Didier MAQUIN Mohammed Chadli

José Ragot
email: josé.ragot@ensem.inpl-nancy.fr

Relaxed stability conditions for Takagi-Sugeno fuzzy systems

This paper discusses conditions on stability and stabilization of continuous T-S fuzzy systems. Stability analysis is derived via non-quadratic Lyapunov function technique and LMIs (Linear Matrix Inequalities) formulation to obtain an efficient solution. The nonquadratic Lyapunov function is built by inference of quadratic Lyapunov function of each local model. We show that stability condition of the open-loop T-S systems is assured under certain restrictions on the rate of change of state variables. Following a similar approach, stabilization of the closed-loop continuous T-S fuzzy systems using the well-known PDC (Parallel Distributed Compensation) technique is investigated. The design methodology is illustrated by numerical examples.

Introduction

Since last years many works have been focused on the stability and the stabilisation of closed loop fuzzy systems including a fuzzy controller. Specially, the approach using T-S fuzzy models [START_REF] Takagi | Fuzzy identification of systems and its application to modelling and control[END_REF], considered like a universal approximator [START_REF] Castro | Fuzzy logic controllers are universal approximator[END_REF], has been considered extensively. The T-S fuzzy models are described by a set of fuzzy "If ... then" rules with fuzzy sets in the antecedents and dynamics LTI systems in the consequent. These sub-models are considered as local linear models, the aggregation of which representing the nonlinear system behaviour. The issue of stability and stabilisation of T-S fuzzy control systems has been considered extensively in nonlinear frameworks. However, the present results are only sufficient and require conservative assumptions. For example, Tanaka and Sugeno presented sufficient conditions for the stability of T-S models [START_REF] Tanaka | Stability and design of fuzzy control systems[END_REF] using a quadratic Lyapunov approach. The stability depends on the existence of a common positive definite matrix guarantying the stability of all local subsystems. These stability conditions may be expressed in linear matrix inequalities (LMIs) form [START_REF] Boyd | Linear matrix inequalities in systems and control theory[END_REF]. The obtaining of a solution is then facilitate by using numerical toolboxes for solving such problems. Recently LMI constraint has been added to compute a decay rate and to guarantee that the control law action doesn't permit an exceeding of a pre-defined normbound of input and output signals [START_REF] Takagi | Fuzzy regulators and fuzzy observers: relaxed stability conditions and LMI-based design[END_REF] [START_REF] Tanaka | Multiobjective fuzzy control of high rise/high speed elevators using LMIs[END_REF]. However, if the number of sub-models is large, it might be difficult to find a common matrix. Moreover, these constraints are often very conservative and it's well known that, in a lot of cases, a common positive definite matrix does not exist, whereas the system is stable.

To overcome this limitation, some works have been developed in order to establish new stability conditions by relaxing some of the previous constraints. So one way for obtaining relaxed stability conditions consists to use a piecewise quadratic Lyapunov function formulated as a set of LMIs [START_REF] Johansson | Piecewise quadratic stability for affine Sugeno systems[END_REF]. Using the PI fuzzy controller and the Lyapunov technique, the authors in [START_REF] Chai | A general fuzzy control scheme for nonlinear processes with stability analysis[END_REF] show that asymptotic stability of the Takagi-Sugeno fuzzy systems can be ensured under certain restriction on the control signal and the rate of change of the output. Jadbabaie in [START_REF] Jadbabaie | A reduction in conservatism in stability and L2 Gain analysis of T-S fuzzy systems via Linear matrix inequalities[END_REF] introduces T-S fuzzy rules that describe the Lyapunov function where each T-S rule has fuzzy sets in the antecedents and quadratic Lyapunov function in the consequent. The global Lyapunov function is state dependant. This method requires fixing a priori bound on the variation of the state, to prove the stability of the T-S fuzzy systems [START_REF] Jadbabaie | A reduction in conservatism in stability and L2 Gain analysis of T-S fuzzy systems via Linear matrix inequalities[END_REF]. The major disadvantage of this approach is the fact that a systematic way to pick the bound of state variable vector has not been derived. The author only proposes to pick initial large number for this bound and then iteratively reducing this value by a trial and error method. The goal of this paper is double. Using the non-quadratic Lyapunov function, we will derive sufficient condition for stability and also stabilisation of the continuous T-S fuzzy system. This paper is organised as follows: section 2 presents an overview of T-S fuzzy systems. In section 3 we recall the LMI formulation of basic quadratic stability conditions. In section 4, first a fuzzy Lyapunov function is built by inference of the quadratic Lyapunov functions designed for each local model. We also develop a new proposition to take into account a finite number of rules activated in each time for less of conservatism. At last, using the LMI technique for efficient way of resolution, we develop a method to estimate an upper bound of the variation of the state to prove the stability of T-S continuous fuzzy systems. In section 5, using the concept of PDC to perform control laws and LMIs formulation, similar approach is used for the stabilization of T-S fuzzy closed loop system. In both cases numerical examples are given to illustrate the effectiveness of the proposed stability conditions.

Takagi-Sugeno fuzzy systems

The T-S fuzzy models are described by a set of fuzzy "if…then" rules with fuzzy sets in the antecedents and dynamic LTI systems in the consequent. These sub-systems represent local linear input-output relations of a nonlinear systems. A general T-S plant rule can be written as follows:

: rule i th ) ( ) ( ) ( THEN F is ) ( and..., F is ) ( IF i n i 1 1 t u B t x A t x t x t x i i n + = &
Where n is the number of rules (sub-models),

p t x ℜ ∈ ) ( is the state vector, l t y ℜ ∈ ) ( is the output vector, m t u ℜ ∈ ) ( is the input vector, pxm i pxp i B A ℜ ∈ ℜ ∈ , .
The global T-S fuzzy systems is inferred as follows:

∑ ∑ = = + = n i i n i i i i t x ù t u B t x A t x ù t x 1 1 )) ( ( )) ( ) ( ))( ( ( ) ( & (1)
where

∏ = = n j j i j i t x f t x ù 1 )) ( ( )) ( ( and )) ( ( t x f j i j is the membership grade of ) (t x j
to the fuzzy set i j F .

Finally we define:

∑ = = n i i i i t x ù t x ù t x 1 )) ( ( )) ( ( )) ( ( µ (2)
))

( ( t x i µ
is the normalised membership function in relation with the i th sub-system such that:

     ∀ ≤ ≤ = ∑ = n i t x t x i n i i .. 1 : 1 )) ( ( 0 1 )) ( ( 1 µ µ (3)
Equation ( 1) can be rewriten as:

∑ = + = n i i i i t u B t x A t x t x 1 )) ( ) ( ))( ( ( ) ( µ & (4)
It is assumed in this paper that the membership functions depend only on the state of the system and that only r locals models are activated for a given x(t) (the another (n-r) local models are ignored).

The minimum and maximum eigenvalues of the matrix X will be respectively denoted by ) ( ) ( X and X M m λ λ .

Basic stability conditions

The open-loop T-S system of ( 4) is defined as:

∑ = = n i i i t x A t x t x 1 ) ( )) ( ( ) ( µ & (5)
Let us recall first the basic stability conditions of the openloop system (5) derived using quadratic Lyapunov function.

Theorem [START_REF] Tanaka | Stability and design of fuzzy control systems[END_REF]: The continuous fuzzy system described by ( 5) is globally asymptotically stable if there exists a common 0 > = T P P such that:

n i PA P A i T i .. 1 : 0 ∀ < + (6) ■
The existence of such a common positive definite matrix described by LMIs ( 6) is a key to check the global stability of T-S continuous fuzzy system.

The authors in [START_REF] Narendra | A common Lyapunov function for stable LTI systems with commuting A-matrices[END_REF] propose an analytic way to finding a common P in the particular case where the matrices A i are asymptotically stable and commute pairwise.

Theorem [START_REF] Narendra | A common Lyapunov function for stable LTI systems with commuting A-matrices[END_REF]: For a given positive definite matrix Q, let P 1 ,…, P n be the unique solution of the following Lyapunov equations:

n i P A P P A Q A P P A i i i i T i T .. 2 : 1 1 1 1 1 ∀ = + - = + - (7)
If the following assumption is made:

n j i A A A A i j j i .. 1 : , ∀ = (8)
i. e. all matrices commute pairwise, then the T-S fuzzy system ( 5) is globally asymptotically stable and have as a common lyapunov function

x P x x V n T = ) ( Proof: see [12] ■
These results mean that a sufficient condition for the existence of a common P is that each A i must be asymptotically stable and commute pairwise.

It is also shown that a common P exists if the following conditions hold [START_REF] Teixeira | Stabilizing controller design for uncertain nonlinear systems using fuzzy models[END_REF]:

n j i A A j i .. 1 : , 0 ∀ < + (9)
The proof can be easily obtained by summing the LMIs [START_REF] Tanaka | Stability and design of fuzzy control systems[END_REF].

This result means that a necessary condition for the existence of a common P is that each A i and (A i +A j ) must be stable. These above conditions show clearly the conservativness of the method, i.e. we can find T-S fuzzy systems that are stable but there is no common P to prove the stability.

Stability analysis using non-quadratic Lyapunov function

So one way for obtaining relaxed stability conditions and overcome these limitations consists to use a non-quadratic Lyapunov function candidate defined as:

: rule i th x P x t x V t x t x i T i n = )) ( ( THEN F is ) ( and..., F is ) ( IF i n i 1 1
where each T-S rule has the same fuzzy sets as the fuzzy system in the antecedents and quadratic Lyapunov functions in the consequent.

