Didier Henrion

Semidefinite characterisation of invariant measures for one-dimensional discrete dynamical systems

Keywords: dynamical systems, measure theory, semidefinite programming

Using recent results on measure theory and algebraic geometry, we show how semidefinite programming can be used to construct invariant measures of onedimensional discrete dynamical systems (iterated maps on a real interval). In particular we show that both discrete measures (corresponding to finite cycles) and continuous measures (corresponding to chaotic behavior) can be recovered using standard software, paving the way for a numerical study of probabilistic properties of dynamical systems.

Introduction

Consider the discrete dynamical system

x k+1 = f (x k ), x 0 ∈ G = [-1, 1] (1)
where f : R → R is a given polynomial map and x k ∈ R is the system state with initial condition x 0 chosen in the real interval G = {x ∈ R : g(x) = 1x2 ≥ 0}.

It is well-known that for simple nonlinear mappings f , a typical state trajectory x 0 , x 1 , x 2 , . . . of system (1) may be erratic, or chaotic, and very sensitive to the choice of the initial state x 0 . The study of such one-dimensional discrete systems can be greatly facilitated by the use of measure theory, see [START_REF] Alligood | Chaos -an introduction to dynamical systems[END_REF]Chapter 6] and [START_REF] Diaconis | Iterated random functions[END_REF] for elementary introductions, and [10, Chapters 1-4] for a more mathematical, yet very accessible treatment. The key idea is the following: if a system operates on a measure as an initial condition, rather than on a single point, then successive measures are given by a linear integral operator called a Markov operator. Fixed points of this operator are called invariant measures, and they convey key information on the long-term behavior of the system, such as the existence of cycles of finite length, or a probabilistic characterization of its chaotic behavior. For example, a chaotic behavior can be chacterized compactly via the knowledge of only a few moments of a measure associated with the dynamical system, in contrast with lengthy computationally demanding time-domain simulations, as illustrated e.g. in [START_REF] Barkley | The moment map: nonlinear dynamics of density evolution via a few moments[END_REF]. Recently, in the context of linear and nonlinear control systems, it has been realized that there is an insightful duality between occupation measures and Lyapunov functions [START_REF] Rantzer | A dual to Lyapunov's stability theorem[END_REF][START_REF] Peyrl | A theorem of the alternative for SOS Lyapunov functions[END_REF][START_REF] Vaidya | Lyapunov measure for almost everywhere stability[END_REF][START_REF] Lasserre | Nonlinear optimal control via occupation measures and LMI relaxations[END_REF].

In this paper we show that recent results mixing measure theory and algebraic geometry can be used to construct invariant measures numerically, with the help of semidefinite programming, as called optimization over linear matrix inequalities (LMIs), a versatile generalization of linear programming to the cone of positive semidefinite matrices [3, Section 4.6.2]. We hope that such computer-based research efforts may contribute to a better understanding of nonlinear phenomena and may suggest directions for theoretical analysis, along the lines sketched in [START_REF] Campbell | Experimental mathematics: the role of computation in nonlinear science[END_REF].

Invariant measures

The material of this introductory section can be found in [START_REF] Hernández-Lerma | Markov chains and invariant probabilities[END_REF].

Consider dynamical system (1) with its polynomial map f on the interval G ⊂ R. Let B(G) be the Borel σ-algebra on G, and let µ be a measure in B(G). Let F (G) denote the set of measurable functions on G. Given x ∈ G and B ∈ B(G), define P (x, B) as a Markov operator such that

• for all x ∈ G, P (x, .) ∈ B(G) is a probability measure supported on G;

• for all B ∈ B(G), x → P (x, B) ∈ F (G) is a measurable function on G.

Markov operator µ → µ P acts on measures as follows:

(µ P )(B) = G P (x, B)µ(dx) (2)
and its adjoint operator φ → P φ acts on functions as follows:

(P φ)(x) = G φ(y)P (x, dy) = G φ(y)δ f (x) (dy) = φ(f (x)) (3)
where δ x is the Dirac measure at x. With respect to dynamical system (1), given x ∈ G and B ∈ B(G), P (x, B) is the probability that x k+1 ∈ B, knowing that x k = x.

We can interpret the Markov operator as a generalization to arbitrary probability measures (including singular measures) of the Frobenius-Perron operator [10, Section 3.2] usually restricted to density functions, i.e. to probability measures which are absolutely continuous w.r.t. the Lebesgue measure. Similarly, the adjoint of the Markov operator is a generalization of the Koopmans operator [10, Section 3.3].

For dynamical system (1) we want to characterize fixed points of Markov operator P , that is, measures µ ∈ B(G) satisfying µ P = µ.

Measures satisfying (4) are called invariant measures.

Semidefinite characterisation of invariant measures

An equivalent characterization of invariance condition (4) is that

µ P φ = µ φ
for all continuous functions φ ∈ F (G). Recalling ( 2) and (3), this relation becomes

G φ(f (x))dµ(x) = G φ(x)dµ(x).
Since function φ is continuous and G is compact, the above relation can be written equivalently

G π α (f (x))dµ(x) = G π α (x)dµ(x), α = 0, 1, 2, . . . (5)
where polynomials π α (x) ∈ R[x], α = 0, 1, . . . , n generate a basis for polynomials of degree at most n in indeterminate x. Basis polynomials are gathered in a column vector π(x) with entries π i (x), i = 0, 1, . . . , n. For example π α (x) = x a , the standard monomial of degree α, or π α (x) = t α (x), the Chebyshev polynomial of degree α, whose definition is recalled in the Appendix.

Let

y α = G π α (x)dµ(x) (6)
denote the moments of measure µ w.r.t. basis π(x). Let y denote the vector with entries y α .

Notice that π α (f (x)) is a polynomial in x, so relation ( 5) is a linear equality constraint on y for each α. All the constraints can be gathered in an infinite-dimensional linear system of equations Ay = b

also including the constraint y 0 = 1 indicating that µ is a probability measure.

Since µ is a measure supported on G, it follows that

G p 2 0 (x)dµ(x) ≥ 0, G g(x)p 2 1 (x)dµ(x) ≥ 0
for all polynomials p 0 (x) and p 1 (x). By expressing polynomials p i (x) = p T i π(x) in basis π(x), with p i a column vector of coefficients, these scalar inequalities can be written as

p T 0 ( G π(x)π(x) T dµ(x))p 0 ≥ 0, p T 1 ( G g(x)π(x)π(x) T dµ(x))p 1 ≥ 0
for all vectors p 0 and p 1 , or, equivalently, as semidefinite constraints

M(y) = G π(x)π(x) T dµ(x) 0, M(gy) = G g(x)π(x)π(x) T dµ(x) 0.
Symmetric linear mappings M(y) and M(gy) are called moment and localising matrices, respectively. Results from algebraic geometry (namely M. Putinar's sum-of-squares representation of polynomials positive on compact semi-algebraic sets) can be invoked as in [START_REF] Lasserre | Global optimization with polynomials and the problem of moments[END_REF] to show that these semidefinite conditions are necessary and sufficient for the entries of vector y to correspond to moments of a probability measure µ supported on G.

Lemma 1

The moments y of an invariant measure for dynamical system (1) satisfy the linear semidefinite constraints

Ay = b, M(y) 0, M(gy) 0. ( 8
)
The set of moments y satisfying constraints ( 8) is called the moment set for later reference. It is a convex set with non-smooth boundary.

Computational issues

Since system (1) is defined on the unit interval [-1, 1], an appropriate choice of polynomial basis are Chebyshev polynomials. The explicit construction of moment and localizing matrices is based on a basic property of these polynomials. From Lemma 2 of the Appendix, entry (α, β) in matrix M(y) is equal to

G t α t β dµ = 1 2 ( G t α+β dµ + G t |α-β| dµ) = 1 2 (y α+β + y |α-β| ).
It follows that

M(y) = 1 2 (T 0 (y) + H 0 (y))
has a mixed Toeplitz-Hankel structure.

Since g(x) = 1x 2 = 1 2 (t 0 (x)t 2 (x)), matrix M(gy) shares a similar mixed Toeplitz-Hankel structure:

M(gy) = 1 8 (2T 0 (y) + 2H 0 (y) -T -2 (y) -T 2 (y) -H -2 (y) -H 2 (y)).
The moment set ( 8) is infinite-dimensional: an infinite number of moments are subject to infinite-dimensional semidefinite constraints. The moment set must be truncated when resorting to numerical computations. Let y = [1 y 1 • • • y 2d ] T now denote the vector of moments up to degree 2d, and truncate all the data in (8) accordingly, yielding truncated conditions

A d y = b d , M d (y) 0, M d (gy) 0. (9)
These finite-dimensional conditions are necessary but not sufficient for the entries of y to correspond to moments of an invariant measure. The idea is therefore to reduce the gap between necessity and sufficiency by increasing d, considering a whole hierarchy of semidefinite relaxations of increasing size for moment set [START_REF] Góra | Absolutely continuous invariant measures that cannot be observed experimentally[END_REF].

