Marine Guigou

Thierry Martin

Adeline Crépieux

Screening of a Luttinger liquid wire by a scanning tunneling microscope tip: II. Transport properties

Keywords:

HAL is

I. INTRODUCTION

Transport properties in one-dimensional interacting systems such as quantum wires or metallic carbon nanotubes have been widely studied both theoretically and experimentally. When the wire or nanotube is well connected to a metallic contact, and another tunnel contact is placed on top of it, the current exhibits a zero bias anomaly 1,2 because of the Coulomb interaction effects in this one dimensional system. On the contrary an isolated wire with bad contacts displays Coulomb blockade effects 3 .

Another interesting tool to characterize the transport is the noise (current-current fluctuations). It allows for instance to obtain information about the charge of the carriers flowing through the conductor. When electrons are injected in an infinite length carbon nanotube from a scanning tunneling microscope (STM) tip, it has been shown 4 that the Fano factor -the ratio between the zerofrequency noise and the current -is equal to (K 2 c +1)e/2, where K c is the Coulomb interaction strength. This Fano factor is tied to the charge of the collective excitations which are present within the wire. However, the presence of adiabatic Fermi liquid contacts at the extremities of the nanotube modifies 5 the Fano factor and effectively erases the information about the Coulomb interactions.

To recover information about Coulomb interactions, finite-frequency noise is needed. Recent experiments 6,7 have shown that the current-current correlator which can be detected is in general related to the non-symmetrized noise rather than the symmetrized one. Calculations of the finite-frequency non-symmetrized noise in the STM+nanotube system showed oscillations due to reflections at the contacts 5 whose period is related to the anomalous charges.

In the above calculations, electrostatic effects such as screening between the wire and tip were typically left out. Screening from the STM tip is likely to affect locally the electronic properties of the wire. Such effects were explicitly computed in Ref. 8, where emphasis was put on the spectral properties of the wire. There, we developed general expressions for the Dyson equations, which allowed to derive non perturbatively the Greens functions of the bosonic fields which are needed to obtain the spectral function of the wire. The tunneling density of states was shown to be enhanced (reduced) for large (weak) Coulomb interaction [START_REF] Guigou | Screening of a Luttinger liquid wire by a scanning tunneling microscope tip: I. Spectral properties[END_REF] .

While the spectral function, the tunneling and the local density of states are quantities of great interest in condensed matter physics problems, they are typically measured in the context of electron transport, by tunneling experiments. In Ref. 9 for instance, the tunneling current between two Luttinger liquids is typically expressed as a convolution of the two tunneling density of states of such materials.

In the present paper, we wish for the first time to address transport and screening effects on the same footing. We argue that because the interaction between the tip and wire mixes the fermionic degrees of freedom of both source and drain, the derivation of the tunneling current and noise has to be revisited. As we shall see, the tunneling current depends on a new class of Greens function which we call "mixed" Green's functions. The use of the bosonization methods allows to compute the tunneling current, as usual, to second order in the tunneling amplitude, albeit "non perturbatively" with respect to the Coulomb interaction between the wire and tip. This work thus suggests plainly that in the presence of screening effect, special care must be taken in the derivation of transport properties. A generalization of our results to carbon nanotubes can be straightforwardly obtained.

The structure of the paper is the following: in Sec. II, we present the system and the Hamiltonian. In Sec. III, we give the definitions of the tunnel current and of the non-symmetrized noise and their general expressions in term of bosonic Green's functions. In Sec. IV, we cal-culate the mixed Green's functions. Next, in Sec. V, we consider a particular form of the screening potential and we calculate the transport properties associated to an infinite length wire. We conclude in Sec. VI.

II. MODEL

The total Hamiltonian we consider is H = H 0 + H V where H 0 is the unperturbed Hamiltonian and H V is the perturbation due to the voltage applied to the tip (see Fig. 1).

The unperturbed Hamiltonian is given by H 0 = H W + H T + H Sc , where H W describes the quantum wire:

H W = j=c,s +∞ -∞ dx v j (x) 2 K j (x) ∂ x φ j (x) 2 +K -1 j (x) ∂ x θ j (x) 2 , (1)
with θ j and φ j are the non-chiral bosonic fields, K c < 1 is the repulsive Coulomb interactions parameter in the charge sector, and K s = 1 is the Coulomb interactions parameter in the spin sector. v j = v F /K j is the velocity of the excitations in each sector j. We put = 1 in the whole paper, excepted in voltage dependent terms and in figure captions.

H T describes the STM tip:

H T = u F 4π σ=±1 +∞ -∞ dy ∂ y ϕ σ (y) 2 , (2)
with ϕ σ , the chiral bosonic field of the STM tip and u F is the Fermi velocity of electrons in the tip. σ is the index which refers to the spin degree of freedom. The x and y axes are defined in Fig. 1. And, H Sc describes the screening:

H Sc = +∞ -∞ dx +∞ -∞ dy ρ W (x)W (x, y)ρ T (y) , ( 3
)
where W is the electrostatic potential caused by the proximity between the STM tip and the quantum wire. ρ W is the density operator associated to the wire and ρ T , the density operator associated to the tip:

ρ W (x) = 2 π ∂ x θ c (x) + ρ 2kF (x) , (4)
ρ T (y) = 1 2π σ ∂ y ϕ σ (y) . (5)
When K c > 1/2, we have shown in Ref. 8 that the 2k F -contribution to the density operator is not a relevant contribution to the screening Hamiltonian. We neglect it in this work.

The perturbation here is caused by a voltage V applied to the tip: electrons can by injected from the tip to the wire or, inversely from the wire to the tip, by tunnel effect:

H V = r=±,σ=± Γ(t)ψ † rσ (0)c σ (0) + Γ * (t)c † σ (0)ψ rσ (0) , (6)
where Γ(t) = Γ 0 exp(ieV t/ ) is the tunnel amplitude and the chirality r refers to the right movers (r = +) or to the left movers (r = -). We assume that the electrons are injected in the wire at the position of the tip, i.e. at position x = 0. c † σ and c σ are the creation and annihilation operators attached to the tip:

c σ (y) = g σ √ 2πa e iϕσ(y) , (7)
where g σ is a Majorana fermion associated to the tip, and a is the ultraviolet cut-off. ψ † rσ and ψ rσ are the creation and annihilation operators attached to the chirality r and to the spin σ of the electrons in the wire:

ψ rσ (x) = f rσ √ 2πa e irkF x+i √ π 2 P r,σ hσ (j)(φj (x)+rθj(x)) , (8)
where f rσ is the Majorana fermion associated to the wire, k F is the Fermi wave vector. The coefficient h σ (j) is equal to 1 when j = c and to σ when j = s.

III. TRANSPORT PROPERTIES

In this section, we give the general expressions of the tunnel current and the non-symmetrized noise as a function of the bosonic Green's functions in the presence of screening.

A. Tunnel current definition

The average tunnel current is defined as:

I T = 1 2 η=± T K I T (t η )e -i R K dt1HV (t1) , (9)
where T K denotes time ordering along the Keldysh contour 10 , η indexes the position of time on it, and

K dt 1 ≡ η1=± dt η1
1 . The tunnel current operator is defined by:

I T (t) = ie r,σ Γ(t)ψ † rσ (0, t)c σ (0, t) -Γ * (t)c † σ (0, t)ψ rσ (0, t) . (10)
Up to the second order of the perturbative calculation with the tunnel amplitude Γ 0 , the average tunnel current can be expressed as

I T = ieΓ 2 0 (2πa) 2 r,σ,η,η1 η 1 +∞ -∞ d t sin eV t e G ϕϕ,ηη 1 σ (0,0, t) × e π 2 P j G φφ,ηη 1 j (0,0, t)+G θθ,ηη 1 j (0,0, t)+rG φθ,ηη 1 j (0,0, t)+rG θφ,ηη 1 j (0,0, t) × e - √ π 2 G φϕ,ηη 1 cσ (0,0, t)+rG θϕ,ηη 1 cσ (0,0, t)+G ϕφ,ηη 1 cσ (0,0, t)+rG ϕθ,ηη 1 cσ (0,0, t) , (11)
where t ≡ tt 1 . G ϕϕ,ηη1 σ is the Keldysh Green's function of the tip in the presence of the electrostatic coupling defined as: Eq. ( 11) differs to what one obtains in the absence of screening, in the sense that mixed Green's functions are not present in this case 4 . We call them mixed Green's functions, because they mix the bosonic fields θ or φ of the wire to the bosonic field ϕ of the tip. The appearance of these new correlators is due to the electrostatic coupling between the wire and the tip, and they will af-fect the transport properties. Notice that these mixed Green's functions do not have any influence on the spectral properties of the wire because in the definitions of the spectral function and of the density of states, it only appears the bosonic fields of the wire 8 . On the contrary, in the transport properties, the bosonic fields of the wire and of the tip are mixed due to the term H V in the Hamiltonian, which corresponds to a tunnel transfer of electrons between the two sub-systems.

