Jean-Philippe Bartier
email: bartier@ceremade.dauphine.fr

Adrien Blanchet
email: adrien.blanchet@univ-tlse1.fr

Jean Dolbeault
email: dolbeaul@ceremade.dauphine.fr

Miguel Escobedo
email: miguel.escobedo@ehu.es

Improved intermediate asymptotics for the heat equation 6,66

Keywords: Heat equation, Fokker-Planck equation, Ornstein-Uhlenbeck equation, intermediate asymptotics, self-similar variables, stationary solutions, large time behavior, rate of convergence, entropy, Poincaré inequality, logarithmic Sobolev inequality, interpolation inequalities 2000 MSC: Primary: 26D10, 35K10; Secondary: 35K15, 47J20

. Results extend to the case of a Fokker-Planck equation with a general confining potential.

Consider the heat equation in the euclidean space,

∂u ∂t = ∆u t > 0 , x ∈ R d (1)
with an initial condition u 0 ∈ L 1 (R d ). By writing u = u +u -where u + and u -are respectively the positive and negative parts of u and solving [START_REF] Arnold | Interpolation between logarithmic Sobolev and Poincaré inequalities[END_REF] with initial data (u 0 ) + and (u 0 ) -, we may reduce the problem to the case of a nonnegative function, corresponding to a nonnegative initial condition u 0 , without restriction. The heat equation being linear, we can assume without loss of generality that u 0 is a probability measure so that in the sequel of this note R d u 0 dx = 1 = R d u(t, x) dx for any t ≥ 0. Getting decay rates and even an asymptotic expansion for large values of t is completely standard, see for instance [START_REF] Duoandikoetxea | Moments, masses de Dirac et décomposition de fonctions[END_REF]. However, a few details and some notations will be useful for later purpose.

First of all, as a straightforward consequence of the expression of the Green function, G(t, x, y) := (4πt) -d/2 e -|x-y| 2 4t , any solution u of (1) can be written as u(t, x) = R d u 0 (y) G(t, x, y) dy and therefore uniformly decays like O(t -d/2 ) since, as t → ∞, u(t, x) ∼ G(t, x, 0). It is also classical to estimate the decay of u(t, •) -G(t, •, 0) in various L p (R d ) norms. Such estimates are called intermediate asymptotics estimates. The point is to determine the first term of an asymptotic expansion of the solution as t → ∞. For instance, as we shall see below, it can be proved that

u(t, •) -G(t, •, 0) L 1 (R d ) = O(t -1/2 ) as t → ∞.
The entropy method can be used among various other approaches to obtain such an estimate. It relies on the logarithmic Sobolev inequality and goes as follows. First consider the time-dependent rescaling

u(t, x) = R -d v log R, x/R with R = R(t) := √ 1 + 2 t , t > 0 , x ∈ R d . ( 2
)
If u is a solution of (1), then v solves the Fokker-Planck equation

∂v ∂t = ∆v + ∇ • (x v) (3) with same initial condition v(t = 0, •) = u 0 . Let v ∞ (x) := (2π) -d/2
e -|x| 2 /2 be the unique stationary solution of (3) with mass 1, and define dµ := v ∞ dx as the Gaussian measure. We denote by L p (R d ) and L p (R d , dµ) the Lebesgue spaces corresponding respectively to Lebesgue's measure and to the Gaussian measure. Understanding the intermediate asymptotics for u amounts to study the convergence of v to v ∞ , as t → ∞. Define the entropy by E 1 [w] := R d w log w dµ. Let v be a solution of (3) and define w(t,

•) := v(t, •)/v ∞ , w 0 := w(t = 0, •). Then d dt E 1 [w(t, •)] = -I 1 [w(t, •)] where I 1 is the Fisher information defined by I 1 [w] := R d w |∇ log w| 2 dµ. Gross' logarithmic Sobolev inequality exactly amounts to E 1 [v/v ∞ ] ≤ 1 2 I 1 [v/v ∞ ]
and so, it follows that

E 1 [w(t, •)] ≤ E 1 [w 0 ] e -2 t ∀ t ≥ 0 .
By the Csiszár-Kullback inequality, see for instance [START_REF] Unterreiter | On generalized Csiszár-Kullback inequalities[END_REF], we get

v(t, •) -v ∞ 2 L 1 (R d ) ≤ 1 4 E 1 [w(t, •)] and deduce that v(t, •) -v ∞ L 1 (R d ) ≤ 1 2 E 1 [w 0 ] e -t ∀ t ≥ 0 .
Undoing the change of variables (2) and observing that u

∞ (t, x) := R(t) -d v ∞ (x/R(t)) = G(t + 1/2, •, 0), we finally get u(t, •) -u ∞ (t, •) L 1 (R d ) ≤ 1 2 E 1 [w 0 ] 1 + 2 t ∀ t ≥ 0 ,
which establishes the claimed estimate, namely:

u(t, •) -G(t, x, 0) L 1 (R d ) ≤ O t -1/2 as t → ∞.
Such an estimate is quite classical. The above method is known as the Bakry-Emery method or entropy / entropy-production method and also provides a proof of the logarithmic Sobolev inequality. See [START_REF] Toscani | Sur l'inégalité logarithmique de Sobolev[END_REF][START_REF] Arnold | On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations[END_REF] for some references on this topic, in the context of partial differential equations.

By combining L 1 (R d ) and L ∞ (R d ) estimates using Hölder's inequality, we get that

u(t, •) -G(t, •, 0) L p (R d ) ≤ O t -1 2 p (1+(p-1) d)
as t → ∞ .

In a L 2 (R d ) framework, a much more detailed description can be achieved using a spectral decomposition. If v is a solution of (3), then w = v/v ∞ is a solution of the Ornstein-Uhlenbeck equation

∂w ∂t = ∆w -x • ∇w (4)
with initial data w 0 = u 0 /v ∞ . Notice that R d w 0 dµ = 1 and, as a consequence, R d w(t, •) dµ = 1 for all t ≥ 0. Define by (H k ) k∈N d the sequence of Hermite type polynomials (see for instance [START_REF] Weisstein | Hermite Polynomial. From MathWorld -A Wolfram Web Resource[END_REF]) acting on x = (x 1 , x 2 . . .

x d ) ∈ R d , such that H k (x) := d j=1 h k j (x j )
where h n (y) := (-1) n (n!) -1/2 e y 2 /2 d n dy n e -y 2 /2 , y ∈ R and k = (k 1 , ..., k d ) ∈ N d . These functions provide an orthonormal family of eigenfunctions in L 2 (R d , dµ) which spans the eigenspaces of the Ornstein-Uhlenbeck operator, that is -

(∆H k -x • ∇H k ) = |k| H k , where |k| := d j=1 k j .
Up to a scaling, (h n ) n∈N is the usual family of Hermite polynomials on R.

If w 0 satisfies the orthogonality condition

R d w 0 H k dµ = 0 ∀ k ∈ N d such that 0 < |k| < n , (5)
then an improved rate of convergence follows, in the sense that

w(t, •) -1 L 2 (R d , dµ) ≤ e -n t w 0 -1 L 2 (R d , dµ) ∀ t ≥ 0 . If (5) initially holds, we indeed have R d w(t, •) H k dµ = 0 for any t ≥ 0 and any k ∈ N d such that 0 < |k| < n. Then, since d dt w(t, •) -1 2 L 2 (R d , dµ) = -2 R d |∇w(t, •)| 2 dµ
, the conclusion holds using the following result.

