Erwan Engel

Jérémy Ratel

Frédéric Mercier

Magaly Angénieux

Christelle Planche

Development of insect farming as an option for urban circular bioeconomy: the chemical safety issue

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

Development of insect farming as an option for urban circular bioeconomy: the chemical safety issue

Erwan Engel 1 , Jérémy Ratel 1 , Frédéric Mercier 1 , Magaly Angénieux 1 and Christelle Planche Background/Objectives: With the growing world population and the rapid urbanization, there is a need to find alternatives to conventional protein sources for food and feed but it is also necessary to find new ways to manage organic waste. To respond to these challenges, farming insects may offer new options with two main scenarios in a circular bioeconomy approach. Firstly, insects like yellow mealworm (Tenebrio molitor) are already exploited for food and feed as sources of high-quality proteins, PUFAs and micronutrients (1). Secondly, some insect species, which can grow on many types of substrates like black soldier fly (Hermetia illucens), could be used in urban areas for biowaste recycling with the production of varied valuable by-products derived from insects like biomaterials (chitin), biofertilizers (frass), biofuels (produced from larvae fat)(2). Learning form the BSE crisis, it is essential to ensure the safety of these insect products for the development of these new practices.

Methods: N/A

Results: Some studies are already available about the microbiological hazards but there is a lack of knowledge concerning the chemical safety. Some chemical contaminants may be found in insect farming environment (persistent organic pollutants, heavy metals. . .), in their feeding substrates (pesticides, mycotoxins, antimicrobials. . .), can be produced during processing methods (process-induced toxicants) or can migrate from packaging material to insect products(3).

Discussion / Conclusion: In order to ensure the chemical safety of insect products, studies need to be conducted with a focus on 1/the transfer of chemical contaminants from the feeding substrate or the environment into the insect organism and their impact on insect survival 2/the bioaccumulation of these contaminants (persistent organic pollutants, heavy metals) or even their biodegradation (mycotoxins, antimicrobials) by insects(3) 3/the generation of process-induced toxicants or the migration of substances from packaging material 4/the potential biomagnification in consuming species or over the successive reuse cycles of insect products.

  1

1

MASS group, UR QuaPA, INRAE, F-63122 Saint Genès Champanelle, France

Proceedings of the Nutrition Societyhttps://doi.org/10.1017/S0029665122000957 Published online by Cambridge University Press

Disclosure of Interest

None Declared


Î T Beno

Rittaud
email: rittaud@math.univ-paris13.fr

00a08

FIBONACCI-LIKE SEQUENCES FOR VARIANTS OF THE TOWER OF HANOI, AND CORRESPONDING GRAPHS AND GRAY CODES

Keywords: 05C20, 11B39, 68R15, 94B25

We modify the rules of the classical Tower of Hanoi puzzle in a quite natural way to get the Fibonacci sequence involved in the optimal algorithm of resolution, and show some nice properties of such a variant. In particular, we deduce from this Tower of Hanoi-Fibonacci a Gray-like code on the set of binary words without the factor 11, which has some properties intersting for itself and from which an iterative algorithm for the Tower of Hanoi-Fibonacci is obtained. Such an algorithm involves the Fibonacci substitution. Eventually, we briefly extend the study to some natural generalizations.

The Tower of Hanoi is a puzzle invented by Édouard Lucas [START_REF] Claus | La Tour d'Hanoï[END_REF][START_REF] Claus | La Tour d'Hanoï, Véritable casse-tête annamite[END_REF] in which a set of n disks of different radius from 1 to n are put on a peg A in decreasing order, thus materializing a triangular tower. Two other pegs, B and C, are empty. The aim of the game is to move all the disks on the peg C (or, in a roughly equivalent version, on either B and C), following the two rules: [START_REF] Atkinson | The Cyclic Tower of Hanoi[END_REF] the disks are moved one at a time, taking a disk on the top of a peg and putting it on the top of another peg, and (2) a disk cannot be put over a smaller disk. This is what we will call the classical Tower of Hanoi puzzle. A set-theoretic description of it is the following. We write either d k or k for the disk of radius k, and ∆ k = {1, . . . , k} for the set of the k smallest disks (with ∆ k = ∅ for k < 1). Any state of the puzzle corresponds to an ordered 3-partition of ∆ n , written as (A, B, C) and referred as a state of the puzzle (also referred as regular state in the litterature, when it is necessary to emphasize on the fact that disks on each peg has to be set up with decreasing size, an assumption that is weakened in some studies). A move from such a state to another one, say (A , B , C ), is allowed if, and only if, the two ordered partitions are equal up to some d ∈ ∆ n such that d ∈ {min(A), min(B), min(C)} ∩ {min(A ), min(B ), min(C )}.

A lot of variants of the puzzle has been proposed since Lucas' orginal one. We report the reader to the highly valuable book [START_REF] Hinz | The Tower of Hanoi -Myths and Maths, Second Edition[END_REF] for a general synthesis on the subject.

Lucas already understood that the Tower of Hanoi was deeply linked to numeration systems. Indeed, he wrote in 1893 [9, p. 58] that Increasing the number of pegs and slightly modifying the rule of the game would easily provide representations of all numeration systems. [En augmentant le nombre de tiges et en modifiant légèrement les règles du jeu, on obtiendrait facilement des représentations de tous les systèmes de numération.]

The optimal algorithm to solve the puzzle with n disks requires 2 n -1 moves (hence passes through 2 n states), and the total number of admissible states is 3 n . (There exists a "worst" algorithm that solves the puzzle passing through all these 3 n state exactly once). Hence, it is not a surprise that there are natural links between the Tower of hanoi and binary and ternary numeration systems. At Lucas' time, only integral numeration systems were known. Since the "worst" solution (i.e. the solution that pass through all possible states) requires 3 n -1 moves, it is sensible to ask for more pegs to represent other numeration systems. But now that non-integer numeration systems are known, we can give to Lucas' sentence a new meaning, keeping the three initial pegs and modifying the rules of the game, to get a Tower of Hanoi version of some nonconventional ways to write integers.

A first possibility consists in restricting the moves allowed between pegs. For example, we can forbid any direct move from A to C and from C to A. It is well-known that this constraint leads to the "worst" algorithm mentioned beforehand, that visits every possible state of the puzzle among the 3 n ones. Each of the possible variants of this kind is linked to some numeration system (as well as to some Gray code) defined by a linear recurring sequence (see [START_REF] Rittaud | A Numeration System and a Gray Code Given by a Variant of the Tower of Hanoi, Numeration 2017[END_REF]).

The initial question that gave rise to the present article was the converse: find natural alternative rules for the Tower of Hanoi such that the minimal number of moves required to solve the puzzle with n disks corresponds to a sequence fixed a priori. One of these sequences for which an answer can be found is the Fibonacci sequence, and we will shw that the answer described here extends to some other linear recurring sequences as well.

Apart from the sequence of minimal moves, other links between the Tower of Hanoi and the Fibonacci sequence can be made. In particular, it is shown in [START_REF] Hinz | Discovering Fibonacci numbers, Fibonacci words, and a Fibonacci fractal in the Tower of Hanoi[END_REF] that, for the classical puzzle with n disks, the number of key states (i.e. for which the minimal number of moves to reach (∅, ∅, ∆ n ) is exactly twice the minimal number of moves to reach (∆ n , ∅, ∅)) is equal to F n-1 . The same article mentions the following other result, due to Merryfield and published in [START_REF] Bennish | The Tower of Hanoi problem and mathematical thinking[END_REF]: for any n, the set of distincts A k (resp. B k , C k ) attained during the optimal algorithm of resolution of the standard puzzle is of cardinality

F n+2 (resp. F n+1 , F n+2 ).
The present paper is organized as follows. In Section 1, we recall the relevant properties of the classical Tower of Hanoi we wish to generalize. Section 2 is devoted to the main variant we are interested in. In this variant is defined Fibonacci moves. We prove that the variant defined by these kind of moves, the Tower of Hanoi-Fibonacci, is optimally solved in a number of moves essentially given by the Fibonacci sequence (Section 2.2). Then, we provide a link with the classical Zeckendorf-Fibonacci numeration system (Section 2.2) and deduce from it an iterative algorithm for the Tower of Hanoi-Fibonacci. We then study a Graylike code associated to this numeration system (Section 2.3), then investigate the general properties of the graph associated to the puzzle (Section 2.4). Eventually, in Section 3 are briefly investigated some generalizations and questions, in two directions. The first one is when the definition of Fibonacci moves is modified so as to get an optimal algorithm that requires a number of moves given by a linear recurring sequence of the form m n = m n-p + m n-q + 1. The second one considers complementary restrictions on moves between pegs, that gives rise to a Tribonacci-like sequence.

The classical Tower of Hanoi

In the following, a subset {d k 1 , . . . ,

d k i } of ∆ n with k 1 < • • • < k i is simply written k 1 • • • k i .
It is known since Lucas that the Tower of Hanoi has a solution for any n 0, and that there is a unique solution with minimal number m n of moves. Such a solution can be described recursively by the following decomposition, valid for any n 1, from which we easily deduce that

m n = 2m n-1 + 1, hence m n = 2 n -1 (since m 0 = 0) . (∆ n , ∅, ∅) m n-1 -→ (n, ∆ n-1 , ∅) 1 -→ (∅, ∆ n-1 , n) m n-1 -→ (∅, ∅, ∆ n ).
Since the number of states during the optimal resolution of the puzzle is 1 + m n = 2 n , it is natural to consider the binary expansion of length exactly n to code the successive states from 0 to 2 n -1. It is easily proved by induction that the index k (between 1 to n) of the leftmost changing digit from the binary expansion of i to the binary expansion of i + 1 corresponds to the disk which is moved when going from the state i to the state i + 1. As a consequence, we have that, for any 1 k n, the number of times the disk d k is moved is 2 n-k . Also, consider the alternative codage of the states of the puzzle with n disks by elements of {0, 1} n given by the following rules: the initial state (∆ n , ∅, ∅) is labelled 0 n , and when we go from partition (A, B, C) to (A , B , C ) by moving the disk d = d k , the label of the state (A , B , C ) is defined as the label of (A, B, C) in which the k-th digit has been switched (that is: this digit becomes a 0 if it was a 1 and a 1 if it was a 0). Then, an induction shows that the sequence of codages of the successive states thus obtained coincides with the reflected binary Gray code, that is: the list G n of all binary words with exactly n letters defined recursively by G 0 = {0} and G n = 0G n-1 + 1G n-1 (where, for a sequence L = {x 1 , . . . , x k } of words and d a letter, dL = {dx 1 , . . . , dx n } and, with L = {y 1 , . . . , y }, the notation L + L stands for {x 1 , . . . , x k , y 1 , . . . y }). The fundamental property of such a list G n is that any two consecutive elements of the list differ by exactly one digit.