Finally, we may want to characterize a particular invariant measure optimal w.r.t. a linear combination of moments. Given a polynomial h(x) = α h α π α (x), the function

h(y) = G h(x)dµ(x) = α h α y α
is linear in y. Its minimization over moment set [START_REF] Góra | Absolutely continuous invariant measures that cannot be observed experimentally[END_REF] can be achieved via a hierarchy of finite-dimensional convex linear semidefinite programming problems

min y h(y) s.t. A d y = b d , M d (y) 0, M d (gy) 0 (10)
that can be solved by interior-point algorithms and off-the-shelf software. When d increases, we obtain a monotonically increasing sequence of lower bounds on the minimum of h(y) achieved over [START_REF] Góra | Absolutely continuous invariant measures that cannot be observed experimentally[END_REF]. Asymptotic convergence of the sequence to the minimum is proved in [START_REF] Lasserre | Global optimization with polynomials and the problem of moments[END_REF].

As will be shown below, for some h(y), the optimal measure can be discrete (corresponding to cycles of finite length) or continuous (corresponding to chaotic behavior), and in general, different choices of objective functions h(y) yield different invariant measures. Finally, the optimal measure can be a linear combination of several measures which are invariant for system (1).

Example

Consider the logistic map xk+1 = 4x k (1 -xk ) defined on the unit interval [0, 1]. Applying the affine change of variables xk = 1 2 (1x k ), we obtain a discrete dynamical system (1) defined on the symmetric interval G = [-1, 1] with

f (x) = 2x 2 -1.
Note that f (x) = t 2 (x), the second Chebyshev polynomial.

Choosing a Chebyshev polynomial basis π α (x) = t α (x), and using Lemma 3 of the Appendix, it follows that relation [START_REF] Dellnitz | On the approximation of complicated dynamical behavior[END_REF] satisfied by moments of an invariant measure becomes

y 2α = y α , α = 0, 1, 2, . . . (11)
Therefore only odd moments y 1 , y 3 , y 5 , . . . are linearly independent.

For illustration, when d = 3, the matrices defining the truncated moment set (9) are given by

M 3 (y) = 1 2     2 2y 1 2y 1 2y 3 2y 1 1 + y 1 y 1 + y 3 2y 1 2y 1 y 1 + y 3 1 + y 1 y 1 + y 5 2y 3 2y 1 y 1 + y 5 1 + y 3     and M 3 (gy) = 1 8   4 -4y 1 2y 1 -2y 3 -2 + 2y 1 2y 1 -2y 3 1 -y 1 y 3 -y 5 -2 + 2y 1 y 3 -y 5 2 -y 1 -y 3   .

Continuous measure

Now suppose we want to find a vector y in the truncated moment set [START_REF] Hernández-Lerma | Markov chains and invariant probabilities[END_REF]. Using the semidefinite programming solver SeDuMi 1.1 for Matlab without specifying an objective function, we obtain an interior point y for which M d (y) ≻ 0 and M d (gy) ≻ 0. For example, when d = 3 (truncation to 6 moments), we obtain (to five significant digits) y 1 = -0.012851, y 3 = -0.057926, y 5 = 0.020568.

The other moments are y 0 = 1, y 2 = y 4 = y 1 , y 6 = y 3 .

By solving a semidefinite programming problem (not described here for conciseness) we can find a polynomial density q(x) generating these moments (up to numerical roundoff errors), i.e. G t α (x)q(x)dx = y α , α = 0, 1, . . . , 6. We obtain q(x) = 6 α=0 q α t α (x) with q 0 = 0.77395, q 1 = -0.084370, q 2 = 0.72051, q 3 = -0.10286, q 4 = 0.42402, q 5 = -0.023627, q 6 = 0.19274. The approximate density q(x) obtained from the 6 moments and the exact density π -1 (1x 2 ) -1 2 corresponding to the continuous measure invariant for the logistic map are represented on Figure 1.

Discrete measures

Our semidefinite characterization of invariant measures also captures discrete measures corresponding to finite cycles. Such measures can be recovered by optimizing a particular linear combination of moments h(y) = α h α y α . In general, moment matrix M d (y) is rank deficient, since the optimum is at the boundary of the semidefinite cone. The corresponding measure is atomic. The discrete support of the measure can be computed via eigenvalue decomposition (not described here for conciseness).

Here are some examples solved with SeDuMi for d = 3 (truncation to 6 moments):

• minimizing h(y) = y 1 returns the moments y 1 = -0.50000, y 3 = 1.0000, y 5 = -0.50000 corresponding to a rank-one moment matrix M 3 (y) of the Dirac measure δ x at x = -1 2 . This is a fixed point for our dynamical systems since it is a root of the polynomial f (x)x = 2x 2x -1 = (2x + 1)(x -1);

• maximizing y 1 yields y 1 = y 3 = y 5 = 1.0000 and the Dirac measure at x = 1, corresponding to the other root of f (x)x;

• a cycle of length three can be found by minimizing y 3 . We obtain y 1 = 0.0000, y 3 = -0.50000, y 5 = 0.0000 and a rank-three moment matrix corresponding to an atomic measure supported at {-0.93969, 0.17365 ,0.76604}. The cycle corresponds to the three roots of the polynomial 8x 3 -6x + 1 dividing the degree-

8 polynomial f (f (f (x))) -x = 128x 8 - 256x 6 + 160x 4 -32x 2 -x + 1 = (2x + 1)(x -1)(8x 3 -6x + 1)(8x 3 + 4x 2 -4x -1);
• by maximizing y 3 we obtain y 1 = -0.095852, y 3 = 1.0000, y 5 = -0.095852 and a rank-two moment matrix corresponding to an atomic measure supported at {-0.50000, 1.0000}. This measure is a linear combination of the two invariant Dirac measures at x = -1 2 and x = 1, already found above;

• by minimizing y 5 we obtain y 1 = 0.20414, y 3 = -0.025163, y 5 = -1.1323 and a rank-three moment matrix. This moment matrix corresponds to an atomic measure supported at {-1.0439, -0.25265,0.86800}, but these values do not correspond to an invariant measure and a finite cycle since x = -1.0439 does not belong to G;

• by maximizing y 5 we obtain y 1 = -0.20658, y 3 = -0.018795, y 5 = 1.04521 and a rank-three moment matrix. This moment matrix corresponds to an atomic measure supported at {-0.77048, 0.29058,1.0450}, but these values do not correspond to an invariant measure and a finite cycle since x = 1.0450 does not belong to G.

The two latter cases illustrate that the moment set ( 8), when truncated, may contain vectors y which do not correspond to moments of an invariant measure. However, invariant measures and finite cycles can be recovered at the price of introducing more moments:

• with d = 4 (truncation to 8 moments) by maximizing y 5 we obtain y 1 = -0.022753, y 3 = -0.022753, y 5 = 1.0000, y 7 = -0.022753 and a rank-three moment matrix. This moment matrix corresponds to an atomic measure supported at {-0.80902, 0.30902,1.0000}. This measure is a linear combination of two invariant measures: the Dirac measure at the fixed point x = 1 and the two-cycle invariant measure supported at the points { 1 4 (-1 -√ 5), 1 4 (-1 + √ 5)}. These points correspond to three roots of the degree-four polynomial f (f (x))x = 8x 4 -8x 2x + 1 = (2x + 1)(x -1)(4x 2 + 2x -1);

• with d = 5 (truncation to 10 moments) by minimizing y 5 we obtain y 1 = 0.12500, y 3 = -0.25000, y 5 = -0.50000, y 7 = 0.12500, y 9 = -0.25000 and a rank-five moment matrix corresponding to an atomic measure supported at {-0.97815, -0.50000, -0.10453, 0.66913, 0.91355}. This measure is a linear combination of two invariant measures: the Dirac measure at the fixed point x = - 1 2 and the four-cycle invariant measure supported at the roots of the polynomial 16x 4 -8x 3 -16x 2 + 8x + 1 dividing the degree-16 polynomial f (f (f (f (x))))x.

Minimizing or maximizing higher order moments can result in longer finite cycles. For example, with d = 20 (truncation to 40 moments), minimizing y 9 yields y 9 = -0.50000, y 27 = 1.0000 and y i = 0.0000 for other values of i between 1 and 40. The 21-by-21 moment matrix has rank 9, and it corresponds (to machine precision) to a cycle of length 9. Our numerical experiments with the solver SeDuMi reveal that semidefinite problems with a few hundreds moments can be solved routinely, to a relative accuracy of the order of 10 -8 , in a few seconds on a standard computer.