Using the same inference as before we obtain the following state dependent Lyapunov function:

x x P x x V T ) ( ) ( = (10)
with

i n i i P t x x P ∑ = = 1 )) ( ( ) ( µ and 0 > = T i i P P (11)
where

)) ( ( t x i µ has the properties (3). The derivative of this Lyapunov function is:

) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )) ( ( t x x P t x t x x P t x t x x P t x t x V T T T & & & & + + = ( 12
)
Proposition 1: given a matrix as in [START_REF] Teixeira | Stabilizing controller design for uncertain nonlinear systems using fuzzy models[END_REF], one can bound the last term of (12) as follows:

∑ = ≤ r i i M T P t x t x t x x P t x 1 2 ) ( ) ( ) ( ) ( ) ( ) ( λ γ & & with:         ∂ ∂ = = ) ( )) ( ( 1 t x t x Max i n i µ γ
r is the number of rules activated simultaneously at each time such that 2≤ r≤ n and the eigenvalues are ordered such that:

) ( ... ) ( ... ) ( 1 n M r M M P P P λ λ λ > > > > Proof:
The proof follows from bounding the derivative of the normalized membership functions and considering the properties of the largest eigenvalue of a symmetric matrix [START_REF] Weinmann | Uncertain models and robust control[END_REF] ■ Theorem 1: Suppose that there exist Q and P i such that the following LMIs hold:

Q A P P A n i i i i T i - < + ∀ .. 1 :
and (13a)

Q A P P A A P P A n j i j i i T j i j j T i - < + + + ≠ ∀ 2 .. 1 : (13b) such that 0 )) ( ( )) ( ( ≠ t x t x j i µ µ
Then the global model ( 5) is exponentially stable if the following constraint on the variation of the state is satisfied:

∑ = ≤ r i i M M P Q t x 1 ) ( ) ( ) ( λ γ λ & ( 14
)
Proof: The proof is obtained by computing the derivative of the Lyapunov function [START_REF] Narendra | A common Lyapunov function for stable LTI systems with commuting A-matrices[END_REF] along the trajectory of the T-S fuzzy system (5) and takes into account the proposition 1.

The exponential stability is proved by considering the properties of the largest and the smallest eigenvalues of symmetric matrix [START_REF] Weinmann | Uncertain models and robust control[END_REF]. ■

Example

In order to appreciate the efficiency of the proposed method, we consider example given in [START_REF] Jadbabaie | A reduction in conservatism in stability and L2 Gain analysis of T-S fuzzy systems via Linear matrix inequalities[END_REF] where the T-S fuzzy model is composed of two rules.

) ( ) ( THEN F is ) ( IF : 1 rule 1 1 1 1 t x A t x t x = & ) ( ) ( THEN F is ) ( IF : 2 rule 2 2 1 1 t x A t x t x = &
The fuzzy sets are described by the following two triangular membership functions:

)) ( ( 1 )) ( ( , 3
) ( 1 )) ( ( 1 2 1 1 t x t x t x t x µ µ µ - = - = and:       - - - - = 2 1 4 5 1 A ,       - - - = 2
) ( ) )) ( ( )) ( ( ( ) ( 2 2 1 1 t x A t x A t x t x µ µ + = & (15)
Note that a common P satisfying (6) that proves the stability of the system (15) does not exist whereas the simulation results indicate that this system is asymptotically stable.

Stability conditions in theorem 1 give the following LMIs:

Q A P P A A P P A Q A P P A Q A P P A T T T T - < + + + - < + - < + 2 2 1 1 2 1 2 2 1 2 2 2 2
. 7 ) ( ≤ t x & (17)
The figure 1 shows that, with respect of the constraint ( 17), all trajectories of the above fuzzy systems converge to zero.

Non-quadratic stabilizability

In this section we utilise the PDC technique [START_REF] Wang | An approach to fuzzy control of nonlinear systems: stability and design: issues[END_REF][2] [START_REF] Tanaka | Robust stabilisation of uncertain non-linear systems via fuzzy control: quadratic stability, H control theory, and LMIs[END_REF] to design fuzzy controllers to stabilize fuzzy system (4). In the PDC approach, the consequent part is local linear feedback law and the antecedent part shares the same fuzzy sets the fuzzy system.

: rule i th ) ( ) ( THEN F is ) ( and..., F is ) ( IF i n i 1 1 t x -K t u t x t x i n =
the resulting global fuzzy controller, which is nonlinear in general, is:

∑ = - = n i i i t x K t x t u 1 ) ( )) ( ( ) ( µ (18) where )) ( ( t x i µ
is defined in (3). Substituting (18) in ( 4), we obtain the closed-loop T-S continuous fuzzy system:

∑∑ = = = n j n i ij i t x A t x t x 1 1 ) ( )) ( ( ) ( µ & ( 19
)
where

j i i ij K B A A - = 5.1

Stabilizability analysis using a non-quadratic Lyapunov function

To study the stability of the closed-loop T-S fuzzy system (19) and find appropriate K i gains, we need the following propositions to prove theorem 2.

Proposition 2 : given normalized membership functions as described in (3), the following inequality holds:

r t x t x j n j i i 1 1 )) ( ( )) ( ( - ≤ ∑ ≠ µ µ n r ≤ ≤ ∀ 2 (20)
where (r) is the number of rules activated simultaneously at each time.

Proof : the proof follows from the properties of the normalized membership function (3) and the corollary 4 in [START_REF] Takagi | Fuzzy regulators and fuzzy observers: relaxed stability conditions and LMI-based design[END_REF].

Proposition 3: given normalized membership functions as described in (3), the following inequality holds:

2 1 3 )) ( ( )) ( ( 1 )) ( ( t x t x r t x j n j i i n i i µ µ µ ∑ ∑ ≠ = - ≥ (21) n r ≤ ≤ ∀ 2
Proof : the proof follows from the proposition 2 and the properties of the normalized membership function [START_REF] Johansson | Piecewise quadratic stability for affine Sugeno systems[END_REF]. ■

In the following, for less of conservatism, it is assumed that only two rules (sub-models) are activated simultaneously at each time, assumption easy to realise in the case of triangular and trapezoidal normalized memberships functions.

Let us now define the following matrices:

ij k k T ij ijk A P P A H + = (22)
Theorem 2: Suppose that there exist Q and P such that the following LMIs hold:

0 0.5 1 1.5 -4 -2 0 2 x 1 ( t ) 0 0.5 1 1.5 -4 -2 0 2 x 2 ( t ) 0 0.5 1 1.5 0 5 10 time N o r m e ( d x / d t
) Q H n i iii - < ∀ .. 1 : and (23a) Q H H H n j i iij jii iji - < + + ∀ .. 1 : , (23b)
such that: 0 )) ( ( )) ( ( ≠ t x t x j i µ µ
Then the global model ( 19) is exponentially stable if the following constraint on the variation of the state is satisfied:

∑ = ≤ r i i M M P r Q t x 1 ) ( ) ( ) ( λ γ λ & (24)
Proof: The proof is obtained by computing the derivative of the Lyapunov function [START_REF] Narendra | A common Lyapunov function for stable LTI systems with commuting A-matrices[END_REF] along the trajectory of the T-S fuzzy system (19) and takes into account the proposition 1 and proposition 3. ■

The LMIs design (23) allows to obtain simultaneously the controller feedback gains K i , Q and P i matrices and then compute the constraint (24) which ensure the exponential stability.

Numerical example

Consider the following T-S fuzzy model composed of two rules: )) ( ( 1)) ( ( , 2

) ( ) ( ) ( THEN F is ) ( IF : 1 rule 1 1 1 1 1 t u B t x A t x t x + = & ) ( ) ( ) ( THEN F is ) ( IF : 2 rule 2 2 2 1 1 t u B t x A t x t x + = &
) ( 1 )) ( ( 1 2 1 1 t x t x t x t x µ µ µ - = + =
The state matrices and input vectors of each T-S local linear model are:

          - = 0 1 0 0 0 1 6 . 227 83 . 71 31 . 2 1 A ,           = 3 0 1 1 B           - = 0 1 0 0 0 1 180 107 604 . 1 2 A ,           - = 3
j i i ij K B A A - = We
know that BMI formulation is not convex problem and can have multiple local solutions. To linearize and solve these BMIs, we propose to decompose them on the following system :

         - < + + + - < + + + ≠ ∀ < + ∀ - < + Q A P P A A P P A Q A P P A A P P A j i A P P A i Q A P P A T T T T ii j j T ii ii i i T ii 12 2 2 12 21 2 2 21 21 1 1 21 12 1 1 12 ~2 , 1 : 0 ~2 , 1 : ~ (26)
Global optimality is ensured through the convex reformulation of the problem. This convexity is achieved through the matrix change of variables:

i i i j i ij i i QX X S X K Y X P = = = - , , 1 where: 2 , 1 :
, 0 j i X X T i i ∀ > =
Constraints (26) can be shown equivalent to the six following LMIs constraints:

2 .. 1 : ) ( ) ( 2 .. 1 : 0 2 .. 1 : j i S Y B B Y Y B B Y X A A A A X j i Y B B Y X A A X i S Y B B Y X A A X i ji i T i T ji ii j T j T ii i i j T j T i i ji j T j T ji i j T j i i ii i T i T ii i i T i i ≠ - < - - - - + + + ≠ < - - + - < - - + (27)
The feedback gain K i , matrix P i and a common matrix Q can be obtained as:

i i i j ij i i i P S P Q P Y K X P = = = - , , 1
Solving the LMIs system (27), we obtain :

          =           = 6 . 2320

Conclusion

The main originality of the proposed method concerns, on one hand, the reduction of the number of constraints guaranteeing the stability of the global system and, on the other hand, the laying down of a systematic method for computing upper bounds of the rates of variation of state variables guaranteeing the stability and the stabilization of T-S continuous fuzzy systems. We have presented new relaxed stability conditions and LMI-based designs for continuous fuzzy control systems. More precisely, the stability is proven if the rates of variations of the states are bounded. This class of Lyapunov function is substantially richer than the globally quadratic functions. By exploiting the non-quadratic stability of T-S continuous fuzzy systems and solving a convex optimisation problem formulated as a set of LMIs, we have shown how to pick systematically upper bounds of the variation of the state to guarantee the stability of the global T-S continuous fuzzy model. These conditions may be used to design fuzzy regulators and/or fuzzy observers.

20 4 2 2 A

 2 the global T-S fuzzy model is:

  Then the inequality[START_REF] Weinmann | Uncertain models and robust control[END_REF] allows to deduce the conditions of stability, namely the upper bound of the state variation of the system:

  64

Figure 1 :

 1 Figure 1: States trajectories of (15) with initials values [ ] 2 5 . 2 ) 0 ( -= x

where 1 1 F and 2 1 F

 11 are fuzzy sets described by the following two triangular membership functions:

  The inequality (24) allows to compute the upper bound of the rate of variation of state variables guaranteeing the exponential stabilization of the global closed loop T-S fuzzy systems.