For a given state, when only the partition is known but not the order of its elements, we write the partition as X Y Z. An expression like X Y Z -→ R S T means a move (or a sequence of moves) in which the position of the element R (resp. S, T ) of the final partition is the same as the position of the element X (resp. Y , Z) in the initial partition. It was observed in [START_REF] Hinz | Pascal's Triangle and the Tower of Hanoi[END_REF] that the graph H n = (V n , E n ) of the classical Tower of Hanoi has a fractal structure similar to the Sierpiński triangle, H n being made of three copies of H n-1 for any n 1, any two of these copies being linked by a single edge corresponding to a move of the form n ∆ n-1 ∅ -→ ∅ ∆ n-1 n (see Figure 1).

Eventually, the optimal solution of the puzzle can be described by the following algorithm: move d 1 (always in the way A → B → C → A if n is odd, and in the way A → C → B → A if n is even), then, while there is a disk d k = d 1 that can be moved, move that disk then then move again d 1 . Definition 2.1. Let X and Y be two different pegs such that, for some k ∈ ∆ n , we have X = k X and Y = ∆ k-1 Ỹ . Write Z for the third peg. We define a k-Fibonacci move as a move that consists in putting simultaneously both k -1 and k onto Z, i.e.:

k X ∆ k-1 Ỹ Z -→ X ∆ k-2 Ỹ (k -1)kZ.
A Fibonacci move is a k-Fibonacci move for some k. The Tower of Hanoi-Fibonacci is the Tower of Hanoi puzzle in which only Fibonacci moves are allowed. (Note that this definition will be slightly modified in Section 2.4.)

Note that the 1-Fibonacci move is the one in which only one disk is moved (the disk d 1 ). Hence, this move is the only one common to the Tower of Hanoi-Fibonacci and the classical Tower of Hanoi.

Theorem 2.2. The Tower of Hanoi-Fibonacci with n disks admits a solution for any n 0. There is only one optimal algorithm for it, that needs exactly F n+2 -1 Fibonacci moves (hence passes through F n+2 different states).

As an example, here is the optimal solution in the case n = 5.

(∆ 5 , ∅, ∅) -→ (2345, ∅, 1) -→ (345, 12, ∅) -→ (45, 1, 23) -→ (45, ∅, 123) -→ (5, 34, 12) -→ (15, 34, 2) -→ (5, 1234, ∅) -→ (∅, 123, 45) -→ (∅, 23, 145) -→ (12, 3, 45) -→ (1, ∅, 2345) -→ (∅, ∅, ∆ 5 )
Proof. The proof is very similar to the classical case. First, the cases n = 0 and n = 1 both admit trivial solutions, with m 0 = 0 = F 2 -1 and m 1 = 1 = F 3 -1, where m n stands for the minimal number of moves to solve the puzzle with n disks. Now, put n 2 and assume that the puzzle with n -1 and n -2 disks are both solvable, with m n-2 = F n -1 and m n-1 = F n+1 -1. Consider the puzzle with n disks. To be moved, the disk of radius n needs to be alone on its peg, and needs the tower ∆ n-1 to be on another peg. To mimimize the number of moves, we can ask d n to be moved only once, hence any solution of the puzzle needs to reach the state (n, ∆ n-1 , ∅). We thus get a recursive description of the optimal solution of the puzzle with n disks:

(∆ n , ∅, ∅) m n-1 -→ (n, ∆ n-1 , ∅) 1 -→ (∅, ∆ n-2 , (n -1)n) m n-2 -→ (∅, ∅, ∆ n ).
Hence, we have that m n = m n-1 + 1 + m n-2 , so, by the induction hypothesis,

m n = (F n+1 -1) + 1 + (F n -1) = F n+2 -1, as required.
2.2. Link with the Zeckendorf-Fibonacci numeration system. The classical link between binary numeration system and the standard Tower of Hanoi extends to the Zeckendorf (or Zeckendorf-Fibonacci) numeration system and the Tower of Hanoi-Fibonacci. This link will provide us an iterative algorithm for the optimal solution of the latter puzzle.

Recall that, as proved in [START_REF] Zeckendorf | Représentation des nombres naturels par une somme de nombres de Fibonacci ou de nombres de Lucas[END_REF], for any fixed n 2 and any integer 0 < k < F n+1 there exists a unique finite sequence (u i ) 2 i n ∈ {0, 1} n-1 such that u i u i+1 = 0 for any i and k =

2 i n u i F i .
Such a sequence is the Zeckendorf-Fibonacci expansion of k. We will also write [u n • • • u 2 ] F for it, and Z((u i ) 2 i n ) for the corresponding k. By convention, we may define Z(0) as the empty sequence. When we need the length of a Zeckendorf-Fibonacci expansion to be of a certain kind (as in the following theorem), we allow ourselves to append leading 0s to it, hence considering

[0u n • • • u 2 ] F as equivalent to [u n • • • u 2 ] F .
In the following, a ZF-sequence (or ZF-word) will denote any binary sequence (or binary word) satisfying the property that it does not contains two successive 1 anywhere in its terms. Two ZF-words like u n • • • u 2 and 0u n • • • u 2 will be regarded as equivalents. Under this equivalence relation, the Zeckendorf-Fibonacci expansion of k is unique. Moreover, this expansion defines a bijection from N * onto the set of (non-empty) ZF-sequences.

The Zeckendorf-Fibonacci expansion of k > 0 can be obtained by the application of the following algorithm:

• r := k, u i := 0 for all i 2, n := max(j 2 :

F j k) • while r > 0: i := max(j : F j r) u i := 1 r := r -F i • return((u i ) 2 i n ).
Theorem 2.3. Let n 0 be some integer, let 0 < k < F n+2 , and let k -

1 = [u n+1 • • • u 2 ] F and k = [v n+1 • • • v 2 ] F .
Let j be the biggest index such that u j = v j . The k-th move of the Tower of Hanoi-Fibonacci with n disks is a (j -1)-Fibonacci move.

Proof. This is a simple induction making use of the recursive description of the algorithm

(∆ n , ∅, ∅) -→ (n, ∆ n-1 , ∅) -→ (∅, ∆ n-2 , (n -1)n) -→ (∅, ∅, ∆ n ).
The property is true for n = 0 and n = 1. Assume it is true for n -2 and n -1 for some n 2. The moves from (∆ n , ∅, ∅) to (n, ∆ n-1 , ∅) are moves from 1 to m n-1 = F n+1 -1, so their Zeckendorf-Fibonacci exansion of length n are all of the form [0u n • • • u 2 ] F . Hence, by induction hypothesis on the puzzle with n -1 disks, the property is true for all these moves. Now, the Fibonacci move (n,

∆ n-1 , ∅) -→ (∅, ∆ n-2 , (n -1)n) is the F n+1 -th one, of Zeckendorf-Fibonacci expansion of length n equal to [10 • • • 0] F .
The biggest moving disk in this move is the n-th one, and the biggest index j as defined in the theorem is equal to n + 1, so the theorem is also valid for this move.

The remaining F n moves are the ones with Zeckendorf-Fibonacci expansion of length n of the form

[10u n-1 • • • u 2 ] F , where [u n-1 • • • u 0 ] F is the Zeckendorf-Fibonacci expansion of length n -2 of k -F n+1 .
Hence, we can apply to it the induction hypothesis made on the puzzle with n -2 disks, and we are done.

Theorem 2.4. Let 0 < k n be two integers. For the Hanoi-Fibonacci puzzle with n disks, the number of k-Fibonacci moves in the optimal algorithm is equal to F n+1-k .

Proof. We procede by induction on n and (decreasing) induction on k. For k = n, the number we are looking for is equal to 1, which corresponds indeed to

F n+1-k = F 1 = 1. For k = n -1,
it is easy to check that there is also exactly one k-Fibonacci move, a number equal to F n+1-k = F 2 . Now, for k < n -1, by induction hypothesis, the part (∆ n , ∅, ∅) -→ (n, ∆ n-1 , ∅) of the algorithm involves a number of k-Fibonacci moves equal to

F (n-1)+1-k = F n-k , and the part (∅, ∆ n-2 , (n -1)n) -→ (∅, ∅, ∆ n ) involves F (n-2)+1-k = F n-k-1 moves, hence a total equal to F n-k + F n-k-1 = F n+1-k moves.
Theorem 2.3 provides a complete iterative algorithm for the Tower of Hanoi-Fibonacci, with the only issue that, for 1-Fibonacci moves (i.e. a move of the single disk d 1 ), one has to determine on which peg the disk d 1 has to move. Here is an answer to this question. Let us say that d 1 is moving to the right (resp. to the left) whenever it moves from A to B, from B to C or from C to A (resp. from A to C, from B to A or from C to B). Thus, by Theorem 2.4, we can code the sequence of 1-Fibonacci moves for the puzzle with n disks as a word µ n ∈ {l, r} Fn , where l denotes a move to the left and r a move to the right. We then have the following result: Theorem 2.5. In the optimal algorithm for the Tower of Hanoi-Fiboonacci:

• if n ∈ 2N * , then the k-th letter of µ n is a r iff Z(k) has an even number of 1s; • if n / ∈ 2N * , then the k-th letter of µ n is a r iff Z(k)
has an odd number of 1s.

Proof. We proceeds by induction on n 2. Write the decomposition of the optimal solution of the puzzle, with the corresponding number of 1-Fibonacci moves (given by Theorem 2.4 with k = 1).

(∆ n , ∅, ∅)

F n-1 -1 -→ (n, ∆ n-1 , ∅) 0 -→ (∅, ∆ n-2 , (n -1)n) F n-2 -1 -→ (∅, ∅, ∆ n ).
Assume for example n ∈ 2N * (the other case would be similar). Consider the k-th letter of µ n corresponding to a 1-Fibonacci move among the F n-1 -1 first ones. By the induction hypothesis, the fact that n -1 is odd and the fact that the 1-Fibonacci moves corresponding to (∆ n , ∅, ∅) -→ (n, ∆ n-1 , ∅) are the same as the one for (∆ n , ∅, ∅) -→ (n, ∅, ∆ n-1 ) but with exchanging the rs and the ls, we have that the considered k-th letter is a r iff Z(k) has an even number of 1s. For a value of k corresponding to a 1-Fibonacci move among the last F n-2 -1 Fibonacci moves, the reasoning is the same, with the additional consideration that the Zeckendorf expansion of k is now of the form [10u n-3

• • • u 2 ] F .
Let us also mention the following qualitative results, that show in particular that the number of 1-Fibonacci moves to the left and to the right are as balanced as possible.

Corollary 2.6. We have µ 0 = ∅, µ 1 = l and, for any n 2,

µ n = (µ n-1 µ n-2 ) * = µ n-2 µ n-3 µ n-3 µ n-4
, where µ * stands for the word in which each l has been replaced by a r and each r by a l. Moreover, denoting by |µ| d the number of letter d ∈ {r, l} in the word µ, we have

|µ 3n | r = |µ 3n | l , |µ 3n+1 | l -|µ 3n+1 | r = 1 and |µ 3n+2 | r -|µ 3n+2 | l = 1.
Proof. The first part is proved by an induction on n and the following decomposition of the optimal solution of the puzzle (for n 2), in which the words on the arrows stand for the sequences of moves of d 1 during 1-Fibonacci moves:

(∆ n , ∅, ∅) µ * n-1 -→ (n, ∆ n-1 , ∅) -→ (∅, ∆ n-2 , (n -1)n) µ * n-2 -→ (∅, ∅, ∆ n ).
The second part is a simple induction on n.