On Figure 2 we represent the projections of the truncated moment set (9) on the plane (y 1 , y 3 ) of first and third order moments, when truncated to 4, 6 and 40 moments (d = 2, 3, 20) respectively. We have not represented projections for intermediate truncations, or for a higher number of moments, because they could not be distinguished on the figure. It seems that the shape of the moment set ( 8) is quickly captured by low-order relaxations.

Further work

Our objective in this paper was to illustrate that semidefinite programming could be an interesting setup for a computer-aided study of nonlinear dynamical systems, and especially of their long-term behavior characterized by invariant measures.

Measures are characterized by their moments with the help of linear constraints in the cone of positive semi-definite matrices. As explained in [START_REF] Góra | Why computers like Lebesgue measure[END_REF], due to numerical round-off errors, histograms of computer simulations of dynamical systems display only the invariant measure that is absolutely continuous w.r.t. to the Lebesgue measure, discarding periodic orbits of finite lengths. Examples for which computer-generated trajectories land on periodic orbits, and for which the absolutely continuous invariant measure is not observable, are exceptional [START_REF] Góra | Absolutely continuous invariant measures that cannot be observed experimentally[END_REF]. It turns out that optimization of a linear function of the moments subject to semidefinite constraints, as proposed in this paper, can produce computationally both types of measures on a unifying ground. Note however that a comparison of our semidefinite programming approach with other numerical techniques lies out of the scope of this short contribution. See e.g. [START_REF] Dellnitz | On the approximation of complicated dynamical behavior[END_REF] or [START_REF] Vaidya | Lyapunov measure for almost everywhere stability[END_REF] for numerical approximations of invariant measures based on discretizations of the Frobenius-Perron operator.

On the negative side, it seems difficult to establish a correspondence between finite cycles and objective functions h(y). For example, in Section 5 if one wants to recover a 9-cycle of the logistic map, what is a suitable choice of h(y) ? Also unclear is the numerical conditioning of the semidefinite programming problem [START_REF] Lasota | Probabilistic properties of deterministic systems[END_REF]. In the logistic map example of Section 5, thanks to the interpretation of the mapping f as a Chebyshev polynomial, the linear system of equations Ay = b became the particularly simple relations [START_REF] Lasserre | Global optimization with polynomials and the problem of moments[END_REF] which allowed for a straightforward substitution of even degree moments. In general, for an arbitrary mapping f , such an explicit substitution is not possible, and it may happen that the linear system of equations Ay = b is poorly scaled, and/or that the semidefinite problem ( 10) is ill-conditioned.

Finally, and maybe most importantly, the methodology allows for (almost direct) extensions to higher dimensional dynamical systems (using e.g. multivariate Chebyshev polynomials) and iterated functional systems (IFS).

yy

  |α| y |α+1| y |α+2| y |α+1| y |α+2| y |α+3| y |α+2| y |α+3| y |α+4| . . . |α| y |α+1| y |α+2| y |α+1| y |α| y |α+1| y |α+2| y |α+1| y |α| . . . symmetric Hankel and Toeplitz matrices, respectively, where α is a possibly negative integer.

Figure 1 :

 1 Figure 1: Approximate density (thick) and exact density (thin) of the continuous measure invariant for the logistic map.

Figure 2 :

 2 Figure 2: Projection on the plane (y 1 , y 3 ) of the moment set (8) truncated to 4, 6 and 40 moments.

CNRS; LAAS; 7 avenue du colonel Roche, F-31077 Toulouse; France. henrion@laas.fr

Université de Toulouse; UPS, INSA, INP, ISAE; UT1, UTM, LAAS; F-31077 Toulouse; France

Faculty of Electrical Engineering, Czech Technical University in Prague, Technická 2, CZ-16626 Prague, Czech Republic

Acknowledgments

This work was initiated by Jean-Bernard Lasserre, whose technical input is gratefully acknowledged. This work was partly supported by research project No. 103/10/0628 of the Grant Agency of the Czech Republic.

Appendix

Given a non-negative integer α, let t α (x) denote the Chebyshev polynomial of first kind satisfying the three-term recurrence relation

Proof: By induction.

Proof: Follows from the trigonometric relation t α (x) = cos(α arccos(x)). Indeed t α (t β (x)) = cos(α arccos(cos(β arccos(x)))) = cos(αβ arccos(x)) = t αβ (x).


Alexander M Stolyarov

Lei Wei

Ofer Sharipa

Fabien Sorin

Song L Chua

John D Joannopoulos

Yoel Fink

Ofer Shapira

Microfluidic directional emission control of an azimuthally polarized radial fibre laser

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Lasers with cylindrically symmetric polarization statesare predominantly basedon whispering-gallerymodes [START_REF] Chang | Optical Processes in Microcavities[END_REF][START_REF] Mccall | Whisperinggallery mode microdisk lasers[END_REF][START_REF] Knight | Core-resonance capillary-fiber whispering-gallery-mode laser[END_REF][START_REF] Kawabe | Whispering-gallery-mode microring laser using a conjugated polymer[END_REF][START_REF] Moon | Cylindrical microcavity laser based on the evanescent-wave-coupled gain[END_REF][START_REF] Malko | From amplified spontaneous emission to microringlasing using nanocrystal quantum dot solids[END_REF][START_REF] Kazes | Lasing from Semiconductor Quantum Rods in a Cylindrical Microcavity[END_REF] , characterized by high angular momentum and dominated by azimuthal emission.Here azero angular momentum laserwith purely radial emission is demonstrated.An axially invariant, cylindrical photonic bandgap fibrecavity [START_REF] Yeh | Theory of Bragg fiber[END_REF] filled with a microfluidic gain medium plug is axially pumped, resulting in a unique radiating field pattern characterized by cylindrical symmetry and a fixed polarization pointed in theazimuthal direction.Encircling the fibre core is anarray ofelectrically contacted and independently addressable liquid crystal microchannelsembedded in the fibrecladding. Thesechannels modulate the polarized wavefront emanating from the fibre coreleading toa laser with a dynamically controlled intensity distribution spanning the full azimuthal angular range. This new capability,implemented monolithicallywithin a single fibre, presents opportunities ranging from flexible multidirectional displaysto minimally invasive directed light delivery systems for medical applications.

Rotationally symmetric resonators capable of omnidirectional emission in the direction perpendicular to the axis of symmetryhave been the subject of recent studies [START_REF] Chang | Optical Processes in Microcavities[END_REF][START_REF] Mccall | Whisperinggallery mode microdisk lasers[END_REF][START_REF] Knight | Core-resonance capillary-fiber whispering-gallery-mode laser[END_REF][START_REF] Kawabe | Whispering-gallery-mode microring laser using a conjugated polymer[END_REF][START_REF] Moon | Cylindrical microcavity laser based on the evanescent-wave-coupled gain[END_REF][START_REF] Malko | From amplified spontaneous emission to microringlasing using nanocrystal quantum dot solids[END_REF][START_REF] Kazes | Lasing from Semiconductor Quantum Rods in a Cylindrical Microcavity[END_REF] .

These cylindrically symmetric sources rely on the excitation of whispering-gallery modes(Fig. 1, bottom), confined near the cavity boundary by total internal reflection.

Lightescapes these resonators only through diffraction losses or scattering from surface roughness,thus limiting control over the output coupling. Here we report on a cylindrical photonic bandgap (PBG)fibrecavitywhich supports purely radial modes (Fig. 1, top).In principle, this structure allows full control over output coupling and the potential for reduction in volume without compromising the quality factor. Additionally, the shorter cavity length of these radial modes could allow for a larger free spectral range and consequently higher finesse as compared to whispering-gallery modes.Importantly, the radially directed emission from the extended cylindrical surface of this fibre laser is in contrast to the axial emission characteristic of conventional Bragg fibre lasers [START_REF] Ball | Continuously tunable single-mode erbium fiber laser[END_REF][START_REF] Dianov | Solid-core photonic bandgap fibers for high-power fiber lasers[END_REF] and planar annular Bragg resonators [START_REF] Wu | Optically pumped surface-emitting DFB GaInAsP/InP lasers with circular grating[END_REF][START_REF] Erdogan | Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum-well semiconductor laser[END_REF][START_REF] Labilloy | High-finesse disk microcavity based on a circular Bragg reflector[END_REF][START_REF] Scheuer | Low-threshold twodimensional annular Bragg lasers[END_REF] .