	K	1 =	[ 7 .	39	. 36	07	. 205	] 38 ,
	K	2	=	[ -	8	34 .	-	36 . 24	-	279	52 .	]
										x &	) ( ≤ t	. 109	40	(28)
								Figure 2: States trajectories with initials
										values	x	(	0	)	=	[ -	1	1	-	2	]
																		. 2	1	7	6 .	. 33	8	3	.	3	9	.	8	. 68	8
																		1 P	7	.	6	142	.	1	. 146	8	,	2 P	. 9	8	. 155	5	. 201	7
																		33	8 .	146	8 .	. 1136	8	68	8 .	201	7 .
																		Q	=	10 3	 13   . . 84 52 0   04 . 3	58 . 3 56	72 04 . 84 .	72 . 52 454 . 13 72 . 58	    


S A Giamini

J Marquez-Velasco

P Tsipas

D Tsoutsou

… G Renaud

A Dimoulas

Molecular Beam Epitaxy of thin HfTe 2 semimetal films; a case of epitaxial layered topological Dirac Semimetal?

Epitaxial thin films of 1T-HfTe 2 semimetal are grown by MBE on AlN(0001) substrates. The measured in-plane lattice parameter indicates an unstrained film which is also azimuthally aligned with the AlN substrate, albeit with an in-plane mosaic spread, as it would be expected for van der Waals epitaxy. Angle resolved photoemission spectroscopy combined with first principles electronic band structure calculations show steep linearly dispersing conduction and valence bands which cross near the Brillouin zone center, providing evidence that HfTe 2 /AlN is an epitaxial topological Dirac semimetal.

Introduction

Atomically thin materials of the metal dichalcogenide family (MX 2 , M=Metal, X=S, Se, Te) crystallized in a stable 2D form have been the subject of intense investigation due to their versatile physical properties [START_REF] Ganesh | Recent Advances in Two-Dimensional Materials beyond Graphene[END_REF]. The molybdenum and tungsten sulfides and selenides are among the best studied 2D MX 2 materials. Stabilized in the hexagonal 2H prismatic crystal structure they are semiconductors with sizable gap between 1 and 2 eV making them suitable for nano and optoelectronic applications.

The corresponding transition metal ditellurides (WTe 2 and MoTe 2 ) have a more complex structural and electronic behavior. In WTe 2 , due to strong W-W intermetallic bonds, there is a lattice distortion which stabilizes this material in the orthorhombic (T d ) semimetallic phase [START_REF] Lee | Tungsten Ditelluride: a layered semimetal[END_REF]. MoTe 2 in its most stable 2H phase is a semiconductor and it has already been grown in its epitaxial form by MBE [START_REF] Roy | Structural and Electrical Properties of MoTe 2 and MoSe 2 Grown by Molecular Beam Epitaxy[END_REF]. However, it can also be prepared in the distorted monoclinic (1T') [START_REF] Clarke | A low-temperature structural phase transition in β-MoTe 2[END_REF] or low temperature (<260K) orthorhombic (T d ) [START_REF] Qi | Superconductivity in Weyl semimetal candidate MoTe 2[END_REF] metastable phases which are both semimetallic. Following predictions WTe 2 , [START_REF] Soluyanov | Andrei Bernevig, Type-II Weyl semimetals[END_REF] and T d -MoTe 2 [START_REF] Sun | Prediction of Weyl semimetal in orthorhombic MoTe 2[END_REF] have been experimentally proven [START_REF] Wu | Observation of Fermi Arcs in Type-II Weyl Semimetal Candidate WTe[END_REF][START_REF] Wang | Spectroscopic Evidence of Type II Weyl Semimetal State in WTe[END_REF][10] very recently to be type II Weyl semimetals (WSM). WSM are new forms of topological quantum matter and have brought much excitement due to analogies with high energy physics and the prospect for radically new applications in electronics and spintronics [START_REF] Nomura | Charge-Induced Spin Torque in Anomalous Hall Ferromagnets[END_REF][START_REF] Kurebayashi | Voltage-driven magnetization switching and spin pumping in Weyl semimetals[END_REF]. A related class of topologically non-trivial matter are the 3D Dirac semimetals (DSM) which are two overlapping WSM in momentum and energy space.

The DSMs, known also as "3D graphene" are formed by conduction and valence band crossing at some point of the Brillouin zone with a linear dispersion in all three directions in k-space. Most of experimentally proven Weyl semimetals (TaAs, NbAs) [START_REF] Yang | Weyl semimetal phase in the non-centrosymmetric compound TaAs[END_REF][START_REF] Xu | Discovery of a Weyl fermion state with Fermi arcs in niobium arsenide[END_REF] and Dirac semimetals ( Na 3 Bi, Cd 3 As 2 ) [START_REF] Liu | Discovery of a Three-Dimensional Topological Dirac Semimetal[END_REF][START_REF] Liu | A stable three-dimensional topological Dirac semimetal Cd 3 As 2[END_REF] are bulk materials and are synthesized by bulk CVT methodologies. On the other hand metal (di)tellurides and other 2D and layered semimetals can be synthesized in stable atomically thin film form either by exfoliation or by epitaxial growth on suitable large area substrates. This permits the investigation of possible interesting nontrivial topology at the very low thickness regime, eventually down to a single layer, and permits the fabrication of devices in compatibility with the scaling trends in current nanoelectronics. T d -MoTe 2 WSM and ZrTe 5 Dirac semimetal [START_REF] Zheng | Transport evidence for the threedimensional Dirac semimetal phase in ZrTe 5[END_REF] are two examples of 2D layered materials with non-trivial topology but so far only bulk crystals have been synthesized by CVT. It is not clear whether thin epitaxial films can be obtained from these materials, especially for the metastable orthorhombic WSM phase of MoTe 2 .

HfTe 2 and ZrTe 2 formed in the stable 1T phase can be alternative 2D di-tellurides whose properties remain unexplored. Early work [START_REF] Klipstein | Electronic properties of HfTe 2[END_REF][START_REF] Hodul | Solid state studies of HfSe 2-x Te x and HfTe 2-x[END_REF][START_REF] Smeggil | The Preparation and X ray Characterization of HfTe 2-x , x = 0.061[END_REF] provided evidence through electronic transport measurements that HfTe 2 is a semimetal but detailed information about its electronic band structure, especially for thin films, is lacking. Since the Hf-Te compounds have been synthesized [START_REF] Kjekshus | The crystal structure of HfTe 5[END_REF][START_REF] Izumi | Anomalous electrical resistivity in HfTe[END_REF] and theoretically predicted [START_REF] Abdulsalama | Structural and electronic properties of MX 3 (M = Ti, Zr and Hf; X = S, Se, Te) from first principles calculations[END_REF][START_REF] Yingdi | Single layer of MX 3 (M = Ti, Zr; X = S, Se, Te): a new platform for nano-electronics and optics[END_REF] to appear in several stoichiometries (HfTe 2 , HfTe 3 , HfTe 5 ), control in the desired stoichiometry by film growth methodologies could be challenging and needs to be investigated.

Here we present the first epitaxial HfTe 2 thin films, grown by MBE on AlN(0001)/Si(111) substrates. The epitaxial quality, the surface morphology, the microstructure and the electronic band structure are investigated in detail. Comparison with first principles calculations suggests the presence of non-trivial topology.

Results and Discussion

The HfTe 2 layers with several thickness from 2 ML to 10 ML are grown on AlN(0001)/Si(111) substrates using Molecular Beam Epitaxy (MBE). A two-step process is employed where in the first step Hf and Te are evaporated using e-gun and effusion cell, respectively, at substrate temperature of 530 °C. In the second step, the sample is annealed in-situ at 650 °C for 15 minutes to improve the crystalline quality.

The ratio of the evaporation rates R Te /R Hf is kept high (~15/1) in order to ensure sufficient incorporation of Te in the compound. The characterization results discussed below reveal that under these growth conditions the compound is produced by MBE in the 1T-HfTe 2 phase. No evidence of HfTe 3 or HfTe 5 phases is obtained.

Electron Diffraction

Reflection high energy electron diffraction (RHEED) patterns for the AlN substrate and HfTe 2 epilayer are given in Fig. 1. The sharp streaky pattern of the epilayer indicates a well ordered and flat HfTe 2 surface. Using the AlN RHEED pattern as reference and knowing that the lattice constant of wurtzite AlN is 3.11 Å, the lattice constant of the HfTe 2 is estimated to be 3.94Å. This is very close to the reported [START_REF] Klipstein | Electronic properties of HfTe 2[END_REF][START_REF] Hodul | Solid state studies of HfSe 2-x Te x and HfTe 2-x[END_REF][START_REF] Smeggil | The Preparation and X ray Characterization of HfTe 2-x , x = 0.061[END_REF] value of bulk, free standing HfTe 2 , which is equal to 3.95Å. Despite the large lattice mismatch (27%) it is remarkable that there is a good azimuthal alignment of the two materials indicating epitaxial growth without strain in agreement with van der Waals heteroepitaxy. The faint additional streaks along the [1-100] azimuth indicates that a small amount of 30 o or 90 o rotated domains may exist in the epilayer. A more detailed picture of the alignment and the domain microstructure can be obtained from high resolution X-Ray data given below.

Surface structure and morphology

The surface of 10ML-thick HfTe 2 epitaxial film is investigated by in-situ room temperature UHV-STM. Figure 2 shows a large area scan of 500x500 nm 2 (Fig. 2a) and a linescan across a layer step (Fig. 2b). The surface morphology (Fig. 2a) shows that the film is grown in a two-dimensional island form in such a way that a layer is almost fully formed before the next one starts to grow. Small darker areas are depressions where the layer is not fully grown. From the linescan, it can be inferred that the height difference between two layers is 6.79Å. This value is very close to the expected [START_REF] Klipstein | Electronic properties of HfTe 2[END_REF][START_REF] Hodul | Solid state studies of HfSe 2-x Te x and HfTe 2-x[END_REF][START_REF] Smeggil | The Preparation and X ray Characterization of HfTe 2-x , x = 0.061[END_REF] A high resolution image and a line scan are given in Fig. 3 along with the theoretically derived structure of 1T-HfTe 2 using Density Functional Theory (DFT).