Proposition 1 (Improved Poincaré inequality). Assume that w

∈ L 2 (R d ) is such that R d w dµ = 1 and the condition R d w H k dµ = 0 holds for any k ∈ N d such that 0 < |k| < n.
Then the following inequality holds, with optimal constant:

w -1 2 L 2 (R d , dµ) ≤ 1 n ∇w 2 L 2 (R d , dµ) .
The proof is no more than a straightforward rewriting of the Rayleigh quotient

∇w 2 L 2 (R d , dµ) / w -1 2 L 2 (R d , dµ
) under the appropriate orthogonality condition. Notice that polynomials H k are of degree |k| so that the Condition (5) can be rephrased in terms of moment conditions. See [START_REF] Duoandikoetxea | Moments, masses de Dirac et décomposition de fonctions[END_REF][START_REF] Kim | Higher Order Approximations in the Heat Equation and the Truncated Moment Problem[END_REF] for further results in this direction.

It is natural to search for improved estimates of convergence also in L p (R d ) with p ∈ [1, 2) by looking for improved functional inequalities whenever condition ( 5) is fulfilled. We may for instance quote [START_REF] Arnold | Improved entropy decay estimates for the heat equation[END_REF] in which improvements on the constant, but not on the rates, have been achieved for p = 1.

For any p ∈ (1, 2], consider the generalized entropy

E p [w] := R d w p -1 p -1 dµ .
This definition is consistent with the definition of E 1 because, under the condition

R d w dµ = 1, E p [w] = R d w p -w p-1 dµ → E 1 [w] as p → 1.
The functional E p controls the convergence in L p (R d , dµ) using a generalized Csiszár-Kullback inequality. In [START_REF] Cáceres | Nonlinear stability in L p for a confined system of charged particles[END_REF][START_REF] Bartier | A qualitative study of linear drift-diffusion equations with time-dependent or degenerate coefficients[END_REF], it has been proved that w -

1 2 L p (R d , dµ) ≤ 1 p 2 2/p max w 2-p L p (R d , dµ) , 1 E p [w], for any p ∈ [1, 2]. Since w L 1 (R d , dµ) = 1, we have 1 ≤ w p L p (R d , dµ) = 1 + (p -1) E p [w]
, and so

w -1 L p (R d , dµ) ≤ A p E p [w] with A p (s) := 2 1/p √ p 1 + (p -1) s 1-p/2 √ s . (6)
Next, assume that R d w H k dµ = 0 for any k ∈ N d such that 0 < |k| < n and consider the generalized Poincaré inequalities, with p ∈ [1, 2], namely

E p [w] ≤ B n,p R d ∇w p/2 2 dµ ∀ w ∈ H 1 (R d , dµ) . (7)
Such inequalities have been established for n = 1 by W. Beckner in [START_REF] Beckner | A generalized Poincaré inequality for Gaussian measures[END_REF] with optimal constant B 1,p = 2/p for the Gaussian measure. By the same method, it has been shown in [START_REF] Arnold | Interpolation between logarithmic Sobolev and Poincaré inequalities[END_REF] that for a larger class of measures dµ, if (7) holds for p = 1 and p = 2, for some positive constants B n,1 and B n,2 respectively, then it also holds for any p ∈ (1, 2) with

B n,p = 1 p-1 1 -((2 -p)/p) B n,1 /(2 B n,2 ) B n,2 . (8)
By the logarithmic Sobolev inequality and the improved Poincaré inequality, see Proposition 1, we know that B n,1 ≤ 2 and

B n,2 = 1/n. Hence it follows that B n,p ≤ 1 p-1 1 -((2 -p)/p) n 1 n .
On the other hand, as in [START_REF] Arnold | On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations[END_REF], if w is a solution of (4), then

d dt E p [w(t, •)] = - 4 p R d ∇w p/2 2 dµ . (9)
If ( 5) is satisfied, we conclude using ( 7) and ( 6) that any solution of (4) with initial data w 0 satisfies

E p [w(t, •)] ≤ E p [w 0 ] e -2 λ(n,p) t and w(t, •) -1 L p (R d , dµ) ≤ A p E p [w 0 ] e -λ(n,p) t ∀ t ≥ 0 , with λ(n, p) := 2 p n (p -1) 1 -((2 -p)/p) n -1 .
The last estimate holds because, for any t ≥ 0,

w(t, •) -1 L p (R d , dµ) ≤ A p E p [w(t, •)] ≤ A p E p [w 0 ] e -2 λ(n,p) t ≤ A p E p [w 0 ] e -λ(n,p) t
Notice that λ(1, p) = 1 and λ(n, 2) = n. Nothing is gained as p → 1, since lim p→1 λ(n, p) = 1 is independent of n.

On the other hand, by Hölder's inequality, we have for free that w -

1 L p (R d , dµ) ≤ w -1 L 2 (R d , dµ) .
Hence, if w is a solution of (4) with initial data w 0 , we know that w(t, •) -

1 L p (R d , dµ) ≤ e -n t w 0 -1 L 2 (R d , dµ) as t → ∞, for any 3 p ∈ [1, 2], if (5)
is satisfied. By interpolation, we recover the rates of [START_REF] Duoandikoetxea | Moments, masses de Dirac et décomposition de fonctions[END_REF][START_REF] Kim | Higher Order Approximations in the Heat Equation and the Truncated Moment Problem[END_REF]. However, this is not satisfactory since neither w 0 -1 L p (R d , dµ) nor E p [w 0 ] are involved in the right hand side of the above estimate.

Consider first the case p = 1. An alternative approach is suggested by the method of [START_REF] Blanchet | Hardy-Poincaré inequalities and applications to nonlinear diffusions[END_REF][START_REF] Blanchet | Asymptotics of the fast diffusion equation via entropy estimates[END_REF], which applies to the fast diffusion equation ∂u ∂t = ∆u m for m < 1. By assuming some uniform bound on the initial data, which is preserved along the evolution, it is possible to relate the asymptotic rate for intermediate asymptotics with the spectrum of the linearized operator. We can indeed observe that w 0 -

1 2 L 2 (R d , dµ) ≤ w 0 -1 L 1 (R d , dµ) w 0 -1 L ∞ (R d , dµ) ≤ 1 2 √ E 1 [w 0 ] w 0 -1 L ∞ (R d , dµ
) using Hölder's inequality and the Csiszár-Kullback inequality. This proves that 5) is satisfied initially. Still, this provides neither an estimate of R d w(t, •) log w(t, •) dµ nor a functional inequality which improves upon the logarithmic Sobolev inequality. To prove such an inequality, we keep following the strategy of [START_REF] Blanchet | Asymptotics of the fast diffusion equation via entropy estimates[END_REF]. A simple but key idea is to observe that the functions defined for any p ∈ [1, 2] by h p (0) = 1, h p (1) = p/2 and, for any s ∈ (0, 1)

w(t, •) -1 2 L 1 (R d , dµ) ≤ 1 2 w 0 -1 L ∞ (R d , dµ) E 1 [w 0 ] e -n t as t → ∞ if (
∪(1, ∞) by h p (s) := [s p -1 -p (s -1)]/[(p -1) |s-1| 2 ] if p > 1, h 1 (s) := [s log s -(s -1)]/|s -1| 2 ,
are continuous, nonnegative, decreasing on R + and achieve their maximum at 0. Define on L ∞ (R d ) the functional

H p [w] := w 2-p L ∞ (R d ) sup x∈R d h p (w(x)) = w 2-p L ∞ (R d ) h p inf x∈R d w(x) .