G ϕϕ,ηη1 σ (0, 0, t) ≡ G ϕϕ,ηη1 σ (0, t η ; 0, t η1 1 ) = ϕ σ (0, t η )ϕ σ (0, t η1 1 ) - 1 2 ϕ σ (0, t η )ϕ σ (0, t η ) - 1 2 ϕ σ (0, t η1 1 )ϕ σ (0, t η1 1 ) . (12)
The calculation of the non perturbative Green's functions associated to H 0 : G θθ j , G φθ j , G θφ j ,G φφ j , and G ϕϕ σ is presented in detail in Ref. 8, and the calculation of the mixed Green's functions is done in Sec. IV.

B. Non-symmetrized noise definition

We define the non-symmetrized noise as

S T (t, t ′ ) = T K I T (t -)I T (t ′+ )e -i R K dt1HV (t1) . ( 13
)
Up to the second order with the tunnel amplitude Γ 0 , the non-symmetrized noise reads

S T (t, t ′ ) = e 2 Γ 2 0 (2πa) 2 r,σ cos eV (t -t ′ ) e G ϕϕ,-+ σ (0,0,t-t ′ ) × e π 2 P j G φφ,-+ j (0,0,t-t ′ )+G θθ,-+ j (0,0,t-t ′ )+rG φθ,-+ j (0,0,t-t ′ )+rG θφ,-+ j (0,0,t-t ′ ) × e - √ π 2 G φϕ,-+ cσ (0,0,t-t ′ )+rG θϕ,-+ cσ (0,0,t-t ′ )+G ϕφ,-+ cσ (0,0,t-t ′ )+rG ϕθ,-+ cσ (0,0,t-t ′ ) . (14)
Since we consider an applied voltage constant in time, the noise depends only on the difference t = tt ′ :

S T (t, t ′ ) = S T (tt ′ ), we can thus define the following Fourier transform for the noise:

S T (Ω) = +∞ -∞ d te iΩ tS T ( t) . (15)
In what follows, we will study both the zero-frequency noise S T (Ω = 0) and the finite-frequency noise S T (Ω = 0). However, we need before to determine the Dyson equations of the mixed Green's functions which appear in the expressions of the current and the noise.

IV. DYSON EQUATIONS FOR THE MIXED GREEN'S FUNCTIONS

The first step is to write the partition function in the Matsubara formalism [START_REF] Peskin | An introduction to quantum field theory[END_REF] :

Z[ϕ, φ, θ] = Dϕ ↑ Dϕ ↓ Dφ c Dφ s Dθ c Dθ s e - R dτ (L0-Laux) , (16)
where L 0 = L W + L T + L Sc is the Lagrangian associated to H 0 and L aux contains auxiliary fields η θj , η φj and η ϕσ needed to extract the Green's functions, such as for example G ϕθ jσ (y, τ ; x ′ , τ ′ ) = T τ {ϕ σ (y, τ )θ j (x ′ , τ ′ )} , through the relation:

G ϕθ jσ (y, τ ; x ′ , τ ′ ) = 1 Z ∂ 2 Z ∂η ϕσ (y, τ )∂η θj (x ′ , τ ′ ) , (17)
where T τ is the time ordering operator. As the Hamiltonian H 0 is quadratic with the bosonic fields θ j , φ j and ϕ σ , it is possible to calculate exactly the full Green's function by integrating out the degrees of freedom of the tip. We obtain

G ϕθ jσ (y, τ ; x ′ , τ ′ ) = -δ jc dτ 1 +∞ -∞ dx 1 +∞ -∞ dy 1 ×G ϕϕ σ (y, τ ; y 1 , τ 1 )G -1 Sc (x 1 , y 1 )G θθ j (x 1 , τ 1 ; x ′ , τ ′ ) , (18)
where

G -1 Sc (x 1 , y 1 ) = ∂ x1 ∂ y1 W (x 1 , y 1 )/(π √ 2π
), and G θθ j the Green's function of the wire without screening. The derivation of the partition function with other auxiliary fields gives access, in a similar way, to the Dyson equations verified by the other mixed correlators:

G θϕ jσ (x, τ ; y ′ , τ ′ ) = -δ jc dτ 1 +∞ -∞ dx 1 +∞ -∞ dy 1 ×G θθ j (x, τ ; x 1 , τ 1 )G -1 Sc (x 1 , y 1 )G ϕϕ σ (y 1 , τ 1 ; y ′ , τ ′ ) , (19)
G ϕφ jσ (y, τ ; x ′ , τ ′ ) = -δ jc dτ 1 +∞ -∞ dx 1 +∞ -∞ dy 1 ×G ϕϕ σ (y, τ ; y 1 , τ 1 )G -1 Sc (x 1 , y 1 )G θφ j (x 1 , τ 1 ; x ′ , τ ′ ) , (20)
and,

G φϕ jσ (x, τ ; y ′ , τ ′ ) = -δ jc dτ 1 +∞ -∞ dx 1 +∞ -∞ dy 1 ×G φθ j (x, τ ; x 1 , τ 1 )G -1 Sc (x 1 , y 1 )G ϕϕ σ (y 1 , τ 1 ; y ′ , τ ′ ) . (21)
In the next section, we present the application of these results to an infinite length wire and we make the approximation, (which is justified in Ref. 8) that the double derivative of the electrostatic potential is represented by a local interaction.

V. APPLICATION TO A LOCAL ELECTROSTATIC POTENTIAL DOUBLE DERIVATIVE ∂x∂yW (x, y)

For the same reasons that are given in Ref. 8, we assume an electrostatic potential of the form

∂ x ∂ y W (x, y) = W 0 δ(y)δ(x) . ( 22
)
With this specific form of the electrostatic potential and for an infinite length wire, we are able to solve the Dyson equations verified by the Green's functions associated to H 0 .

The Keldysh Green's function associated to the bosonic field ϕ of the tip that we need in Eqs. ( 11) and ( 14) reads 8 :

G ϕϕ,-+ σ (0, 0, t) = G ϕϕ,-+ σ (0, 0, t) + 1 2 2 ln(1 + iω c t) -e ωSc t-i ω Sc ωc Ei -ω Sc t + i ω Sc ω c + e -i ω Sc ωc Ei i ω Sc ω c -e -ωSc t+i ω Sc ωc Ei ω Sc t -i ω Sc ω c + e i ω Sc ωc Ei -i ω Sc ω c , (23)
where ω c = v F /a is the frequency cut-off, ω Sc = (W 0 /π) K c /2 is the screening frequency, Ei is the exponential integral function, and G ϕϕ σ is the bare Green's function of the tip in the absence of screening. Notice that Eq. ( 23) was derived exactly since no restriction has been imposed to the strength of the screening potential. The only assumption we have made is to take K c > 1/2 in order to neglect the 2k F -contribution in ρ W .

The Keldysh Green's function attached to the bosonic field θ of the wire is:

G θθ,-+ c (0, 0, t) = G θθ,-+ c (0, 0, t) + K c 4π 2 ln(1 + iω c t) -e ωSc t-i ω Sc ωc Ei -ω Sc t + i ω Sc ω c + e -i ω Sc ωc Ei i ω Sc ω c -e -ωSc t+i ω Sc ωc Ei ω Sc t -i ω Sc ω c + e i ω Sc ωc Ei -i ω Sc ω c . (24)
For the spin sector (j = s), the Green's function is not affected by the screening: G θθ,ηη1 s (0, 0, t) = G θθ,ηη1 s (0, 0, t). The other Keldysh Green's functions associated to the wire are aslo unchanged:

G θφ,ηη1 j (0, 0, t) = G θφ,ηη1 j (0, 0, t) , (25)
G φθ,ηη1 j (0, 0, t) = G φθ,ηη1 j (0, 0, t) ,

G φφ,ηη1 j (0, 0, t) = G φφ,ηη1 j (0, 0, t) .