The fibre fabrication method is based on the thermal drawing of a macroscopic preform (Fig. 2a) assembled with all the solid materials and geometryof the resulting fibre (see Methods section for details). [START_REF] Abouraddy | Towards multimaterial multifunctional fibres that see, hear, sense and communicate[END_REF] The preform consists of a hollow annular multilayered cavity comprised of a 35-layerstructure of chalcogenide glass (As 25 S 75 ) and polycarbonate (PC). [START_REF] Ruff | Polymer-composite fibers for transmitting high peak power pulses at 1.55 microns[END_REF] Further out in the PC cladding are several pairs of conductive carbon-loaded polyethylene (CPE) electrodes which flank hollow channels.The composite structure is drawn under a high stress regime (~400 g/mm 2 ) yielding an axially invariant, rotationally symmetric, photonic bandgapfibrecavitywith a transmission bandgap centered near550 nm and a bilayer periodicity of 160 nm (Fig. 2b, top-right).

The fibrecore is surrounded by four electrically addressablemicrofluidic channels embedded in the fibre cladding (Fig. 2b, bottom).The challenge in drawing such a structure which is reduced in size by a factor of 100 lies in maintaining the integrity and symmetry of the multilayers down to the final scale while preventing the deformation of the hollow channels and electrodes.

Recent studies have demonstratedoptofluidicfibre devices with dynamically tunable properties [START_REF] Alkeskjold | Integrating liquid crystal based optical devices in photonic crystal fibers[END_REF][START_REF] Kuhlmey | Fluid-Filled Solid-Core Photonic Bandgap Fibers[END_REF] . In this work, amicrofluidic pump (shown in Fig. 3a anddescribed in Methods section) is utilized to transport organic dye-doped water plugs inside the fibre core. The internal surface of the fibre core is hydrophopic and notwet by the aqueousplug, which remains intactduring rapid movement at velocities of 35mm/s (Fig. 3b). The pressure field is mediated through atransparent and immiscible fluid (e.g., silicone oil),allowingreversible transport of the lasing plug to any position within the fibre (Fig. 3b,c).

The laser wavefrontemitted from the fibre core appears as an axially collimated ring-like beam in the far-field (Fig. 4a).The axial collimation results from the low diffraction facilitated by the smooth interfaces of the water plug, and the continuous translational symmetry of the laser cavity along the fibre axis. The penetration depth of the pump beam, combined with the circular cross section of the core, defines a cylinder from which laser light is emitted. (Note that the actual plug length, which is on the order of a cm, does not influence the laser emission). For the 1mM R590 waterplug used in our experiments, the penetration depth is ~100 µm. Therefore, the axial divergence angle, which is proportional to the ratio between the wavelength and the penetration depth, is expected to be ~10 mrad.

Thecylindrical symmetryof the emission stems from the isotropic fluorescence of the dye and its coupling to the fibrecavity modes. Below threshold, the measured isotropic and unpolarized spontaneous emission populates all the available fibre modes. However, above threshold, a purely azimuthally polarized beam emerges from the PBG fibre cavity (Fig. 4b) which indicates coupling to specific low-threshold fibre modes.

These modes are the high-Q TE 0n modes with an electric field polarization identical to what we measure in the radial laser emission.This isotropic emission is in contrast to the anisotropic emission observed fromdye-doped solid plugs. [START_REF] Shapira | Surface-emitting fiber lasers[END_REF] In a low viscosity solvent such as water, fluorescence randomization occurs dueto the fast molecular reorientationof the dye, which happens on a picosecond scale [START_REF] Eichler | Measurement of orientational relaxation times of Rhodamine 6G with a streak camera[END_REF] , much faster than the nanosecond scale of the fluorescence lifetime [START_REF] Duarte | Dye Laser Principles[END_REF] . Thus fluorescence anisotropy is not expected, which is consistent with our experimental observations. A 1mMRhodamine 590 (R590)-dopedwater plugis introduced into the fibre core and pumped by a linearly polarized Nd:YAG laser at 532 nm (see Methods section for details). A typical measurement depicting the dependence of output energy on input energy shows a clear threshold at 165 nJ (Fig. 4c). The spacing between individual lasing peaks reveals an average separation of 1.78 nm for a fibre with an 80 µm core (Fig. 4d).

Considering the free spectral range of the longitudinal modes in a Fabry-Perot resonator,

∆λ ≈ λ 0 2 /2nd
, where λ 0 is the central lasing wavelength, and n is the refractive index of the dye solution (n = 1.33), a cavity length d = 78 µm is calculated, which agrees well with the fibre core diameter andfurther confirms the radial nature of the emission.

So how can we use the intrinsic radial emission andazimuthal polarization state to achieve a useful effect? The problem of directing coherent light is recognized as an important one for a variety of applications. Current solutions could be classified into ones that utilize moving mechanical parts and ones that rely on non-mechanical mechanisms 23- 28 . While having advantages such as high angular precision, the non-mechanical solutions are nevertheless constrained to a narrow angular range. Here we demonstrate an approach that facilitates directional intensity control withinafull 2π radians.

Thus far the hollow microchannels (Fig. 2b) have been empty. But now, we exploit them by infiltrating them with liquid crystals (LCs) in order to create a novel hybrid fibre device capable of modulating the coherent laser emission as a function of angle. The linearly polarized annular wavefront escaping the PBG cavity simultaneously interacts with the multiple LCmicrochannels surrounding it, each of which can be independently controlled. In the absence of an applied electric field, the LC director points along the microchannel axis.Applying a potential difference across an electrode pair flanking an LC-infiltrated microchannel generates an electric field perpendicular to the LC director.Above a threshold voltage, the applied field produces a torque on the LC moleculeswhich leads to their rotation [START_REF] Yeh | Optics of Liquid Crystal Displays[END_REF] . The optical anisotropy of LC molecules together with the control on their spatial orientation provides a mechanism to continuously modify the linearly polarized state of the laser light passing through each microchannel. Fixing a thin external linear polarizer around the fibre, this mechanism is used to modulate the azimuthal laser intensity distribution. We note that although a whispering-gallery-like emission with a precise polarization state could also in principle be modulated by an annular array of LC channels, the non-radial nature of this emission in the vicinity of the cavity would pose significant practical challenges to enable this oblique transmission to be efficiently modulated.

We demonstrate the principle of directional emission control by contacting two oppositely facing LC microchannels toallow for simultaneous and independent tuning of the light intensity emitted from opposite sides of the same fibre (shown in Fig. 5a and described in Methods section).A R590-doped water plug is positioned in the core of the fibre and axially pumped as described previously. A thin polarizer sheet is wrapped around the fibre with its transmission axis aligned along the fibre axis such that the emitted light is blocked in the absence of an applied electric field or below the threshold voltage. Above the threshold voltage, independent intensity modulation of laser light passing through each LC-filled microchannel is clearly visiblein the images (left and right sides of Fig. 5a) and is further substantiated by measuring the laser spectra as a function of driving voltage (Fig. 5b). The measurements reveal a maximum extinction ratio of ~9 dB.

The precise control over the omnidirectional intensity distribution from the surface of a thin and flexible fibre presents a wide range ofintriguing opportunities.The ability to introduce a directionally emitting laser catheter into a blood vessel could enable spatial and angularly selective irradiation of diseased sections while sparing healthy tissue(see Supplementary Fig. S1) in conjunction with treatments such as photodynamic therapy [START_REF] Dolmans | Photodynamic therapy for cancer[END_REF] . A lab-in-a-fibre application would involve utilizing the fibre laser to interrogate fluids in the adjacent microchannels. Finally, theintegration of a modulator and light source in a single fibre represents an important step towards the realization of flexible and directional fabric displays (see Supplementary Fig. S2). (The green pump light is also seen along the fibre axis. This is due to the transmission bandgap shift, which arises from filling the core with the silicone oil. [START_REF] Rowland | Bragg Waveguides with Low-Index Liquid Cores[END_REF] )

Methods

:

  Fibre fabrication.The multilayer film is fabricated by thermal evaporation of a 4 µm thick As 25 S 75 layer onto each side of a 10 µm thick PC (Lexan) film. This film is subsequently rolled onto a Teflon lined mandrel and additional thicker PC films are added to increase the outer diameter to 30 mm, after which this preform is consolidated under vacuum at ~190 °C until the individual layers fuse together into one solid part.After consolidation, groups of three pockets with prescribed dimensions are machined along the length of the preform. The two outer pockets are filled with CPE strips while the center pocket is left empty. Additional PC films are rolled around the preform to a 40 mm diameter and a second consolidation step ensues. During the subsequent thermal drawing process, the preform is scaled down by a factor of ~100, resulting in the vacant pockets being pulled into hollow microchannels, the CPE strips drawn into continuous electrodes which are contacted post draw, and the multilayer glass / polymer microstructure drawn into a nanostructured optical cavity lining the inner core of the fibre.Microfluidic controls. Bidirectional pumping of fluid into and out of the fibre is achieved by alternating the electrical actuation of microdispensing valves (Lee Company VHS M/2) connected to pressurized and depressurized silicone oil reservoirs.Depressurization is achieved using a standard Venturi-based vacuum generator.Laser characterization. The organic laser dye Rhodamine 590 (Exciton) dissolved in DI water at 1 mM concentration is infiltrated into the fibre core with a syringe. The pump beam from the second harmonic (532 nm) of a linearly polarized Nd:YAG laser (Continuum Minilite II) with a nominal pulse duration of 9 ns and 10 Hz repetition rate is coupled into the fibre core through a 10 cm focal length lens. A small fraction of the pump beam is diverted with a beam splitter to monitor the input pump energy using a laser energy meter (Molectron EPM 1000 with the J4-09 probe). A 10x objective lens is used to collect a fraction of the light emitted radially from the fibre and focused directly onto the entrance slit of a spectrometer (Ocean Optics USB4000-UV-VIS). The polarization of the emitted light is measured by rotating a linear polarizer positioned between the fibre and the spectrometer. Intensity modulation using LCs.LCs (Merck MLC-2058) are introduced into themicrochannelsby capillary action, resulting in the LC director aligning parallel to the fibre axis. A function generator (Stanford Research Systems DS345) operated at a sinusoidal 100 Hz is used to drive a high voltage amplifier (A. A. Lab Systems A-301) which is connected to the electrodes. A 10x objective lens is used to image each LC-filled microchannel to a CCD camera (MoticMoticam 1000) and a series of images are recorded as the voltage is changed. The CCD camera is replaced with a spectrometer to measure the laser spectra.

Figure 1

 1 Figure 1 |Comparison between radial and whispering-gallery modes.(top) Schematic

Figure 2

 2 Figure 2 |Preform and fibre structure.a, Photograph of the drawn preform. The black

Figure 3 |

 3 Figure 3 | Microfluidic laser system.a, Schematic of the microfluidic system used to

Figure 4 |

 4 Figure 4 | Laser characterization.a, (top) Schematic of the fibre laser threaded through

Figure 5 |

 5 Figure 5 | Azimuthally controlled laser emission.a, Schematic of the fibre device

Acknowledgments: A.M.S. is grateful to Prof. Federico Capasso, Prof. Ayman F. Abouraddy, Prof. Howard Stone, Prof. Zheng Wang, Dana Shemuly, Sylvain Danto, Daosheng Deng, Zachary Ruff, Nicholas Orf, and Alex Nemiroski for fruitful discussions. We appreciate Mark Stolyar for his help in engineering the first rapid prototype version of the fluidic /optical fibre coupler, and we thank Andrew Gallant at the MIT Central Machine Shop for producing the final version. Jelena Ryvkina is credited for the artwork in Supplementary Fig. S2. A.M.S. is thankful for the support from the US National Science Foundation Graduate Research Fellowship. L.W. acknowledges the support from Technical University of Denmark. This work was supported in part by the Materials Research Science andEngineering Program of the US National Science Foundation under award numberDMR-0819762 and also in part by the US Army Research Office throughthe Institute for SoldierNanotechnologies under contract number W911NF-07-D-0004.

Additional information:

The authors declare no competing financial interests.


Nicolas Camara

Benoit Jouault
email: jouault@ges.univ-montp2.fr

Bilal Jabakhanji

Alessandra Caboni

Antoine Tiberj

Christophe Consejo

Philipe Godignon

Jean Camassel

Multidimensional characterization, Landau levels and Density of States in epitaxial graphene grown on SiC substrates

teaching and research institutions in France or abroad, or from public or private research centers.   Multidimensional characterization, Landau levels and Density of States in epitaxial graphene grown on SiC substrates

Introduction

It is now widely accepted that graphene-based devices are promising candidates to complement silicon in the future generations of high-frequency microelectronic devices. To this end, the most favourable technique to produce graphene for industrial scale applications seems to be epitaxial graphene (EG) growth. This can be done by chemical vapour deposition on a metal [START_REF] Sutter | Epitaxial graphene on ruthenium[END_REF][START_REF] Coraux | Structural Coherency of Graphene on Ir(111)[END_REF] or by heating a SiC wafer up to the graphitisation temperature [START_REF] Forbeaux | Heteroepitaxial graphite on 6H-SiC (0001): Interface formation through conduction-band electronic structure[END_REF][START_REF] Berger | Ultrathin Epitaxial Graphite: 2D Electron Gas Properties and a Route toward Graphenebased Nanoelectronics[END_REF][START_REF] De Heer | Epitaxial graphene[END_REF][START_REF] Kedzierski | Epitaxial Graphene Transistors on SiC Substrates[END_REF]. In the first case, the disadvantage is the need to transfer the graphene film on an insulating wafer. In the second case, the SiC wafer plays the role of the insulating substrate without any need for further manipulation. Of course, to be suitable for the microelectronics industry, these EG layers must be continuous and homogeneous at the full wafer scale or, at least, on surfaces large enough to process devices.

On the Si-face of 6H or 4H SiC substrates, graphitisation at high temperature in an Ar atmosphere close to atmospheric pressure shows promising results for on-axis substrates. In this way, single-layer epitaxial graphene (SLEG) has already been grown at the full wafer scale [START_REF] Emtsev | Towards wafer-size graphene layers by atmospheric pressure graphitization of silicon carbide[END_REF][START_REF] Virojanadara | Homogeneous large-area graphene layer growth on 6H-SiC[END_REF] but an open issue remains the 6√3 SiC surface reconstruction which is a C-rich buffer monolayer on top of the SiC substrate. The first "real" graphene layer on top of this buffer layer is strained, not at all free-standing, strongly coupled to the C-rich buffer, heavily n-type doped, with a low-carrier mobility. On the contrary, on the C-face of the same SiC substrates, there is no need of a C-rich buffer layer at the interface before growing the first graphene layer [START_REF] Hass | Why Multilayer Graphene on 4H-SiC (000-1) Behaves Like a Single Sheet of Graphene[END_REF][START_REF] Sprinkle | First Direct Observation of a Nearly Ideal Graphene Band Structure[END_REF][START_REF] Camara | Current status of self-organized epitaxial graphene ribbons on the C face of 6H-SiC substrates[END_REF][START_REF] Camara | Anisotropic growth of long isolated graphene ribbons on the C face of graphite-capped 6H-SiC[END_REF]. In this way, the mobility could reach 30,000 cm 2 /V s in the work of Ref. [START_REF] Berger | Electronic Confinement and Coherence in Patterned Epitaxial Graphene[END_REF].

For a long time, whatever the growth technique, the uniformity and quality of the EG was not good enough to find evidence of the so-called "half integer" quantum Hall effect (QHE). However, recently, large SLEG areas have been produced on the C-face of on-axis SiC substrates and, on such monolayer graphene, the carriers were holes with mobility close to the one found in mechanically exfoliated graphene films on SiO 2 /Si [START_REF] Novoselov | Two-dimensional gas of massless Dirac fermions in graphene[END_REF]. Consequently, the QHE could be demonstrated [START_REF] Wu | Half integer quantum Hall effect in high mobility single layer epitaxial graphene[END_REF]. This shows clearly the advantage and quality of SLEG grown on the on-axis C-face of a SiC wafer over the onaxis Si-face. However, for further integration of graphene with current SiC technology, 8°off-axis substrates should be also considered since they constitute the standard in modern SiC industry [START_REF] Si | Investigations of 3C-SiC inclusions in 4H-SiC epilayers on 4H-SiC single crystal substrates[END_REF].

In this work, we compare the results of graphene growth on semi-insulating, on axis and and 8°off-axis, 4H-SiC substrates. The quality, uniformity and size of the growth products will be compared using optical microscopy (OM), scanning electron microscopy (SEM), atomic force microscopy (AFM) and micro-Raman spectroscopy (μR). Then, Hall effect measurements will be done at different temperature in order to extract the density of states in the epitaxial monolayers.

Graphene growth, microscopy and Raman studies

To produce SLEG, in both cases of on axis and 8°off-axis SiC substrates, we used the recipes of Ref. [START_REF] Camara | Anisotropic growth of long isolated graphene ribbons on the C face of graphite-capped 6H-SiC[END_REF]. On the on-axis material, this produces long, self-ordered, graphene ribbons which are typically 5 μm wide and several 100 μm long. This has been described at length in the work of Ref. [START_REF] Si | Investigations of 3C-SiC inclusions in 4H-SiC epilayers on 4H-SiC single crystal substrates[END_REF]. On the off-axis substrates, this resulted also on SLEG islands but the morphology is completely different This is shown in Figure 1. Instead of narrow ribbons, after 30 min graphitisation at 1700°C, large SLEG islands can be obtained which can reach 300 μm long and 50 μm wide for the biggest ones. See Figure 1a and1b. They can have a trapezoidal or triangular shape, see Figure 1a-c and 1f and, usually, nucleate from a defect on the surface. See Figure 1e and 1f. This may be either an unintentional particle remaining on the surface, a crystallographic defect such as a threading dislocation or a simple scratch made by a diamond tip. Whatever the origin, the growth starts from one nucleating centre and expands in a two-dimension carpet-like way. All resulting triangles are then self-oriented, with the longest side following the (11-20) plane direction.

