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Température effective d’un système hors équilibre : fluctuations thermiques d’un microlevier soumis à un flux de chaleur

Mickael Geitner, PhD Thesis, ENS de Lyon (2015)

hal: tel-01235762

A l’aide d’un interféromètre différentiel à quadrature de phase nous mesurons les fluctuations thermiques de la déflexion d’un micro-levier. Il est alors possible de déduire différentes propriétés mécaniques du levier telles que raideur, fréquences de résonance, facteurs de qualité etc. Dans un tel système, la précision maximale sur les mesures est limitée par le bruit de grenaille des photodiodes (shot-noise). Afin d’augmenter le rapport signal sur bruit, nous augmentons l’intensité lumineuse du laser de mesure, diminuant ainsi le bruit de fond des spectres de fluctuations thermique. En revanche, l’augmentation de l’intensité du laser a pour effet de décaler vers les basses fréquences les résonances du levier. Une première partie de ce travail de thèse a pour objectif la compréhension de ce phénomène. Ainsi, nous associons le décalage en fréquence à un échauffement du levier par le laser de l’interféromètre et au flux de chaleur associé le long du levier. Nous développons alors un modèle permettant de relier cet effet à la température de l’extrémité du levier en se basant sur un profil de température linéaire. Une seconde partie de ce travail vise à mesurer la température effective d’un levier à l’aide d’une extension du théorème fluctuation-dissipation. Nous montrons que les fluctuations de ce système hors équilibre sont plus faibles que celles attendues compte tenu du profil de température. Nous cherchons alors à identifier l’origine de ce déficit de fluctuations. Dans une dernière partie nous estimons les profils de température sur des leviers en faisant varier leurs paramètres géométriques ou leur coefficient d’absorption, ainsi que la position du laser chauffant le levier.

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Resonance frequency shift of strongly heated micro-cantilevers

Felipe Aguilar Sandoval, Mickael Geitner, Éric Bertin, and Ludovic Bellon, Journal of Applied Physics 117, 234503 (2015)

doi: 10.1063/1.4922785

In optical detection setups to measure the deflection of micro-cantilevers, part of the sensing light is absorbed, heating the mechanical probe. We present experimental evidences of a frequency shift of the resonant modes of a cantilever when the light power of the optical measurement set-up is increased. This frequency shift is a signature of the temperature rise and presents a dependence on the mode number. An analytical model is derived to take into account the temperature profile along the cantilever; it shows that the frequency shifts are given by an average of the profile weighted by the local curvature for each resonant mode. We apply this framework to measurements in vacuum and demonstrate that huge temperatures can be reached with moderate light intensities: a 1000 °C with little more than 10 mW. We finally present some insight into the physical phenomena when the cantilever is in air instead of vacuum.

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Adhesion energy of single wall carbon nanotube loops on various substrates

Tianjun Li, Anthony Ayari and Ludovic Bellon,  Journal of Applied Physics 117, 164309 (2015)

doi:10.1063/1.4919355

The physics of adhesion of one-dimensional nano structures such as nanotubes, nano wires, and biopolymers on different substrates is of great interest for the study of biological adhesion and the development of nano electronics and nano mechanics. In this paper, we present force spectroscopy experiments of individual single wall carbon nanotube loops using a home-made interferometric atomic force microscope. Characteristic force plateaus during the peeling process allow the quantitative measurement of the adhesion energy per unit length on various substrates: graphite, mica, platinum, gold, and silicon. Moreover, using a time-frequency analysis of the deflection of the cantilever, we estimate the dynamic stiffness of the contact, providing more information on the nanotube configurations and its intrinsic mechanical properties.

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Simultaneous and accurate measurement of the dielectric constant at many frequencies spanning a wide range

Roberto Pérez-Aparicio, Caroline Crauste-Thibierge, Marius Tanase, Pascal Metz, Ludovic Bellon, Antoine Naert, and Sergio Ciliberto,  Rev. Sci. Instrum. 86, 044702 (2015)

doi: 10.1063/1.4916260

We present an innovative technique which allows the simultaneous measurement of the dielectric constant of a material at many frequencies, spanning a four orders of magnitude range chosen between 10-2 Hz and 104 Hz. The sensitivity and accuracy are comparable to those obtained using standard single frequency techniques. The technique is based on three new and simple features: a) the precise real time correction of the amplification of a current amplifier; b) the specific shape of the excitation signal and its frequency spectrum; and c) the precise synchronization between the generation of the excitation signal and the acquisition of the dielectric response signal. This technique is useful in the case of relatively fast dynamical measurements when the knowledge of the time evolution of the dielectric constant is needed.