The mixed Green's functions G ϕθ,-+ cσ and G θϕ,-+ cσ are calculated in Appendix A, we obtain:

G ϕθ,-+ cσ (0, 0, t) = G θϕ,-+ cσ (0, 0, t) = 1 4
K c 2 sgn(1 + iω c t) sin ω Sc ω c (1 + iω c t) . ( 28
)
The two other mixed Green's functions, which appear in Eqs. ( 11) and ( 14), are equal to zero even in the presence of screening:

G φϕ,ηη1 jσ (0, 0, t) = 0 , (29)
G ϕφ,ηη1 jσ (0, 0, t) = 0 , (30)
because the Hamiltonian H Sc couples only the θ and ϕ fields. Now, we have all the ingredients to calculate the transport properties of the wire in the presence of screening by the tip. 0 /(avF 2 π 3 ), for Kc = 0.7. Bottom panel: tunnel current for Kc = 0.2. On both graphics, the screening strength is ωSc = 0 (solid line), ωSc/ωc = 0.001 (dotted line), and ωSc/ωc = 0.002 (dashed line).

A. Tunnel current

When we report these Green's functions in Eq. ( 11), we obtain up to second order in Γ 0 :

I T = - 2eΓ 2 0 (2πa) 2 σ +∞ -∞ d t sin eV t × Im cosh 2 π 2 G θϕ,-+ cσ (0, 0, t) e G ϕϕ,-+ σ (0,0, t)
× e π 2 P j G φφ,-+ j (0,0, t)+G θθ,-+ j (0,0, t)

, ( 31
)
where we have performed the explicit sums over r and over the Keldysh indices: only the η = η 1 terms contribute to the integral, because for η = η 1 , the function we have to integrate is odd in t. After some algebra, we obtain:

I T = - 2eΓ 2 0 π 2 a 2 +∞ 0 d t sin eV t Im cosh √ πK c 4 sgn(1 + iω c t) sin ω Sc ω c (1 + iω c t) e -(ν+1) ln(1+iωc t) × e ( Kc 8 + 1 2 ) " 2 ln(1+iωc t)-e ω Sc t-i ω Sc ωc Ei(-ωSc t+i ω Sc ωc )+e -i ω Sc ωc Ei(i ω Sc ωc )-e -ω Sc t+i ω Sc ωc Ei(ωSc t-i ω Sc ωc )+e i ω Sc ωc Ei(-i ω Sc ωc ) « , (32)
where ν = (K c + 1/K c + 2)/4, and where the summation over σ has been made assuming a non-magnetic STM tip. To simplify the notations, we assume identical Fermi velocities in the wire and in the tip: v F = u F . In Fig. 2 the tunneling current is plotted as a function of applied voltage. For weak Coulomb interactions (top panel of Fig. 2) and in the absence of screening, we see that the I-V characteristic is linear (solid line). When the tip is approached close to the wire, the I-V characteristic is modified: the amplitude of the current is reduced and the I-V curve follows a power law. For strong Coulomb interactions (bottom panel of Fig. 2), the I-V characteristic follows a power law in the absence of screening. When the tip is approached close to the wire, the tunnel current stays a power law but is strongly reduced.

The interpretation of these results is the following: electrostatic interactions between the tip and the wire plays a similar role than Coulomb interactions inside the wire: they both reduce the tunnel current and change the linear behavior of the I-V curve to a power law behavior. The tunnel current is strictly linear only in the absence of Coulomb interactions and electrostatic screening.

B. Zero-frequency noise

Up to second order with the tunnel amplitude Γ 0 , the zero-frequency noise reads:

S T (Ω = 0) = e 2 Γ 2 0 (2πa) 2 r,σ +∞ -∞ d t cos eV t e 2πG ϕϕ,-+ σ (0,0, t)
× e π 2 P j G φφ,-+ j (0,0, t)+G θθ,-+ j (0,0, t)+rG φθ,-+ j (0,0, t)+rG θφ,-+ j (0,0, t) 0 /(avF 2 π 3 ), for Kc = 0.7. Bottom panel: zero-frequency noise for Kc = 0.2. On both graphics, the screening strength is ωSc = 0 (solid line), ωSc/ωc = 0.001 (dotted line), and ωSc/ωc = 0.002 (dashed line).

× e - √ π 2 G φϕ,-+ cσ (0,0, t)+rG θϕ,-+ cσ (0,0, t)+G ϕφ,-+ cσ (0,0, t)+rG ϕθ,-+ cσ (0,0, t) . (33
When one reports the expressions of the Green's functions, it reduces to:

S T (Ω = 0) = e 2 Γ 2 0 2π 2 a 2 σ +∞ -∞ d t cos eV t × cosh 2 π 2 G θϕ,-+ cσ (0, 0, t) e 2πG ϕϕ,-+ σ (0,0, t) × e π 2 P j G φφ,-+ j (0,0, t)+G θθ,-+ j (0,0, t) , (34)
which leads to

S T (Ω = 0) = 2e 2 Γ 2 0 π 2 a 2 +∞ 0 d t cos eV t Re cosh √ πK c 4 sgn(1 + iω c t) sin ω Sc ω c (1 + iω c t) e -(ν+1) ln(1+iωc t) × e ( Kc 8 + 1 2 ) " 2 ln(1+iωc t)-e ω Sc t-i ω Sc ωc Ei(-ωSc t+i ω Sc ωc )+e -i ω Sc ωc Ei(i ω Sc ωc )-e -ω Sc t+i ω Sc ωc Ei(ωSc t-i ω Sc ωc )+e i ω Sc ωc Ei(-i ω Sc ωc ) « , (35)
where the summation over σ has been made assuming a non-magnetic STM tip.

In the top panel of Fig. 3, we plot the zero-frequency noise as a function of applied voltage for weak Coulomb interactions. The zero-frequency noise, which is linear in the absence of screening (solid line), follows a power law in the presence of screening, similarly to what we obtain for the tunnel current. For strong Coulomb interactions, the zero-frequency noise follows a power law whatever the screening strength is (bottom panel of Fig. 3). The effect of the screening is to reduce strongly the zero-frequency noise.

An interesting quantity to consider is the Fano factor defined as the ratio between the zero-frequency noise and the average current:

F ≡ S T (Ω = 0) | I T | . ( 36
)
This Fano factor is related to the charge of the carrier which is transferred trough the conductor: for example, F equals e for a junction between normal metals, whereas F equals 2e for a junction between a normal metal and a superconductor [START_REF] Lesovik | [END_REF] . In a two-dimensional electron gas in the fractional quantum Hall regime, simultaneous measurements of the current and noise at a constriction were performed and a Fano factor equals to e/3, which corre-sponds to the fractional charge, was obtained 13,14 .

Here, the Fano factor stays equal to the electron charge e even in the presence of screening. The reason is that only electron can tunnel from the tip to the wire and we end up with the standart Schottky relation 15 :

S T (Ω = 0) = e| I T | . (37)
At zero frequency, the non-symmetrized noise and the symmetrized noise are identical because of the time translation invariance. It is not more the case when one studies the finite-frequency noise 16,17 . In the next section, we focus our interest to the finite-frequency non-symmetrized noise S T (Ω), knowing that the symmetric noise can be deduced simply by taking (S T (Ω) + S T (-Ω))/2.