2.3.

Gray-like code. To complete notations set up for lists in Section 1, for any list L of elements of {0, 1} n , write L for the list made of all elements of L in which the leftmost letter is removed. Then, set N 0 = ∅, N 1 := {1} and N n := 10N n-1 + 10N n-2 (here writing each word always with a 1 as leftmost digit). Eventually, let G := i 0 N i := {g 1 , g 2 , . . .}. Such a construction may be seen as a mirroring process analogous to the classical one for binary Gray codes, as shown in Table 1.

G 6                                                                      g 1 = 0 0 0 0 0 1 N 1 g 2 = 0 0 0 0 1 0 N 2 g 3 = 0 0 0 1 0 0 N 3 g 4 = 0 0 0 1 0 1 g 5 = 0 0 1 0 0 1    N 4 g 6 = 0 0 1 0 0 0 g 7 = 0 0 1 0 1 0 g 8 = 0 1 0 0 1 0            N 5
g 9 = 0 1 0 0 0 0 g 10 = 0 1 0 0 0 1

g 11 = 0 1 0 1 0 1 g 12 = 0 1 0 1 0 0 g 13 = 1 0 0 1 0 0                        N 6
g 14 = 1 0 0 1 0 1 g 15 = 1 0 0 0 0 1 g 16 = 1 0 0 0 0 0 g 17 = 1 0 0 0 1 0 g 18 = 1 0 1 0 1 0 g 19 = 1 0 1 0 0 0 g 20 = 1 0 1 0 0 1 Table 1. The Gray-like code of the Tower of Fibonacci-Hanoi with n = 6 disks.

Recall that the Hamming distance between w and w ∈ {0, 1} n , written h(w, w ), is the number of their differents digits, that is: for w = w 1 • • • w n and w = w 0 • • • w n , we have h(w, w ) = card(1 i n : w i = w i ). When w and w do not have the same number of letters, we append as many 0 as necessary to the shortest one to make it of the same length as the other.

Theorem 2.7. The application g n -→ Z(n) is a bijection from the set of all non-empty ZFwords (assuming the equivalence between

u n • • • u 2 and 0u n • • • u 2 ) onto N * . Moreover, for any n 1 we have h(g n , g n+1 ) = 2 if n + 1 = F k for some k 3; 1 otherwise.
Proof. By induction, assume that N n-1 ∪ N n-2 contains all ZF-words of {0, 1} n-2 exactly once, N n-1 (resp. N n-2 ) containing those with a 0 (resp. a 1) as leftmost digit. Hence, by definition of N n , N n contains all ZF-words in {0, 1} n with 10 as leftmost digits, each exactly once. As a consequence, by the induction hypothesis, N n ∪ N n-1 contains all ZF-words of {0, 1} n-1 exactly once, with N n (resp. N n-1 ) containing those with a 0 (resp. a 1) as leftmost digit. Hence, G is in bijection with the set of all ZF-words (but the empty one). The theory of Zeckendorf-Fibonacci recalled in Section 2.2 thus gives us that g n -→ Z(n) is a bijection onto N * . Now for the property of Hamming distances. By induction, assume the property satisfied for G n-1 and that the Hamming distance among two successive elements of N k is always equal to 1. We have

G n = 0G n-1 + N n = 0G n-1 + 10N n-1 + 10N n-2 .
Hence, by induction hypothesis, the Hamming distance of any pair of successive elements of G n is equal to 1, apart, possibly, for the pairs (0g F n+1 -1 , g F n+1 ) (made by the last element of G n-1 and the following one in G n ) and (g F n+1 +F n-1 -1 , g F n+1 +F n-1 ) (made by the last element of G n-1 + 10N n-1 and the following one in G n ).

To prove that the induction hypothesis is valid also for n, it only remains to prove that the Hamming distance of the first of these pairs is equal to 2 and the one of the second pair is equal to 1. For the first one, by definition of N n , the two different digits of 0g F n+1 -1 and g F n+1 are the two leftmost ones, that is:

writing 0g F n+1 -1 in the form 010u n-3 • • • u 1 , we have g F n+1 = 100u n-3 • • • u 1 .
Hence their Hamming distance is indeed equal to 2. For the second pair, by construction we have (writing w for the word w from which the leftmost digit has been removed) g

F n+1 +F n-1 -1 = 10g F n+1 -F n-1 = 10g Fn and g F n+1 +F n-1 = 10g Fn-1 . As already noticed, g Fn = 0100u n-4 • • • u 1 and g Fn-1 0010u n-4 • • • u 1 , so 10g Fn = 1000u n-4 • • • u 1 and 10g Fn-1 = 1010u n-4 • • • u 1 , hence their Hamming distance is equal to 1.
Strictly speaking, a Gray code has the property that two consecutive elements are always of Hamming distance equal to 1, hence our list G is only a Gray-like code. One may wonder if we could recover a real Gray code that lists all the ZF-words. In itself, such a question is too large, and a natural restriction on it is to ask for such a Gray code to be length-increasing, that is: the list (g n ) n 1 should order the words in such a way that the leftmost 1 of g n is of increasing index with n. It is quite easy to check that such a natural condition cannot be satisfied, since for w the last ZF-word of length n -1 and w the first one of length n we necessarily have h(w, w ) 2. This remark leads to the following result: Theorem 2.8. Let (u n ) n>0 be a length-increasing sequence made of all non-null ZF-words (each of them appearing exactly once). For any n > 0, we have h(u n , u n+1 ) h(g n , g n+1 ).

In this sense, our Gray-like code G is as close as possible to a true Gray code length-increasing and containing all the ZF-words.

As it is done in [10, Theorem 5] for another Gray-like code linked to Fibonacci combinatorics given in [START_REF] Bernini | Two Gray Codes for q-ary k-generalized Fibonacci Strings[END_REF], it is possible to "de-mirror" the construction of G n , in the following way: Theorem 2.9. Let N 0 = ∅, N 1 = {1}, N 2 = {10} and N 3 = {100, 101}. For any n 4, all the following is well-defined and correct by induction. First, N n-1 has cardinality F n-1 . Set N n-1 = {w 0 , . . . , w F n-1 -1 }. For any 0 m < F n-1 , there exists a unique q = q n (m) ∈ {-2, -1, 0, 1, 2} of smallest absolute value such that w m+q ends with a 1. Eventually, for any

0 m < F n-1 , set V m =      {w i 0} if q n (m) = 0, {w i 0, w i 1} if q n (m) ∈ {-1, 2}, {w i 1, w i 0} if q n (m) ∈ {-2, 1}.
We then have

N n = 0 m<Fn V m .
Practically speaking, the previous result can be enforced in the following way: to get N n from N n-1 , write each element of the list N n-1 , each element being written twice in a row iff it ends with a 0. Then, concatenate (on the right) to each element of this new list either a 0 or a 1 so as to get a word that differs from the previous one by exactly one digit (or, for the first of the list, exactly two digits from the last word of N n-1 ).

For the proof, even if the Gray code in [START_REF] Bernini | Two Gray Codes for q-ary k-generalized Fibonacci Strings[END_REF] does not exactly match the one considered in the present article, the two situations are close enough to allow us to invoke here the proof given in [START_REF] Rittaud | Reflected Gray codes with Fibonacci constraints, and generalizations[END_REF] for Theorem 2.9 since, as can be easily checked, the sequence of last digits (starting at U 2 ) are exactly opposite (i.e. with 1s and 0s exchanged) in both definitions. Following [START_REF] Rittaud | Reflected Gray codes with Fibonacci constraints, and generalizations[END_REF] also leads to the following description that makes use of the Fibonacci substitution to describe the way digits are to be added to the elements of N n-1 to get N n . (Hence we do not provide the details here either, see [10, Corollary 1].) Corollary 2.10. Let σ be the Fibonacci substitution on the alphabet {α, β}, defined by σ(α) = αβ and σ(β) = α, and let (σ n ) n∈N := σ ∞ (α) be its fixed point. Let (τ n ) n∈N * be the word on the alphabet {0, 1} defined by τ 1 = 1 and, for any n 2,

τ n = 1 iff σ n/2 -1 = β. To get N n+1 from N n = {g F n+1 , . . . , g F n+2 -1 }
in Theorem 2.9 (with n 2), we can proceed by the following algorithm:

• Initialization:

L := ∅ • for i from F n+1 to F n+2 -1: if g i ends with a 1 then L := L + {g i } else L := L + {g i } + {g i } • write L =: {g F n+2 , . . . , g F n+3 -1 } • for i from F n+2 to F n+3 -1: in L, replace g i by g i τ i • return(L).
2.4. The Hanoi-Fibonacci graph. In the present section, our aim is to investigate how to represent the Tower of Hanoi-Fibonacci by a graph F n = (V n , E n ), in which V n as the 3 n possible states of the puzzle, and E n as the set of edges (e, e ) such that the move from e to e is a Fibonacci move. Note that, contarily to the graph of the classical Tower of Hanoi, this graph is oriented since Fibonacci moves are not reversible (apart for 1-Fibonacci moves).

Theorem 2.11. For any n 2, the un-oriented graph that corresponds to F n is non-planar.

Proof. Since F n ⊂ F n+1 for any n, it is sufficient to prove that F 2 is non-planar.

Let us merge the vertices (∅, 1, 2) and (∅, 2, 1) in a single vertex a, then the vertices (1, ∅, 2) and (2, ∅, 1) to get another single vertex b, and eventually the vertices (1, 2, ∅) and (2, 1, ∅) to get a third vertex c. The graph thus obtained can be split into two subsets of vertices, V = {a, b, c} and V = {(12, ∅, ∅), (∅, 12, ∅), (∅, ∅, 12)}. The set of edges of this new graph is the set of all possible edges between V and V , hence it is isomorphic to the complete bipartite graph K 3,3 . Hence, K 3,3 is a minor of F 2 , so, by Wagner's theorem, F 2 is not planar.

Theorem 2.12. For any n 0, F n is strongly connected. In other words, any possible state of the puzzle can be attained from any other under the rule of the Tower of Hanoi-Fibonacci.

Proof. We proceed by induction on n. Assume F n is strongly connected, and consider F n+1 . This latter graph contains exactly three copies of F n , that we denote by F A n , F B n and F C n depending on the peg on which is located d n+1 in each. By the induction hypothesis, each of these three copies of F n is strongly connected. Hence, to obtain the desired result it is sufficient to prove that there exists an edge from some vertex of F X n to some vertex of F Y n for any different pegs X and Y . The Fibonacci move (n + 1) ∆ n ∅ -→ ∅ ∆ n-1 n(n + 1) provides such an edge. The previous drawing of F 2 seems quite difficult to extend in a natural way to larger values of n, and a slight modification of Figure 1 seems more interesting for visualization purposes, even if it needs some specific codage to makes the arrow diagram handy. Also, as announced in Definition 2.1, it will be here easier to work with a slightly modified version of a Fibonacci move, hereafter defined as

k X ∆ k-1 Ỹ Z -→ ∆ k-2 X Ỹ (k -1)kZ.
In this new version, the tower ∆ k-2 ends up on X instead of remaining on Ỹ . This does not fundamentally change what precedes, and it is easy to check that all the results obtained under the initial Definition 2.1 of Fibonacci moves remain unchanged under the present variant. (In particular, since the graph F 2 remains the same for this variant, the new graph is still nonplanar.)