The image shows the top Te atoms in a hexagonal arrangement with an in-plane lattice parameter of 3.97Å in good agreement with the experimental values [START_REF] Klipstein | Electronic properties of HfTe 2[END_REF][START_REF] Hodul | Solid state studies of HfSe 2-x Te x and HfTe 2-x[END_REF][START_REF] Smeggil | The Preparation and X ray Characterization of HfTe 2-x , x = 0.061[END_REF] as well as the theoretically estimated value of 3.94Å derived by our DFT calculations shown in Fig. 3c and3d.

Epitaxial quality and microstructure

Al-capped HfTe 2 layers with thickness of 2ML grown on AlN are investigated in a UHV-MBE-CVD diffractometer [START_REF] Cantelli | The In situ growth of Nanostructures on Surfaces (INS) endstation of the ESRF BM32 beamline: a combined UHV-CVD and MBE reactor for in situ X-ray scattering investigations of growing nanoparticles and semiconductor nanowiresJ[END_REF] at the European Synchrotron Radiation Facility (ESRF) using two different X-ray energies/wavelengths of 11 keV (1.127 Å) and 9.5 keV (1.305 Å). The incident beam, of energy resolution 10 -4 , had a very small (5 x10 -5 rad) divergence parallel to the surface. The incident angle was set at 0.22°, which is slightly below the critical angle for total external reflection perpendicular to the surface, in order to enhance the thin film signal while minimizing the background from the AlN buffer layer.

Figure 4a shows a reciprocal space map (RSM) covering one sixth of the in-plane reciprocal space. As shown in the Supplementary Information, precise radial scans were performed along the high symmetry directions as well as measurements along the rods of scattering perpendicular to the surface on all measurable Bragg peaks/rings. Rocking scans (see Fig. 4b) were also performed on all of them. The the quantity of 30° rotated domains that may exist (see also discussion of RHEED data, Fig. 1) is negligibly small. The in-plane lattice parameter of the HfTe 2 layer is deduced by fitting the positions of the corresponding Bragg peaks along radial scans, which yields a lattice constant of 3.967Å. This again is in very good agreement with the reported value for HfTe 2 [START_REF] Klipstein | Electronic properties of HfTe 2[END_REF][START_REF] Hodul | Solid state studies of HfSe 2-x Te x and HfTe 2-x[END_REF][START_REF] Smeggil | The Preparation and X ray Characterization of HfTe 2-x , x = 0.061[END_REF] as well as with what was estimated in this work by the analysis of RHEED and STM images and the theoretical value derived from our DFT calculations. Furthermore, the Full Width at Half Maximum (FWHM) of the HfTe 2 peaks along radial scans are also used to estimate the average in-plane domain size D defining areas of single crystalline quality which is found to be between 10 and 20 nm. It also showed the existence of small inhomogeneous strains in the HfTe 2 layer, with a FWHM of 0.04 Å for the corresponding in-plane lattice parameter distribution. Moreover, the in-plane mosaic spreads were deduced from rocking scan measurements across the AlN and HfTe 2 Bragg peaks (see Fig. 4b) yielding a mosaic spread of 11 ± 0.5° for HfTe 2 , significantly larger than the mosaicity of AlN which is only 0.7 o . Finally, the out-off plane thickness and structure, and in particular the stacking sequence, were determined by simulating the rods of scattering of the HfTe 2 layer (Supplementary). The position of the out-off plane allowed and forbidden Bragg peaks demonstrates that the HfTe 2 layer is of 1T structure, as opposed to 2H (Supplementary). The structural parameters deduced from the different X-ray measurements are summarized in Table 1.

Sample

AlN / HfTe The HfTe 2 electronic band structure of 4ML film is imaged along high symmetry crystallographic directions in the surface Brillouin zone by using in-situ angle resolved photoelectron spectroscopy (ARPES) and the results are presented in Fig. 5c.

In general there is a fairly good agreement between theory (Fig. 5a, and supplementary) and experiment (Fig. 5c), especially regarding the steep variation of the valence band showing a maximum at  and the appearance of conduction band minima at the M points as predicted by theory. The experimental results indicate a semimetal but, compared to theory (Fig. 5a), there is a smaller energy overlap between conduction and valence bands (Fig. 5c). This is mainly due to an apparent shift of the valence band at  to lower energies in such a way that its maximum is located exactly on the Fermi level, instead of being 0.4 eV above it as predicted by theory (Fig. 5a and Supplementary). This could be interpreted as a result of charge exchange with the substrate or unintentional defect-related doping which could raise the Fermi level closer to the apex of the cone where VB and CB nearly touch each other at the  point. However in this case we would expect that the Fermi level is positioned well above the conduction band minima at M, which is not supported by the ARPES images (Fig. 5c) since the Fermi level is clearly located near the bottom of the CB at M. Although misfit or thermal strain could modify the band structure potentially explaining the discrepancy with theory near , the presence of strain is not supported by HRXRD, STM and RHEED results all of which suggest a relaxed (unstrained) film with lattice parameter near the bulk value.

A faint band is observed (Fig. 5c) connecting the valence band near  with the conduction band at M. Since it does not appear in our DFT calculations of the bulk bands (Fig. 5a), we attribute this feature to a surface band with possible non-trivial topology in connection to the band inversion predicted in Fig. 5b. It should be noted that similar (topological) surface states connecting electron and hole pockets have been observed recently [START_REF] Xu | Discovery of Weyl semimetal state violating Lorentz invariance in MoTe[END_REF][START_REF] Liang | Electronic Evidence for Type II Weyl Semimetal State in MoTe[END_REF][START_REF] Deng | Experimental observation of topological Fermi arcs in type-II Weyl semimetal[END_REF] in the orthorhombic (T d ) phase of the WSM MoTe 2 .

Finally, the faint feature appearing at k x,// = 0.8Å -1 along K in Fig. 5c, not present in theoretical calculations (Fig. 5a), originates from the CB minima at the M points in neighboring BZs as better explained in the discussion of the Fermi surface below.

The measured Fermi surface in a portion of the k-space is shown in the k x -k y constant energy contour plots (CECP) in Fig. 6a,6b. With the help of the Brillouin zone diagram in Fig. 6c, we identify the electron pockets around the M points which are clearly observed in both CECP (Fig. 6a and6b); a narrower hole pocket at the  point is also visible. The two electron pockets around M, seen in the CECP in Fig. 6b, have a finite spread and produce the faint signal in the E-k x,// image of Fig. 7a which is not expected from theoretical simulations (Fig. 5a). We further investigate the band dispersion at the BZ center close to the Fermi level as shown in Fig. 7. The E-k // dispersion (Fig. 7a) and the CECPs (Fig. 7b) show a structure consisting of two cone-like bands with hexagonal distortions, one within another. The outer band exhibits a parabolic behavior with a maximum ~ 0.

Figure 1 .

 1 Figure 1. RHEED pattern of HfTe 2 grown on AlN(0001)/Si(111) substrates along (a)

Figure 2 .

 2 Figure 2. Large area scans of 10ML of HfTe 2 grown on AlN/Si(111). (a) A scan area of 500x500 nm 2 shows a two-dimensional island growth mode. (b) The step between

Figure 3 .

 3 Figure 3. (a) High resolution scan of 10ML of HfTe 2 grown on AlN/Si(111) in 3x3nm 2 area. (b) The linescan from (a) showing a measured in-plane lattice constant

  diffraction from three hexagonal lattices is clearly visible on the in-plane RSM, the sharp but hardly visible peaks of the Si(111) substrate, the peaks from the 200 nmthick hexagonal AlN layer and those of the HfTe 2 thin layer. No other feature is visible, except a wide ring attributed to the (2-20) Bragg reflection of the aluminum capping layer. The positions of these peaks (together with out-off plane ones, see Supplementary Information), yield the epitaxial relationships: Si[1-10](111)//AlN[10-10](0001)// HfTe 2 [10-10](0001)//Al[1-10](111). These results are in fairly good agreement with the RHEED images and they further prove that the material is azimuthally aligned with the AlN substrate despite the large lattice mismatch, consistent with van der Waals epitaxial growth. It should be noted that no peaks are detected at 30° positions within the sensitivity limits of the technique indicating that

Figure 4 .

 4 Figure 4. (a) In-plane reciprocal space map (RSM) of a 2ML thick HfTe 2 sample,

Figure 5 .

 5 Figure 5. (a) DFT band structure calculation of bulk 1T-HfTe 2 with spin-orbit

Figure 6 .

 6 Figure 6. (a) constant energy k x -k y contour plots of the Fermi surface along the high

  2 eV below E F . On the other hand the inner band shows a steep linear dispersion crossing the outer band at E b ~-0.4 eV and forming a cone with its apex located exactly at the Fermi level. The linear variation of the inner cone indicates Dirac-like dispersion which is characteristic of massless particles. The Fermi velocity v F is calculated to be ~0.76 10 6 m/sec which is by 25% smaller compared to that of graphene. The behavior is similar to the dispersion reported in 3D topological Dirac semimetal Na 3 Bi. Taking into account the similarities with the Na 3 Bi and the predicted near crossing behavior in Figs.5a and 5bas discussed above, it is anticipated here that the linear dispersion near  in Fig.7provides evidence that HfTe 2 /AlN is an epitaxial topological Dirac semimetal. Unlike the case of Na 3 Bi and Cr 3 As 2 Dirac semimetals which exist in bulk form, HfTe 2 is a 2D material and can be scaled down to a single monolayer either by epitaxial growth or by exfoliation thus providing an excellent opportunity to investigate the transition of topological 3D Dirac semimetal properties to the 2D limit.

Figure 7 .

 7 Figure 7. (a) Detail of the band structure around the  point. The inner bands form a

Table 1 .

 1 Summary of results for the calculated lattice parameter and mosaic spread of a AlN/HfTe 2 /Al(capping) sample for the three materials. The average domain size is also shown for HfTe 2 , calculated to be between 10 and 20 nm.

