Thermal Noise Spectrum Research Articles

Implications of in-ice volume scattering for radio-frequency neutrino experiments

Over the last three decades, several experimental initiatives have been launched with the goal of observing radio-frequency signals produced by ultra-high energy neutrinos (UHEN) interacting in solid media. Observed neutrino event signatures comprise impulsive signals with duration of order the inverse of the antenna+system bandwidth (∼10 ns), superimposed upon an incoherent (typically white noise) thermal noise spectrum. Whereas bulk volume scattering (VS) of radio-frequency (RF) signals is well-studied within the radio-glaciological communities, polar ice-based neutrino-detection experiments have thus far neglected VS in their signal projections. As discussed herein, coherent volume scattering (CVS, for which the phase of the incident signal is preserved during scattering) generated by in-ice neutrino interactions may similarly produce short-duration signal-like power, albeit with a slightly extended time structure, and thereby enhance neutrino detection rates, whereas incoherent (randomized phase) volume scattering (IVS) will persist for O(100 ns), appearing similar to thermal white noise and therefore reducing the measured Signal-to-Noise Ratio (SNR) of neutrino signals. Herein, we present the expected voltage profiles resulting from in-ice volume scattering as a function of the molecular scattering cross-section, for both CVS and IVS, and assess their impact on UHEN experiments. VS contributions are currently only weakly constrained by extant data; stronger limits may be obtained with dedicated calibration experiments.

Observation of thermal acoustic modes of a droplet coupled to an optomechanical sensor

The bulk acoustic modes of liquid droplets, well understood from a theoretical perspective, have rarely been observed experimentally. Here, we report the indirect observation of acoustic vibrational modes in a picoliter-scale droplet, extending up to ∼40 MHz. This was achieved by coupling the droplet to an ultra-sensitive optomechanical sensor, which operates in a thermal-noise limited regime and with a substantial contribution from acoustic noise in the ambient medium. The droplet vibrational modes manifest as Fano resonances in the thermal noise spectrum of the sensor. This is among the few reported observations of droplet acoustic modes and of Fano interactions in a coupled mechanical oscillator system driven only by thermal Brownian motion.

Monitoring magnetic nanoparticle clustering and immobilization with thermal noise magnetometry using optically pumped magnetometers.

Thermal noise magnetometry (TNM) is a recently developed magnetic characterization technique where thermally induced fluctuations in magnetization are measured to gain insight into nanomagnetic structures like magnetic nanoparticles (MNPs). Due to the stochastic nature of the method, its signal amplitude scales with the square of the volume of the individual fluctuators, which makes the method therefore extra attractive to study MNP clustering and aggregation processes. Until now, TNM signals have exclusively been detected by using a superconducting quantum interference device (SQUID) sensor. In contrast, we present here a tabletop setup using optically pumped magnetometers (OPMs) in a compact magnetic shield, as a flexible alternative. The agreement between results obtained with both measurement systems is shown for different commercially available MNP samples. We argue that the OPM setup with low complexity complements the SQUID setup with high sensitivity and bandwidth. Furthermore, the OPM tabletop setup is well suited to monitor aggregation processes because of its excellent sensitivity in lower frequencies. As a proof of concept, we show the changes in the noise spectrum for three different MNP immobilization and clustering processes. From our results, we conclude that the tabletop setup offers a flexible and widely adoptable measurement unit to monitor the immobilization, aggregation, and clustering of MNPs for different applications, including interactions of the particles with biological systems and the long-term stability of magnetic samples.

Characterization of Dry-Contact EEG Electrodes and an Empirical Comparison of Ag/AgCl and IrO2 Electrodes.