Theorem 2 (Improved logarithmic Sobolev inequality). Assume that w

∈ L ∞ + (R d ) is such that R d w dµ = 1 and satisfies the condition R d w H k dµ = 0 for any k ∈ N d such that 0 < |k| < n.
Then the following inequality holds, with optimal constant:

R d w log w dµ ≤ H 1 [w] n R d |∇w| 2 w dµ .
Proof. We may indeed observe that by the Poincaré inequality and using the definition of H 1 , we get

R d |∇w| 2 w dµ ≥ 1 w L ∞ (R d ) R d |∇w| 2 dµ ≥ n w L ∞ (R d ) R d |w -1| 2 dµ ≥ n H 1 [w] R d w log w dµ .
The optimality of the constant can be checked by a lengthy but elementary computation using the functions w k ε := H k (x) χ x ε 1/(2n) + C k ε for some smooth truncation function χ such that 0 ≤ χ ≤ 1, χ ≡ 1 on B(0, 1) and χ ≡ 0 in R d \ B(0, 2). Here for k ∈ N d is such that |k| = n and the constant

C k ε is chosen so that R d w k ε dµ = 1.
As a consequence of the Maximum Principle applied to the heat equation ( 1) and the fact that to u 0 = v ∞ corresponds a self-similar solution of (1), namely u(t, x) = G(t + 1 2 , x, 0), we have the estimate

H 1 [w(t, •)] ≤ H 1 [w 0 ] ∀ t ≥ 0 .
By applying Theorem 2, we obtain a new result of decay for E 1 [w(t, •)] with a constant which is exactly E 1 [w 0 ], to the price of a rate which is less than 2 n.

Corollary 3 (Improved decay rate of the entropy). Let w be a solution of (4) with a nonnegative bounded initial data w

0 ∈ L 1 (R d , dµ) such that R d w 0 dµ = 1 and (5) is satisfied. Then E 1 [w(t, •)] ≤ E 1 [w 0 ] e -n t/ H 1 [w 0 ] ∀ t ≥ 0 .
This result is actually equivalent to Theorem 2, as follows by differentiating the above inequality at t = 0 (for which equality is trivially satisfied) and using the fact that -

R d |∇w 0 | 2 /w 0 dµ = d dt E 1 [w(t, •)] |t=0 ≤ E 1 [w 0 ] d dt e -n t/ H 1 [w 0 ]
|t=0 . What we have achieved is a global, improved exponential decay of the entropy E 1 in a restricted class of functions. To simplify even further, for any ε ∈ (0, 1) and n ∈ N * , consider the set X n ε := {w ∈ L 1 (R d , dµ) :

1ε ≤ w ≤ 1 + ε a.e. and the condition R d w H k dµ = 0 holds for any k ∈ N d such that 0 < |k| < n }, which is appropriate to handle the optimality case corresponding to ε → 0 + . The best constant in Theorem 2 is indeed asymptotically equivalent to the sharp rate of convergence in Corollary 3, in the sense that lim ε→0

+ inf w∈X n ε n/H 1 [w] = lim ε→0 + n/[(1 + ε) h(1 -ε)] = 2 n.
For simplicity, we have considered only the case p = 1, but the method also applies to any p ∈ (1, 2). We obtain an improved version of (7) under the restriction that w ∈ L 1 (R d , dµ) is bounded nonnegative and the condition [START_REF] Bonforte | Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities[END_REF]. Using the entropy / entropy-production identity (9), the fact that K[n, p, w(t, •)] ≤ K[n, p, w 0 ] and the generalized Csiszár-Kullback inequality (6), we obtain

R d w H k dµ = 0 holds for any k ∈ N d such that 0 < |k| < n. With B n,1 = 4 H 1 [w]/n and B n,2 = 1/n, we get B n,p ≤ K[n, p, w] := (n (p -1)) -1 1 -((2 -p)/p) 2 H 1 [w] by
E p [w(t, •)] ≤ E p [w 0 ] e - 4 t p K[n,p,w 0 ] and w -1 L p (R d , dµ) ≤ A p E p [w 0 ] e - 2 t p K[n,p,w 0 ] ∀ t ≥ 0 . (10)
Alternatively, an elementary computation as in the proof of Theorem 2 gives a similar result:

4 p 2 R d ∇w p/2 2 dµ = R d w p-2 |∇w| 2 dµ ≥ 1 w 2-p L ∞ (R d ) R d |∇w| 2 dµ ≥ n w 2-p L ∞ (R d ) R d |w -1| 2 dµ ≥ n H p [w] E p [w] if R d w dµ = 1 and the condition R d w H k dµ = 0 holds for any k ∈ N d such that 0 < |k| < n. This proves that E p [w] ≤ 4 p 2 H p [w] n R d ∇w p/2 2 dµ .
Using ( 9) and ( 6), this proves that any solution of (4) with initial data in

w 0 ∈ L 1 ∩ L ∞ (R d , dµ) satisfies E p [w(t, •)] ≤ E p [w 0 ] e -n p t/ H p [w 0 ] and w -1 L p (R d , dµ) ≤ A p E p [w 0 ] e -n p t/(2 H p [w 0 ]) ∀ t ≥ 0 . (11)
Comparing the rates of ( 10) and ( 11) is a natural question. In the limit ε → 0, inf . Hence, at least in the regime ε → 0, (11) is a better estimate in terms of rates than [START_REF] Carrillo | Entropy-energy inequalities and improved convergence rates for nonlinear parabolic equations[END_REF]. Undoing the change of variables (2), we have achieved a detailed result on improved u 0 .

Corollary 4 (Improved intermediate asymptotics for the heat equation).