In Figure 2a we show a typical AFM image of such a SLEG islands. When zooming, wrinkles become clearly visible in Figure 2 and show evidence of the continuity and strain-free character of the monolayers. Below the graphene islands, the step-bunched areas of the SiC surface are also clearly visible in both SEM and AFM pictures. The corresponding terraces are typically 100 nm wide and less than 2 nm high. A last evidence of the fact that the first layer of graphene is not coupled with the substrate and continuous despite the step-bunched surface is the facility with which we can remove the SLEG layer with an AFM tip. The result presented with the AFM picture of Figure 2c demonstrates the almost free-standing and continuous character of the grown SLEG. Tens of similar monolayer islands grown on, both, on axis and off-axis substrates were probed by Raman spectroscopy. We used the 514 nm laser line of an Ar-ion laser for excitation and got very similar features. At the micrometer size, all spectra reveal that the islands are of the same nature and very homogeneous. First, the D-band, which usually indicates the presence of disorder or edges defects, is very weak and the Raman signature is extremely close to the one found for exfoliated graphene on SiO 2 /Si [START_REF] Camara | Current status of self-organized epitaxial graphene ribbons on the C face of 6H-SiC substrates[END_REF]. Second, the 2D-band appears at low frequency (2685 cm -1 ) which is strong evidence that there is no strain at the layer to substrate interface (i.e. almost a free-standing SLEG layer). Third, this 2D-band can be fitted with a single Lorentzian shape with a FWHM of 30 cm -1 [START_REF] Ferrari | Raman Spectrum of Graphene and Graphene Layers[END_REF]. Fourth, the ratio I 2D /I G between the integrated intensities of the 2D-band and the G-band is high, which suggests weak residual doping in the order of 3 to 6 × 10 12 cm -2 [START_REF] Basko | Electron-electron interactions and doping dependence of the two-phonon Raman intensity in graphene[END_REF]. Altogether, these Raman and microscopy measurements tend to demonstrate the almost free-standing low-doped and continuous character of the grown layers [START_REF] Camara | Anisotropic growth of long isolated graphene ribbons on the C face of graphite-capped 6H-SiC[END_REF][START_REF] Camara | Growth of monolayer graphene on 8°o ff-axis 4H-SiC (000-1) substrates with application to quantum transport devices[END_REF].

Electrical transport measurements

Gold alignment marks were used to select some SLEG position by OM. Then, they were contacted by e-beam lithography and subsequent deposition of a contact layer made of Cr/Au in Hall bar configuration. A typical example is shown in Figure 3.

Then transport measurements were done at low temperature on the different samples, using a maximum magnetic field of 13.5 T. The contact geometry allowed simultaneous measurement of, both, the longitudinal and transverse voltages with the current flowing between two injection contacts at the flake extremities. In both series of samples, from the sign of the Hall voltage, we found that the carriers were holes (in agreement with other results published on the C-face [START_REF] Berger | Electronic Confinement and Coherence in Patterned Epitaxial Graphene[END_REF][START_REF] Novoselov | Two-dimensional gas of massless Dirac fermions in graphene[END_REF]). The holes concentration ranged from 1 × 10 12 to 1 × 10 13 cm -2 at low temperature, with a weak temperature dependence.

For carrier concentrations larger than 3 × 10 12 cm -2 , no QHE could be detected and only Shubnikov-de Haas (SdH) oscillations were found. This is shown in Figure 4 for an off-axis sample and, as usual, the plot of the inverse field at which the oscillations maxima occur versus the Landau level index shows a clear linear dependence going down to the origin. This is the usual signature of the heavily doped graphene. For the low doped layers, the transverse resistance exhibits now quantized Hall plateaus, clearly governed by the sequence R K /4(N + 1/2) in which R K = h/e 2 is the Von Klitzing constant [START_REF] Klitzing | The quantized Hall Effect[END_REF] and N = 0, 1, 2... As already known, this peculiar sequence of resistance values is the well-known quantum transport signature of the monolayer graphene Landau levels [START_REF] Novoselov | Two-dimensional gas of massless Dirac fermions in graphene[END_REF]. In Figure 4(b).we show the longitudinal and Hall resistance values for such a low-doped SLEG device with hole concentration n s = 1.2 × 10 12 cm -2 and mobility μ ~5000 cm 2 /V s at T = 1.6 K. At B = 12 T, the longitudinal resistance cancels while the transverse resistance tends to 12.9 kΩ which is the expected value for the N = 0 plateau (R K /2).

In Figure 5a, we present similar resistance measurements obtained with a lower doped sample with a hole concentration n s = 8 × 10 11 cm -2 and a mobility μ ~11,000 cm 2 /V s. The mobility is high enough and the concentration low enough to make the N = 0 and N = 1 plateaus well resolved and stable up to 13.5 T. The experimental results of Figure 5a have been obtained in a three probes configuration with low resistance contacts (40 Ω). The Hall resistance corresponds to the symmetric part of the signal: ρ xy ~(V(B)+V(-B))/2I, where the voltage V is measured between a lateral probe and the current drain. At high magnetic fields, we identify V(+B)/IG as r xx , where G~4 is the geometric factor and I is the current.

The temperature dependence of r xx (B) is shown in Figure 5a, between 1.6 and 44 K. In this temperature range, an activated behaviour is found for the resistivity: r xx ~exp(-E a /k B T) of the N = 0 plateau. This activation energy E a is the energy separation between the Fermi energy E F and the delocalised states of the N = 1 Landau level. In Figure 5b we plot the resistivities values ρ xx taken at different magnetic fields in the vicinity of the R K /2 plateau. The activation energy E a varies from 0.7 to 3.3 meV between B = 10 and 13 T, which remains much smaller than the distance between the first and the second Landau level (~120 meV at B = 10 T). This indicates that the Fermi energy is firmly pinned by localised states. E a has been calculated by taking into account only temperatures above 6 K. At lower temperatures, there is an additional contribution to the conductivity, which is visible in Figure 5b as a change in the slope. We attribute this additional contribution to hopping.

In principle, from the activation energy, we can reconstruct the density of state ρ(E). The filling factor is calculated from B = 10 to 13 T, each filling factor change ∆ν at a given magnetic field corresponding to a density variation ∆n s = n s ∆ν/ν. The Fermi energy shifts by ∆E a to compensate for the density variation and the mean value for the density of states at energy ~Ea is given by ρ(E) = ∆n s /∆E a .

Following this procedure, already used in the early times after the discovery of the integer QHE [START_REF] Weiss | Density of States in Landau Level Tails of GaAs-AlxGa1-xAs Heterostructures[END_REF], we find the density of states plotted in Figure 6. The formation of the Landau level is evidenced as, when E a decreases, the density of states ρ(E) increases and becomes one order of magnitude larger than the density of states ρ 0 (E) without magnetic field at a comparable energy E F ~100 meV: ρ 0 (E) ~15 × 10 9 cm -2 meV -1 . The shape of ρ(E) gives a rough upper bound of the halfwidth at half-maximum (HWHM) of the N = 1 Landau Level. We find HWHM ≤ 3 meV. This value is in good agreement with results obtained recently on EG by STM [22]. However, the extracted density is systematically larger than ρ 0 over the whole investigated energy range. This observation, combined with the fact that hopping was neglected, indicates that more detailed investigations are still needed.

Finally, since EG has recently been proposed for metrological application, we plot, in Figure 7, the longitudinal resistance as a function of the current at B = 13.5 T. This magnetic field is far from the filling factor υ = 2 and; therefore, the breakdown occurs at relatively low current: I = 0.5 μA, which corresponds to a current density j = 0.025 A/m. By comparison, for III-V heterostructures, critical current values of 1 A/m are reported.

Conclusion

To summarize, we have shown the possibility to grow large islands of monolayer graphene on the C-face of on-axis and 8°off-axis commercial 4H-SiC wafers. The graphene layers are continuous, almost free-standing and show quantum transport properties comparable with high-quality, low-doped, exfoliated graphene. We show evidence of half-integer QHE specific of graphene monolayer and give a first estimate of the density of states in the magnetic field.

Figure 1

 1 Figure 1 SEM images of a monolayer graphene islands grown on the C-face of an 8°off-axis 4H-SiC substrate. (a, b) Images of the largest homogeneous SLEG islands, (c) early growth, (d) zoomed image with visible wrinkles, (e, f) example of starting nucleation point by a surface defect with step bunching clearly visible in (f).

Figure 2

 2 Figure 2 AFM images of continuous and almost free standing monolayer graphene islands grown on the C-face of an 8°off-axis 4H-SiC substrate. (a) at a large scale, the zoom in (b) showing the wrinkle and the step bunched character of the SiC surface below and (c) a layer scratched by an AFM tip.