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Progress and challenges in advanced ground-based gravitational-wave detectors

M. Adier, F. Aguilar et al.,  General Relativity and Gravitation 46, 1749 (2014)

doi: 10.1007/s10714-014-1749-4

The Amaldi 10 Parallel Session C3 on Advanced Gravitational Wave detectors gave an overview of the status and several specific challenges and solutions relevant to the instruments planned for a mid-decade start of observation. Invited overview talks for the Virgo, LIGO, and KAGRA instruments were complemented by more detailed discussions in presentations and posters of some instrument features and designs. This article is a collection of the abstracts of this session, including the presentation by my collaborator Massimo Granata :

Results from Raman spectroscopy and direct thermal noise measurements on tantala and silica coatings (Presenter: M. Granata)

In order to isolate the mechanisms behind thermal-noise fluctuations in optical coatings, the Laboratoire des Matériaux Avancés has started a collaboration with the Institut Lumière Matière to investigate the Raman spectra of ion-beam sputtered tantala samples with different annealing history. Work is presently ongoing to understand the observed behaviours, in order to correlate the mechanical loss to the evolutions of the Raman spectra. The same analysis will be shortly carried out on fused silica coatings too.

A novel technique of direct thermal noise measurements, developed by the École Normale Supérieure de Lyon, is presented. Measured samples are tipless cantilevers with ion-beam sputtered coatings of tantala or fused silica, annealed before and after the coating. In all cases, the power spectral density of the measured noise is inversely proportional to the frequency, as predicted by the structural noise model with constant loss.

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Mode coupling in a hanging-fiber AFM used as a rheological probe

C. Devailly, J. Laurent, A. Steinberger, L. Bellon, S. Ciliberto,  EPL 106, 54005 (2014)

doi: 10.1209/0295-5075/106/54005

We analyze the advantages and drawbacks of a method which measures the viscosity of liquids at microscales, using a thin glass fiber fixed on the tip of a cantilever of an ultra-low-noise Atomic Force Microscope (AFM). When the fiber is dipped into a liquid, the dissipation of the cantilever-fiber system, which is linked to the liquid viscosity, can be computed from the power spectral density of the thermal fluctuations of the cantilever deflection. The high sensitivity of the AFM allows us to show the existence and to develop a model of the coupling between the dynamics of the fiber and that of the cantilever. This model, which accurately fits the experimental data, gives also more insights into the dynamics of coupled microdevices in a viscous environment.

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Measurements of mechanical thermal noise and energy dissipation in optical dielectric coatings

Tianjun Li, Felipe A. Aguilar Sandoval, Mickael Geitner, Gianpietro Cagnoli, Vincent Dolique, Jérôme Degallaix, Raffaele Flaminio, Danièle Forest, Massimo Granata, Christophe Michel, Nazario Morgado, Laurent Pinard, and Ludovic Bellon,  Phys. Rev. D 89, 092004 (2014)

doi: 10.1103/PhysRevD.89.092004

In recent years, an increasing number of devices and experiments are shown to be limited by mechanical thermal noise. In particular, subhertz laser frequency stabilization and gravitational wave detectors that are able to measure fluctuations of 10-18 m/rtHz or less are being limited by thermal noise in the dielectric coatings deposited on mirrors. In this paper, we present a new measurement of thermal noise in low absorption dielectric coatings deposited on microcantilevers, and we compare it with the results obtained from the mechanical loss measurements. The coating thermal noise is measured on the widest range of frequencies with the highest signal-to-noise ratio ever achieved. In addition, we present a novel technique to deduce the coating mechanical losses from the measurement of the mechanical quality factor which does not rely on the knowledge of the coating and substrate Young’s moduli. The dielectric coatings are deposited by ion beam sputtering. The results presented here give a frequency-independent loss angle of (4.7±0.2)x10-4 with a Young’s modulus of 118 GPa for annealed tantala from 10 Hz to 20 kHz. For as- deposited silica, a weak frequency dependence (~ f-0.025) is observed in this frequency range, with a Young’s modulus of 70 GPa and an internal damping of (6.0±0.3)x10-4 at 16 kHz, but this value decreases by one order of magnitude after annealing, and the frequency dependence disappears.

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Adhesion and dissipation at nanoscale

Tianjun Li, PhD Thesis, École Normale Supérieure de Lyon & East China Normal University – Shanghai (2013)

hal: tel-00907812

In this thesis, we test some interactions involving surfaces processes at the nanometer scale. The experiments are conducted with a highly sensitive interferometric Atomic Force Microscope (AFM), achieving a resolution down to 10-28 m2/Hz for the measurement of deflection. Combined with original thermal noise analysis, this tool allows quantitative characterization of the mechanical response of micrometer and nanometer sized systems, such as microcantilevers or carbon nanotubes, on a large frequency range.The first part of my work deals with the viscoelasticity of the coating of AFM cantilevers. Evidenced by a 1/f thermal noise at low frequency, this phenomenon is present when a cantilever is coated with a metallic layer (gold, aluminium, platinium, etc…). Using the fluctuation dissipation theorem and Kramers Kronig relations, we extract the frequency dependance of this viscoelastic damping on a wide range of frequency (1 Hz to 20 kHz). We find a generic power law dependence in frequency for this dissipation process, with a small negative coefficient that depends on materials. The amplitude of this phenomenon is shown to be linear in the coating thickness, demonstrating that the damping mechanism takes its roots in the bulk of the metallic layer.The second part of my work tackles new experiments on the interaction of carbon nanotubes with flat surfaces. Using our AFM, we perform a true mechanical response measurement of the rigidity and dissipation of the contact between the nanotube and the surface, in a peeling configuration (the nanotube is partially absorbed to the substrate). The results of this protocol are in line with the dynamic stiffness deduced from the thermal noise analysis, showing an unexpected power law dependence in frequency for the contact stiffness. We suggest some possible physical origins to explain this behavior, such as an amorphous carbon layer around the nanotube.