Now, with this variant of Fibonacci moves, we can make use of the classical graph of Figure 1 to represent the Hanoi-Fibonacci puzzle. We preserve in this graph the edges that represent 1-Fibonacci moves. The other edges are also preserved, but in a form that we will call here pseudo-edges. More precisely: the graph G n of the classical puzzle is made of three copies of G n-1 , together with three edges that make G n connected. In F n we define these oriented edges as n-pseudo-edges, represented as arrows labelled by 2 n-2 + 1 (for n 2). Let v be a vertex of F n which is the origin of a k-pseudo-edge e (hence with 1 < k n). The k-Fibonacci move for the vertex v ends up on the vertex v obtained by a jump of length 2 k-2 + 1 in the direction of the edge, i.e. v is the vertex of the graph at a distance 2 k-2 + 1 from v (each edge or pseudo-edge counting for 1) attained by following the path of length 2 k-2 + 1 defined by the geometrical direction defined by e. Theorem 2.13. Under the previous definition of F n , if the vertex v is the origin of a kpseudo-edge (k 2), then the vertex v is the state of the puzzle attained by the (only) possible k-Fibonacci move from the state v. Proof. Assume the result until n -1. The graph F n contains three copies of F n-1 , in each of which the property is true by induction. Therefore, it remains only to prove that the property is true also for the n-pseudo-edges of F n . By symmetry, it is enough to consider the case of the n-pseudo-edge of origin (n, ∆ n-1 , ∅). The n-Fibonacci move from this state leads to (∆ n-2 , ∅, (n -1)n). Also, under the classical rules of the Tower of Hanoi puzzle, going from (n, ∆ n-1 , ∅) to (∆ n-2 , ∅, (n-1)n) with the optimal algorithm requires exactly 2 n-2 +1 moves, which are all on the same geometrical direction on H n , so we are done.

We deduce from this a combinatorial proof of the following equality.

Corollary 2.14. For any n 0, we have

2 n = F n+2 + n-2 k=0 2 k F n-1-k .
Proof. With the notation of Section 1, we have 2 n -1 = m n . Also, by Theorem 2.4 and the proof of Theorem 2.13, we have Corollary 2.14 can be related to the classical fact that the sum of the n-th row of Pascal's triangle is 2 n and the sum of its n-th diagonal is F n .

m n = F n + n+1 k=3 (2 k-3 + 1)F n+2-k = n+1 k=2 F n+2-k + n+1 k=3 2 k-3 F n+2-k = F n+2 -1 + n-2 k=0 2 k F n-1-k .
3. Some generalizations and questions 3.1. Modifying the Fibonacci moves. Here, we briefly consider alternative ways of defining the allowed moves, extending in a natural way the Fibonacci moves. We write ∆ n n for the set of disks d k with n k n (so ∆ n n = ∆ 1 n = ∆ n for n 1 and ∆ n n = ∅ for n > n). Definition 3.1. Let p 1 and q 0 be two integers. Let X and Y be two different pegs of some state such that, for some k ∈ ∆ n , we have X = ∆ k-p+1 k X and Y = ∆ k-p Y . Write Z for the third peg of the state. We define a (p, q)-move as a move that consists in putting simultaneously all the disks of ∆ k-p+1-q k onto Z, i.e.:

∆ k-p+1 k X ∆ k-p Y Z -→ X ∆ k-p-q Y ∆ k-p+1-q k Z.
We will talk about the (p, q)-Tower of Hanoi for the Tower of Hanoi puzzle in which only (p, q)-moves are allowed.

Note that the (1, 0) case is the classical puzzle, and that the (1, 1) one is the Tower of Hanoi-Fibonacci puzzle.

Theorem 3.2. The (p, q)-Tower of Hanoi puzzle with n disks admits a solution for any n 0. There exists only one optimal algorithm for it, that needs exactly m n (p, q)-moves, where the sequence (m n ) n∈Z is defined by

m n = 0 for n 0 m n-p + m n-p-q + 1 for n > 0.
Proof. For n ≤ p, the (p, q)-move (∆ n , ∅, ∅) -→ (∅, ∅, ∆ n ) is allowed, so we have m n = 1 for any n p, which correspond to the formula stated in the theorem.

For n > p, the optimal solution is provided by the sequence of critical states, each of which needing, by induction, the number of moves written on its arrow:

(∆ n , ∅, ∅) m n-p -→ (∆ n-p+1 n , ∆ n-p , ∅) 1 -→ (∅, ∆ n-p-q , ∆ n-p+1-q n ) m n-p-q -→ (∅, ∅, ∆ n ).
By summing the moves we get the expected formula.

There is no serious doubt that generalization of Zeckendorf-Fibonacci, Gray-like codes and pseudo-edges of the graph H n can be given for (p, q)-moves, but some additional technicalities may be quite hard to overcome. For example, the case p = q = 2 provides a sequence m n which is not strictly increasing (since m 2n-1 = m 2n ), hence a convenient numeration system derived from it is probably not as simple as the Zeckendorf one for the Fibonacci sequence of the case p = q = 1. The study of the corresponding graph may be a little bit tricky as well to be extended to (p, q)-moves.

3.2.

Restricting the moves between pegs. Possible variants on the classical puzzle consist in allowing moves only between some pegs. For example, in the clockwise-cyclic variant introduced in [START_REF] Atkinson | The Cyclic Tower of Hanoi[END_REF], additionally to the classical rules of Section 1, a disk can move only from A to B, from B to C or from C to A.

Any variant of this type can be defined by an oriented graph with set of vertices {A, B, C}, an arrow XY standing for the fact that moves from the peg X to the peg Y are allowed. The sensible variants of this kind (i.e. for which the puzzle is solvable for any n) are the ones for which the corresponding graph is strongly connected [START_REF] Hinz | Discovering Fibonacci numbers, Fibonacci words, and a Fibonacci fractal in the Tower of Hanoi[END_REF]Theorem 8.4]. We will not consider all possible cases here, but only mention briefly the linear variant, in which the allowed moves are those from A to B, from B to A, from B to C and from C to B. It is well-known that, for such a restriction, the optimal algorithm for the classical puzzle needs 3 n -1 moves, so, since the number of distincts states is 3 n , the linear puzzle also provides the "worst solution", that is: the longest solution that does not come back to any state already met. Now, consider the linear variant for the Tower of Hanoi-Fibonacci, in which a k-Fibonacci move is allowed iff it makes d k going from A to B, from B to A, from B to C or from C to B. Write again m n for the minimal number of moves to solve ths variant with n disks. The optimal solution is then given by the following recursive description (for n 3):

(∆ n , ∅, ∅) m n-1 -→ (n, ∅, ∆ n-1 ) 1 -→ (∅, ∆ n-1 n , ∆ n-2 ) 1 -→ (∆ n-2 n-1 , n, ∆ n-3 ) m n-3 -→ (∆ n-1 , n, ∅) 1 -→ (∆ n-2 , ∅, ∆ n-1 n ) m n-2 -→ (∅, ∅, ∆ n ).
Hence, the sequence (m n ) n is given by m 0 = 0, m 1 = 2, m 2 = 5 and, for any n 3, m n = m n-1 + m n-2 + m n-3 + 3 (a kind of a Tribonacci sequence).

As regards the other variants derived from the restriction of moves between pegs, there is probably no specific difficulty to deal with them in the context of Fibonacci moves (or (p, q)-moves), apart from the fact that some of these variants already involve linear recurring sequences of order 6 in the classical Tower of Hanoi, so are possibly tiresome to describe in our even more technical context.

More interesting would be to find a general way to derive the sequence of moves (or at least of number of moves) from the conjunction of the two kinds of rules. For example, is it possible to deduce the previous Tribonacci sequence directly from what we separately know from the linear variant of the classical puzzle and from our study of the Tower of Hanoi-Fibonacci, instead of the recursive description we presented?

Figure 1 .

 1 Figure 1. The graph H 3 of the classical Tower of Hanoi with 3 disks.

Figure 2 .

 2 Figure 2. The 3-Fibonacci move from (35, ∆ 2 4, 6) to (5, ∆ 1 4, 236).

2 .

 2 The Tower of Hanoi-Fibonacci 2.1. Definition and optimal algorithm.

Figure 3 .

 3 Figure 3. The graph F 2 of the Tower of Hanoi-Fibonacci with 2 disks. (Line segments stand for edges going both ways.)

Figure 4 .

 4 Figure 4. The graph F 3 of the Tower of Hanoi-Fibonacci with 3 disks (under the variant of the Fibonacci moves) with its pseudo-edges.

Figure 5 .

 5 Figure 5. Vizualization of Corollary 2.14 for n = 5.

3.3. Further questions. We could also consider even more general rules for the moves. For example, we could allow moves of the form kX ∆ k-1 Y Z -→ X ∆ n-3 (n-1)Y (n-2)nZ, and so on. One may wonder if two different rules can lead to the same sequence, hence asking for the links between these rules.

Eventually, a deeper work would be to obtain a theoretical way to find from a linear recurring sequence some natural rules for the Tower of Hanoi for which the number of moves of the optimal algorithm would be given by the sequence. This will probably involve a more precise definition of a "natural rule". (For example, we may ask whether we can always restricts the study to markovian moves, i.e. moves for which their legality depends only on the initial and final states.) In a sense, answering this question would truly complete Lucas' original assertion.


A A Sinchenko

P Lejay

O Leynaud

P Monceau

Dynamical properties of bidirectional charge-density-waves in ErTe 3

Keywords: numbers: 72.15.Nj, 71.45.Lr, 61.44.Fw

We report a strong difference in the sliding properties of the bidirectional charge density wave (CDW) in the two-dimensional rare earth tritelluride ErTe3 which occurs below TCDW 1 = 265 K with a wave vector along c-axis and below TCDW 2 = 165 K with a wave vector along a-axis; the excess current carried by the motion of the CDW is 10 times less for the lower CDW compared with the value of the upper one. We tentatively explain this result by a stronger pinning of the lower temperature CDW intricated with the upper one, which inhibits its motion and may generate a phase slippage lattice.

I. INTRODUCTION

Symmetry-breaking phase transitions with interplay between multiple degrees of freedom are intensively studied in strongly correlated systems such as high-T c superconductors, modulated charge or spin structures (charge/spin density waves) and another ordered states. A charge density wave (CDW) is characterized by a spatial periodic modulation ∼ cos(Qx + φ) of the electronic density concomitant with a lattice distortion with the same periodicity inducing the opening of a gap, ∆, in the electronic dispersion. The collective properties of the CDW quantum state can be described by a complex order parameter Ψ ∼ ∆e iφ with the collective excitations of ∆ and φ being the amplitude and phase modes.

