminal resistance of the graphene layer as a function of back gate voltage V g for different values of θ at room temperature. Near θ = 0, additional satellite resistance peaks appear symmetrically in density around the charge neutrality point (CNP). This is consistent with the emergence of satellite Dirac points induced by scattering from the Moiré superlattice potential [18,[20][21][22].

V CNP -V Sat (V) θ (degrees) E k Δ Δ Sat E k F -3 -2 -1 0 1 2 3
As θ increases away from zero, the satellite peaks diminish in intensity and moves further from the CNP to higher gate values. To analyze this behavior more quantitatively, Figure 3b plots the satellite peak position in V g versus θ determined from the AFM imaging. The measured position shows excellent agreement with the values of carrier density at which the full filling of the miniband occurs, n = 8/ √ 3λ 2 , where the carrier density and gate voltage are related by n = C g (V g -V CNP ) and λ is given by expression (S1) [18,[20][21][22]. The error bars in Fig. 3b, approximately ±0.2 degrees, reflects the precision with which θ can be determined from the AFM topographic images. Determining θ from the gate voltage position of the satellite peak at low temperature provides a more accurate measurement with uncertainty less than < ±0.1 degrees, however this method of determining the angle is limited to the only a few degrees where the satellite peak remains within an accessible density range.

Plotting the resistance of the satellite peak versus λ for a single device reveals an apparently linear variation in the magnitude of the satellite peak resistance (Fig. 3c). Interestingly, the resistance of the CNP does not change linearly with θ in this range (see supplementary information). The origin of this linear dependence is not known at present. The striking observation however highlights an example of a physical phenomenon previously obscured [18,[20][21][22] by sample-to-sample variations but which becomes clear when able to measure the effect of varying rotation in single sample.

We measured the energy gap versus angle at both the central and satellite Dirac points by thermal activation. The gap magnitudes, shown in Fig. 3d, are in good agree- ment with electronic transport measurements in encapsulated [34] and non-encapsulated devices [19,20], optical measurements made in epitaxial BN/graphene structures [35], and theoretical calculations [36]. As shown in Figure 3d, the energy gap of the satellite peak decreases smoothly away from θ = 0. In contrast, the energy gap at the CNP displays a more complex behavior near θ = 0 and only decreases significantly for |θ| > 2 degrees. The origin of the extra structure observed for the CNP remains to be understood. However, this does not appear to be simple experimental noise since several gap values were confirmed to be reproducible when measured nonconsecutively (i.e, after thermally cycling and rotating through different angles and back). The difference in behavior of the two energy gaps, impossible to observe in study utilizing multiple samples at fixed angles [34], reflects their different physical origins and highlights the importance of our new technique in the fully understanding the band structure modifications resulting from variations in angular alignment, Fig. 3e-f. The persistance of an energy gap at the CNP in encapsulated devices for angles beyond the angle at which a commensurateincommensurate transition was previously identified [19] suggests that this energy gap is not related solely to the presence of a commensurate state.

Fig. 4 shows a direct comparison of the optical, mechanical and electrical response versus angle measured in the same device. The four probe resistance (Fig. 4a), the maximum tip deflection signal (friction, Fig. 4b) and the maximum FWHM of the Raman 2D peak (Fig. 4c) coincides exactly, confirming the relationship between these properties. The four-probe resistance (Fig. 4a) and friction response (Fig. 4b) were acquired simultaneously at a fixed carrier density of 1.9x10 12 cm -1 (corresponding to the dashed line in Fig. S2) while continuously rotating the BN layer. We note that at this relatively large carrier density the bulk resistance is modulated by more than an order of magnitude over less than 2 degrees of rotation (at cryogenic temperatures this increases is of more than two orders of magnitude, see supplementary information). Our demonstration that rotatable heterostructures with dynamically tunable device characteristics can be realized provides a new opportunity in device engineering. While here we have investigated BN encapsulated graphene as a model tunable system, this technique is readily extended to generic heterostructures fabricated from 2D materials where in addition to band structure tunability, emergent phases such as superconductivity and magnetism may be controllably varied with rotation.

Initial layers stack

FIG. 1 .

 1 FIG. 1. Rotatable heterostructures a, Schematic cartoon of the device structure and the experimental technique. b-d, AFM image of a fabricated device showing three different orientations of the top BN. The angles identified in each panel is the absolute angle referenced to the AFM coordinate system (labelled θA in the text). The images were acquired by the same AFM used to rotate the BN layer. e Schematic illustration of the Moire superlattice arising between graphene (Red) and BN (Blue) at zero angle. The moire wavelength is identified by λ. f, Raman spectrum of the device shown in (b-c) for θA between 34.2±0.2 degrees and 36±0.2. Black curve shows an additional measurement acquired at ≈ 64 degrees. g, FWHM of the 2D peak as a function of the absolute angle. All Raman measurements were taken with the gate bias held at VG = 0 V. The peak FWHM position identifies zero angle crystallographic alignment (see text). Dashed line represents the FWHM measured for all angles larger than approximately 2 degrees away from perfect alignment with the shaded region representing the associated uncertainty.