	2 2ML / Al capping

Acknowledgements: We acknowledge financial support from ERC Advanced Grant SMARTGATE-291260


Dr P Tsipas

Dr S Fragkos

Dr D Tsoutsou

A Dimoulas
email: a.dimoulas@inn.demokritos.gr

Dr C Alvarez

Dr R Sant

Dr G Renaud

Dr H Okuno

Keywords: 2D materials, molecular beam epitaxy, orthorhombic MoTe 2, transmission electron microscopy, Weyl semimetal

FULL PAPERS

The room temperature observation of the noncentrosymmetric orthorhombic phase of epitaxial MoTe 2 on InAs(111) substrate is reported. Due to enlarged lattice parameters of 1-3 ML films a type II topological Weyl semimetal is predicted with Weyl nodes located a few meV below the Fermi level, accessible to transport. The direct observation at room temperature of the noncentrosymmetric orthorhombic topological Weyl semimetal phase in epitaxial thin films of MoTe 2 grown on InAs(111)/Si(111) substrates by molecular beam epitaxy is reported. The orthorhombic phase is typically found at lower temperatures but its observation at room temperature in this work is attributed to the enlarged lattice parameters, influenced by the substrate, which stabilize an interlayer antibonding state compatible with the orthorhombic stacking. First-principles calculations predict eight type II Weyl nodes which are located below (but near) the Fermi energy making them accessible to charge transport and creating the prospect for practical applications exploiting the nontrivial topological properties. The orthorhombic phase coexists with an unconventional triclinic layer stacking which is different than the monoclinic or orthorhombic structures but it is centrosymmetric and topologically trivial. DOI: 10.1002/adfm.201802084 Q3 Q4 Q5 1T′ is the most stable phase of WTe 2 at room temperature (RT). On the other hand, for MoTe 2 the most stable phase at RT is the 2H, while the 1T′ is metastable lying only about a few tens of meV higher in energy. [5] The small difference in energy allows the RT stabilization of 1T′ utilizing bulk growth methodologies [6][7][8][9][10] at high temperature typically followed by rapid cooling to RT. The metastability of 1T′ implies that reversible transformations from the semiconducting to the metallic phase through a small energy barrier could be possible [5] by applying external stimuli. This has already been demonstrated experimentally at RT by applying strain [11] or electrostatic doping by a gate bias, [12] rendering MoTe 2 a promising candidate material for the realization of useful electronic devices. Moreover, MoTe 2 and WTe 2 exhibit exciting physical properties adding functionality and significantly extending the range of applications. In the limit of a monolayer with a centrosymmetric structure these materials have been predicted to be 2D topological insulators (or quantum spin Hall insulators) [13] which have been experimentally verified recently for the case of WTe 2 . [14] In the few layer regime, 1T′ WTe 2 adopts at RT an orthorhombic Pmn21(T d ) structure (γ-phase) which lacks inversion symmetry thus becoming a topological type II Weyl semimetal as first predicted [15] and later verified [16] for bulk WTe 2 material. On the other hand, few layer or bulk 1T′ MoTe 2 , at RT is typically found in a centrosymmetric monoclinic P21/m β-phase which is proposed to be a quantum spin Hall state. [17] However MoTe 2 undergoes a structural phase transition from the β-phase to the noncentrosymmetric orthorhombic (T d ) γ-phase at lower temperatures, typically below 250 or 150 K, in which case MoTe 2 is predicted [18] to become a type II topological Weyl semimetal (similar to γ-WTe 2 ), already verified by ARPES in bulk MoTe 2 . [8,9,19,20] The number of Weyl nodes, their position in k-space and energy, and the nature of surface states (topological or trivial) in γ-MoTe 2 depend sensitively on the lattice parameters and the strain. Theoretical values [21] of the lattice parameters differ a lot depending on the density functional used and deviate from experimental values, [6][7][8][9] which also depend a lot on the fabrication method. Most of the works find 8 Weyl nodes [8,9,18,20,22] while for certain choice of small lattice parameters only 4 Weyl nodes are predicted, [23] the xxxx 1. Introduction 2D group VI (Mo,W) transition metal disulphide and diselenide materials, typically stabilized in the hexagonal 2H prismatic structure, are semiconductors and have been extensively studied due to their great potential for diverse and versatile applications. [1][2][3] The Mo and W ditellurides (WTe 2 and MoTe 2 ) are notable exceptions because structural and electronic instabilities drive a transformation to a lower symmetry, distorted 1T (often referred to as 1T′) octahedral crystal structure with metallic or semimetallic properties. [4] Q6 [8,9,18,20,22] are closer to Fermi energy (E F ) compared to WTe 2 , [15,16,24,25] however they are always above E F , including those in Mo x W 1-x Te 2 , [26,27] thus inhibiting both their observation by ARPES and their effective use in functional electronic devices.

Adv. Funct. Mater. 2018, 1802084 U N C O R R E C T E D P R O O F
To shed light on the electronic structure of γ-MoTe 2, the Weyl node configuration in particular, and to exploit the topological properties in practical applications, it is important to control the lattice parameter and to stabilize the γ-phase of MoTe 2 at RT by suitable epitaxial methods and substrates. More generally, the ability to epitaxially grow MoTe 2 thin films by synthetic methods on cm-scale crystalline substrates is essential to enable device processing. Epitaxial processes are typically nonequilibrium and are influenced by the substrate so the most stable structural phases, layer stacking, and physical properties of epitaxial thin films may be different compared to those produced by equilibrium bulk methodologies. In most cases, CVD-grown MoTe 2 thin films are found in the RT most stable 2H structure although 1T′ structures are also obtained by postgrowth temperature treatments. [28][29][30] Growing epitaxial thin films of MoTe 2 (and WTe 2 ) by MBE is challenging because Mo (W) and Te have very similar electronegativities [31] so during coevaporation of Mo (W) and Te, the material tends to form intermetallic MoMo or WW bonds which do not favor the incorporation of Te in the material resulting in poor Te-deficient and highly nonstoichiometric films. Following the successful growth of the stable 1T′-WTe 2 by MBE on a variety of 2D substrates by employing interrupted growth methodology, [31] a few works [32][33][34] have reported the stable 2H MoTe 2 phase by MBE but the metastable 1T′ MoTe 2 phase was elusive until recently when two independent groups [35,36] showed evidence of room temperature 1T′ MBE films mixed with 2H on epi-graphene/SiC substrates. [35] However, they lack of information about the epitaxial quality and the thin film structure; it is not known in particular whether they adopt the monoclinic (β-MoTe 2 ) or orthorhombic (γ-MoTe 2 ) structure. Stabilization of the latter Weyl semimetal phase is highly desirable but difficult to obtain at RT as already mentioned above. An interesting approach is to substitute ≈8% of Mo by W to engineer the γ-phase as it has already been shown using bulk single crystals. [37] Recently, indirect evidence of RT γ-MoTe 2 has been reported in thin films exfoliated from bulk where low dimensionality is thought to play a role. [38] Direct observation of the γ-MoTe 2 phase at RT either on bulk or epitaxial material has not been reported.

In this paper we report on the direct RT observation of the important orthorhombic noncentrosymmetric Weyl semimetal γ-phase in epitaxial 3 ML MoTe 2 films grown layer by layer on InAs(111)/Si(111) substrates by MBE. Moreover, we observe an unconventional triclinic layer stacking, which is however centrosymmetric, topologically predicted to be trivial with no Weyl points. The measured lattice constants of our epitaxial orthorhombic phase are found to be notably larger than the previously reported experimental values from bulk γ-phase MoTe 2 . We argue in this paper that the enlarged lattice parameters have important consequences on the energy position of the Weyl points as well on the stabilization of the γ-phase at room temperature in epitaxial thin films.

Results

Crystal Structure and Reciprocal Lattice

The structure details and the reciprocal lattice are described in brief to facilitate the data interpretation. As already mentioned, the 1T distorted (1T′) structure of MoTe 2 is typically found either in the RT monoclinic P21/m β-phase or the low temperature (T < 250 K) orthorhombic (T d ) Pmn21 γ-phase (Figure 1). Here we use the notation as follows. 1T′ refers to the distorted 1T without distinguising between orthorhombic and monoclinic stacking. T d -MoTe 2 refers to orhorhombic stacking while β-MoTe 2 refers to monoclinic stacking. A characteristic difference between the two phases is the interatomic distance d (Figure 1b,c) which becomes larger in the orthorhombic structure compared to the monoclinic (d o > d m ) due to a parallel shift of the second (top) layer with respect to the bottom one. [21] The undistorted 1T lattice can be described either by a primitive hexagonal unit cell (solid lines in Figure 1d) or by a rectangular surface nonprimitive unit cell (dashed line) which is double the hexagonal primitive unit cell (solid line), with lattice parameters a T and b T such that a T /b T = √3. On the other hand, the surface rectangular primitive unit cell of 1T′ has a T′ /b T′ > √3 deviating from the "ideal" value of √3. Similarly, the reciprocal space or 1T′ (Figure 1e) has a rectangular primitive unit cell with the reciprocal lattice constant ratio b* T′ /a* T′ > √3. The deviation of these ratios from ideal values produces characteristic signatures in our data as will be discussed below facilitating the nanostructure characterization of our films.

Sequential Layer by Layer Thin Film Epitaxial Growth and Surface Structure

In-terminated InAs(111)/Si(111) substrates are prepared by ex situ chemical cleaning followed by in situ thermal annealing and, where required, by a mild Ar + sputtering (see the Experimental Section) prior to annealing, until a 2 × 2 reconstruction pattern is obtained in reflection high energy electron diffraction (RHEED) (Figure S1, Supporting Information), indicative of a clean, oxide-free surface. Thin 1-3 ML MoTe 2 films are grown (see the Experimental Section) in a layer-by-layer sequence, with each layer grown in a two-step process. First, a single layer MoTe 2 is grown at a low temperature of 280 °C and Te/Mo ratio of 100/1, adopting a beam-interrupted epitaxy method similar to that used for the MBE growth of WTe 2 . [31] After 30 s of W and Te codeposition, the growth is interrupted by closing the Mo shutter for about 60 s, while the film is continually exposed to Te and then the growth continues again by opening the Mo shutter for another 30 s. The process is repeated until 1 ML of MoTe 2 is completed. The growth rate is estimated to be 0.5 ML min -1 . Subsequently, the sample is annealed at 400 °C in the presence of Te flux (0.5 Å s -1 ) for 30 min to improve the crystallinity and the surface ordering monitored by RHEED (Figure S1, Supporting Information). Once the first layer is fully formed and optimized, the second and third layers are grown by repeating the same process, thus enabling the sequential layer-by-layer growth of MoTe 2 thin films.