Although theoretically predicted by Kohn 1 , Overhauser 2 , Peierls [START_REF] Peierls | Quantum Theory of Solids[END_REF] and Fröhlich [START_REF] Fröhlich | [END_REF] in the 1950s and 1960s, the first Peierls-Fröhlich transition was experimentally observed nearly simultaneously in the beginning of 1970s in quasi-one dimensional systems -linear Pt-chain material [START_REF] Comes | Highly Conducting One-Dimensional Solids[END_REF] [K 2 Pt(CN) 4 Br 0.30 xH 2 O], organic tetrathiafulvalene-tetracyanoquinodimethane salt (TTF-TCNQ)(see for review 6 ), and quasi-two dimensional transition metal dichalcogenides MX 2 (M: Nb, Ta; X: S, Se) 7 . Then a few years later, CDW transitions were found in transition metal trichalcogenides, NbSe 3 , TaS 3 8 and in molybdenium bronze K 0.3 MoO 3 [START_REF]Low-Dimensional Electronic Properties of Molybdenum Bronzes and Oxides[END_REF] . In one hand, from 1D weak coupling mean field theories, with ∆/E F ≪ 1, the Peierls instability is driven by the electronic energy gain which originates mostly from the Fermi surface nesting with Q = 2k F . In the other hand, in the strong limit coupling, ∆/E F ≃ 1, as recognized by McMillan 10 and Varma and Simons 11 the transition is driven by the entropy of the lattice; the energy gain is then spread over the entire Brillouin zone as recently observed by inelastic neutron scattering 12 . As the electron phonon coupling is increased the importance of the electronic structure in k-space is reduced and a local chemical bonding picture in real space is more appropriate.

One of the most representative features of onedimensional (1D) compounds with a charge-density wave (CDW) is the possibility of collective electron transport first predicted by H. Fröhlich 4 , as a model for superconductivity in 1D. The extraconductivity which results from the CDW sliding is a collective motion of electrons, a mechanism totally different from the classical one of elastic scattering of individual electrons which leads to the residual resistance. However various mechanisms such as impurities, defects, interchain interaction or commensurability pin the phase of the CDW and at low electric field the conductivity exhibits a constant ohmic behavior due to quasi-particle excitations only. To overcome the pinning energy and to initiate the CDW sliding it is necessary to apply an electric field of a sufficient strength larger than some characteristic threshold electric field, E t 8,[START_REF] Grüner | Grüner Charge Density Waves in Solids[END_REF] . At the present time sliding CDW properties have been observed and well studied in many inorganic as well as in organic one-dimensional compounds (for recent review, see Ref. 8).

Since many years, the search of a possible CDW sliding in two dimensional (2D)compounds was unavailing. Only recently it was succeeded in observing collective CDW motion in quasi-2D rare-earth tritellurides compounds [START_REF] Sinchenko | [END_REF]15 . This new family of quasi-2D compounds has raised an intense research activity last time [16][17][18] , because it is considered as a model system for which the structure of the CDW ground state can be theoretically studied 19 . Thus a phase diagram as a function of the electron-phonon parameter was derived with a bidirectional (checkerboard) state if the CDW transition temperature is sufficiently low whereas a unidirectional stripe state, as observed experimentally, occurs when the transition temperature is higher. This result is relevant for a deeper understanding of the charge pattern in highly correlated materials, and particularly to the recent determination of the biaxial CDW in underdoped cuprates 20 .

RTe 3 (R=Y, La, Ce, Nd, Sm, Gd, Tb, Ho, Dy, Er, Tm) layered compounds have a weakly orthorhombic crystal structure (space group Cmcm). They are formed of dou-ble layers of nominally square-planar Te sheets, separated by corrugated RTe slabs. In this space group, the long b axis is perpendicular to the Te planes. These systems exhibit an incommensurate CDW below the Peierls transition temperature T CDW 1 through the whole R series 18,21 , with a wave vector Q CDW 1 = (0, 0, ∼ 2/7c * ) , with T CDW 1 above 300K for the light atoms (La, Ce, Nd). For the heavier R (Tb, Dy, Ho, Er, Tm) atoms a second CDW occurs at low temperature T CDW 2 with the wave vector Q CDW 2 = (∼ 2/7a * , 0, 0) perpendicular to Q CDW 1 . The superlattice peaks measured from X-ray diffraction are very sharp and indicate a long range 3D CDW order 18 .

Below the Peierls transition, in all RTe 3 compounds, the Fermi surface is partially gapped resulting in a metallic behavior at low temperature. The layered RTe 3 compounds exhibit a large anisotropy between the resistivity along the b-axis and that in the (a, c) plane, typically ∼ 10 2 below T CDW 1 and much higher at low temperature 22 . Because the unidirectional character of the upper CDW 19,21,23 , a conductivity anisotropy in the (a, c) plane arises in the CDW state as was observed experimentally and explained theoretically in Ref. 24. The effect of the upper CDW on the in-plane resistivity observed in experiments is very weak, no more than a few percents of the total resistance 18,22,24 .

Amplitude CDW excitations were probed by Raman spectroscopy 21 , time-resolved ARPES 25 and femtosecond pump-probe spectroscopy 26,27 . This later technique has allowed to study the disentanglement of the electronic and lattice path of the CDW order parameter, the collective vibrations being assigned to amplitude modes. On the other hand, collective charge phase excitations could not be observed in far-infrared measurements due to screening by the residual metallic component of the Fermi surface. But the phase collective mode is accessible through non-linear transport properties.

Thus, effect of collective electron transport was observed for the high temperature CDW in single crystals of DyTe 3 [START_REF] Sinchenko | [END_REF] , TbTe 3 and GdTe 3 15 but only when the electric field is applied along the Q CDW 1 -direction, namely caxis. Sliding effect is completely absent when the current is applied in the perpendicular, a-axis direction, demonstrating the unidirectional character of the high-T CDW. No attempt up to now was made at observing the possible sliding of the low-T CDW in RTe 3 .

Among all CDW compounds, only a few exhibit multiple CDWs. In MX 2 compounds, a triple Q structure is formed with three wave vectors of equal amplitude, 120 o apart 7 . In TTF-TCNQ a Peierls transition occurs first on TCNQ stacks at 54 K, a second CDW transition on parallel TTF stacks at 49 K which drives the transverse modulation along a from 2a to a locked value 4a at 38 K 5 . Both CDWs in NbSe 3 are formed on two different parallel chains 8 .

Hereafter we report measurements of CDW sliding properties of ErTe 3 which exhibit two CDWs at T CDW 1 = 270 K and T CDW 2 = 165 K. The situation is unique. Unlike NbSe 3 where both CDWs slide along parallel different chains, both CDWs in ErTe 3 exist within the same Te-planes, thus both CDWs modulate the positions of the same Te atoms. The main questions we would like to answer are the following:

(i) is it possible to detect the sliding of the low-T CDW, besides the fact that the anomaly of resistivity at T CDW 2 is barely visible? (ii) is there any change of the sliding of the high-T CDW below T CDW 2 ?

(iii) are these orthogonal CDWs totally independent or are they interacting one with the other? If yes, what is the result and the mechanism of this interaction?

II. EXPERIMENTAL

Single crystals of ErTe 3 were grown by a self-flux technique under purified argon atmosphere as described previously [START_REF] Sinchenko | [END_REF] . For experiments we chose good quality single crystals. Thin single-crystal samples with a thickness less than 2 µm were prepared by micromechanical exfoliation of relatively thick crystals glued on a sapphire substrate. The quality of selected crystals and the spatial arrangement of crystallographic axis were controlled using X-ray diffraction techniques. X-ray diffraction measurements show that single crystals with a thickness more than 2 µm as a rule are twinned with the change between c and a axis in neighboring layers. However, for thinner samples it was possible to select untwinned single crystals with well defined crystallographic axis positions.

From untwinned single crystals with a thickness typically 0.2-2.0 µm, we cut stripes with a length of a few mm and a width 50 -80 µm in well defined orientation, namely [100] and [001]. In the following we call c-axis stripes samples with the length along the [001] direction and a-axis stripes those with the length along [100]. In some cases it was succeeded to select native crystals with a shape of a narrow stripe with a width 50-100 µm and length near 1 mm, oriented along c-axis. Measurements of current-voltage characteristics (IVs) and their derivatives have been performed with a conventional 4probe configuration. For contacts preparation we used gold evaporation and cold soldering by In. The current was applied along the length of the stripes. For studying nonstationary effects a radiofrequency (rf) current was superposed on the dc current using current contacts connected with the generator via two capacitors. All measurements have been performed in the temperature range 4.2-340 K.

Example of temperature dependencies of the normalized resistance, R(T )/R(300K), for c-and a-axis stripes and conductivity anisotropy, R a (T )/R c (T ) is shown in Fig. 1. The difference between the T-dependence of R a and R c is a clear indication that the sample is untwinned. The dependencies shown in Fig. 1 corresponds qualitatively to those reported in Ref. 24 for DyTe 3 , HoTe 3 and TbTe 3 . However, one can note the decrease of anisotropy R a /R c below 40 K the origin of which needs further investigations.

III. EXPERIMENTAL RESULTS

A. high-T CDW

As reported before, the collective CDW motion for the high-T CDW is observed only in c-axis stripes. We were able to measure characteristic nonlinearity in IV curves below T CDW 2 down to 100 K; the conductivity of the stripe increases sharply above a threshold electric field. However, the shape of IV characteristics change qualitatively for temperatures slightly above and below T CDW 2 as demonstrated in Fig. 2. As can be seen, at temperatures below T CDW 2 IV curves become hysteretic and much more noisy. Additionally, sharp maxima of dV /dI appear at electric field close to E t .

The temperature evolution of differential IV curves in the temperature range 270-110 K for one of the stripes is shown in Figure 3. Neighbour curves differ by ∆T = 5 K. The same behavior was observed for another stripes. Note that the relative change of the differential resistance from static to sliding state is very small, no more than 3% from the total resistance, indicating a very low contribution of the CDW to the electron transport.

It can be seen as drawn in Fig. 8 that the threshold electric field increases monotonically with the decrease of temperature but in the temperature range T ≈ 190 -160 K E t increases more rapidly. Nearly the same effect was observed in the quasi-one dimensional NbSe 3 for high-T CDW at temperatures close to the second CDW transition 30 . Note that in these c-axis stripes (with the current applied along the c-axis) below T CDW 2 there are no indication of sliding of the low-T CDW the Q-vector of which being along a-axis.

CDW sliding is accompanied by low-frequency broadband electric noise (BBN) and narrowband noise (NBN). The collective electronic transport and the NBN generation can be characterized in terms of the CDW coherence and homogeneity of the CDW in space and in time. The coherence can be affected by external rf (or hf) irradiation. The most widely studied effect of the irradiation is the synchronization of the CDW, known also as the interference effect, mode locking, or Shapiro steps. 8 . In the present study we have observed Shapiro steps at such experimental conditions. Fig. 4 shows dV /dI(I) dependencies at T = 230 K under application of a rf field with different frequencies from 2 MHz up to 11 MHz with the rf ac field amplitude of 900 mV. Shapiro steps are clearly observed in the dV /dI(I) characteristics at all frequencies as sharp maxima in the differential resistance demonstrating high level coherency of the CDW.