Figure 3

 3 Figure 3 Optical microscopy of a SLEG grown on 8°off-axis semi-insulating SiC substrate. (a) before contact and (b) after contacting in a Hall Bar configuration for Hall Effect measurement.

Figure 4

 4 Figure 4 Typical magnetoresistance measurements for low doped and highly doped epitaxial graphene-based Hall Bars. (a) Longitudinal resistance of highly p-type doped epitaxial monolayer versus the magnetic field B, measured at 1.6 K. The resistance increases linearly with B with the superimposed SdH oscillations clearly resolved. Index of Landau levels (8-14) is also reported. Inset: the Landau plot indicates a phase equal to 0°, as expected for Dirac electrons. (b) Longitudinal and transverse resistance of low p-type doped epitaxial monolayer versus applied magnetic field B, at T = 1.6 K. The Hall resistance approaches the integer plateau R xy ~12.9 kΩ at B ~13 T. The second plateau at 4 kΩ is hardly visible.

Figure 5

 5 Figure 5 Magnetoresistance measurements of the best sample at different temperatures. (a) Longitudinal and transverse resistances of low p-type doped (n s = 8 × 10 11 cm -2 ) epitaxial monolayer versus applied magnetic field B, at different temperatures. (b) Temperature dependence of the resistivity r xx of a graphene ribbon at different magnetic field values close to the filling factor v = 3. The slope in the semilog scale gives the activation energy E a , which is the energy difference between the Fermi energy and the mobility edge of the second (N = 1) Landau level.

Figure 6

 6 Figure 6 Density of states r(E) as a function of the energy E a . For comparison, the density of states without magnetic field at E F = 100 meV is indicated by an arrow.

Figure 7

 7 Figure 7 Longitudinal resistance (in ohms) as a function of the injected current. Breakdown of the quantization occurs at I = 0.5 μA.

Acknowledgements

This work was supported by the French ANR ("GraphSiC" Project No. ANR-07-BLAN-0161). We acknowledge the EC for partial support through the RTN ManSiC Project, and the Spanish Government through a grant Juan de la Cierva.

Abbreviations AFM: atomic force microscopy; EG: epitaxial graphene; HWHM: half-width at half-maximum; QHE: quantum Hall effect; SEM: scanning electron microscopy; SdH: Shubnikov-de Haas; SLEG: single-layer epitaxial graphene.

Author details

Competing interests

The authors declare that they have no competing interests.


Xianqin Meng

Valerie Depauw

Guillaume Gomard

Ounsi El Daif

Christos Trompoukis

Emmanuel Drouard

Alain Fave

Frederic Dross

Ivan Gordon

Christian Seassal

De- Sign

Design and fabrication of photonic crystals in epitaxy-free silicon for ultrathin solar cells

Keywords: Photonic crystals, Epitaxial crystalline silicon, Thin-film devices and applications, Photovoltaics, Holographic lithography

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Design and fabrication of photonic crystals in epitaxial free silicon for ultrathin solar cells

Xianqin Meng * 1,2 , Valerie Depauw 3 , Guillaume Gomard * 1,2 , Ounsi El Daif 3 , Christos Trompoukis 3 , Emmanuel Drouard * 1 , Alain Fave * 2 , Frederic Dross 3 , Ivan Gordon 3 and Christian Seassal * 1

Introduction

In order to overcome the limitations of classical thin film silicon solar cells, integrating planar Photonic Crystals (PCs) structures [START_REF] Park | Absorption enhancement using photonic crystals for silicon thin film solar cells[END_REF][START_REF] Daif | Proc. "Photonic Crystal Enhanced Absorption of Amorphous Silicon for Solar Cells[END_REF][START_REF] Daif | Absorbing one-dimensional planar photonic crystal for amorphous silicon solar cell[END_REF] has appeared as an attractive solution. The PCs have been considered to create either back reflectors [START_REF] Zeng | Demonstration of enhanced absorption in thin film Si solar cells with textured photonic crystal back reflector[END_REF] or selective filters for tandem solar cells [START_REF] Bielawny | Intermediate reflectors for enhanced top cell performance in photovoltaic thin-film tandem cells[END_REF] . Meanwhile, thin film crystalline-silicon (c-Si) solar cells are one of the promising candidates for low cost photovoltaic applications thanks to commercially compatible mass production processes [START_REF] Depauw | Large-area monocrystalline silicon thin films by annealing of macroporous arrays: Understanding and tackling defects in the material[END_REF] . Considering both fabrication costs and bulk recombination issues, the thickness of c-Si layer should be limited as much as possible. Increasing its absorption through PCs light trapping techniques is then essential to achieve reasonable conversion efficiencies on such devices. More precisely, to enable efficient current generation, light trapping schemes are essential to capture the red and near-infrared part of the solar spectrum [START_REF] Franken | Understanding light trapping by light scattering textured back electrodes in thin film n-i-p-type silicon solar cells[END_REF][START_REF] Müller | TCO and light trapping in silicon thin film solar cells[END_REF] . We recently proposed to pattern the absorbing layer itself as a planar PC. This approach constitutes an efficient light trapping technique and can lead to a substantial increase of the solar light absorption over a wide spectral range [START_REF] Daif | Absorbing one-dimensional planar photonic crystal for amorphous silicon solar cell[END_REF][START_REF] Goetzberger | Light trapping, a new approach to spectrum splitting[END_REF][START_REF] Ferrier | Surface emitting microlaser based on 2D photonic crystal rod lattices[END_REF][START_REF] Gomard | Two-dimensional photonic crystal for absorption enhancement in hydrogenated amorphous silicon thin film solar cells[END_REF] . Out of this concept, and considering c-Si based structures, we proposed to control light capture and absorption using planar PCs Bloch modes that stand over the light-line. Such nanopatterns allow thin c-Si absorbing layers (with thicknesses of around 1000 nm), i.e. much lower than the diffusion length of the minority carriers. The absorption increase related to light trapping is then expected to lead to short-circuit current densities exceeding those similar unpatterned cells.

In order to generate such PC patterns in a compliance with industrial requirements, relevant techniques then include nano-imprint [START_REF] Chen | Enhancement of the light conversion efficiency of silicon solar cells by using nanoimprint anti-reflection layer[END_REF] and holographic lithography; they are essential to develop a high throughput process compatible with large surfaces. Additionally, holographic lithography is a maskless process, particularly wellsuited to the fabrication of periodic patterns like PCs [START_REF] Campbell | Turberfield, Fabrication of photonic crystals for the visible spectrum by holographic lithography[END_REF][START_REF] Cho | Photonic Crystal Slab Waveguides Fabricated by the Combination of Holography and Photolithography[END_REF][START_REF] Kim | Enhanced light extraction from GaN-based light-emitting diodes with holographically generated two-dimensional photonic crystal patterns[END_REF][START_REF] Bogaerts | SOI nanophotonic waveguide structures fabricated with deep UV lithography[END_REF] . Its parameters have been carefully adjusted and controlled from our previous investigation based on the desired stack and pattern [START_REF] Meng | Absorbing photonic crystals for silicon thin-film solar cells: Design, fabrication and experimental investigation[END_REF] .

In this paper, we report on a way of patterning the epitaxial free c-Si [START_REF] Depauw | Large-area monocrystalline silicon thin films by annealing of macroporous arrays: Understanding and tackling defects in the material[END_REF] as a 1D and 2D PC. The global design of the ultra-thin film solar cell, which has been optimized optically by simulation, is introduced in section 2. Its optical properties as well as the parameters of the PC membrane which maximize the absorption in the active layer are also discussed in this section, which defines the objectives to attain for further experimental development. This last point is developed in section 3. After presenting the general fabrication process of our patterned cells, we finally generate the targeted pattern by combining holographic lithography with an ICP and RIE steps.