C. Finite-frequency non-symmetrized noise

Up to the second order with the tunnel amplitude Γ 0 , the finite-frequency non-symmetrized noise is:

S T (Ω) = e 2 Γ 2 0 (2πa) 2 r,σ +∞ -∞ d te iΩ t cos eV t e 2πG ϕϕ,-+ σ (0,0, t) × e π 2 P j G φφ,-+ j (0,0, t)+G θθ,-+ j (0,0, t)+rG φθ,ηη 1 j (0,0, t)+rG θφ,-+ j (0,0, t) × e - √ π 2 G φϕ,-+ cσ (0,0, t)+rG θϕ,-+ cσ (0,0, t)+G ϕφ,-+ cσ (0,0, t)+rG ϕθ,-+ cσ (0,0, t) . (38)
When one reports the expressions of the Green's functions, it reduces to:

S T (Ω) = e 2 Γ 2 0 2π 2 a 2 σ +∞ -∞ d te iΩ t cos eV t × cosh 2 π 2 G θϕ,-+ cσ (0, 0, t) e 2πG ϕϕ,-+ σ (0,0, t) × e π 2 P j G φφ,-+ j (0,0, t)+G θθ,-+ j (0,0, t) . (39)
Finally,

S T (Ω) = 2e 2 Γ 2 0 π 2 a 2 +∞ 0 d t cos eV t Re cosh √ πK c 4 sgn(1 + iω c t) sin ω Sc ω c (1 + iω c t) e iΩ t-(ν+1) ln(1+iωc t) × e ( Kc 8 + 1 2 ) " 2 ln(1+iωc t)-e ω Sc t-i ω Sc ωc Ei(-ωSc t+i ω Sc ωc )+e -i ω Sc ωc Ei(i ω Sc ωc )-e -ω Sc t+i ω Sc ωc Ei(ωSc t-i ω Sc ωc )+e i ω Sc ωc Ei(-i ω Sc ωc ) « . (40)
In experiments, we have access to the total noise which contains many contributions, such as zero-point fluctua-tions, 1/f noise 18 or thermal fluctuations 19,20 . If one Bottom panel: the same quantity in the presence of Coulomb interactions (Kc = 0.5). On both graphics, the screening strength is ωSc/ωc = 0 (solid line), ωSc/ωc = 0.0001 (dotted line), and ωSc/ωc = 0.0002 (dashed line), and the voltage value is eV / ωc = 0.001.

wants to isolate the contribution due to the applied voltage (shot noise), one has to look at the non-symmetrized excess noise defined as:

∆S T (Ω) ≡ S T (Ω) -S T (Ω)| V =0 . (41)
In the top panel of Fig. 4, we plot the non-symmetrized excess noise, divided by the tunnel current times the electron charge, in the absence of Coulomb interactions. In the unscreened case, the emission noise (Ω < 0) cancels when Ω < -eV / = -0.001, and the absorption noise (Ω > 0) cancels when Ω > eV / = 0.001 (solid line in the top panel of Fig. 4). In the screened case, the emission noise cancels when Ω < -eV / , whereas the absorption noise does not cancel any more when Ω > eV / (dotted and dashed lines in the top panel of Fig. 4).

In the presence of Coulomb interactions (bottom panel of Fig. 4), the non-symmetrized excess noise is always asymmetric, whatever the value of the screening strength is. From these graphics, we conclude that the Coulomb interactions inside the wire and the electrostatic potential due to the tip have a similar effect on the asymmetry of the finite-frequency excess noise. Moreover, the excess noise is symmetric only when both Coulomb interactions in the wire and screening by the STM tip are turned off.

D. Conductance

In Refs. 17 and 21, it has been shown that the difference between the absorption noise and the emission noise is related to the real part of the conductance:

S T (Ω) -S T (-Ω) = 2 ΩRe [G(Ω)] . (42)
As a consequence, the real part of the excess conductance can easily be obtained by taking the difference between the excess noise at positive frequency and the excess noise at negative frequency:

Re [∆G(Ω)] = ∆S T (Ω) -∆S T (-Ω) 2 Ω , (43)
where Ω > 0. 5: Top panel: real part of the excess conductance as a function of frequency, normalized by its value at zero frequency, in the absence of screening. Bottom panel: the same quantity in the presence of screening (ωSc/ωc = 0.0002). On both graphics, the Coulomb interactions is Kc = 0.9 (solid line), Kc = 0.7 (dotted line), and Kc = 0.5 (dashed line), and the voltage value is eV / ωc = 0.001.

In Fig. 5, we plot the real part of the excess conductance normalized by its value at zero frequency. The top panel of the Fig. 5 is obtained when screening is absent: we see that the excess conductance decreases rapidly and converges through its value in the absence of Coulomb interactions, which is zero. Indeed, in the absence of Coulomb interactions (i.e., for K c = 1), the current is linear with the voltage and as a consequence, the conductance is constant in voltage. Thus, the excess conductance is simply zero in this case. We also see a cusp when the frequency equals the voltage, here at Ω/ω c = eV / ω c = 0.001, which is attenuated when the Coulomb interactions strength increases.

The bottom panel of the Fig. 5 shows the excess conductance in the presence of screening. We observe a weaker decreasing at low frequency in comparison to the unscreened case. More interestingly, the excess conductance becomes almost independent of the Coulomb interactions strength. This is due to the fact that the effect of screening by the STM tip dominates over the effect of the Coulomb interactions in the quantum wire.

VI. CONCLUSION

In this paper, we have calculated the transport properties of a Luttinger liquid wire in the presence of electrostatic screening by a STM tip. Whereas the spectral properties of the wire depend only on the Green's functions of the wire 8 , the transport properties depend also on the Green's function of the tip and on new correlators that we call mixed Green's functions because they mix the bosonic fields of the wire with the bosonic field of the tip. These new correlators are non-zero due to the electrostatic tip-wire interaction. We have solved the Dyson equations associated to them in the case where the double derivative of the electrostatic potential can be approximated as a local interaction and we have calculated the tunnel current, the noise and the conductance.

We have shown that the tunnel current and the zerofrequency noise are strongly reduced in comparison to the unscreened situation. In addition, the I-V characteristic, which is linear in the absence of Coulomb interactions in the wire, follows a power law behavior when screening is turned on. We can conclude that the electrostatic interactions between the wire and the tip have a similar effect on the I-V curve than the electron-electron interactions inside the wire.

Next, we have studied the finite-frequency nonsymmetrized excess noise and we obtain an interesting result. Whereas in the absence of screening, the nonsymmetrized noise associated to a non-interacting wire is symmetric in frequency, it becomes asymmetric in the presence of screening. For an interacting wire, the nonsymmetrized excess noise is always asymmetric in frequency regardless of the screening strength is. The explanation of this asymmetric signal for a non-interacting wire is again the fact that the electrostatic potential between the wire and the tip has similar effects as electronelectron interactions. The subtle difference here is that the effective local density-density interaction which is induced by the presence of the tip contains retardation effects: they involve the dynamics of the tip (via its Green's function).

A useful tool to test the asymmetry of the excess noise is to calculate the excess conductance: indeed, when the excess noise is symmetric in frequency, the excess conductance is simply zero. We have calculated the excess conductance in different situations. In the absence of Coulomb interactions and screening, this quantity cancels. On the contrary, in the presence of Coulomb interactions, or of screening, the excess conductance does not cancel any more.

The results that we have obtained in this paper are preliminaries since we assume an infinite length wire. However, with the help of the Dyson equations that we have derived, it is now possible to calculate the transport properties for more realistic situations: in particular, in the presence of electrical contacts, at the extremities of the wire, which are needed to measure transport properties. We can estimate that current and noise will be affected by the contacts only for short length wire, due to finite size effects. Moreover, it would be interesting to calculate the current and its fluctuations in the wire, and not only the tunnel current and tunnel noise. It has been shown that the electrical contacts induce oscillating behavior of the finite-frequency noise in the wire due to reflections at the contacts 22,23 . The question of how these oscillations are modified by the screening, in the case where the electrostatic potential between the wire and the tip has a finite length extension, would also be interesting to consider.

FIG. 1 :

 1 FIG.1: Electron injection by a STM tip in a quantum wire. V is the applied voltage to the tip and Γ is the tunnel amplitude.

Similary

  's functions of the wire in the presence of the electrostatic coupling. And finally, G φϕ,ηη1 cσ , G θϕ,ηη1 cσ ,G ϕφ,ηη1 cσ and G ϕθ,ηη1 cσ are the mixed Keldysh Green's functions which are non-zero only in the presence of screening.

FIG. 2 :

 2 FIG.2: Top panel: tunnel current as a function of voltage, in unit of eΓ 2 0 /(avF 2 π 3 ), for Kc = 0.7. Bottom panel: tunnel current for Kc = 0.2. On both graphics, the screening strength is ωSc = 0 (solid line), ωSc/ωc = 0.001 (dotted line), and ωSc/ωc = 0.002 (dashed line).