FIG. 2 .

 2 Photodetector

FIG. 3 .

 3 FIG.3. Electronic transport properties. a, Four-terminal resistance as a function of the gate voltage for different alignments of the graphene/BN structure, acquired at room temperature. b, Position of the satellite peak in gate voltage as a function of the relative angle. The 0.2 degrees error bar in the angle reports the precision achieved with the AFM imaging in tapping mode. c, Linear dependence of the maximum value of the four-terminal resistance at the satellite peak at 300 K (red) and 1.7 K (blue) as a function of the Moiré length. d Energy gap, measured by thermal activation, for the satellite peak (circles) and the charge neutrality point (diamonds) as a function of the relative angle. Open symbols represent a repeated measurement at a given angle after moving through other angles and thermally cycling. e, Schematic band structure for native graphene and f for a graphene-hBN heterostructure with a small twist angle.

FIG. 4 .

 4 FIG. 4. Compilation of the angle control technique.a Four-terminal resistance as a function of the relative angle measured at a carrier density of -1.9x10 12 cm -1 . b, Tip deflection in friction measured simultaneously as the electronic transport. c, FWHM of the 2D peak of the Raman spectrum as a function of the angle. All measurements were performed in the same device.

  FIG. S1. Fabrication process. a, Initial graphene/BN stack deposited on a Si/SiO2 substrate b, etching of the graphene flake to give a Hall bar shape. c, Dry transfer of a pre-shaped BN structure. d, electrical connection with Cr/Pd/Au top surface contacts.

  FIG. S2.Raman spectrum. a, FWHM of the 2D peak as a function of the Moiré length, notice that the slope of the curve is the same reported in[28], see main text. Position of the 2D c and G d peaks as a function of the absolute angle. d, Room temperature measurement of the four-terminal resistance as a function of gate voltage for each Raman spectrum of this figure and of figure1of the main text. e, Maximum of the resistance at the satellite peak as a function of the Moiré wavelength and energy calculated using n = 8/ √ 3λ 2 and E = hvF/ √ 3λ, where the capacitive coupling of Cg/e = 6.72 × 10 14 V -1 m -2 was obtained experimentally from Hall measurements.

T - 1 (K - 1 )

 11 FIG. S4. Linear resistance dependence. Value of the four probe resistance at the charge neutrality point and satellite peaks at 300 K a and 1.6 K b for a second sample. The linear fits for the second sample can be described by R max-h(300K) = 275.26λ -2376, R max-e(1.6K) = 81.8λ -711 and R max-h(1.6K) = 10040λ -95141.9

ACKNOWLEDGEMENTS

We acknowledge discussions with Matthew Yankowitz and Andres Botello-Mendez as well as Juan Huerta, Shaowen Chen and Martin Gustafsson for technical support. This research was supported by the NSF MRSEC programme through Columbia in the Center for Precision Assembly of Superstratic and Superatomic Solids (DMR-1420634). C.R.D. acknowledges partial support by the National Science Foundation (DMR-1462383).

AUTHOR CONTRIBUTIONS

R.R.-P. and C.R.D. designed the experiment. Ch.Z. and R.R.-P. fabricated the samples, performed the experiments, analyzed the data and wrote the paper. T.T. and K.W. grew the crystals of hexagonal boron nitride. J.H. and C.R.D. advised on experiments, data analysis and writing the paper.

SUPPLEMENTARY INFORMATION RAMAN SPECTRUM OF ALIGNED STRUCTURES

The inset in Fig. S7a shows a plot of the FWHM of the 2D peak versus calculated Moiré wavelegth. A remarkably linear dependence, described by the linear fit FWHM 2D = 2.7λ -0.77, is observed. The slope of this linear fit is identical, within experimental uncertainty, to previous observations [28]. Previous theoretical studies of the modification of the Raman spectra by a superlattice potential associated this changes with the folding of the phonon structure due to the Moiré pattern [37,38], which could also be the case in graphene.

ANGLE AND MOIR É LENGTH RELATION

The angle between layers, θ, and the Moiré length, λ are related by

where δ = 0.017 is the lattice mismatch, a is the lattice constant of graphene [18]. As explained in the main text the energy of this satellite peaks is given by:

As previously reported in [18] we found that the satellite peak is weaker for the conduction band (positive V g ) than the one of the valence band (negative V g ). This asymmetry in strength has been associated with the breaking of the electron-hole symmetry induced by a modulated hopping between different graphene sublattices.