Adv. Funct. Mater. 2018, 1802084 U N C O R R E C T E D P R O O F
© 2018 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
The surface nanostructure of 1 ML 1T′ MoTe 2 is imaged in Figure 2a by in-situ ultrahigh vacuum scanning tunneling microscopy (UHV-STM) at RT. The domains have an average lateral size ≈40 nm and they differ in height about 3.5 Å (Figure 2b), which is the same as the height of monoatomic steps of the underlying InAs substrate. It is therefore concluded that the domain formation in MoTe 2 originates from the monatomic step nanostructure of the InAs surface, consistent with scanning transmission electron microscopy (STEM) observations shown in subsequent section (Figure 3). Deposition of a second layer results in the formation of 2D islands (Figure 2c) with a height of ≈7 Å (Figure 2d) which is roughly equal to the thickness of a single 1T′ MoTe 2 layer including the van der Waals gap between the top and the bottom layers. The islands coalesce to form a complete second layer.

Atomic resolution STM images of a single domain are consistent with the symmetry of 1T′ MoTe 2 surface (Figure 2e). The Fourier transform in Figure 2f yields the surface reciprocal space in similarity with the schematic of Figure 1f. In a different part of the surface, two domains A and B rotated 60° with each other are imaged (Figure 2g) resulting in a more complex reciprocal space (see the Fourier transform of Figure 2h) where additional periodicities (1 and 2 in Figure 2h), define a reciprocal lattice for domain B (white dashed rectangles) rotated by 60° with respect to the one corresponding to domain A (red dashed rectangles). Consideration of the complex reciprocal space composed of superimposed rotated lattices is essential Adv. Funct. Mater. 2018, 1802084 for the interpretation of X-ray diffraction (XRD), angle resolved photoelectron spectroscopy (ARPES) and RHEED measurements reported in the subsequent sections and the Supporting Information.

U N C O R R E C T E D P R O O F

Orthorhombic and Unconventional Triclinic Layer Stacking Structure

The MoTe 2 thin films are evaluated in cross-section by STEM at RT to identify the film layer stacking. Figure 3 shows crosssection images for both1 and 3 ML MoTe 2 . In Figure 3a, two regions of 1 ML of 1T′ MoTe 2 are shown interrupted by an atomic step in the InAs, ≈3.5 Å, hence forming domains with different heights, similar to STM observations in Figure 2a. It is worth mentioning that there is no evidence for the 2H prismatic phase which is the most stable at RT, therefore the thin films are in a pure 1T′ phase.

As the growth continues from 1 to 3 ML 1T′ MoTe 2 , two dominant phases are observed, namely the orthorhombic T d -MoTe 2 phase (Figure 3c), comprising two layers in the unit cell, which typically appears at lower temperatures (<250 K) and an unconventional triclinic stacking, containing one layer per unit cell (Figure 3e), not previously reported. A fundamental difference between the triclinic stacking of Figure 3e and the conventional room temperature stable β-MoTe 2 monoclinic phase is that the second layer is rotated by 180° with respect to the first layer. The lattice parameter along the c-axis in the T d orthorhombic structure (Figure 3b,c) is measured to be between 14.1 and 14.2 Å (Figure 3d), which is significantly larger than the reported experimental c values from bulk T d -MoTe 2 crystals [6][7][8][9] found to be in the range between 13.861 and 13.8935 Å. Moreover, among the different 1T′ structures (Figure 1), the most stable room temperature monoclinic stacking, known as β-MoTe 2 (Figure 1c) is absent in our films. Additionally, fast Fourier transform (FFT) patterns were calculated from the regions containing only the MoTe 2 film in the STEM images; they are shown in Figure 3f-h the on-axis (400) peak and two off-axis (220) peaks from rotated domains as well as the triplet involving (200) and two (110) peaks although the latter triplet is not resolved very well. As a consequence, the radial scans along h and h = k directions in Figure 4b show doublets for certain diffractions.

U N C O R R E C T E D P R O O F
U N C O R R E C T E D P R O O F
U N C O R R E C T E D P R O O F
In the doublet the peak due to the off axis signals superposition have larger transversal width as seen from the rocking curves of Figure 4c and d because they have the contributions from two off-axis diffraction peaks (see the deconvolution of (310) diffraction peak in Figure 3d). From the FWHM of the on-line single peak (020) in Figure 3c an in-plane mosaicity of 3.71° is estimated which is substantially higher than that of the InAs substrate (0.79°); also higher than the mosaicity of 1.79° observed in epitaxial ZrTe 2 on InAs(111) substrate. [39] Considering only the well resolved on-axis single peaks (020), (300), and (400), the reciprocal (a*, b*) and direct (a, b) lattice parameters can be accurately determined from the RSM and the radial scans as follows. The position of the 020 peak in the q space is q = 3.5894 Å -1 and it is equal to two times the reciprocal space unit cell lattice constant b (Figure 4b). Therefore b* = q/2 = 1.7947 Å -1 and b = 2π/b* = 3.501 (±0.003) Å. Looking along the h and k directions in Figure 4a, the (300) and (400) peaks are 3a* and 4a*, respectively, and by considering their location in k-space from Figure 4b, a* = 0.99105 Å -1 (averaged over the four peaks, two along each direction h and k) and a = 2π/a* = 6.340 (±0.005) Å are obtained. These values are in reasonably good agreement with the ones deduced from RHEED (Figure S1, Supporting Information).

It is worth noting that the values a, b (determined from GIXD) and c (determined from STEM) in our epitaxial thin films are larger than the ones measured from bulk T d -MoTe 2 crystals. [6][7][8][9] This is attributed to the influence of the substrate and the layer-by-layer sequential growth adopted in the present work.

Electronic Band Structure of 1T′ MoTe 2 Epitaxial Thin Films

The electronic band structure of 1 ML 1T′ MoTe 2 imaged by in situ ARPES is given in Figure 5, in direct comparison with 1 ML density functional theory (DFT) first-principles calculations using the experimental lattice parameters deduced from our epitaxial thin films. In contrast to exfoliated single crystal 1T′ MoTe 2 , our epitaxial material contains rotated domains differing by 60° between each other (see also discussion of STM, XRD, and RHEED in Figures 2 and4 and Figure S1, Supporting Information). This gives rise to a complex Brillouin zone (BZ) configuration (Figure 5d) resulting in an overlap of the electronic band structure along two different directions in the BZ, as for example ΓY and ΓS directions, in the ARPES spectrum of Figure 5a, yielding a complex dispersion adversely impacting energy and momentum resolution. By employing second derivative of the energy dispersion the resolution is improved (Figure 5b,c) yielding a reasonably good agreement with DFT simulations along the high symmetry lines in the BZ which are indicated by thick (blue and red) arrows in Figure 5d. Notably, the feature at around k // ≈ 1 Å -1 in Figure 5b is located at the Γ point of the neighboring BZ (indicated by broken line in Figure 5d) and is effectively a replica of the band at k // = 0 Å -1 .

There is also evidence for the existence of electron pockets near the Γ point along ΓX as predicted by DFT in Figure 5c (blue line).

In principle, by applying ARPES to thicker films, it should be possible to distinguish between orthorhombic T d -phase and triclinic stacking and therefore identify the most dominant configuration however the ARPES spectra for 2 and 3 ML (not shown here) are very similar to the ones for 1 ML (Figure 5). This could be explained by considering that 21.22 eV ARPES known to be a surface sensitive technique probes the topmost layer which is coupled weakly to the bottom layers showing the behavior of a single layer in ARPES.

Using the measured set of lattice parameters in our epitaxial thin films (a = 6.340 Å, b = 3.501 Å, c = 14.15 Å) we have performed DFT bulk band structure calculations for the noncentrosymmetric orthorhombic (T d ) structure with the aim to predict the existence of Weyl nodes and find their location in energy and k-space. By considering the topmost valence band and the next (conduction) band we find eight Weyl nodes as a result of broken inversion symmetry (Figure 6), which are all located in the k z = 0 plane and are symmetrically placed in k-space with respect to the high symmetry directions X-Γ-X and Y-Γ-Y as shown in the schematic of Figure 6a. Their exact coordinates in k-space and energy are as follows:

W1 -: (k x = 0.18707 Å -1 , k y = 0.0456 Å -1 , k z = 0 Å -1 ) at E-E F = -1.1 meV, W2 + (k x = 0.1871 Å -1 , k y = 0.0351 Å -1 , k z = 0 Å -1
) at E-E F = -12.9 meV. The Weyl nodes in each pair have opposite chirality and are located below the Fermi level but very close to it (Figure 6b). The Fermi arc connecting the Weyl nodes with opposite chirality (+, -) is shown as part of the surface state passing through them in Figure 6b. It should be noted that this set of lattice parameters deduced from our epitaxial thin films yields Weyl nodes which are closer to the Fermi level than any other reported [8,9,18,20,23,24] Weyl nodes for MoTe 2 in the literature. While in all of these cases, they are typically located a few tens of meV above E F , in our epitaxial thin films the Weyl nodes are predicted just below the Fermi level which facilitates observations by ARPES and makes them accessible to transport, thus suitable for electronic applications where topological properties can play a critical role.

The unconventional triclinic layer stacking as evidenced by STEM in Figure 3d, yields a calculated bulk band structure (Figure S4, Supporting Information) with no Weyl points, a result which is expected since this structure is centrosymmetric.

To gain more insight about the possible role of the enlarged lattice parameters on the stabilization of the orthorhombic T d -phase at room temperature in our epitaxial thin films we analyze our first-principles calculations which have been performed using two different sets of experimental lattice parameters (Figure 7) following similar methodology and reasoning as in ref. [21] Set #1 (a = 6.3341 Å, b = 3.4751 Å, c = 13.8816 Å) in Figure 7a is an average obtained from experimentally reported values for bulk, free standing T d -MoTe 2 , [6][7][8][9] 7c,d, it is inferred that interlayer coupling occurs through an antibonding configuration along interlayer distance d between inner Te atoms having large and opposite sign wavefunction values. Since B2 is fully occupied in both cases the degree of occupancy of the interlayer antibonding band B1 along ΓA and near Γ determines the tendency of MoTe 2 to stabilize in the T d phase. For Set #1 of the experimental bulk parameters obtained from the literature (Figure 7a; Figure S4, Supporting Information), a significant portion of the uppermost valence band B1 near Γ point along ΓA is unoccupied. In contrast, for Set #2 of our epitaxial thin film enlarged experimental parameters in Figure 7b, B1 is fully occupied thus strengthening the interlayer antibonding state which favors an elongated interlayer distance d (Figure 7c). This elongation stabilizes T d stacking configuration at RT since d( = d o ) for orthorhombic stacking is large and indeed larger than the corresponding monoclinic distance d m (see Figure 1b,c).