)FIG. 3 :

 3 FIG.3: Top panel: zero-frequency noise as a function of voltage, in unit of e 2 Γ 2 0 /(avF 2 π 3 ), for Kc = 0.7. Bottom panel: zero-frequency noise for Kc = 0.2. On both graphics, the screening strength is ωSc = 0 (solid line), ωSc/ωc = 0.001 (dotted line), and ωSc/ωc = 0.002 (dashed line).

FIG. 4 :

 4 FIG.4: Top panel: non-symmetrized excess noise as a function of frequency, divided by the tunnel current times the electron charge, in the absence of Coulomb interactions (Kc = 1). Bottom panel: the same quantity in the presence of Coulomb interactions (Kc = 0.5). On both graphics, the screening strength is ωSc/ωc = 0 (solid line), ωSc/ωc = 0.0001 (dotted line), and ωSc/ωc = 0.0002 (dashed line), and the voltage value is eV / ωc = 0.001.

  FIG.5: Top panel: real part of the excess conductance as a function of frequency, normalized by its value at zero frequency, in the absence of screening. Bottom panel: the same quantity in the presence of screening (ωSc/ωc = 0.0002). On both graphics, the Coulomb interactions is Kc = 0.9 (solid line), Kc = 0.7 (dotted line), and Kc = 0.5 (dashed line), and the voltage value is eV / ωc = 0.001.

Z. Yao, H.W.Ch. Postma, L. Balents, and C. Dekker, Nature 402

, 273 (1999).2 C. Kane, L. Balents, and M.P.A. Fisher, Phys. Rev. Lett.

VII. ACKNOWLEDGMENTS

We thank D. Bercioux, F. Dolcini, H. Grabert, F. Hekking and I. Safi for valuable discussions.

APPENDIX A: EXPRESSIONS OF THE MIXED GREEN'S FUNCTIONS IN THE PRESENCE OF SCREENING

The starting point is the Dyson equation given by Eq. ( 18) that verifies the mixed Green's function G ϕθ jσ in imaginary time:

where G ϕϕ σ is the Green's function of the tip in the presence of screening, G -1 Sc (x 1 , y 1 ) = W 0 δ(x 1 )δ(y 1 )/(π √ 2π), and G θθ j is the Green's function of the wire in the absence of screening. Since we assume that the tunnel transfert takes place at x ′ = y = 0, we only need G ϕθ jσ (0, τ ; 0, τ ′ ) which obeys to:

Using the time translation invariance, we perform a Fourier transform and obtain:

G ϕϕ σ (0, 0, ω)G θθ j (0, 0, ω) .

(A3)

Reporting the expression of G ϕϕ σ which was determined in Ref. 8:

we obtain

whose Fourier transform is:

Finally, performing an analytic continuation τ → i t + τ 0 where τ 0 = 1/ω c , we get the Keldysh mixed Green's function

Next, we consider the Dyson equation verified by G θϕ jσ which is given by Eq. ( 19):

The resolution of the equation is similar to the previous one, it leads to:

Now, we look at the Dyson equation verified by G ϕφ jσ which is given by Eq. ( 20):

At positions x ′ = y = 0, it reduces to:

×G θφ j (0, τ 1 ; 0, τ ′ ) .

(A11) Since we have G θφ j (0, 0, ω) = 0 (see Appendix B), it leads to G θφ j (0, τ 1 ; 0, τ ′ ) = 0 and as a consequence:

Finally, we look at the Dyson equation for G φϕ jσ which is given by Eq. ( 21):

At positions x ′ = y = 0, it reduces to:

Since we have G φθ j (0, 0, ω) = 0 (see Appendix B), it leads to G φθ j (0, τ 1 ; 0, τ ′ ) = 0 and as a consequence:

APPENDIX B: EXPRESSIONS OF THE BARE GREEN'S FUNCTIONS

For an infinite length wire and in the absence of screening, the Matsubara bosonic Green's functions at zero temperature and at positions x = x ′ = 0 are:

G θθ j (0, 0, ω) =

G θφ j (0, 0, ω) = G φθ,0 j (0, 0, ω) = 0 . (B3)

The bosonic Green's function of the tip that we need in the calculation is the one located at y = y ′ = 0 which is given at zero temperature by: )


Marie-Marthe Groz
email: marie-marthe.groz@u-bordeaux.fr

Alain Sommier

Emmanuelle Abisset-Chavanne

Stéphane Chevalier

Anissa Meziane

Jean-Christophe Batsale

Christophe Pradere

Stephane Chevalier

Contactless thermal profilometry of carbon-resin materials by IR thermography

come

Contactless thermal profilometry of carbon-resin materials by IR thermography

Introduction

A profilometer is an instrument that is used to measure the surface condition, the relief of a surface, in order to evaluate its roughness or microgeometry. The two main devices currently used for profilometry are contact profilometers, based on the physical contact between a diamond tip and the surface to be measured [START_REF] Sayers | The little book of profiling: basic information about measuring and interpreting road profiles[END_REF], and contactless optical profilometers [START_REF] Whitefield | Noncontact optical profilometer[END_REF] that allow measuring the distance between the sensor and the surface being studied. The optical coherence tomography technique (OCT), for example, use the echo principle [3] and is widely employed in bio-medical domain and art conservation [4]. It is a very useful technique for sub-surface imaging but is limited in depth. The principle of the opto-acoustic imaging is another contactless optical technique. It is based on the photoacoustic effect. A non-ionizing laser pulse is delivered and in some media, a part of the laser energy is absorbed and converted into heat, leading to transient thermoelastic expansion and thus wideband ultrasonic emission [5,6]. This method is particularly well-suited for bio-medical media, because in the biological tissues the thermoelastic expansion is high enough to generate a measurable ultrasonic emission [7]. In this paper, a contactless thermal profilometer is proposed to be complementary to these techniques. The principle is schematized by figure 1. Concretely, this approach corresponds to measuring the interface between two media of a material. As already used in the pioneering work of Burgholzer et al. [8, 9, 10] and Mendioroz et al. [START_REF] Waters | 3d reconstruction of tilted cracks using infrared thermography and the virtual wave concept[END_REF]12,[START_REF] Mendioroz | Defect characterization from lock-in vibrothermography data[END_REF][START_REF] Mendioroz | Characterization and spatial resolution of cracks using lock-in vibrothermography[END_REF]15] for source shape reconstruction and super-resolution approaches, the principle of the method presented here is to create a buried point such as a heat source that becomes a source of energy in the material. From this heat generation, 3D transient thermal diffusion will occur into the sample, and the resulting surface temperature may be monitored using an IR camera. From these transient temperature fields, an inverse processing method based on the model presented by Groz et al. [START_REF] Groz | Bayesian inference for 3d volumetric heat sources reconstruction from surfacic ir imaging[END_REF][START_REF] Groz | Threedimensional reconstruction of thermal volumetric sources from surface temperature fields measured by infrared thermography[END_REF] is used. The complete 3D transient model can be adapted in order to find the position of the source and therefore the thickness between this point source and the surface of the material. The complete profile of the buried interface can then be measured with the flying spot setup proposed by [START_REF] Gaverina | Pulsed flying spot with the logarithmic parabolas method for the estimation of in-plane thermal diffusivity fields on heterogeneous and anisotropic materials[END_REF] and depicted in figure 2. This method enables simply repeating the overall process to elucidate the complete sample surface by laser scanning.

Contrary to the previous work of Burgholzer et al. and Mendioroz et al., the objective here is not to reconstruct the source or the shape of the buried sources but to estimate the exact depth of the source in order to perform contactless profilometry. Moreover, the idea is to use the separability of the impulse response [START_REF] Beck | Heat conduction using Green's function[END_REF] and to use the 1D transient problem (relative to the maximum 3D temperature field) as already depicted in the work of Holland [START_REF] Holland | Thermographic signal reconstruction for vibrothermography[END_REF].