It is interesting also to trace the evolution of Shapiro steps with temperature at fixed frequency. Fig. 5 shows dV /dI(I) curves under application of a rf field with F = 4.5 MHz and amplitude V = 0.9 V at different tem-peratures. As can be seen, Shapiro steps are clearly observed only down to temperatures close to T CDW 2 . Close to this temperature the amplitude of Shapiro steps starts to decrease that indicates the loss of coherency of the high-T CDW in the temperature range where the low CDW occurs.

As shown in Fig. 4 the separation ∆I between Shapiro steps increases when the rf-frequency increases. Having two contributions to the electric current: normal electrons and collective CDW transport, ∆I CDW can be easy calculated using dc IV characteristics:

∆I CDW = I total (1 - R R N ) (1)
where I total -total current, R -actual resistance of the sample and R N -normal state resistance.

In Fig. 9 we have plotted the excess current density ∆J CDW = ∆I CDW /S with the cross-section of the sample S = 2.1 × 10 -6 cm -2 as a function of rf frequency. As can be seen, as in quasi-1D systems 8 , ∆J CDW linearly increases with frequency.

B. low-T CDW

In a-axis stripes prepared from untwinned singlecrystals we succeeded to observe collective motion for the low-T CDW. A typical IV curve demonstrating characteristic CDW non-linearity at T = 140 K well below T CDW 2 is shown in Fig. 6(a). As can be seen, at E > E t = 0.44 V/cm, the differential resistance sharply decreases.

The temperature evolution of differential IV curves in the temperature range 90-170 K for the same sample is shown in Figure 6(b). The curves are shifted for clarity and the difference between each of them is ∆T = 5 K. As can be seen, the threshold electric field weakly decreases with the increase of T in the range 90-150 K. In contrast to the high-T CDW, at temperatures close to T CDW 2 ≈ 165 K E t starts to increase indicating divergency at T CDW 2 . Such a behavior has been previously observed in quasi-1D compounds with a CDW 8 . The Tdependence of the threshold field E t for the low-T CDW is plotted in Fig. 8. There are no any non-linearity in the IV-characteristics at temperatures above T CDW 2 = 165 K except a very weak Joule heating. Note that the observed contribution to the electron transport from the low-T CDW sliding is nearly 4 times less compared with the high-T CDW and its amplitude is no more than 0.7% from the total current.

To confirm that the observed non-linearity of IV curves is the real sliding of the low-T CDW we measured IV characteristics under application of dc and rf electric field. As in the case of the high-T CDW, in spite of the low amplitude in the sliding effect, we observed pronounced Shapiro steps however. Figure 7(a) shows dV /dI(I) dependencies at T = 130 K under application of a rf field with a frequency of 28 MHz and amplitude 0.9 V. For comparison, the static (without rf field) differential IV curve (blue) measured at this temperature is also shown. Note that the application of a rf electric field leads to a reduction of the threshold electric field E t . At the same time, Shapiro steps appear in the dV /dI(I) characteristics as sharp maxima in the differential resistance. With increasing frequency the distance between neighboring maxima increases proportionally to the frequency; that is illustrated in Fig. 7(b) where we show IV curves under application of a rf field with a frequency of 7, 14, 21, 35 and 55 MHz at the same temperature and with the same rf power. The curves are shifted relative to each over for clarity. The excess current density calculated from Eq.1 (with the cross-section of the sample S = 0.25 × 10 -6 cm -2 ) is plotted in Fig. 9 as a function of rf frequency. As for the upper CDW ∆J CDW linearly increases with frequency.

IV. DISCUSSION

The RTe 3 family (R: La, Ce, Pr, Nd, Gd, Tb, Dy, Er, Tm) is a model system for studying the effect of chemical pressure on the Peierls transition due to a reduction of the in-plane lattice constant from light R (La, Ce, ...) towards heavier R ions (Dy, Er, Tm). While the Q CDW 1 wave vector along c-axis (which results from charge transfer between RTe buckled planes and Te planes) is nearly the same for each member of the family, the Peierls transition temperature is much larger than room temperature for light R and decreases below room temperature for heavier R 18 . From optical spectroscopy 31 and ARPES measurements 17 it was shown that the CDW gap scales with the lattice parameters. In addition, optical measurements 40 have revealed that the remaining fraction of ungapped FS in the CDW state is larger for compounds with smaller lattice parameters. Thus phenomenologically, it was suggested that a second CDW is formed only when the first CDW is weakened with the decrease of the lattice parameter, making larger FS sections available for the new nesting condition in the transverse a-axis.

While ARPES measurements 17 are interpreted as a strong evidence for a FS nesting scenario, inelastic xray scattering 32 and Raman experiments 29 emphasize the strongly momentum-dependent electron-phonon coupling. In that case, a local chemical description of the distortions is more realistic, as proposed in Ref. 39 where the distortion of the Te net is viewed as an oligomer sequence of Te trimers and tetramers.

As far as ErTe 3 is concerned, only a few publications are available. The low-T CDW was discovered by Ru et al. 18 .The superstructures Q CDW 1 = 0.298c * and Q CDW 2 = 0.313a * (measured at T = 10 K) are sharp. Both wave vectors are present in the same crystallite. The integrated intensity of Q CDW 2 exhibit large fluctuations above T CDW 2 , at least up to 180 K. The CDW gaps were determined by ARPES 28 with ∆ CDW 1 = 175 meV and ∆ CDW 2 = 50 meV with the temperature dependence of ∆ CDW 1 slightly suppressed from the mean-field variation. These values are in agreement with those measured by Raman scattering 29 . The ratio 2∆ CDW 1 /k B T CDW 1 ≃ 15 is much larger than the BCS mean field 3.52 value, similarly to many one-dimensional systems 8 indicating strong coupling effect or the role of fluctuations.The observation of the amplitude mode by Raman scattering 29 such ω AM = √ λω 2kF with ω 2kF the frequency of the unnormalized CDW phonon energy yields λ = 0.4 indicating strong coupling.

It has to be noted 28 that the lower CDW is weaker with

2∆ CDW 1 k B T CDW 1 ∼ 2 2∆ CDW 2 k B T CDW 2 (2)
The density of states was calculated from the interacting tight-binding model. The onset of the high-T CDW suppresses N (E F ) by ∼ 77% of the unmodulated value while the low-T CDW further suppresses N (E F ) by ∼ 74% revealing that the gain in the second CDW is really modest 28 .

Many models for the RTe 3 electronic structure consider only Te planes, sometimes a single one; the buckled Te slabs are viewed as a simple charge reservoir that determines the Fermi level with the f states of the rare earth localized away from the Fermi surface and expected to play no role. However the hybridization between the rare-earth 4f electrons and Te 2p electrons was revealed 33 by the observation of a diffraction peak near the M 5 (3d-4f) absorption edge of rare-earth ions with a wave vector identical to that of the CDW.

We have experimentally measured nonlinear transport properties in ErTe 3 associated to each CDW, the upper one with current I applied parallel to Q CDW 1 along caxis, and the lower one with I Q CDW 2 along a-axis. Are these nonlinearities the signature of the Fröhlich-type conductivity as demonstrated 8 in quasi one-dimensional compounds?

As was mentioned in section I, it is assumed that both CDWs exist within the Te planes and that both CDWs modulate the position of the same Te atoms. The possibility that CDWs occur on a different Te plane of the Te bilayer is very unlikely, although evoked in Ref. 19. However it is worth to note that by Raman scattering experiments, the amplitude mode in the high-T CDW develops as a succession of two mean field transitions with different critical temperatures; that was associated to the Te bilayers 21 .

The main results we have obtained can be summarized as follows: in cooling, when approaching the bidirectional CDW ground state at T CDW 2 ≃ 165 K, the threshold field of the high-T CDW increases more than linearly, noise appears in the differential IV characteristics and the high-T CDW coherence is lost by the disappearance of Shapiro steps. In Fig. 8 we have plotted the temperature dependence of the threshold electric field for both CDWs. Taking into account the fact that the absolute value of E t is sample dependent and that the threshold characteristics of high-and low-T CDWs were measured in different stripes, exfoliated however from the same single crystal, the curves in Fig. 8 demonstrate only the qualitative behavior of E t .

As can be seen, in the temperature range 270-200 K the dependence E t (T ) of the upper CDW is linear and can be well described by the expression:

E(T ) = E(0)(1 - T T 0 ) (3)
with E(0) = 235 mV/cm and T 0 = 1.07T CDW 1 similarly to the behavior of E t for the upper CDW reported previously 15 . In the range 200-165 K we observe a deviation from the linear dependence and E t increases more rapidly in this temperature range. At temperature T < T CDW 2 = 165 K, E t (T ) resumes again a linear dependence. We understand such a behavior as the result of interaction between high-T and low-T CDWs. Indeed it has been shown that fluctuations of the low-T CDW extend up to nearly 200 K 18 .

The depinning process for the unidirectional CDW, namely in TbTe 3 at room temperature, has been probed by coherent x-ray diffraction 34 .Contrary to onedimensional systems (such as NbSe 3 and K 0.3 MoO 3 ) 8 the CDW remains undeformed below threshold and suddenly rotates to overcome pinning centers and reorders by motion above threshold.

In a superfluid there are a macroscopic occupation of a quantum state that picks out a unique reference frame which describes the velocity v s of the superfluid. In a superconductor it is the common momentum of Cooper pairs that defines v s . In the Fröhlich model, as stated by Allender et al. 35 , v s is determined by the velocity of the macroscopically occupied lattice wave which produces energy gaps. Then the extracurrent carried by the CDW into motion is j = nev. The frequency of Shapiro steps in differential IV characteristic was identified as the signature of the CDW velocity with v = λf with λ: the CDW wavelength. In Fig. 9 we have plotted together the excess current density in the nonlinear state of ErTe 3 as a function of the frequency of Shapiro steps for the high-T CDW at T = 230 K and the low-T CDW at T = 130 K. For the high-T CDW, in Fig. 9 we can evaluate the ratio J CDW 1 / f 0 = 28 A/MHz cm 2 . This value is very similar to the average value obtained on 14 samples for the upper CDW on NbSe 3 : 40.2 A/MHz cm 2 and 24 A/MHz cm 2 for the lower one 36 . Similar values were also obtained for o-TaS 3 37,38 . In a one-band model (or for one chain in 1D description), λ = 2π/2k F and the condensate density is 2k F /π. Then

J CDW f 0 = 2e (4)
This relation is well satisfied in 1D systems 8 and demonstrates the Fröhlich-type of conductivity.

Determining a similar value of J CDW /f 0 for the high-T CDW in ErTe 3 indicates that the same Fröhlich process is operating. Let consider the number of unit cells in the cross section of our sample with the lattice parameters b = 25.02 Å and a = 4.29 Å, one get 1.96 × 10 8 unit cells. Consider also that along the c-axis, one have one Te chain per unit cell. Then we recover the value of J CDW 1 /f 0 = 28 A/MHz cm 2 .