U N C O R R E C T E D P R O O F

Discussion

In this work we use MBE to grow the distorted 1T (1T′) phase of MoTe 2 epitaxially on InAs(111)/Si(111) substrates. Adopting an interrupted growth methodology at low growth temperatures (≈280 °C) first applied to the MBE growth of WTe 2 on MoS 2 and HOPG substrates, [31] we show by STEM and GIXD that a Adv. Funct. Mater. 2018, 1802084 pure 1T′ MoTe 2 phase can be obtained with improved crystallinity after postgrowth annealing at 400 °C. Despite the fact the 2H prismatic phase of MoTe 2 is more stable at RT, a pure 1T′ MoTe 2 phase is observed here attributed to the influence of the substrate due to tensile biaxial strain and/or electron doping. Tensile strain between 0.3% and 3% under uniaxial conditions as theoretically predicted [5] or 0.2% as experimentally demonstrated [11] is thought to drive the transition from the stable 2H to the metastable 1T′ MoTe 2 phase at or near RT. In our case, the in-plane a and b lattice parameters of the epitaxial films are accurately measured by synchrotron GIXD (Figure 4) and found to be larger than those obtained in free standing films exfoliated from bulk, [6,9] indicating a relatively small epitaxial tensile strain configuration with ε xx = δa/a ≈ +0.1% and ε yy = δb/b ≈ +0.7%, which may partly explain the transformation to 1T′ phase at RT in our epitaxial thin films. It has also been predicted [17] that electron doping ≈0.1e -per unit cell induces the 2H → 1T′ transition in MoTe 2 . Electron doping could be a possible scenario in our case, since the electronic charge can be provided by the InAs substrate which is known to have strong n-type semiconductor character at the surface due to the accumulation of a dense electron gas. [40] MBE, a typically far from equilibrium process, can produce thin film structures which cannot be obtained under equilibrium conditions. Beyond the substrate effects (strain, doping) already discussed above, the ability to sequentially grow one layer on top of another as in the present work, allows different stacking configurations which cannot be obtained in films exfoliated from bulk. Such a layer stacking, namely a triclinic structure is observed here by STEM (Figure 3e; Figure S2, Supporting Information), where all layers are in phase presenting a fundamental difference compared to the typical monoclinic or orthorhombic structures where two consecutive layers are 180° out of phase (Figure 1). Hints of such in-phase layer stacking are found also in MBE-grown WTe 2 [31] but not sufficiently emphasized. Although the triclinic structure is centrosymmetric with trivial topology (no Weyl nodes found), it is very interesting that here the important noncentrosymmetric orthorhombic (T d ) Weyl semimetal phase is observed by STEM coexisting with the triclinic structure. This orthorhombic phase is typically reported in the literature below 250 or 150 K so questions are raised about the possible origin of its stability at RT as observed in our work. The reduced dimensionality analysis previously proposed to explain the indirect observation [38] of RT T d -MoTe 2 in 12 nm thick films exfoliated from bulk does not directly apply to our ultrathin (1-3 ML) films. Our DFT analysis for T d -MoTe 2 (Figure 7) shows that the enlarged lattice parameters of our epitaxial thin films result in a characteristic band structure where the uppermost valence band B along ΓΑ is fully occupied dispersing well below the Fermi level similar to what is observed for T d -WTe 2 . [21] It should be noted that in the case of T d -WTe 2 , the full occupation of this band is correlated with the absence of a barrier in the energy profile for the transition from β to T d -phase [21] which explains the stability of T d -phase at RT. A similar situation could apply for the epitaxial MoTe 2 in this paper. Expressing it in a different way, it can be said that this fully occupied band B (Figure 7) is related to an antibonding state [21] along the interlayer distance d (see Figure 1), then the T d -MoTe 2 is favored because this phase is compatible with an enlarged d (d o > d m in Figure 1) as it would be expected for an antibonding state. This analysis shows that the enlarged lattice parameters in the epitaxial thin films could explain the observation at RT of the orthorhombic T d -MoTe 2 instead of the monoclinic phase.

U N C O R R E C T E D P R O O F
The RT orthorhombic MoTe 2 reported here lacks inversion symmetry so it is expected to be a topological Weyl semimetal. Using the enlarged experimental lattice parameters found in this work, a DFT analysis (Figure 6) for the T d -MoTe 2 predicts eight type II Weyl nodes which are located very close to the Fermi level and just below it, in notable difference with previous results where all Weyl points were found above E F . The occupation of states near the Weyl nodes makes them visible in ARPES electronic band imaging and accessible to transport which means that epitaxial thin films of T d -MoTe 2 on InAs(111) substrates can be used to fabricate devices which could exploit the nontrivial topological properties of these films in practical RT applications.

Future work should focus on improving the epitaxial quality of these films. Although there is a very good in-plane epitaxial alignment between MoTe 2 and InAs substrate such that [100] MoTe2 // [11-2] InAs , and [010] MoTe2 //[01-1] InAs (Figure 4), with no signs of polycrystallinity, the in-plane mosaicity of 3.71° is substantially higher than the InAs one (0.79°) and higher than the 1.79° mosaicity of epitaxial ZrTe 2 on InAs. [39] Moreover, the MoTe 2 epilayer, similar to the case of ZrTe 2 , [39] follows the landscape of the InAs surface (STEM in Figure 3) which shows monatomic steps of 3.5 Å, producing a MoTe 2 nanostructure with domain size in the order of a few tens of nm as also seen by STM in Figure 2. The presence of 60° and 120° rotated domains is widely evidenced by STM, RHEED, and GIXD data, which is a distinct characteristic of our thin epitaxial films compared to single crystals obtained from bulk where such domains are absent. While these rotated domains in the hexagonal undistorted 1T or 2H MoTe 2 structure are equivalent, in 1T′ MoTe 2 result in a complex k-space and Brillouin zone, mixing the energy dispersion along different directions thus adversely affecting resolution, inhibiting clear observation of Weyl node configuration in ARPES. Domain rotation may be facilitated at the step edges, where also the continuity of the film may be interrupted (Figure 3), which could be harmful for electrical transport. By improving the surface treatment of InAs prior to growth we may be able to minimize the density of monatomic steps and increase the domain size in an attempt to obtain a single crystalline epitaxial film. Although the use of single crystal InAs(111) could resolve some of the issues related to the step and domain formation and improve the mosaicity of MoTe 2 , the effort should focus on the improvement of epitaxial InAs on Si substrates since the availability of large area wafers (200 and 300 mm) of this kind is essential for a manufacturable technology of MoTe 2 and other 2D materials in the future. Using STEM cross-sectional measurements in 3 ML MoTe 2 we identify two different phases coexisting in the layer. The first phase shows an orthorhombic (T d ) stacking (or γ-MoTe 2 ) and the second phase is a triclinic stacking not reported before. From GIXD and STEM we measure lattice parameters values a = 6.340 Å, b = 3.501 Å and out of plane parameter for the orthorhombic T d -phase c = 14.1-14.2 Å which are larger compared to freestanding thin films exfoliated from bulk as a result of the influence from the substrate.

Conclusion

The MoTe 2 orthorhombic T d -phase typically found at temperatures lower than 250 K, here is observed at room temperature which is attributed to an interlayer antibonding state compatible with the orthorhombic T d -phase as a result of the enlarged lattice parameters of our epitaxial thin films. The orthorhombic T d -MoTe 2 is noncentrosymmetric yielding a topological type II Weyl semimetal phase. Eight Weyl nodes calculated by DFT a few meV below the Fermi level are accessible to transport creating the prospect for room temperature electronic applications where the topologically nontrivial properties of MoTe 2 epitaxial layers could play a role.

Experimental Section

Surface Preparation and Film Growth Details: The InAs(111)/Si(111) substrates were chemically cleaned in a 5N HF solution in isopropyl alcohol for 5 min to etch the surface oxide and subsequently rinsed in isopropyl alcohol for 30 s in order to avoid reoxidation of the substrate. An annealing step at 400° C in UHV follows to get a clean and flat InAs(111) surface as evidenced by RHEED (Figure S1, Supporting Information) and X-ray photoelectron spectroscopy. Where appropriate, mild Ar+ sputtering was used (E ≈ 1.5 keV, p ≈ 2 × 10 -5 mbar, t ≈ 30 s) prior the annealing step to obtain a clean surface as evidenced by a 2 × 2 reconstruction in RHEED pattern (Figure S1a,b, Supporting Information) attributed to In surface vacancies. [41] ARPES Measurements: In-situ ARPES was conducted at room temperature in a µ-metal analytical chamber equipped with a 100 mm hemispherical electron analyzer (SPECS PHOIBOS 100) and a 2D CCD detector. The energy resolution of the detection system was better than 40 meV using a 21.22 eV photons from a He discharge source (SPECS UVS35/10), although, since the measurement is performed at RT, the resolution is limited by thermal effects and is about 100 meV.

STM Characterization: After MoTe 2 films growth, the samples were transferred to the STM (OMICRON) chamber without breaking the vacuum for in situ STM characterization. STM images were obtained in UHV conditions (base pressure ≈ 10 -9 mbar) at room temperature using a Pt/Ir tip at the following conditions: V = 0.1 mV, I = 1 nA for high resolution images (Figure 2e-h) and V = 0.4 mV, I = 400 pA for the rest.

STEM Characterization: STEM measurements have been carried out using a Cs-corrected FEI Themis at 200 keV. HAADF-STEM images were acquired using a convergence semiangle of 18 mrad and collecting scattering >65 mrad. STEM specimens were prepared by the FIB lift-out technique using an FEI dual-beam Strata 400S at 30 kV.

Synchrotron GIXD Measurements: Diffraction measurements were performed at the European Synchrotron Radiation Facility by means of the UHV-MBE CVD diffractometer installed at the BM32 CRG/IF beamline and optimized for GIXD. The experimental setup energy and incident angle were set at 11 keV (1.13) and 0.2°, the latter set slightly below the critical angle value for total reflection in order to enhance the 2D film signal while minimizing the background.