Experimental setup

The utilized experimental setup illustrated in figure 2 was originally developed for in-plane thermal diffusivities mapping by Gaverina et al. [START_REF] Gaverina | Pulsed flying spot with the logarithmic parabolas method for the estimation of in-plane thermal diffusivity fields on heterogeneous and anisotropic materials[END_REF]. This consists of a laser diode (wavelength 976 nm) with 330 mW theoretical maximum power. To collimate the laser beam, an optical collimator system (Thorlabs) was used. A dual-axis scanning galvanometric system (Thorlabs GVS112/M) was used to control the spatial displacement of the laser spot. The principle of the laser beam deviation towards the sample is shown in figure 2. To focus the laser beam on the surface, an f-theta scan lens was used with a focusing length of 160 mm. With this focal length, the scanning area is equal to a square of 11 cm by 11 cm, and the resulting diameter of the focused spot is 26 µm, with a minimum displacement of 4.5 µm. In practice, the angular positions of the galvanometric system are related to a couple of coordinates expressed as voltage (V). This laser diode is mounted horizontally, and the beam is reflected with a dichroic mirror (MD, treated to reflect 95% of the visible light from 700 nm to 1000 nm and to transmit 95% of the infrared radiation between 2 and 16 µm). To measure the temperature fields, an MCT IR camera (FLIR SC7000, 320 by 256 pixels, pitch 30 µm, spectral band from 9 to 11 µm) was used with an infrared objective lens (focal length 25 mm). Finally, the resulting spatial resolution with this setup can be between 100 to 300 µm approximately [21]. In this paper, the spatial resolution was 320 µm.

The exposed surface of the studied sample can be of any geometry, but this topography must be known to compute the proposed method. Indeed, in each area of measure, the IR-Camera must be parallel to the studied surface, but the positioning of the sample in order to achieve it for each measure can be made automatically with the help of a robot [21].

In a bilayer, as illustrated by figure 1, the only real constraint is to have a laser with a wavelength that is semitransparent to the surface material layer of the sample (layer 1 in figure 1). On the other hand, the buried layer for which the profile will be measured must effectively absorb the incident source wavelength. One of the advantages of this technique is that it can work in media that are opaque to visible. In industrial fields, this system can be generalized in a wide range of applications. For example, in aeronautical domain, some composite are coated with a resin. Some materials, like the silicium, vastly used in electronic devices, are opaque to visible, but can be semi-transparent in the IR.

Different types of laser scanning can be used, as illustrated in figure 3. These examples are important because, as will be analyzed in section 3.2.1, one configuration could be better than another depending on the need. Finally, an important point is the ability of the system to be calibrated in space and time. Indeed, during the different scans shown in figure 3, the positions and times relative to each laser pulse 4/21 are recorded in a file that will be used in postprocessing. This crucial step allows for a perfect spatial and temporal localization of the laser sources.

3 Description of the profilometry method 3.1 Presentation of the method for a unique point source

The proposed inverse method is based on the method of source reconstruction proposed in [START_REF] Groz | Threedimensional reconstruction of thermal volumetric sources from surface temperature fields measured by infrared thermography[END_REF], which is based on the following 3D impulse thermal response at each point r = (x, y, z) at time t from a heat source located at r = (x , y , z ) delivering a pulse of heat quantity Q (J) at time t = t 0 in an orthotropic homogeneous medium:

θ Dirac (x, y, z, t) = Q (x , y , z ) ρC p exp -(x-x ) 2 4ax(t-t 0 ) 4πa x (t -t 0 ) exp -(y-y ) 2 4ay(t-t 0 ) 4πa y (t -t 0 ) exp -(z-z ) 2 4az(t-t 0 ) 4πa z (t -t 0 ) (1)
where a x , a y and a z (m 2 •s -1 ) are the diffusivities along each of the x, y or z directions of space, respectively. ρ (kg•m -3 ) is the density and C p (J•K -1 •kg -1 ) the specific heat of the material.

The method of source reconstruction presented does not involve any prior information on the sources, and in particular, it enables finding several sources in depth. However, in the case of thermal profilometry, it is known that there is only one source in depth due to the laser absorption at the interface between the semitransparent resin layer and the opaque composite layer. Thus, the goal is only to estimate the position of the source along the z-axis. With this assumption, the whole model of equation 1 can be simplified according to the schema of figure 1 to obtain the surface temperature resulting from the heat source located at (x , y , z ).

The laser impact, shown in figure 3 and schematized in figure 1, creates a heat point source pulsed at the interface between the laser-transparent layer and the absorbent layer. If there is only one impact on the sample, at the point (x , y , z ), the temperature at any point on the material is then given by equation 1. Since there is only one source point to be determined, the vectorial temperature (or thermogram) at the surface of the impact occurring at (x , y , z ) is given by:

θ x y 0 (t) = θ Dirac (x = x , y = y , z = 0, t) = Q 8ρC p exp -z 2 4az(t-t 0 ) a x a y a z (π(t -t 0 )) 3 (2)
It is noted that equation 2 is very close to the 1D impulse thermal response from a point source.

The main difference arises from the diffusions along the x and y directions that are taken into account, which comes from the fact that the model chosen is 3D. Equation 2 can then be written as:

5/21

3.2 Generalization for n point sources

-log a x a y a z (π(t -t 0 )) 3 θ x y 0 (t) = z 2 4a z (t -t 0 ) -log Q 8ρC p (3)
By writing the following variable changes

Y (t -t 0 ) = -log a x a y a z (π(t -t 0 )) 3 θ x y 0 (t)
and X(t -t 0 ) = 1/(t -t 0 ), equation 3 becomes a linear formulation, depicted by equation 4.

Y (t -t 0 ) = z 2 4a z X(t -t 0 ) -log Q 8ρC p (4)
By the knowledge of the thermal diffusivity along the z-direction and from the slope estimation, the depth z can be determined. This depth corresponds to the profile of the studied interface.

It is important to note that this proposed method requires the knowledge of the diffusivity of the studied layer. If this parameter is not given, it can be determined by front-or rear-face flash methods for istropic materials [22] and logarithmical parabolic method for orthotropic ones [START_REF] Gaverina | Pulsed flying spot with the logarithmic parabolas method for the estimation of in-plane thermal diffusivity fields on heterogeneous and anisotropic materials[END_REF].

It is noted that instead of just taking the temperature above the impact, one can also average (in space) the temperature at the surface of the material; equation 2 would then be slightly modified, but the inversion method would remain the same.

Generalization for n point sources

The objective of this part is to detail the inverse process used to robustly, automatically and rapidly estimate large quantities of data from the surface-measured IR films. Depending on the type of scanning, as illustrated in figure 3 and already modeled with respect to constant velocity in a previous paper [23], equation 1, available for only one point-like pulse thermal response, can be generalized as follows:

θ Scan (x, y, z, t) = t 0 Q (x , y , z ) ρC p exp -(x-x ) 2 4ax(t-τ ) 4πa x (t -τ ) exp -(y-y ) 2 4ay(t-τ ) 4πa y (t -τ ) exp -(z-z ) 2 4az(t-τ ) 4πa z (t -τ ) dτ (5)
To obtain this model, the first assumption is to assume that the sum of equation 5 is obtained with complete relaxation or sufficient distance or repetitive time between the laser pulses to ensure that the baseline is null so that no influence from the neighboring source can occur. A detailed study of the scanning parameters and influences on the simplified proposed method that may occur if this assumption is not verified is discussed in the following section.

Influence of the scanning parameters

In this section, it is important to study the influences of pulse spatial and temporal distances. As already discussed in our previous paper [START_REF] Groz | Bayesian inference for 3d volumetric heat sources reconstruction from surfacic ir imaging[END_REF], one way to overcome the source superposition problem is to apply the superposition principle by subtracting the first estimated source from the set of measurements. Here, this approach requires knowledge of both space-time positions (x , y , τ ) and trust in the position e(z ) estimated by the proposed method. In this study, considering the possibilities offered by laser scanning methods, it is proposed to study the alternatives to the superposition, especially the influence of the latter on the method of estimating the parameters of equation 4. The final objective is to provide a criterion guaranteeing good robustness. To this aim, and as shown in figure 4, the distance between two successive spots must be compared with the rise time of the temperature on the surface of the object and the τ (s) time between two impacts. To begin, let us study the influence of a unique impact on the whole sample. The thermal response measured at the surface, given by equation 2, depends on the depth of the point source correlated with the thermal properties and, in particular, on the diffusion coefficient a z of the studied layer. To take into account these two parameters, the Fourier number along the z direction, F o = a z t/[e(z)] 2 , is introduced [START_REF] Groz | Threedimensional reconstruction of thermal volumetric sources from surface temperature fields measured by infrared thermography[END_REF]. Then, for a given medium (with a given diffusivity), the deeper the source is, the smaller the F o parameter is. In order to see the influence of this parameter on the thermal response at the surface, the results obtained for one pulse centered in an image are presented in figure 5 Thus, as seen in figure 5, if the point source is just below the surface (small thickness of layer 1), the maximum temperature measured at the surface at time t max (time value corresponding to the maximum of temperature) ) is high and is quickly reached, whereas if the point source is deep (small Fourier number), the temperature elevation at the surface is low and takes longer to be reached, and the influence on the neighboring pixels is high. Thus, the higher the Fourier number is, the greater the influence of a source on its neighbors.