Optical design of 1D and 2D PC c-Si thin film solar cells

The basic solar cell structure consists, from the back to the front, of a glass substrate, a silver (Ag) layer acts as a reflector and metallic conductor, a p-doped c-Si layer, an real absorbing c-Si layer, a n-doped a-Si:H and a top transparent conductive oxide (TCO) layer deposited at the end of the pattering process to be an anti-reflector layer. As described in Fig. 1, the top layers are patterned as planar PCs. The finite-difference time-domain (FDTD) approach numerically solves the differential form of Maxwell's equations, so that complex device design can be molded and optimized device designs can be determined [START_REF] Dewan | Light-trapping in thin-film silicon solar cells with integrated diffraction grating[END_REF] . To investigate the wave propagation in the patterned c-Si membrane and the thin film c-Si solar cells, the AM1.5G solar spectrum intensity distribution was taken into account over the 300-1100 nm spectral range; these bounds roughly and respectively corresponds to the lower limit of the solar spectrum and the c-Si gap. The boundary conditions for x and y directions are periodic and Perfectly Matched Layers (PML) for 1D PCs. For the 2D PC simulation, the x, y directions are set as Periodic, and z is PML. The first test is based on a single 1 µm c-Si layer, which is patterned by 1D PC and 2D PC with the same period, air filling faction (ff) and etching depth, in order to compare the absorption spectra with the planar c-Si layer, as shown on Fig. 1. It indicates that the absorption spectra in the patterned 2D and 1D PC layers is respectively 57% and 50% under normal incident, which is higher than 31% from the planar c-Si layer. The main is achieved over 0.61µm, which corresponds to the lattice parameter of the PCs. Moreover, under normal incidence, the absorption efficiency of 2D PC layer does not depend on the polarization of the incident light. In the planar c-Si layer, from about 450 nm to 1100 nm, the clearly visible absorption resonances correspond to Fabry-Perot modes thanks to the large index contrast between air and c-Si layer. If these resonances can lead to a complete absorption of the incident light, the spectral density of modes is limited by the low thickness of the layer, leading to low integrated absorption on the whole spectral range. Moreover, the large index contrast leading to a high reflection is also responsible for the low absorption (~50%). Then, the envisaged roles of the PC are twofold: to reduce the reflection at short wavelengths, and to increase the spectral density of modes at large wavelengths. The absorption spectra of the whole stack and c-Si layer, which are displayed on Fig. 3, indicate the patterned stacks have higher absorption than the reference stack both in the stack and in the sole c-Si layer. Moreover, a maximal value for the integrated absorption of the whole 1D patterned cell is about 71 % over for L = 0.61 µm, D/L = 65 % and etching depth = 0.14 µm. Looking for the useful part of this integrated absorption, namely the one in the sole active layer, we found that optimized parameters of 1D patterned stack could be determined yielding to 46%, a theoretical improvement of 37% of the absolute absorption efficiency in the sole c-Si, compared to 33.7% of unpatterned configuration in normal incidence. The important increase of absorption efficiency in the c-Si layer is expected to give rise to a subsequent gain of the conversion efficiency of such a 1 µm thick c-Si solar cell.

As evidenced in our former work for a different configuration on a-Si:H [START_REF] Gomard | Two-dimensional photonic crystal for absorption enhancement in hydrogenated amorphous silicon thin film solar cells[END_REF] , the absorption efficiency of thin film solar cells integrating 2D PC should lead to higher gains than their 1D counterpart. As an example, by choosing the same lattice parameter, surface filling fraction and etching depth as in the case of the 1D PC, the absorption in the sole c-Si obtained within the 2D PC patterned stack is 56.5%, which is higher than the one from the 1D PC stack and the unpatterned stack to 23% and 68%, as well as independent of the polarization. When the wavelength is lower than the period, the absorption spectral mainly depends on the Fabry-Perot resonance thanks to the high index of membrane, therefore, which can be seen both the unpatterned and patterned cells. Over 610nm, multiple absorption peaks, which exist both for 1D and 2D PCs, are related to the PCs Bloch modes.

Implementation the PC patterns in the fabrication cells

In this section, we discuss on the steps leading to the patterning of our cells, using holographic lithography and RIE as well as ICP techniques. A more detailed description of those techniques will be developed in the following part. In particular, we will introduce the main techniques involved in this process. These can be tuned in order to reach the targeted parameters of the 1D and 2D PC (its period L, ff and its etching depth) and to transfer the obtained pattern in the different layers, while keeping the roughness of the etched sidewalls as low as possible.

Holographic lithography consists in irradiating a photosensitive resist using an interference pattern [START_REF] Meng | Absorbing photonic crystals for silicon thin-film solar cells: Design, fabrication and experimental investigation[END_REF] . The sample, covered with a UV sensitive resist, is placed on a sample holder, associated to a mirror placed at 90° of the sample. This system is illuminated by a laser beam; this generates fringes which irradiate the resist, with an inter-fringe distance L directly related to the angle θ through the relation L= λ laser /(2sinθ). In order to generate a 2D pattern with a square symmetry at the surface of the sample, the irradiation needs to be repeated after rotating the sample by 90°. The resist is then exposed as schematically shown in Fig. 4.a. We implemented this generic process using a laser source at 266 nm with a power around 8 mW and a NEB 22 negative tone Chemically Amplified Resist (CAR). While the selected wavelength enables the patterning of structures with lattice parameters around 600 nm, using negative resist is well-suited to the generation of holes in the resist, as in the case of our design. With this system, the processing sequence associated to holographic lithography includes sample cleaning and preparation, hexamethyldisilizane (HMDS) adhesion promoter spinning; the resist and HMDS are spun at 3000 rpm for 30s. The post-apply bake (PAB) is performed at a temperature of 110°C, followed by UV exposure, for 120s, and post-exposure bake (PEB), at 92°C for 180s. The development is performed in MF702 solution (supplied by Shipley Co.) for 20s; see Fig. 4 b). The next step is a descum step in oxygen plasma for 8s [START_REF]Field Guide to Optical Lithography[END_REF] . The PC pattern defined by holographic lithography is first transferred into a 100 nm thick SiO 2 hard mask (as shown in Fig. 4 c)), and then into the underlying ITO and a-Si layers, using RIE processes. Transfer into the SiO 2 hard mask is achieved using a CHF 3 plasma. The ITO layer was then etched by CH 4 : H 2 plasma, and the a-Si:H layer was etched with a SF 6 :Ar by RIE (as Fig. 4 d)). In order to obtain vertical etched sidewalls, ICP is used to transfer the pattern in the c-Si layer using ICP with a high etch rates by high ion density and high radical density, which is displayed on Fig. 4 e); finally, the resist and mask are removed by RIE (shown in Fig. 4f)).

In our first test, another process is considered based on the following stack: glass/Al(1µm)/c-Si(1µm)/ SiO 2 (0.1µm). This stack, which is derived from the design presented above, is based on the established epi-free technology. It should be noted that the actual thicknesses of these layers may be slightly different due to technological uncertainties. A 100 nm thick silica layer was added on the top of the stack and acts as a hard mask during the etching of the c-Si layer. Starting from such samples, it is possible to first pattern the hard mask as a 1D or 2D PC (with a square symmetry) via holographic lithography, and then to transfer this pattern into the mask layer with RIE with a 16sccm flow, at 15mT, and with a power of 100W for 900s. The c-Si layer is finally etched by ICP using 50sccm for Cl 2 , radio frequency RF1 and RF2 with 500W and 100W for 45s though the SiO 2 mask to get vertical sidewalls, as shown on the SEM profile views, Fig. 5 b) and d). Then the resist and SiO 2 are removed at the end of etching process by RIE, which is shown in the front view Fig. 5 a) and c). The lines and holes with a reasonable regularity and roughness could be transferred. The achieved period of 1D and 2D stacks are in the targeted range 550 -650 nm; the ff is just in the range corresponding to the optimum configuration (40 %-65 %), and etching depth is in the 100 -140 nm range. The following steps are the deposition of the a-Si and ITO layers on the top of these patterned stacks.

Conclusion and further work

Further work will consist in the measurement of the optical properties of these 1D and 2D completed structures. It is noticeable that similar PC structures have already experimentally exhibited an improved absorption in 1 µm thick c-Si layers. Finally, opto-electrical measurements are planned on complete PC structured solar cell samples, to determine their yield and to compare them to the one of non structured samples. Provided efficient electrical passivation, the yield improvement is expected to be close to the optical absorption improvement.

Fig. 1 .Fig. 2 .

 12 Fig.1. Absorption spectra in the patterned 2D and 1D PC c-Si layer with the same period, ff and etching depth (period = 0.61µm, ff = 65% and etching depth = 0.14µm), compared to the planar layer

Fig. 3 .

 3 Fig.3. Absorption spectra in a) the whole stack and b) the c-Si layer for the 1D and 2D PC patterned stacks (L, D/L and etching depth set to 610 nm, 35% and 140 nm respectively) compared to the unpatterned stack

Fig. 4 .

 4 Fig.4. Main steps of the processes enabling to pattern the active layer as a 1D and 2D PC through holographic lithography (a, b), RIE(c, d, f) and ICP (e)

Fig. 5 .

 5 Fig.5. Top (a, c) and profile (b, d) view of a c-Si layer patterned as a 1D network (period around ≈ 550 nm, D ≈ 280 nm) and a 2D network of round holes (period ≈ 550 nm, D ≈ around 350 nm) into a 250 nm depth

Acknowledgements

PC patterning was performed at the NanoLyon Technology Platform. Regis Orobtchouk is acknowledged for his support on laser holography. X.Meng acknowledges the China Scholarship Council (CSC).