First-Principles Calculation: The first-principles calculations were performed using the Vienna Ab initio Simulation Package [42,43] and projector-augmented waves. [44] The generalized-gradient approximation with Perdew-Burke-Ernzerhof [45] parameterization was used as exchange correlation functional. The kinetic energy cutoff was set at 450 eV, using the Monkhorst-Pack scheme [46] employing a 12 × 10 × 6 k-point mesh for bulk calculations. The experimental lattice constants were used and the atomic positions were fully optimized by conjugate gradient, using a force threshold of 1 × 10 -3 eV Å -1 . Van der Waals corrections were used by applying the semiempirical DFT-D3 Grimme's method. [47] The Maximally-Localized Wannier functions are fitted based on Mo's s and d and Te's p orbitals using the Wannier90 code [48] and the surface states, topological Fermi arcs, and Weyl points calculations were carried out by the WannierTools software. [49] Spin-orbit coupling was included in band structure calculations.

U N C O R R E C T E

Direct

  P. Tsipas, S. Fragkos, D. Tsoutsou, C. Alvarez, R. Sant, G. Renaud, H. Okuno, A. Dimoulas* ......... 1802084

Figure 1 .

 1 Figure 1. Crystal phases for the 1T distorted (1T′) MoTe 2 a) along [1 100] direction and b) orthorhombic T d -phase along the [1120] direction, c) monoclinic β-phase along the [1120] direction. d-f) Top view of the direct and reciprocal lattices. Lattice parameters are indicated as a, b (direct) and a*, b*(reciprocal): (d) 1T, (e) 1T′, (f) 1T′ reciprocal lattice. The solid lines represent the primitive unit cells. The broken line represents the nonprimitive unit cell for 1T [1 100] and [1120] of 1T correspond to [100] and [010] of the distorted (1T′) lattice, respectively.

  . The FFT patterns were analyzed and compared with corresponding simulated electron diffraction patterns to verify the different MoTe 2 crystals structures. The simulated electron diffractions patterns and this analysis are presented in FigureS3of the Supporting Information.Adv. Funct. Mater. 2018, 1802084

Figure 2 .

 2 Figure 2. In situ room temperature UHV-STM of 1T′-MoTe 2 . a,b) 1 ML MoTe 2 and the corresponding line scan, showing the influence of the substrate. c,d) Surface morphology after the deposition of an incomplete second layer MoTe 2 and the corresponding linescan showing the thickness of the top layer of about 7 Å, including the van der Waals gap with the bottom layer. e,f) Surface and Fourier pattern showing a single domain 1ML 1T′-MoTe 2 . The dashed rectangles show the unit cells of real (e) and reciprocal space (f) lattices. g) Image of two domains A and B rotated by 60° between each other h) The additional intense spots marked by 1 and 2 originate from the rotated pattern B and defines the superimposed reciprocal lattice (white dotted rectangles).

2. 4 .

 4 In-Plane Epitaxial Orientation and Rotated Domain Microstructure Investigated by Synchrotron Grazing Incidence X-Ray Diffraction (GIXD) Single-layer and three-layer MoTe 2 are evaluated by GIXD and the data for 3 ML films are presented in Figure 4. The in-plane reciprocal space map (RSM) in Figure 4a reveals that the MoTe 2 film is epitaxially grown on InAs in such a way that [100] MoTe2 // [11-2] InAs , and [010] MoTe2 //[01-1] InAs in the plane. We observe no 30° or 90° rotated domains or polycrystallinity other than that arising from the Al protecting cap (rings in Figure 4a). The positions of the diffraction peaks in k-space are in agreement Adv. Funct. Mater. 2018, 1802084

Figure 3 .

 3 Figure 3. STEM images from epitaxial MoTe 2 thin films. a) 1 ML of 1T′ MoTe 2 showing film interruption at the edge of an InAs monoatomic step b) 3 ML 1T′ phase along [1 100] zone axis, c) 3ML orthorhombic (T d ) stacking along [1120] zone axis, d) intensity line plot in the c direction from images (b) and (c) in red and green, respectively, e) 3 ML of unconventional triclinic stacking along [1120] zone axis. f-h) FFT patterns from respective STEM images. The equivalent [uvw] notation for the zone axis directions are [100] for [1 100], and [010] for [1120].

  with a complex reciprocal space (open circles in Figure4a) which is a superposition from 0°, 60°, and 120° rotated domains. The most characteristic configuration is the triplet along h = k ([01-1] InAs ) consisting of the on-axis (020) peak and the two off-axis (310) peaks all belonging to different domains rotated between each other by 60°. It should be noted that in the ideal case where b*/a* = √3 applying to the hexagonal symmetry of the undistorted 1T structure (see Figure1d), all three closely spaced reciprocal lattice points would be expected to coincide giving a single diffraction peak. In our case though of a distorted 1T′ MoTe 2 lattice, where the ratio deviates from ideal (b*/a* > √3), the peak splitting into a triplet configuration is clearly detected by GIXD (Figure4a). Similar situation arises along h ([11-2] InAs ) direction with the triplet consisting of Adv. Funct. Mater. 2018, 1802084

Figure 4 .

 4 Figure 4. Synchrotron GIXD from epitaxial 3 ML MoTe 2 a) reciprocal space map (RSM). The diffraction peak and crystallographic orientations correspond to bulk Miller indices notation. The h, k and h = k axis are high symmetry directions in hexagonal surface k-space. The open circles and rectangles denote reciprocal lattice points and unit cells, respectively, originating from three rotated domains: 0° (red), 60° (blue), 120° (black). CTR denotes crystal truncation rods. Polycrystalline rings are due to Al cap layer b) Radial scans along the h and h = k crystallographic directions shown in (a). c) In-plane rocking curves of the on-axis (020) single peak and d) the off-axis (310) peak (symbol curve) deconvoluted into two peaks (solid blue and red lines) originating from two different domains rotated by 60°.

  detected at low temperatures and Set #2 (a = 6.340 Å, b = 3.501 Å, c = 14.15 Å) in Figure 7b is from our measured T d -MoTe 2 epitaxial films at RT. In both band structures (Figure 7a,b), the uppermost valence band B1 along ΓA consists of hybridized Te p x and Mo d xz orbitals while the lower band B2 consists of hybridized Te p z and Mo d z2 orbitals. From the plot of the wavefunctions of B1 and B2 at Γ for Set #2 (our experimental Adv. Funct. Mater. 2018, 1802084

Figure 5 .

 5 Figure 5. a) Electronic band structure of 1 ML 1T′ MoTe 2 imaged by ARPES at room temperature a) ARPES along ΓY, b) 2nd derivative of spectra in (a) compared with DFT simulations for 1 ML 1T′ MoTe 2 . Blue line is along ΓY, red line is along ΓS c) 2nd derivative of spectra along ΓX compared with DFT simulations for 1 ML 1T′ MoTe 2 . e) Schematic illustration of k-space constructed by the superposition of orthorhombic Brillouin zones (BZ) originated from of 0° (blue solid line) 60° or 120° rotated domains (red solid lines). The dashed lines denote neighboring Brillouin zones. The Γ point of the neighboring 0° BZ nearly coincides with the S point of the rotated BZs. The horizontal thick arrows shows the paths along which the ARPES spectra in (a) and (b) are recorded. Note that ARPES images simultaneously the bands along ΓY (blue arrow) and ΓS (red arrow) paths since these two paths are parallel due to the overlapping of rotated domains. The thick vertical arrow shows the paths along which the ARPES in (c) is recorded.

Figure 6 .

 6 Figure 6. Calculated Weyl node configuration for the bulk orthorhombic T d -MoTe 2 using the experimental parameter sets deduced from our epitaxial films a) Positions of Weyl nodes in the first Brillouin zone. The blue and red points closely spaced between each other indicate pair of Weyl nodes with opposite chirality. b) Theoretical energy dispersion along a path KK′ shown by broken line in (a), passing through the pair of Weyl points. Theoretical constant energy contour plots for two different energies c) E = -1.1 meV and d) E = -12.9 meV corresponding to the energy location of the Weyl node pairs. e -and h + denote the touching electron and hole pockets, respectively.

  In this work epitaxial thin films of MoTe 2 with the distorted 1T′ structure are grown by molecular beam epitaxy on InAs(111)/ Si(111) substrates. The films are rotationally commensurate with the substrate such that [100] MoTe2 //[11-2] InAs and [010] MoTe2 //[01-1] InAs although 60° and 120° rotational domains are present, detected by STM, synchrotron GIXD, and RHEED.

Figure 7 .

 7 Figure 7. DFT calculations for bulk T d -MoTe 2 for two different lattice parameter sets: a) averaged experimental lattice parameters from bulk T d -MoTe 2 taken from refs. [6-9], b) experimental lattice parameters from epitaxial thin films in this work. The symbols in both (a) and (b) represent the projections of the bands in the Te and Mo orbitals. c,d) Wavefunctions at the Γ point are plotted for valence bands B1 and B2, respectively, corresponding to T d -MoTe 2 calculated with the experimental parameters reported in this work (Set #2). The wavefunction sign (+) and (-) is indicated by blue and green color, respectively.

of 12) Direct Observation at Room Temperature of the Orthorhombic Weyl Semimetal Phase in Thin Epitaxial MoTe 2

  Polychronis Tsipas, Sotirios Fragkos, Dimitra Tsoutsou, Carlos Alvarez, Roberto Sant, Gilles Renaud, Hanako Okuno, and Athanasios Dimoulas*

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Acknowledgements

This work was financially supported by the LANEF Chair of Excellence program of U. Grenoble Alpes and CEA (A.D.), the Greek State Scholarships Foundation (IKY) Program for the strengthening of postdoctoral research (P.T.), the French state fundsANR-10-LABX-51-01 (Labex LANEF du Programmed'Investissementsd'Avenir) and Equipex ANR-11-EQPX-0010 (G.R.). The authors thank Dr. T. Baron of Université Grenoble Alpes, CNRS, CEA/LetiMinatec, LTM for providing us with the InAs/Si substrates. The authors thank Dr. A. Kontos of INN/NCSRD for his support in the structural characterization.

Supporting Information

Supporting Information is available from the Wiley Online Library or from the author.

Conflict of Interest

The authors declare no conflict of interest.

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