Let p ∈ [START_REF] Arnold | Interpolation between logarithmic Sobolev and Poincaré inequalities[END_REF][START_REF] Arnold | Improved entropy decay estimates for the heat equation[END_REF] and assume that u 0 is a probability measure such that w 0 = u 0 /v ∞ is bounded and satisfies the condition R d u 0 H k dx = 0 for any k ∈ N d such that 0 < |k| < n. If u is the solution of (1) with initial condition u 0 , then

u(t, •) -u ∞ (t, •) L p (R d ) ≤ (2π) -d 2 (1-1 p ) A p E p [w 0 ] (1 + 2 t) -n p 4 Hp [w 0 ] -d 2 (1-1 p ) ∀ t ≥ 0 .
The proof relies on the remark that u(t,

•) -u ∞ (t, •) L p (R d ) ≤ u ∞ (t, •) 1-1 p L ∞ (R d ) w(t, •) -1 L p (R d , dµ) where u ∞ (t, •) := G(t + 1/2, •, 0). The conclusion holds using u ∞ (t, •) L ∞ (R d ) = (2π R 2 ) -d/2 with R = √ 1 + 2 t.
Up to now, we have considered the simple case of the harmonic potential, V(x) = 1 2 |x| 2 . As in [START_REF] Arnold | Interpolation between logarithmic Sobolev and Poincaré inequalities[END_REF], the previous results can be extended to more general potentials as follows. Consider V ∈ W 1,2 loc ∩W 2,2 loc (R d ) such that R d e -V(x) dx = 1, and define the probability measure dµ(x) := e -V(x) dx in R d , which generalizes the Gaussian measure. Under the above conditions on V, the logarithmic Sobolev inequality holds (resp. [START_REF] Blanchet | Hardy-Poincaré inequalities and applications to nonlinear diffusions[END_REF] for p = 1) for some positive constant (resp. for B 1,1 > 0). The Ornstein-Uhlenbeck operator N := -∆ + ∇V • ∇ is essentially self-adjoint on L 2 (dµ), has a nondegenerate eigenvalue λ 0 = 0 and a spectral gap λ 1 > 0. According to [18, Theorem 2.1], N has a pure point spectrum without accumulation points. Since lim k→∞ λ k = ∞, then by [15, Theorem XIII.64], the eigenfunctions of N form a complete basis of L 2 (R d , dµ). We shall denote the eigenvalues by λ k , k ∈ N, and by E k the corresponding eigenspaces.

Theorem 2 adapts without changes. Assume that w

∈ L ∞ + (R d ) is such that R d w dµ = 1. Then R d w log w dµ ≤ H 1 [w] λ n R d |∇w| 2 w dµ under the orthogonality condition: w ∈ n-1 k=1 E k ⊥ , that is R d w f k dµ = 0 for any f k ∈ E k , k = 1, 2,. . . n -1.
Next, consider the solution w of the Ornstein-Uhlenbeck equation

∂w ∂t = -N w = ∆w -∇V • ∇w , (12)
with initial condition

w 0 ∈ n-1 k=1 E k ⊥ ∩ L ∞ (R d ) is such that R d w 0 dµ = 1.
With the same definition as above for E p , for any solution of [START_REF] Pino | Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions[END_REF] with initial data w 0 , ( 11) is now replaced by

E p [w(t, •)] ≤ E p [w 0 ] e -λ n p t/ H p [w 0 ] and w -1 L p (R d , dµ) ≤ A p E p [w 0 ] e -λ n p t/ (2 H p [w 0 ]) ∀ t ≥ 0 .
Let us conclude this letter by some comments and open questions. It is standard in entropy / entropy-production methods that determining sharp rates of convergence in an evolution equation is equivalent to finding sharp constants in functional inequalities, as we have seen in the case of the heat equation: the rate of convergence in L 2 (R d , dµ) is given by the Poincaré inequality, while the rate of convergence in entropy, which controls the L 1 (R d , dµ) norm, is related with the logarithmic Sobolev inequality. This is also true for nonlinear diffusion equations, see for instance [START_REF] Pino | Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions[END_REF]. In this case, a breakthrough came from the observation that uniform norms can also be used, see [START_REF] Carrillo | Entropy-energy inequalities and improved convergence rates for nonlinear parabolic equations[END_REF][START_REF] Blanchet | Hardy-Poincaré inequalities and applications to nonlinear diffusions[END_REF][START_REF] Blanchet | Asymptotics of the fast diffusion equation via entropy estimates[END_REF], to the price of a restricted functional framework. This allows to relate nonlinear quantities of entropy type with spectral properties of the linearized problem, in an appropriate functional space and, again, to relate sharp rates with best constants, see [START_REF] Bonforte | Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities[END_REF]. As long as nonlinear evolution problems are concerned, only a few invariant quantities are usually available: the mass and the position of the center of mass of the solution, for instance. In linear evolution problems, we can impose an arbitrary number of orthogonality conditions, which are preserved along the evolution. Improved rates of convergence are then expected, even when measured with nonlinear quantities like the entropy. Various attempts have been done, see for instance [START_REF] Arnold | Improved entropy decay estimates for the heat equation[END_REF], but the question has been left open for many years. Such ideas have been partially explored by R.J. McCann, including in the linear case (see [START_REF] Cordero-Erauquin | Accelereated diffusion to minimum entropy[END_REF]), based on considerations on an appropriate Hessian matrix. Our approach provides a simpler and elementary answer under restrictions which are natural in view of [START_REF] Blanchet | Asymptotics of the fast diffusion equation via entropy estimates[END_REF]. It also raises a number of questions concerning the optimality of the new functional inequalities from a variational point of view, the convergence of minimizing sequences and the symmetry of the eventual minimizers.

 [START_REF] Blanchet | Asymptotics of the fast diffusion equation via entropy estimates[END_REF]This work has been partially supported by the Fondation Sciences Mathématiques de Paris and by the ANR projects CBDif-Fr and Evol. 66 c 2009 by the authors.


Pierre Joyot
email: pjoyot@estia.fr

Nicolas Verdon
email: nverdon@estia.fr

Ga Ël Bonithon
email: gael.bonithon@ec-nantes.fr

Francisco Chinesta
email: francisco.chinesta@ec-nantes.fr

Pierre Villon
email: pierre.villon@utc.fr

PGD-BEM APPLIED TO THE NONLINEAR HEAT EQUATION

The Boundary Element Method (BEM) allows efficient solution of partial differential equations whose kernel functions are known. The heat equation is one of these candidates when the thermal parameters are assumed constant (linear model). When the model involves large physical domains and time simulation intervals the amount of information that must be stored increases significantly. This drawback can be circumvented by using advanced strategies, as for example the multi-poles technique. We propose radically different approach that leads to a separated solution of the space and time problems within a nonincremental integration strategy. The technique is based on the use of a space-time separated representation of the unknown field that, introduced in the residual weighting formulation, allows to define a separated solution of the resulting weak form. The spatial step can be then treated by invoking the standard BEM for solving the resulting steady state problem defined in the physical space. Then, the time problem that results in an ordinary first order differential equation is solved using any standard appro- * Address all correspondence to this author.

priate integration technique (e.g. backward finite differences). When considering the nonlinear heat equation, the BEM cannot be easily applied because its Green's kernel is generally not known but the use of the PGD presents the advantage of rewriting the problem in such a way that the kernel is now clearly known. Indeed, the system obtained by the PGD is composed of a Poisson equation in space coupled with an ODE in time so that the use of the BEM for solving the spatial part of the problem is efficient. During the solving, we must however separate the nonlinear term into a space-time representation that can limit the method in terms of CPU time and storage, that is why we introduce in the second part of the paper a new approach combining the PGD and the Asymptotic Numerical Method (ANM) in order to efficiently treat the nonlinearity.