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Interferometría de alta resolución para AFM

Felipe Aguilar Sandoval, PhD Thesis, Universidad de Santiago de Chile (2013)

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El manuscrito se compone de cinco capítulos. Después de una introducción corta al AFM y el objetivo del trabajo, el primer capítulo describe el principio de funcionamiento del dispositivo interferométrico de cuadratura de fase, su diseño y implementación. Destaca como el instrumento permite medir deflexiones importantes (hasta varias veces la longitud de onda del láser utilizado), con una sensibilidad constante.

El segundo capítulo se enfoca en la descripción del comportamiento mecánico del “microcantilever”, y el ruido térmico de éste: las fluctuaciones de deflexión debidas a la temperatura del sistema se pueden medir con buena precisión usando el interferómetro, ofreciendo un método de calibración de su rigidez. En este capitulo, Felipe vuelve a las bases de la descripción de una viga empotrada con el modelo de Euler-Bernoulli, demostrando experimentalmente la validez de este modelo para el microcantilever, y luego desarrolla una demostración del teorema de fluctuación-disipación por este sistema, partiendo de la equipartición de la energía.  Finalmente, usando el modelo de Sader para describir la amortiguación del sistema, logra tener una descripción completa del ruido térmico del “microcantilever”.

En el tercer capítulo, se explica como se puede acondicionar las salidas del interferómetro a cuadratura de fase para usar lo en un dispositivo de microscopia de sonda de barrido clásico. De hecho, el interferómetro tiene dos salidas que permite seguir la deflexión en un rango amplio con sensibilidad constante, cuando se necesita una sola señal de retroalimentación para hacer imagines en modo contacto o dinámico con un AFM. Un dispositivo analógica ha sido desarrollado para esa tarea, y Felipe lo caracteriza ampliamente, demostrando su capacitad y limitaciones.

El cuarto capitulo se enfoca en entender las varias fuentes de ruido del sistema de detección, demostrando que la mayor parte se debe al ruido de disparo o “shot noise”, que se traduce por un ruido blanco del orden de 10-14 m/√Hz en los espectros de fluctuaciones de la deflexión con el láser He-Ne inicial. Después de una derivación de la expresión de este “shot noise”, Felipe concluye que el ruido de detección del interferómetro se puede mejorar subiendo la potencia del láser y bajando su longitud de onda. Usando un láser de estado solido verde de potencia 40 veces mayor que la del He-Ne, logra bajar el ruido de fondo de un orden de magnitud, llegando a una resolución del orden del femtómetro por raíz Hertz.

En un último capítulo, Felipe Aguilar estudia como la mayor potencia de iluminación afecta la respuesta del “cantiléver”. De hecho, se detecta un corrimiento de sus frecuencias de resonancia a medida que sube la potencia del láser. Ese efecto se entiende por la subida de la temperatura del cantilever que absorbe una parte de la luz enfocada en su extremo. Para entender como ese efecto depende del modo considerado, Felipe desarrolla una análisis completo del fenómeno considerando el efecto de un perfil de temperatura sobre las ecuaciones de Euler-Bernoulli. Este modelo describe con buen éxito las observaciones experimentales.

Al final del manuscrito, una conclusión corta resuma el trabajo y abre unas perspectivas a explorar para el futuro.

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Quadrature phase interferometer for high resolution force spectroscopy

Pierdomenico Paolino, Felipe A. Aguilar Sandoval and Ludovic Bellon,  Rev. Sci. Instrum. 84, 095001 (2013)

doi: 10.1063/1.4819743

In this article, we present a deflection measurement setup for Atomic Force Microscopy (AFM). It is based on a quadrature phase differential interferometer: we measure the optical path difference between a laser beam reflecting above the cantilever tip and a reference beam reflecting on the static base of the sensor. A design with very low environmental susceptibility and another allowing cal- ibrated measurements on a wide spectral range are described. Both enable a very high resolution (down to 2.5 × 10-15 m/√Hz), illustrated by thermal noise measurements on AFM cantilevers. They present an excellent long-term stability and a constant sensitivity independent of the optical phase of the interferometer. A quick review shows that our precision is equaling or out-performing the best results reported in the literature, but for a much larger deflection range, up to a few μm.

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