Regardless of the depth of the source, the diffusion of the heat in the entire layer causes a temperature increase at the surface of the neighboring points, particularly for high Fourier numbers.

Therefore, the study of the influences of the distance and time between several impacts must be conducted. Then, for a given sample (i.e., a fixed Fourier number), three different cases can occur:

• τ < t max : "fast" scanning

• τ ≈ t max : "moderate" scanning

• τ > t max : "slow" scanning

The example of 9 point sources located at the same depth (1 mm) in the layer and scanned in a square is studied here. Three constant scanning velocities are simulated, leading to three different time parameters τ between two impacts: τ = 0.1 s, τ = 1 s, and τ = 100 s. In this example, t max = 1.9 s and the thermal diffusivity was fixed at a z = a y = a x = 1.10 -7 m 2 .s -1 . The displacement of the laser is made from point 1 to point 9 in ascending order (figure 6.a) with a distance of 1.6 mm between two impacts. The dimensionless temperatures measured at the surface of each of the 9 points are illustrated in figures 6.b, 6.c and 6.d. For the three velocities used, the results enable the determination that the presence of several point sources influences the surface temperature. For "fast" scanning (τ = 0.1 s), it is found that three groups can be differentiated at long times (t > t max ). The temperature of point 5, which has 8 neighboring sources, is higher than that of the points. The temperatures of the point sources 1, 3, 7 and 9, which have "only" three neighboring sources each, are the lowest, and the temperatures of the point sources 2, 4, 6 and 8, each with 5 neighbors, are in the middle. The differences within groups are due to the laser displacement direction. The same observations can be made for "slow" scanning (τ = 100 s), but to a lesser extent. For "moderate" scanning (τ = 1 s), the influences of neighboring sources are important at long times, but they are also present at very short times due to thermal diffusion phenomena.

Thus, a methodology is proposed in the following part in order to estimate the depth of each source despite the presence and influence of the neighboring sources.

Methodology for parameter estimation

The objective of this part is to detail the inverse process for n point sources. To do so, the main idea is to use the maximum sensitivity of the depth pulse response and to use the same reasoning as that of the back face flash (cf. work of Parker [22] and Degiovanni [START_REF] Fercher | Optical coherence tomography-principles and applications[END_REF]). For each pixel (source), the measured temperature is then derived, and the point corresponding to the maximum of the derivative t M is retained as the upper limit to estimate the slope of equation 4. Thus, the inverse process can be performed within the time interval [t 0 , t M ], where t 0 is the time of the laser impact for the studied source. In the same way, to lessen the effect of neighboring sources at short times, the method can be performed within interval [t m , t M ], where t m is adjusted to be within the optimal area of sensitivity to the slope of equation 4. For all the time delays used (τ = 0.1, 1 and 100 s), one can find a linear behavior of Y (t).

As discussed previously, linearity is not present at very short times (high values of 1/t ) or at long times (small values of 1/t). Indeed, the influence of the neighboring pixels is strong both at short times (before the laser pulse) and at long times (heat rise of the neighboring pixels), particularly when τ = 1 s.

In light of the above, for each measure, the calculation of the statistical correlation between the magnitude Y (t) and X(t) of equation 4 can be performed to optimize and determine the best values of t m and t M .

The focus on the case where the time between two laser pulses is small in front of the rise time, as illustrated by figure 6.a, is made here. The statistical correlation between the magnitude Y ( Tab. 1: Estimated slope when applying the inverse processing and associated error.

In this case, the error of the estimated slope for each of the 9 pixels is very small (<0.2%).

Influence of Laser source absorption

The laser source is assumed to be collimated and totally absorbed at the interface. Nevertheless, the laser energy will be absorbed during its path through the studied layer according to the Beer-Lambert law:

Q(z) = I 0 (1 -R)e -βz (6)
where I 0 (W) is the incident optical power of the laser, β (m -1 ) is the absorption coefficient of the studied layer and R is the reflection coefficient.

The thermal response issued from equation 2 was calculated for the source as a function of z given by equation 6, as illustrated in figure 9.a. At short times, there is a significant difference between the theoretical model and the model with absorption. However, this difference quickly decreases and becomes negligible for long times. This result can be explained by the fact that at short times, the temperature due to shallow thermal sources is dominant over the temperature of the deepest (and most intense) sources. Indeed, although not very intense, these sources are close enough to the surface to manifest a significant temperature elevation. On the other hand, 11/21 at long times, sources close to the surface exert negligible action, and only the temperature due to the most intense source is significant. From the temperature in figure 9.a, the model of equation 4 can be calculated, as represented in figure 9.b, where we can clearly observe the effect of the absorption. In fact, for βz < 0.01, this influence occurs only at short times, and the linear behavior is reached after a few thousand ms. In contrast, for βz ≥ 0.1, the linear behavior is never reached. The interesting point here is the limiting nature of this study. Indeed, the study shows that for layers with βz < 0.01, it will be possible to apply the contactless profilometry proposed here. Another interesting point is that with the advent of IR laser diodes [25], we can consider this approach in environments that are completely opaque to visible light. Thus, the multispectral nature of these sources can also allow controlled absorption in heterogeneous multilayer media.

Validation of the proposed method 4.1 Validation on a numerical case

To illustrate the thermal profilometry method, a bilayer of two homogeneous materials has been modeled. The lower layer is composed of a rough material (carbon epoxy), whereas the upper layer is composed of a resin that is applied on the material to obtain a smooth surface. In this example, the interface is modeled by a 2D-sinus function, which translates the roughness of the material by the formation of bumps and hollows. The amplitude variation of the sinus is chosen as 1 mm. The thickness average of the resin layer is 1 mm; it therefore varies between 0.5 and 1.5 mm due to the roughness of the lower layer. Here, the resin is considered to be an isotropic material, with a diffusion coefficient given by a = 1.10 -7 m 2 .s -1 .

A laser will scan each pixel of this material (from bottom to top and from left to right) with a constant speed chosen such that 0 < τ < t max . The studied surface is a square with dimensions 5 cm by 5 cm. The lower layer is considered opaque, and the upper layer is semitransparent to the wavelength of the laser. The energy deposited at the interface between the two layers is therefore considered constant, with a value set here as 7.5 × 10 -8 J.m 2 . An IR camera located above the sample will measure the resulting temperature at the surface of the material. The spatial periodicity of the heat source in this numerical example is 1.6 mm along each directions.

The temperature data are computed in the first instance without the addition of noise. A study on the influence of the noise will be performed at the end of this section.

For each pixel of the surface, the proposed method, depicted in the previous sections, enables estimating the thickness of the material. Figure 10 illustrates the results obtained on the whole surface of the studied material. Figure 10.a illustrates the temperature measured at the surface during the process. The temperature evolution as a function of time is illustrated for five points on figure 10.b. Figure 10.c represents the maximum temperature amplitude for each pixel, and figure 10.d depicts its corresponding time. We can see that these data provide a first glimpse of the topography of the interface studied. Indeed, the highest temperature amplitudes correspond to small thicknesses, unlike "cold" points that correspond to greater thicknesses. We note that for the time of the maximum, the tendency is reversed: the temperature of the deepest points arrives later at the surface because of the greater distance to be covered. The gradient observed in figure 10.d is due to the laser scanning: the points furthest to the left are excited at short times by the laser, while those to the right are excited much later. It can be seen that the highest errors are made for the deepest sources, however, they are minimal. This difference between theory and the estimation for the deepest source can be explained by the diffusive nature of the temperature and the influence of the neighboring sources closer to the surface. The temperatures of these sources reach the surface faster than those of the deepest sources, and with much greater intensity, and diffuse in all directions.