INTRODUCTION

Solving the nonlinear heat equation with the Boundary Element Method is not straightforward because it is generally not possible to determine its Green's kernel. A direct resolution method based on the kernel of linear heat equation has been proposed in [START_REF] Goto | A boundary integral equation method for nonlinear heat conduction problems with temperature-dependent material properties[END_REF] and some other approaches have been used, we could just mention here the homotopy method used quite recently in [START_REF] Shijun | An optimal homotopy-analysis approach for strongly nonlinear differential equations[END_REF]. The approach we use in this paper is radically different because it involves Reduced Order Modelling techniques, from which we can obtain a separated representation of the solution. One of the pioneer ROM methods, the Proper Orthogonal Decomposition, allows to obtain a space-time decomposition of the solution from a set of samples of the solution called snapshots. But this sampling step limits the performances of the method that is why the Proper Generalized Decomposition is nowadays preferred. Based on a a priori separated representations of the unknown, the PGD allows to treat high dimensional problems arising from multidimensional physics and its range of application is very large, see [START_REF] Chinesta | A short review on model order reduction based on proper generalized decomposition[END_REF] for a recent review. For the application in the nonlinear heat equation, it will consist in determining a space-time separated representation with an iterative procedure: the algorithm computes at each iteration a couple of modes from the modes that have already been calculated. More details about the PGD can be found in [START_REF] Ladevèze | The latin multiscale computational method and the proper orthogonal decomposition[END_REF][START_REF] Chinesta | On the reduction of stochastic kinetic theory models of complex fluids[END_REF][START_REF] Nouy | A priori model reduction through proper generalized decomposition for solving time-dependent partial differential equations[END_REF]. The advantage of coupling the PGD and the BEM lies on the fact that is no more necessary to know the space-time kernel of the heat equation but only the Green's kernel of the Poisson equation. The method, called in the later PGD-BEM, has been first introduced in [START_REF] Bonithon | Non-incremental boundary element discretization of parabolic models based on the use of the proper generalized decompositions[END_REF] for the linear heat equation and we show in this paper that its algorithm is easily adapted to the nonlinear case. Even if the developed algorithm shows impressive results, it remains some limitations due to the cost of the space-time separation of the nonlinear term, that is why we focus our work in the second part of the paper on the Asymptotic Numerical Method, see for instance [START_REF] Abichou | Asymptotic numerical method for problems coupling several nonlinearities[END_REF]. Originally designed for tracking bifurcation points, it has been lately considered as a good alternative to circumvent the problems due to high nonlinearity. Combined with the POD, Yvonnet et al. [START_REF] Yvonnet | A model reduction method for the post-buckling analysis of cellular microstructures[END_REF] and more recently Niroomandi et al. [START_REF] Niroomandi | Model order reduction for hyperelastic materials[END_REF] applied with success a POD-ANM strategy for respectively post-buckling analysis and large deformations of hyperelastic materials. In this paper, we will present an approach coupling PGD and ANM that takes advantage of both methods. The performances of the different approaches will be outlined with a numerical example.

THE PGD-BEM METHOD

In this section, we present the PGD-BEM method for solving the nonlinear heat equation. Here, the nonlinearity comes from the thermal conductivity function which depends on the temperature. Hence, the problem that will be treated throughout this paper is recalled here:

     u ,t -(k(u)u ,i ) ,i = b in Ω × τ u = 0 in ∂ Ω × τ u(t = 0) = 0 in Ω ( 1
)
where Ω is the spatial domain, ∂ Ω its boundary, τ the time interval, u the temperature field, b the source term and k the thermal conductivity function. The main idea of the PGD is to decompose the solution of the problem (1) as a sum of products of spatial functions X α (x) and temporal functions T α (t), namely we seek u as follows:

u(x,t) ≈ N ∑ α=1 X α (x)T α (t) (2)
In order to explain what we will call in the following the direct PGD, let us assume the p-1 first couples of X α T α known. The solution u (p) at iteration p is then expressed u (p) = u (p-1) + XT and the functions X and T are obtained as solutions of the following nonlinear system:

τ u (p) ,t T dτ - τ k(u (p) )u (p) ,i ,i T dτ = τ b T dτ (3) Ω u (p) ,t X dx - Ω k(u (p) )u (p) ,i ,i X dx = Ω b X dx (4)
Equation ( 3) is a PDE in space that will be solved using the BEM whereas equation ( 4) is an ODE in time. This system of coupled equations is solved using a fixed-point method that will be discussed later.

Spatial resolution with the BEM

As the spatial equation of the system is solved using the BEM, we need to rewrite (3) in a way that allows to compute Green's functions. Hence, the system (3)-( 4) is rewritten as follows:

- τ k(u (p) ) T T dτ X ,ii = τ B T dτ - τ T ,t T dτ X - p-1 ∑ α=1 τ T α ,t T dτ X α - Ω k(u (p) ) X ,ii X dx T = Ω B X dx - Ω X X dx T ,t - p-1 ∑ α=1 Ω X α X dx T α ,t (5) where
B = b + k ′ (u (p) )u (p) ,i u (p) ,i + k(u (p) )u (p-1) ,ii
. For the case of homogeneous boundary conditions, we can write τ T ,t T dτ

= 1 2 T 2 f where T f = T (τ f ). The functions k(u (p)
) and B depend on x and t that is why they must be decomposed into separated forms for computing the integrals in system [START_REF] Chinesta | On the reduction of stochastic kinetic theory models of complex fluids[END_REF]. This is practically achieved by using a SVD, so that if we note:

k(u (p) ) ≈ p ∑ α=1 K α x K α t and B ≈ p ∑ α=1 B α x B α t ( 6
)
the system can finally be written as:

- p-1 ∑ α=1 K α x τ K α t T T dτ X ,ii - p-1 ∑ α=1 B α x τ B α t T dτ + 1 2 T 2 f X + p-1 ∑ α=1 τ T α ,t T dτ X α = 0 (7) - p-1 ∑ α=1 K α t Ω K α x X ,ii X dx T - p-1 ∑ α=1 B α t Ω B α x X dx + Ω X X dx T ,t + p-1 ∑ α=1 Ω X α X dx T α ,t = 0 (8)
The fixed-point algorithm The system of equations ( 7)-( 8) is then solved using a fixedpoint method. Assuming that all fields are known at iteration q, the way of computing them at iteration q + 1 is described hereafter. We first determine X q+1 by solving equation [START_REF] Bonithon | Non-incremental boundary element discretization of parabolic models based on the use of the proper generalized decompositions[END_REF], note that it is a Poisson-like equation and hence is efficiently solved by the BEM. So, we have to solve:

-X q+1 ,ii = 1 p-1 ∑ α=1 K α x τ K α t T q T q dτ p-1 ∑ α=1 B α x τ B α t T q dτ - 1 2 (T q f ) 2 X - p-1 ∑ α=1 τ T α ,t T q dτ X α (9)
The terms B α x and B α t are then evaluated by achieving a SVD over the following form of B:

B = b + k ′ u (p-1),q,q u (p-1),q,q ,i u (p-1),q,q ,i + k u (p-1),q,q u (p-1)
,ii [START_REF] Niroomandi | Model order reduction for hyperelastic materials[END_REF] where u (p-1),q,q = u (p-1) + X q T q . In a similar maner, K α x and K α t are obtained from k u (p-1),q,q . In a second step, T q+1 is obtained by solving [START_REF] Abichou | Asymptotic numerical method for problems coupling several nonlinearities[END_REF], which gives at iteration q + 1:

- p-1 ∑ α=1 K α t Ω K α x X q+1
,ii

X q+1 dx T q+1 - p-1 ∑ α=1 B α t Ω B α x X q+1 dx + Ω X q+1 X q+1 dx T q+1 ,t + p-1 ∑ α=1 Ω X α X q+1 dx T α ,t = 0 (11) Once again, B α
x and B α t are evaluated from the following form:

B = b +k ′ u (p-1),q+1,q u (p-1),q+1,q ,i u (p-1),q+1,q ,i +k u (p-1),q+1,q u (p-1) ,ii (12)
whereas K α x and K α t are obtained from k u (p-1),q+1,q , where u (p-1),q+1,q = u (p-1) + X q+1 T q . Contrary to the linear PGD-BEM presented in [START_REF] Bonithon | Non-incremental boundary element discretization of parabolic models based on the use of the proper generalized decompositions[END_REF], it is here necessary to compute the separated forms of B and k(u) during the fixed-point loop. The way of evaluating the nonlinear term is very important for the efficiency of the algorithm but also affects the computational effort needed for the whole method. This aspect has been widely discussed in [START_REF] Pruliere | On the deterministic solution of multidimensional parametric models using the proper generalized decomposition[END_REF] where they test different linearization using the solution at previous iteration. Here, we linearize in a more implicit way because we separate the nonlinear term inside the fixed-point loop with a SVD. It involves then a number of SVD which is equal to the number of iterations until convergence of the fixed-point algorithm times the number of PGD modes we keep in the solution. The CPU time and storage can hence become non negligeable so that we propose in the following another approach that avoids dealing with these separations.

AN ALTERNATIVE: THE PGD-ANM APPROACH

In this section we present the methodology we have employed in order to circumvent the need of separating the conductivity function into a product of space and time functions.

The Asymptotic Numerical Approach

The Asymptotic Numerical Method (ANM) has been introduced in the 90's to deal with strong nonlinear problems. It consists in transforming the nonlinear problem into a recursive sequence of well posed linear problems having the same tangent operator. In comparison with classical methods that are used for solving nonlinear problems, e.g. Newton-Raphson methods, the ANM has proved to be much more efficient in terms of computational costs. More details about the wide range of applications can be found in [START_REF] Abichou | Asymptotic numerical method for problems coupling several nonlinearities[END_REF]. For generality's sake, let us briefly explain the principle of the method by considering the following nonlinear problem written in quadratic form:

L (u) + Q(u, u) = λ b in Ω u = 0 in ∂ Ω (13)
where L (u) represents a linear operator and Q(u, u) a quadratic one, b is the second member and λ a load parameter. The main idea of the ANM is the asymptotic expansion around an known solution (u n , λ 0 ) -in our case λ 0 is always zero -at step n into power series with respect to a path parameter a:

u n+1 (a) = u n + au 1 + a 2 u 2 + • • • λ (a) = aλ 1 + a 2 λ 2 + • • • (14)
When truncating the series at order N, the problem (13) gives:

L u n + N ∑ p=1 a p u p + Q u n + N ∑ p=1 a p u p , u n + N ∑ p=1 a p u p = N ∑ p=1 a p λ p b
(15) and by identifying the terms of same power p, we obtain a recursive sequence of linear problems:

L p (u p ) = λ p b + F p (16)
where F p is a nonlinear second member term that depends on terms until order p -1. An additional condition is obtained when defining a as a pseudo arc-length:

a = (u n+1 (a) -u n ) • u 1 + λ (a)λ 1 (17)
Relations ( 16) and (17) after a given discretization procedure allow to write a sequence of linear problems in the form:

order 1 K t u 1 = λ 1 f u T 1 , u 1 + λ 2 1 = 1 (18) order p K t u p = λ p f + f p u T p u 1 + λ p λ 1 = 0 (19)
where K t is the tangent stiffness matrix, f is the loading vector and f p is the second term vector obtained from the discretisation of F p . These problems are solved using a predictioncorrection approach, their solutions are obtained as follows:

order 1          û1 = K -1 t f λ 1 = 1 ( û1 , û1 ) + 1 u 1 = λ 1 û1 (20) order p        ûp = K -1 t f p λ p = -λ 1 ( ûp , u 1 ) u p = λ p λ 1 u 1 + ûp (21)

Coupling between PGD and ANM

There are obviously several ways of coupling the PGD and the ANM. A first strategy consists in applying the PGD on the suite of linear systems that would be obtained by the ANM on the nonlinear heat equation. The other approach takes advantage of the ANM as a nonlinear solver, so that the order of the steps is different: the ANM is applied inside the PGD algorithm. The first approach is a work currently in progress and we will here briefly present the most important features of the second approach we have implemented. The first steps of the algorithm remain unchanged, so that we deal with the system of coupled equations ( 3)-( 4) that are obtained thanks to the PGD procedure. Here, instead of separating the nonlinear term k(u) into a couple of space-time functions, we propose to consider it as a new unknown that will be expressed as an asymptotic expansion. If we note v = k(u), the expansion will be now written around the known solution (u n , v n , λ n ) at step n:

  u n+1 v n+1 λ   =   u n v n 0   + N ∑ p=1 a p   u p v p λ p   (22)
so that the PGD method will determine at the same time the separated representation of u and k(u). Typically, knowing u and v at iteration p -1, we seek their expression at iteration p as:

u (p) = u (p-1) + XT (23) v (p) = v (p-1) + RS (24)
and the ANM will be applied inside the PGD step over X and R so that u (p) and v (p) will be then expressed as:

u (p) = u (p-1) + N ∑ α=1 a α X α T (25) v (p) = v (p-1) + N ∑ α=1 a α R α S (26)
Rewriting these expressions into the direct PGD algorithm leads finally to the PGD-ANM method.

RESULTS

In the later, the nonlinear heat equation will be solved in the spatial domain Ω = [0, 1] × [0, 1] and over the time domain τ = [0, 1]. For comparison reasons, the analytic solution is imposed to be u ex = x(x -1)y(y -1)t and the second member is computed from this solution, depending on the value of the nonlinear term k(u). For the PGD-BEM, we consider the nonlinear term of the following form k(u) = (u 2 + 1) . Each side of the domain Ω is discretized using 8 nodes and the time interval is decomposed into 129 time steps.

The figure 1 shows the results obtained with the PGD-BEM approach compared to the analytic solution at times t = {0s, 7.8 × 10 -3 s, 0.25s, 0.5s, 0.75s, 1s} and figure 2 represents the corresponding absolute error computed from the analytic solution:

e(x,t) = u(x,t) -u ex (x,t) u ex (x,t) ∀(x,t) ∈ Ω × τ (27)
At last, figure 3 shows the first three couples of modes that are obtained at the end of the algorithm, note here that the first mode is able to represent the structure of the solution whereas the additional modes are useful for catching accurately the details. Now, we present the results obtained by applying the PGD-ANM method compared to the direct PGD, recalling here that this method avoids achieving a large number of SVD which is the case when using the direct PGD. We keep the same value of the analytic solution but now we consider the nonlinear term defined as k(u) = (u + ε) n where n takes the values n = 1 or 2. Figure 4 presents the evolution of the error in a L 2

Ω×τ sense with the number of PGD modes for the direct PGD compared to the PGD-ANM method at respectively order 1 and 2. We observe in this figure that the PGD-ANM at first order slightly improves the results in comparison with the direct PGD whereas they are significantly better at second order, even for the most nonlinear case. For this case, with 5 PGD modes, the error is divided by two when using the PGD-ANM instead of the direct PGD, which is already a quite interesting results.