We can also estimate the CDW velocity. At T = 240 K, at the total current applied I = 10 mA, we have evaluated I CDW = 0.27 mA. With the electron density ∼ 10 23 cm -3 and taking into account that no more than 30%-40% of the FS is affected by the formation of the CDW 21 , we get a very small value of the CDW velocity ∼ 10 -2 cm/s. However the number of carriers condensed below the CDW gap needs to be verified.

While the sliding properties of the high-T CDW appear to correspond to the Fröhlich mechanism, the situation is totally different for the low-T CDW. From Fig. 9 we deduce the slope of J CDW 2 /f 0 10 times lower than the value for the high-T CDW. Additionally the threshold field is higher. This result is opposite to NbSe 3 with both CDWs sliding along parallel chains and for which the J CDW 2 /f 0 is nearly the same for both CDWs 36 .

We can only speculate on a possible explanation of these results. The bidirectional CDW is formed by two orthogonal modulations of Te atoms which naturally interact and are imbricated. Because the sliding is along c-axis for the high-T CDW and along a-axis for the low-T CDW, the coupling between them, say bonds, should be broken. The crystallographic structure should have an important role. It was shown 18 that the formation of the high-T CDW (in TbTe 3 ) appears to "stretch" the lattice from its expected value along the direction of the modulation wave vector (c-axis). It is also the direction of the glide between the two Te planes. Then the depinning along c-axis may appear to be easier, even if the high-T CDW looses its coherence when crossing T CDW 2 .

the general Fröhlich mechanism, with the current carried by the motion of the CDW as J CDW = nev and with v identified from Shapiro steps as v = λf 0 , then J CDW /f 0 = neλ. In ErTe 3 the slop J CDW /f 0 is 10 times less for the lower CDW with respect to the upper one. One may then suggest that the full electronic density condensed below the gap in the band of low CDW, as in the Fröhlich model, does not participate to the conductivity but only a part of it, namely arround 1/10. Pinning along a-axis may be stronger and the bond between both modulations anisotropic. Nonlinearity may result from the motion of a lattice of phase defects which is formed in the checkerboard lattice of both CDWs.

This interpretation needs naturally to face some experimental results. It appears to determine the modulated structure of ErTe 3 in the unidirectional CDW state as well as in the bidirectional one. STM measurements at low temperature are also crucial as well as a theoretical model for sliding of a bidirectional CDW. However, STM images may be difficult to be interpreted because possibly blurred by the disorder 41 of the bidirectional superstructure.

In conclusion we have observed the nonlinearity in transport properties of the bidirectional charge density wave ground state of ErTe 3 . While the sliding properties of the upper CDW appear to be similar to those previously found in quasi one-dimensional systems, the nonlinearity for the lower CDW may involve a phase defect lattice. More works are naturally needed to ascertain or weaken the present interpretation.

FIG. 1 .FIG. 2 .FIG. 3 .

 123 FIG. 1. (color online) ErTe3: temperature dependencies of normalized resistance, R(T )/R(300K), and conductivity anisotropy, Ra(T )/Rc(T ), for c and a-axis stripes

FIG. 4 .

 4 FIG. 4. (color online) dV /dI(I) dependencies of ErTe3 at T = 230 K under application of a rf field with different frequencies from 2 MHz up to 11 MHz with the rf ac field amplitude of 900 mV for the same sample as shown in Fig.2. The curves are shifted for clarity.

FIG. 5 .

 5 FIG. 5. ErTe3: dV /dI(I) dependencies under application of a rf field with frequency 4.5 MHz at different temperatures for a ErTe3 stripe oriented along c-axis.

  FIG. 6. (color online) a) Differential current-voltage characteristics, dV /dI(E) of a ErTe3 stripe oriented along a-axis at T = 140 K. b) Temperature evolution of normalized differential IV curves in the temperature range 90-170 K; the difference between each curve is 5 K.

  FIG. 7. (color online) a) dV /dI(I) dependencies at T = 130 K under application of a rf field with a frequency of 28 MHz (red) and without rf electric field (blue) for a ErTe3 sample oriented along a-axis. b) dV /dI(I) dependencies at T = 130 K under application of a rf field with a frequency of 7, 14, 21, 35 and 55 MHz at the same temperature and with rf power 0.9V. The curves are shifted for clarity.

FIG. 8 .

 8 FIG. 8. (color online) Temperature dependence of the threshold field ET , for high-and low-T CDW in ErTe3.

FIG. 9 .

 9 FIG. 9. (color online) Excess current density in the nonlinear state of ErTe3 as a function of the frequency of Shapiro steps for the high-T CDW at T = 230 K and the low-T CDW at T = 130 K.

ACKNOWLEDGMENTS

The work has been supported by Russian State Fund for the Basic Research (No. 14-02-01126-a), and partially performed in the frame of the CNRS-RAS Associated International Laboratory between CRTBT and IRE "Physical properties of coherent electronic states in coherent matter".


Marc Ganzhorn

Wolfgang Wernsdorfer

Dynamics and dissipation induced by single-electron tunneling in carbon nanotube nanoelectromechanical systems

We demonstrate the effect of single-electron tunneling (SET) through a carbon nanotube quantum dot on its nanomechanical motion. We find that the frequency response and the dissipation of the nanoelectromechanical system (NEMS) to SET strongly depends on the electronic environment of the quantum dot, in particular on the total dot capacitance and the tunnel coupling to the metal contacts. Our findings suggest that one could achieve quality factors of 10 6 or higher by choosing appropriate gate dielectrics and/or by improving the tunnel coupling to the leads.

Carbon nanotubes (CNT) have become an essential building block for nanoelectromechanical systems (NEMS). Their low mass and high Young's modulus allow for instance ultrasensitive mass [1][2][3] or force detection [4,5] (electric and magnetic) over a wide range of frequencies and its small diameter enables even single-molecule detection [5][6][7]. Moreover, CNT devices exhibit remarkable electronic transport properties, ranging from Kondo physics [8] to Coulomb blockade at high temperature [9]. It was recently demonstrated that a CNT NEMS' nanomechanical motion at very low temperature (i.e. in Coulomb blockade regime) is strongly affected by the electronic transport through the CNT quantum dot (QD), and vice versa: For instance, single-electron tunneling (SET) caused a frequency softening and increased dissipation when tuning the CNT dot's potential through a Coulomb peak [10][11][12].

Here we demonstrate that the response and dissipation of a CNT NEMS at very low temperature induced by zero bias SET through the CNT NEMS-QD critically depends on the dot capacitance, the tunnel coupling to the metal leads and temperature. We studied the frequency and dissipation response of nanomechanical bending modes to zero bias SET in suspended CNT devices with tunable tunnel couplings and different gate dielectrics, i.e. different dot capacitance. We observe that SET causes a frequency softening for small dot capacitance and/or tunnel coupling, whereas a frequency hardening or no frequency modulation is observed for large dot capacitance and/or tunnel coupling. We show that the dissipation of the CNT NEMS is mainly dominated by the capacitance, when electron tunneling through the dot is suppressed (i.e. in the Coulomb valley), whereas it is limited by the mean tunneling rate Γ and the conductance, when electron tunneling through the dot is allowed (i.e. on a Coulomb peak). Finally we demonstrate that the tunnel current is the dominant dissipation mechanism in CNT NEMS at low temperature. Our findings are in fair agreement with a theoretical model provided previously [10,11]. The measurements are carried out in a 3 He/ 4 He dilution refrigerator with a base temperature of 30 mK. The NEMS actuation and detection scheme used in our experiment is similar to the one used by Steele and co-workers [11,12]. The RF actuation signal is injected into the gate electrode via a home-built bias T. As the induced mechanical motion changes the charge flow through the CNT quantum dot and vice versa, we can detect the CNT resonance through a change in zero bias conductance. The actuation power is kept to a minimum (P RF ≈ -100 dBm) in order to ensure a quasi-linear regime of the oscillator and a high Q lorentzian resonance shape [Fig. 1(c)]. All measurements were done under zero bias with a standard lock-in technique.

SET in a CNT NEMS-QD can be considered as an external perturbation to the CNT mechanical motion, and vice versa. This perturbation can be described as an electrodynamic force acting on the CNT. The contribution of this force, which is in phase with the mechanical motion, is responsible for a frequency modulation ∆f , whereas a contribution of the force being out of phase with mechanical motion induces a modification of the dissipation and the quality factor Q. ∆f and Q can be expressed as follows [10] ∆f = -

f 0 2 C 2 g k V 2 g C dot 2G C dot Γ -1 (1) 1 Q = 2πf C 2 g k V 2 g 2 ΓC dot 2 G ( 2
)
Where Γ is the mean tunnel coupling of the dot to the leads, yielding Γ = (Γ d + Γ s ) /2, and C dot is the total dot capacitance. We can estimate the tunnel coupling Γ for each device from the line shape of the Coulomb blockade peaks (see Supporting Information). Eqs. ( 1)

and ( 2) are valid in the regime of Coulomb blockade and if Γ f 0 .

First, we compare the frequency response ∆f (scaled with f • V 2 g ) of devices with different dot capacitances C dot . We studied the frequency response for comparable Coulomb peaks height, i.e. for comparable conductance G (V g ) ≈ 20 µS. The tunnel coupling are on the order of 10 GHz. As depicted in Fig. 2(a), we observe a strong frequency softening for small dot capacitances, whereas for high dot capacitances a frequency hardening or no modulation is observed. As shown in Fig. 2(b), the experimental data are in fair agreement with the model. A fit with Eq. 1 yields fitting parameters C '2 g /k = 8 • 10 -22 F 2 /Nm and Γ ≈ 10 GHz, which are in rather good agreement with previous experiments [10,11,13]. Fig. 2(c) shows the (scaled) frequency modulation for different Coulomb peaks and tunneling rates Γ on a given device (device 1). It was shown previously that one can tune a CNT quantum dot from a Coulomb blockade regime to a regime of strong tunnel coupling by simply changing the CNT's gate potential [14,15]. Hence, we can change the tunnel coupling Γ by tuning the gate voltage of our CNT device and estimate the values of Γ from the Coulomb peak line experiments [10,11,13]. Despite the good agreement between theory and experiment, we observe deviations from the model for some devices in Fig. 2(b) and 3(b), which we attribute to variations in C '2 g /k (see Supporting Information). Moreover, we expect from Eq. 2 and previous experiments [10,11] an enhanced electromechanical dissipation in a given device if the conductance G (i.e. the tunnel current) and the tunnel resistance at the nanotubeelectrode interface R ∼ 1/Γ d are increased, in analogy to the current dissipation in a simple resistance. Indeed we observe increased dissipation (decreased Q factor) when tuning the gate voltage through a Coulomb peak [Fig. 3 becomes more pronouced as we move to gate voltage regions with a smaller tunneling rate Γ, i.e. larger tunnel resistance at the nanotube-electrode interface. The dot capacitance C dot yields 160 aF for this device and the height of the Coulomb peak are comparable for all traces (G ≈ 7 µS). Therefore, the CNT capacitance is the limiting factor for the dissipation in regions of suppressed SET, whereas the mean tunnel coupling and the tunneling current itself define the dissipation in regions of strong SET. We conclude that the electron transport through the CNT is the main dissipation mechanism in our nanoelectromechanical system.