Study on the influence of the noise

The numerical example and the results illustrated by figure 11 were performed in an ideal case where the measured temperatures were noiseless. In this part, the influence of the noise on the method is studied. The focus is made on five contiguous excitation positions on the previous sample, corresponding to different thicknesses, as illustrated by figure 12.a. Three different noise levels are studied (SNR = Signal to Noise Ratio) and compared:

• SNR = +∞ : it corresponds to the previous case without noise.

• SNR = 200 : it corresponds to a moderate noise level.

• SNR = 20 : it corresponds to a high noise level.

The implemented noise in this part of study is a "white noise", because it is the type of noise observed in thermal measurements. The SNR in thermal domain generally range between 10 to 100 000.

The five thermograms are illustrated in figure 12 For the five pixels, the same linear behaviour between Y and 1/t is retrieved, regardless of the noise level. The values of the corresponding thicknesses are then calculated and the absolute errors (in percentage) are given for each pixel in table 2. The errors are minimal (less than 2%), even in the worst case, with a very high noise level (SNR = 20). Tab. 2: Relative error (in percentage) when applying the inverse processing for the 5 adjacent spots with the three different noise levels depicted by figure 12.

The low sensitivity to the noise is due to the choice of the time interval of study [t m , t M ]. The estimation of the depth parameter is indeed always performed during the rise time of temperature. On this local area, the temperature variation is very high and therefore the sensitivity to noise is minimal.

For the setup presented in part 2 and through the use of the laser that can be very focalised, the noise level is very low and the SNR can easily go up to 1000. Thus, this example allows us to validate the proposed method and its ability to realize contactless thermal profilometry through a semitransparent layer added to an absorbent layer.

Application to Carbon-Resin composite materials

In this part, the results obtained on a carbon-resin composite pictured in figure 13 will be presented. Here, the studied composite exhibits groove-type defects in the resin-coated composite.

The measurements were carried out on the rectangle delimited by the adhesive tape, which is located between typical defects. This defect was a groove in the composite media, perpendicular to the surface of the material. As the surface of the bilayer (composite + resin) is uniform, the thickness of the upper material (resin) is therefore greater at the location of the defect. After the estimation by the proposed thermal profilometer, a destructive testing was performed on the sample at the location of the defect : the resin layer has been scrapped of and the defect was measured with a mecanical comparator. The thickness of the resin was theoretically 300 µm.

The objective here is to show that the thermal method proposed in this article allows us to detect and quantify the defect. To carry out these measurements, the laser diode presented in section 2 is used at full power with a very fast laser, with all pixels solicited. It should be recalled here that the experimental procedure undergoes a calibration phase that consists of converting the voltages (V) of the galvanometer mirrors into pixels. Then, the laser excitation is programmed in a loop on the number of pairs (i.e., pixels). For each pair, the laser power is given, the pair of voltages for the mirrors is determined, and a pause time is employed, which allows for control of the time interval between two impacts. The resulting time τ between two laser impacts was then 9.1×10 -4 s. The distance between two impact was 320 µm. The camera frequency rate was fixed to 500 Hz, and the final acquisition time was fixed at 30 s. The studied area forms a rectangle of dimensions 3.5 by 3.8 cm 2 . The diffusion coefficient of the resin was measured by using the classical rear-face flash method [22] and was found to be 5.2 × 10 -6 m 2 .s -1 . To illustrate the measurements, the temperature fields as well as the thermogram of several pixels are represented in figure 14 for several time steps. In the same way as for the synthetic case, the study of the maximum amplitudes and their corresponding times allows for a first glimpse of the studied profile. The two figures 14.a, 14.b enable observation of the laser displacement with increasing temperature. In particular, in figures 14.b and 14.c, the presence of a heterogeneity/crack in the carbon layer is indeed evident because it results in lower amplitudes of temperature and rising time.

Thus, at each point (pixel) of the surface, the thickness of the resin layer is calculated using the method proposed here, and illustrated in figure 14.d for the pixels 2, 3 and 4. Figure 15 shows the thickness (profile of fields) found at each point. The profile of the studied material is effectively retrieved in just a couple of minutes: the average thickness of the upper layer is 300 µm, except at the place of the crack, where the thickness is almost doubled. It can also be noted that the roughness of the carbon is retrieved, even where there is no defect.

The computer used is a laptop with 16 Go of RAM with a 3.1 GHz Intel Core i7 processor. However, any laptop is enough to execute this methodology because it does not require a special computing power.

In conclusion, the method of contactless thermal profilometry presented in this paper is fast and robust with respect to noise. This approach enables estimation of the profile of a material with accuracy.

Conclusion

In this paper, a method of contactless thermal profilometry is proposed. This approach enables estimation of the relief of different surfaces, which can be the interfaces between two media of a material. The proposed method is performed in real time, is very insensitive to noise and is therefore robust.

Fig. 1 :

 1 Fig. 1: Scheme of the contactless thermal profilometry principle

Fig. 2 :

 2 Fig. 2: a) Schematic representation of the flying spot experimental setup and b) photograph of the bench.

Fig. 3 :

 3 Fig. 3: Example of the temperature field measured after different types of laser scanning: a) pulsed grid, b) random and c) constant velocity.

Fig. 4 :

 4 Fig. 4: Schema of the superposition principle with the important parameters

Fig. 5 :

 5 Fig. 5: Results obtained for one pulse centered in an image for four different Fourier number. The data are presented along the normalized x-direction for the line located at y = 0: a) normalized time corresponding to the maximum value of temperature b) value of the maximum temperature.

Fig. 6 :

 6 Fig. 6: Example of scanning parameter influences on the constant velocity flying spot technique. a) Schema of the 9 point sources located at the same depth in the layer. Dimensionless temperature T * for different τ for the 9 pixels: b) τ = 0.1 s, c) τ = 1 s, d) τ = 100 s.

ForFig. 7 :

 7 Fig. 7: Example of scanning parameter influences on the constant velocity flying spot technique. Representation of the expression of equation 4 for several τ values equal to: a) 0.1 s, b) 1 s and c) 100 s.

Fig. 8 :

 8 Fig. 8: Presentation of inverse processing principle: a) derivative of the temperature response illustrated in figure 6.a, b) calculated correlation from t 0 to the time corresponding to the maximum sensitivity and c) linear evolution of the temperature calculated with equation 4 and estimated slope in the range of correlation close to 1.

Fig. 9 :

 9 Fig. 9: a) Normalized front-face temperature as a function of several incident laser source absorptions in semitransparent resin and b) inverse of the Y (t) expression of equation 4 as a function of time.

Fig. 10 Fig. 11 :

 1011 Fig. 10: a) Image of the temperature at the surface during the process, b) temperature as function of time for the five pixels highlighted on figure (a), c) maximum temperature values at each point of the surface and d) time corresponding with the maximum temperature value at each point of the surface.

Fig. 12 :

 12 Fig. 12: Study on the influence of noise. a) Focus on five contiguous points corresponding to different thicknesses. Resulting temperatures at the surface for the five points for : b) no noise (SNR = +∞), c) a moderate noise level (SNR = 200) and d) a high noise level (SNR = 20). Y as a function of 1/t on the time interval [t m , t M ] and their fit for each of the five points : e) for SNR = +∞, f) for SNR = 200 and g) for SNR = 20.

Fig. 13 :

 13 Fig. 13: Photography of the standard composite used in this study

Fig. 14 :

 14 Fig. 14: Measured temperature fields as function of time: a) time step t = 2 s, b) time step t = 5.8 s. c) Extracted temperature as function of time for the pixels reported in figure 14.a. and d) Y as function of 1/t for the pixels 2, 3 and 4.

Fig. 15 :

 15 Fig. 15: Estimated fields of carbon profilometry (i.e., resin thickness): a) 2D mapping and b) 3D mapping.

Acknowledgments

The authors acknowledge Epsilon Groupe Alcen for this study.