CONCLUSIONS

The work presented in this paper addresses some issues or at least interesting perspectives in the field of efficient solving of nonlinear equations. Indeed, the PGD-BEM algorithm has been successfully adapted for solving the nonlinear heat equation, without increasing the level of complexity in the methodology for the linear case. The main advantage of this approach lies on the fact that only the Green's kernel of Poisson equations is required to solve the nonlinear heat equation with the Boundary Element Method. Furthermore, coupling the PGD with the ANM allows to take advantage of both techniques, and the gains we can expect are really encouraging. Optimizations in the algorithm are currently in progress in order to make it even more accurate and CPU time efficient.

FIGURE 4 .

 4 FIGURE 4. COMPARISON BETWEEN PGD DIRECT AND PGD-ANM FOR DIFFERENT VALUES OF k(u).

[min =0.00, max =0.00] × 0.0e +00 t =0.0e +00


George Chimowa
email: chimowag@biust.ac.bw

Emmanuel Flahaut

Venkateswara Rao Sodisetti

Somnath Bhattacharyya

Converting Ambipolar Double Walled Nanotube bundles to Unipolar Carbon Nanotube Field Effect Transistor devices by inner functionalization

Keywords: nanoelectronics, functionalisation, endohedral filling, nanowires

Carbon Nanotube (CNT) based transistor devices have been demonstrated and are seen as the next generation alternative to replace the current generation electronic devices based on silicon MOS technology. Their limitations, however, are related to their instability in ambient conditions and technical challenges in scaling up production of individual CNT devices. This work demonstrates a simple technique to convert ambipolar (both n & p-carrier) individual CNT based transistors into unipolar (p-type carrier) transistors that carry large currents approximately ±7 mA before saturation. Such large current carrying capability from an individual or a single bundle CNTs will be ideal for applications that require high currents such as output stages of radio frequency (RF) and audio amplifiers. Our work highlights the ability to tune CNT based transistor devices to suit specific applications which is essential for the development of next generation of nanoelectronics.

Introduction

Morden electronics has benefitted significantly from the scaling down the channel length of metal oxide semiconductor field effect transistors (MOSFET) from micro scale into the nano-scale regime. This has enabled the packaging of thousands of transistors into small microprocessors almost the size of one's thumb nail, that consume low energy and have low operating voltages. However, challenges, arise due to short channels, such as high leakage currents because of quantum tunnelling [1], Drain induced Barrier lowering (DIBL) [2] and the Fermi -Dirac limited sub threshold (SS) swing of the metal oxide semiconductor (MOS) technology [3]. New materials for transistor manufacturing are therefore required if the scaling down of device size is to be continued further beyond 30 nm. Silicon tunnelling field effect transistors have shown promising fast switching, but their limiting factor is the high bandgap of the material [4].

It is against this background that carbon nanotubes have emerged as potential transistor devices due to their nano-scale dimension, excellent electrical properties, which include high current mobility, and low bandgap (0.6 -0.8 eV) which depends on their diameters and chirality [START_REF] Ph | Nanotube electronics, Physics World, physicsweb.org[END_REF][START_REF] Appenzeller | [END_REF]7].

Semiconducting double walled nanotubes (DWNTs) exhibit both p-type and n-type (ambipolar) characteristics which is not so much observed with single walled nanotubes (SWNTs) [8]. This ambipolar behaviour of DWNTs devices makes them ideal for light emitting transistors as the holes and electrons recombine to emit a photon. There are however other applications which require unipolar devices such as fast switching transistors for high frequency applications. It is therefore essential that ambipolar devices be made into stable unipolar devices.

Another limiting factor of the semiconducting SWNTs transistor devices that have been reported so far, is the low ON currents normally in the nano-ampere (nA) range [9,10, 11,]. Such low currents will not be sufficient for the output stages of sound amplifiers and some RF output amplifiers that require high currents to produce enough energy to vibrate speaker diaphragm or power the antennas of telecommunications devices. In this light, individual CNT bundles or double walled nanotubes capable of carrying higher currents will be ideal for these applications. The DWNT individual bundle transistor devices reported in this article show great promise in this aspect as they can carry current in the mA regime in agreement with previous predictions for large diameter DWNTs with low array density, which is not observed in SWNTs [12].

Often SWNT based transistors performance is influenced significantly by the ambient air as it makes the SWNT into p-type doped material [13], such behaviour is not ideal for stable and reliable devices.

Unlike SWNTs, DWNTs have also been reported to have better sub threshold swing (SS) =

𝑖𝑛𝑣𝑒𝑟𝑠𝑒 (𝑑𝑙𝑜𝑔 𝐼 𝑑 𝑑𝑉 𝑔𝑠

) [14]. It is against this background that this work sought to find ways of making high current DWNT based transistors and explore the important parameters such the transconductance (gm), sub-threshold swing, and the Ion / Ioff current ratio. The operation of these transistors was also investigated at liquid nitrogen temperatures, and it was shown that the overall device performance improves highlighting the significance of suppressing thermal processes. Such This article is protected by copyright. All rights reserved fundamental studies are important as they set a framework for innovative ideas in future application.

By using inner functionalization of DWNTs, a technique not commonly used, ambipolar pristine

DWNTs can be converted into unipolar (p-type) transistors that are stable. Similar attempts were made with graphene field effect transistors in which titanium deposition on the graphene surface induced p-type doping [15]. While there are several chemicals techniques that can be applied to induce p-type doping of carbon-based materials, inner functionalisation has added advantages.

Namely, inner functionalization retains the CNT's one dimensionality property, as the outer surface area is not altered and secondly the density of states is improved as was shown previously [16]. So, to some extent the important fundamental properties of CNTs are preserved and some properties such as sensitivity to gases is improved as the outer surface area is not affected.

The current work reported here, lays a strong foundation for high current DWNT based unipolar transistor devices that can be used in applications which require relatively high currents such as the ones mentioned earlier.

Methodology

Double walled carbon nanotubes were synthesized using a chemical vapour deposition technique as outlined in the article Communication [17]. The DWNTs were then opened at their ends by mixing 1 g of DWNTs, with 3.5 mL distilled water and 25 mL of 37 % HCL and sonicated for 20 minutes. The solution was filtered and neutralized using 1.5 L of distilled water. The DWNTS were then heated at 400 o C for 30 mins followed by another heating in a vacuum at 200 o C for 4 hours to ensure maximum drying. The inner functionalization with ZnI2 was done as outlined in the article [18]. The mixture was then washed with warm ethanol several times to remove the excess of ZnI2 outside the nanotubes and ensure that all remaining ZnI2 was present as filling inside the nanotubes.

The inner functionalized DWNTs were then taken for transmission electron microscopy to check the success of the filling, TEM images are shown in the results section. After this, the functionalized DWNTs were made into back gated transistor devices by aligning individual bundles using dielectrophoresis method on prefabricated electrodes whose structure dimensions are shown in The transistor characteristics were measured using the Agilent semiconductor parameter analyzer system B 1500 A. The Instrument is attached to a Janis Cryogenic system, which enables low temperature measurements, performed by flowing liquid nitrogen through an evacuated system.