Finally we study the temperature dependance of the frequency response and the dissipation. Fig. 2(d) depicts the frequency shift ∆f max for G max = G (∆V g = 0) whereas Fig. 3(d behavior [16][17][18], where the conductance follows a characteristic power law dependence as a function of temperature G(T ) ∼ T α , α being related to the Luttinger parameter g by α = (g -1 + g -2)/2 [16,17]. We find α ≈ 1 and a Luttinger parameter of g = 0, 1, which is close to previous measurements [16] and theoretical predictions [17,18]. Comparing the temperature dependance of the conductance, the quality factor and the frequency shift we finally obtain ∆f max ∼ -G max and Q max ∼ 1/G min , which is in full agreement with Eqs. 1 and 2. We can conclude that the current is the dominant dissipation mechanism for carbon nanotube based NEMS in a Coulomb blockade regime at cryogenic temperatures.

We have demonstrated that the response and the dissipation to single-electron tunneling of carbon nanotube NEMS at low temperature depends on the dot capacitance and the tunnel coupling to the leads, the tunnel current itself being the main dissipation mechanism in the system. By choosing high-κ dielectrics (HfO 2 or ZrO 2 ) and improving the tunnel contact to the metal leads, one could significantly enhance the quality factor Q to values exceeding 10 6 . It was proposed in theoretical calculation that one can use such high-Q CNT NEMS as magnetic torque or force detectors for nanoparticles [5] or single molecule magnets [19,20] grafted to the CNT NEMS. In principle one can achieve a sensitivity of one µ B at low temperature, whereas the best magnetometers, for instance the micro-SQUID, only have a sensitivity of 10 3 µ B [21]. Finally, such molecular quantum spintronic device would allow single spin manipulation on a molecular level.

Estimating quantum dot capacitances of CNT NEMS

The total dot capacitance is obtained from bias spectroscopy measurements. The total dot capacitance C dot is related to the charging energy E c of the dot via C dot = e/E c . The charging energy is deduced from the size of the Coulomb diamonds (fig. S1a). Alternatively one can determine the capacitances C s and C d from the slope of the diamond edges and the gate capacitance C g from the distance between 2 degeneracy points (Fig S1b). The total dot capacitance then yields

C dot = C s + C d + C g .
For SiO 2 based devices we obtain capacitance between 20 and 40 aF, whereas for Al 2 O 3 based devices the values range from 160 aF to 260 aF. The spread in capacitance values for nominally identically fabricated devices can mainly be attributed to different device lengths (ranging from 700 nm to 850 nm) and therefore to variations in charging energies. Electron microscopy revealed slack on most devices, suggesting a low built-in tension which is characteristic for CVD grown CNT's. 1,2 As the slack may be a different for each device, the distance between the CNT and the gate can vary from one device to another, which would also contribute to the spread in capacitance values.

Estimating the tunnel coupling Γ in CNT quantum dot

In the regime of Coulomb blockade, the energy level broadening in the carbon nanotube quantum dot is given by the electronic temperature and the tunnel coupling to the leads. The electronic temperature in our system is around 150 mK, therefore at least an order of magnitude smaller than the expected tunnel couplings in carbon nanotube junctions Γ>> kT. In this limit, the Coulomb peak line shape can approximated as follows 3 :

      2 2 2 2 max ,0 / 2 / 2 g g G G V V       (S1)
Where G max is the maximum of the Coulomb peak, Γ = (Γ s + Γ d )/2 the mean tunneling rate through the quantum dot, V g,0 the center of the peak and α an energy scaling factor given by:

1 1 1.36 1.36 g sd g s d g C dV dV C C C      (S2)
We used Eq. S1 to fit the Coulomb peaks and estimate the mean tunnel coupling Γ for devices 1 and 2 studied in Fig. Although Eq. ( 1) is only an approximation (with an error estimated to 7%), it is sufficiently accurate to show a dependence of the frequency shift and the dissipation on the tunnel coupling in qualitative agreement with Eq. 1 and 2 of the main text.

Accuracy of fitting parameter C g '2 /k

In Fig. 2b and 3b of main text, we fitted the experimental data with equation 1 and 2, using Γ and C g ' 2 /k as fitting parameters. We obtain average value for C g ' 2 /k = 10 -21 F 2 /Nm and Γ= 10 GHz, which is in fair agreement with previous experiments. 4,5 Although the model fits the experimental data with good accuracy, we can observe certain deviation from the model for some devices, which could indeed be attributed to variations in C g ' 2 The above estimation shows that we indeed have variations in C g ' 2 /k, caused by a small spread in capacitance values C g and CNT-gate distance Z, which could ultimately lead to deviations of some devices from the model fit.

Power law temperature dependence of the zero bias conductance

As temperature increases the Coulomb blockade peak broadens and becomes smaller (see Fig. S2). As a result G max = G(ΔV g = 0) decreases whereas G min = G(|ΔV g | = -1 mV) increases with temperature, which is depicted in the insets of Figs.

  FIG. 1. False color SEM image of (a) a low capacitance device based on a SiO 2 covered Si backgate (grey color) and (b) a high capacitance device based on an Al 2 O 3 covered local metallic gate (blue). The RF actuation signal is injected into the local metallic gate and Si backgate, respectively, through a home-built bias-T. As the induced mechanical motion changes the charge flow through the CNT quantum dot and vice versa, we can detect the CNT resonance through a change in zero bias conductance. (c) Mechanical resonance of a typical CNT NEMS at low driving power P RF = -100 dBm. The resonance width δf = 2.5 kHz leads to a quality factor of Q ≈ 140000.

SiO 2 .

 2 A layer of 100 nm of Al 2 O 3 is then deposited by atomic layer deposition. Using optical DUV lithography and e-beam evaporation of Mo (20 nm) and Pt (160 nm), source-drain electrodes are aligned above the local gate. Suspended CNT's are finally grown by chemical vapor deposition at 800 • C from a CH 4 feedstock and Fe/Mo catalyst spots patterned on the source-drain electrodes next to the junction. The CNT device length is approximately 800 nm (± 50 nm) and the dot capacitances are deduced from the dot's charging energy at low temperature. The spread in capacitance values of nominally identically fabricated devices is due to variations in the nanotubes length and slack (see Supporting Information).

FIG. 2 .

 2 FIG. 2. Modulation of the resonance frequency due to SET: (a) Frequency shift ∆f (scaled with f • V 2 g ) for different dot capacitance C dot , with comparable tunneling rates Γ. inset shows a typical Coulomb peak centered at ∆V g = 0; (b) Maximum scaled frequency shift at maximum tunnel current G max (∆V g = 0) vs. the dot capacitance for all measured devices. The solid red line corresponds to the fit using eq. 1 with C '2 g /k = 8 • 10 -22 F 2 /Nm and Γ d ≈ 10 GHz. The data are in fair agreement with the model. (c) Scaled frequency shift for different Coulomb peaks i.e. tunneling rates Γ, with a fixed dot capacitance C dot = 260 aF (device 1). The tunneling rate Γ is estimated from the lineshape of the respective Coulomb peaks (see Supporting Information). (d) Maximum frequency shift ∆f max at maximum tunnel current G max (∆V g = 0) vs. temperature for a device yielding C dot = 160 aF (device 2). The inset shows the evolution of G max as a function of temperature.

  (a) and 3(c)]. In Fig 3(c) (device 2), this effect

  ) shows the quality factor Q max for G min = G (|∆V g | 0) as function of temperature. A fit of the data yields ∆f max ∼ T and Q max ∼ 1/T . As temperature increases the Coulomb blockade peak broadens and becomes smaller (see supporting information Fig. S3). As a result G max = G (∆V g = 0) decreases whereas G min = G (|∆V g | 0) increases with temperature, as depicted in the insets of Figs. 2(d) and 3(d). From the data fit we obtain G max ∼ -T and G min ∼ T . It was previously demonstrated that carbon nanotubes show Luttinger liquid

Fig. S1 :

 S1 Fig. S1: Estimating the dot capacitance via bias spectroscopy: (a)(c) Coulomb diamonds for two Al 2 O 3 based devices with a length of 780 nm and 820 nm, respectively. (b)(d) Coulomb diamonds for two SiO 2 based devices.

  2 (c) and Fig.3(c) of the main text. The results are depicted in Fig.S2with the same color code than in the main text figures. For device 1, studied with respect to the frequency shift Δf, we can tune the tunnel coupling Γ from 30 µeV to 55 µeV(7.25 GHz to 13.3 GHz) [Fig.S2 (a)], leading to a stiffening of the CNT [Fig.2(c)]. For device 2, studied with respect the Q factor modulation, we can change Γ from 20 µeV to 90 µeV (4.8 GHz to 21.5 GHz), leading to an increase of the quality factor [Fig.3(c)].

  Fig. S2: Estimating the tunnel coupling Γ in the limit of tunnel broadened Coulomb blockade. a) Device 1 studied in Fig. 2(c) of the main text with respect to the frequency shift Δf. b) Device 2 studied in Fig. 3(c) of the main text with respect to the quality factor Q. The color code is identical to the one of the main text figures. The estimation is accurate enough to

  /k:According to ref.4, C g '2 /k is given by C g '2 /k = (C g /Z)2 /k in good approximation, with C g being the gate capacitance, Z the CNT-gate distance and k the CNT's spring constant. We studied two different device layouts: SiO 2 based CNT QD with a gate capacitance C g = 2±1 aF and Z = 700 nm and Al 2 O 3 based CNT QD with C g = 3±1 aF and Z =300 nm. C g is estimated from the voltage separation of two adjacent Coulomb peaks (see Fig.S1) and Z is given by the sum of the oxide thickness and thickness of the metal electrodes. With an average spring constant k = 10 -3 N/m (value extracted from ref.4 and 5) we obtain values for C g ' 2 /k = (6±2)•10 -21 F 2 /Nm, which is close to values obtained from the fit in Fig.2b and 3b.

  2(d) and 3(d) of the main text. Both follow a power law dependence, for instance G max ~ -T and G min ~ T, as expected for Luttinger liquids like carbon nanotubes.6,7,8

Fig. S3 :

 S3 Fig. S3: Evolution of the zero-bias conductance as a function of temperature (from 20 mK to 1 K) for the device studied in Fig. 2(d) and 3(d) of the main text.

This work is partially supported by the ANR-PNANO project MolNanoSpin No. ANR-08-NANO-002 and ERC Advanced Grant MolNanoSpin No. 226558. M.G. acknowledges the financial support from the RTRA Nanosciences Foundation. Samples were fabricated in the NANOFAB facility of the Neel Institute. We thank F. Balestro, E. Bonet, T. Crozes, J.P. Cleuziou, E. Eyraud, T. Fournier, R. Haettel, C. Hoarau, D. Lepoittevin, V. Nguyen, V. Reita, A. Reserbat-Plantey, C. Thirion, M. Urdampilleta.