Results and Discussions

As aforementioned the filled /inner functionalised DWNTs were analysed using high resolution transmission electron microscope (HRTEM) in both bright and dark field modes, to check the effectiveness of the filling. This article is protected by copyright. All rights reserved From the HRTEM dark field images in Figure 2, it is evident that the filling method used in this work was effective, with some CNTs even getting filled over the entire 30 to 40 nm nanotube length see Fig. 2 (a). Figure 2 (c andd) shows EDX spectra of the filled nanotube confirming that the filling material is a composition of Zinc, iodine and/or Zinc iodide.

After material characterisation, the filled DWNTs were made into back gated transistor on the prefabricated devices as mentioned earlier. The FE-SEM images in Fig. 3 shows an illustration of some of the devices. DWNTs bundles. What is significant is the high current that the device can carry, maximum of 8 mA at 77 K and about 6 mA at 295 K for Vds = 2 V and Vgs = -1V. This is in accordance with the extensive numerical study by J. Zh. Huang et. al., which predicted large on currents, when the DWNT diameters are large, and the array density is low [12]. The current begins to get significantly pinched only when the gate voltage is above 2 V for measurements done at 295 K, see Fig. 4 (a), this is an indication that the gate oxide layer is too thick at 300 nm. Reducing this oxide layer thickness will predictably Accepted Article This article is protected by copyright. All rights reserved While this work has demonstrated, large current CNT transistor devices, that can be manipulated from ambipolar to unipolar devices, it is evident that the switching capabilities are still very poor.

The sub threshold swing (SS) for the hole current in the pristine device is about 2.3 V/dec and for the filled DWNTs is about 4.8 V/dec, which are very far from a target of less than 60 mV/dec. This might not be surprising because the gate oxide is about 300 nm, meaning that gate voltage effect on the charge carrier transport in the CNT channel device is weak. Furthermore, the devices are typical Schottky barrier devices as the CNTs were just placed on top of the electrodes making poor van der Waals contacts with the source -drain electrodes. The Ion / Ioff current ratio for pristine DWNT devices is ~ 10 3 A for the electron current, which is 10 2 lower than the ideal value demonstrating high leakage currents and hence there is need to further improve device physical geometry. The filled DWNTs devices have even lower Ion / Ioff current ratio.

The transconductance (gm) for the hole current for both devices are not very different -2.5 mS for the Pristine and -2.2 mS for the filled DWNTs. What is evident is that for the filled DWNTs the transconductance has some temperature dependence (-3.2 mS at 76 K and -2.2 mS at 295 K) which is not the case with pristine DWNTs. A closer look at Fig. 6 (b), shows that the hole current saturates at -0.5 V, suggesting a limited hole carrier concentration. To understand the above discussed observations, we proposed a simple band diagram shown in Fig. 7. that can be used in fields requiring high currents such as power amplifiers and RF output stages of electronic transmission equipment that power antennas. Improvement of the device architecture will improve some of the poor observed parameters.

Accepted Article

Fig. 1 (

 1 Fig.1 (a & b). The figure 1 (a) shows the electrode top layout and figure 1(b) shows the crosssectional view. The back side is heavily p-doped Silicon, which was used as the gate terminal, while two of the top electrodes were used as source and drain depending on which terminals the DWNTS will be aligned (either the inner or outer electrodes). To ensure that we investigated DWNTs of

Fig. 1 :

 1 Fig. 1: (a) Top-view of electrode layout for the transistor devices (b) Cross-sectional view showing that a common back gated is below the top gold source and Drain terminals.

Fig. 2 :

 2 Fig.2: (a) Dark field HRTEM image of the filled DWNTs, some CNTs showing filling over the entire shown area. (b) Bright field HRTEM image of the image shown in (a), showing the double walls of the CNTs. (c) Dark field HRTEM image showing one filled individual CNT. (d) EDX spectra taken on the segment of the CNT in (c), showing the presence of Zinc, Iodine and Copper from the sample holder.

  , Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/pssa.202200669 by Portail BibCNRS INC, Wiley Online Library on [18/03/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License This article is protected by copyright. All rights reserved

Fig. 3 :

 3 Fig.3: FE-SEM images of the individual DWNT bundles deposited on gold contacts which make the source and drain contacts making the transistor devices, (a) Device made from pristine DWNTS, (b) Device made from ZnI filled DWNTs.The scanning electron microscopy images shows individual bundles of DWNTs aligned across the source and drain contacts. No additional top metal was evaporated to improve the contacts, resulting in most likely Schottky barrier contacts, and this has a significant effect on the device performance which shall be explained later.

Fig. 4 :

 4 Fig. 4: (a) Output characteristics of Pristine DWNTs devices at 295 K as the gate voltage was varied from -1 to 5 V. (b) Output characteristic of the same device at 77 K, with the gate voltage being varied from -1 to 5 V.

Figure 4 (

 4 Figure 4 (a and b) shows the output characteristics of the CNT transistor device made from pristine

Fig. 5 :

 5 Fig. 5: (a) Output characteristics of ZnI filled DWNTs devices at 293 K as the gate voltage was varied from -1 to 5 V. (b) Output characteristic of the same device at 76 K, with the gate voltage being varied from -1 to 5 V. The device transfer characteristics of the two-transistor shown in Figure 6, clearly show the unique differences between the two types of DWNT devices. The pristine DWNTs device, Fig.6 (a), demonstrates ambipolar transport, with hole current dominating when Vgs is negative and electron current dominates for positive Vgs values. The transition between the two types of transport occurs when Vgs = Vds. The DWNTs filled with Zinc Iodide, Fig. 6 (b), shows that the electron current gets suppressed, indicating that the DWNTs have become p doped. This agrees with other previous studies which have shown that Iodine, causes p-doping of CNTs [20].

Fig. 6 :

 6 Fig. 6: (a) Transfer characteristics of the pristine DWNTs showing ambipolar transport. (b) Transfer characteristics of zinc iodide filled/functionalized DWNTs, showing that electron current is suppressed.

Fig. 7 :

 7 Fig.7: A schematic illustration of the band diagram for both the pristine DWNTs and ZnI filled DWNTs.In the schematic diagram in Fig.7, for pristine DWNTs, the carriers (holes or electrons) Fermi levels are close to the chemical potential (µ 𝑠 𝑎𝑛𝑑 µ 𝐷 ) of the contact electrodes. The Source-drain bias voltage (Vds) will result in a slight tilting of the CNT energy band towards the drain terminal, and the chemical potential at the Drain will be pushed down by energy equal to qVds, where q is the charge on an electron. The varying gate voltage modulates both the Schottky barrier potentials at the contacts as well the positions of the carrier Fermi level, since it is over the entire length of CNT device including the contacts. When Vgs is positive, electron current dominates, and the reverse is true when Vgs is negative, i.e., hole current dominates.

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Acknowledgements

The authors would like Mr Franc Carcenac who assisted with E-beam lithography during the fabrication of the gated transistor electrodes. We also like to thank BIUST for the staff initiation grant that funded part of the work and the University of the Witwatersrand for the honorary research position which gave the lead author access to the equipment in the Nanoscale transport laboratory to do the transistor measurements.