Dry-contact electrodes are increasingly being used for EEG recordings in both research studies and consumer products. They are more user-friendly and better suited for long-term recordings. However, dry-contact electrodes also bring challenges with respect to the stability and impedance of the electrode-skin interface. We propose a methodology to characterize and compare dry-contact electrodes. The characterization is based on measuring the electrode-skin impedance spectrum, fit a parametric model of the electrode-skin interface to the measured spectrum, and calculate the resulting thermal noise spectrum. Thereby it is possible to relate the noise of an EEG recording to the theoretical noise contribution from the electrode-skin interface. To demonstrate the methodology, we performed an empirical study comparing two types of dry-contact electrodes in an ear-EEG setup. The electrodes were IrO2, previously used for ear-EEG, and a new design based on Ag/AgCl. Here, we related the noise floor of an auditory steady-state response (ASSR) to the thermal noise spectrum of the electrode-skin interface. The study showed similar impedance and EEG recording quality for the two electrode types, and the thermal noise of the electrode-skin interface was below the noise floor of the EEG recordings for both electrode types. Dry-contact EEG is an enabling technology for long-term brain monitoring of patients. This may be relevant for example for monitoring of neurodegenerative diseases, stroke patients, patients with traumatic brain injuries, and psychiatric patients.

Thermal Fluctuations and Electromagnetic Noise Spectra in Quantum Statistical Mechanics

We derive the thermal noise spectrum of the longitudinal and transverse electric field operator of a given wave vector starting from the quantum-statistical definitions and relate it to the frequency and wave vector dependent complex conductivity in a homogeneous, isotropic system of electromagnetic interacting charged particles in the frame of the non-relativistic QED. No additional assumptions except the validity of linear response are used in the proof. The Nyquist formula for vanishing frequency, as well as the noise spectral density of Callen-Welton follow as byproduct. Furthermore we discuss also the noise of the photon occupation numbers.

Effect of Bulk Acoustic Wave in Piezo Driven Nanomechanical Motion

Piezo actuation of mechanical resonators is widely adapted because of its simplicity and versatility. Piezo-driven atomic force microscopy cantilevers in air or liquid have a substantial drawback in that they produce spurious resonances that conceal the cantilever resonance peak. Bulk acoustic wave propagation via the piezo-shaker and device substrate causes these undesired peaks. Such restrictions of piezo actuation are rarely reported in nanomechanical resonant sensing. Because most NEMS (nanoelectromechanical systems) experiments are carried out at low pressure to achieve a higher quality factor ) and hence increased sensitivity, spurious resonances are frequently overlooked due to their insignificance. However, this piezo-driven issue may affect NEMS responses at higher pressures (lower Q) and must be addressed carefully. This study reveals spurious resonances from high vacuum to the atmosphere while investigating piezo-driven nanoscale doubly clamped beam responses. At all pressures, spurious peaks with a characteristic frequency span independent of air damping exist, and at higher pressures, they squeeze the mechanical peak. Such squeezing provides a larger derived from the driven phase responses by order of magnitude than the mechanical computed from the measured thermal noise spectra. Interestingly, the characteristic frequency span, not air damping, is revealed to dominate driven .

Quantum Backaction Cancellation in the Audio Band

We report on the cancellation of quantum back action noise in an optomechanical cavity. We perform two measurements of the displacement of the microresonator, one in reflection of the cavity, and one in transmission of the cavity. We show that measuring the amplitude quadrature of the light in transmission of the optomechanical cavity allows us to cancel the back action noise between 1 kHz and 50 kHz, and obtain a more sensitive measurement of the microresonator's position. To confirm that the back action is eliminated, we measure the noise in the transmission signal as a function of circulating power. By splitting the transmitted light onto two photodetectors and cross correlating the two signals, we remove the contributon from shot noise and measure a quantum noise free thermal noise spectrum. Eliminating the effects of back action in this frequency regime is an important demonstration of a technique that could be used to mitigate the effects of back action in interferometric gravitational wave detectors such as Advanced LIGO.

Calibration of T-shaped atomic force microscope cantilevers using the thermal noise method.

The tip-sample interaction force measurements in atomic force microscopy (AFM) provide information about materials' properties with nanoscale resolution. The T-shaped cantilevers used in Torsional-Harmonic AFM allow measuring the rapidly changing tip-sample interaction forces using the torsional (twisting) deflections of the cantilever due to the off-axis placement of the sharp tip. However, it has been difficult to calibrate these cantilevers using the commonly used thermal noise-based calibration method as the mechanical coupling between flexural and torsional deflections makes it challenging to determine the deflection sensitivities from force-distance curves. Here, we present thermal noise-based calibration of these T-shaped AFM cantilevers by simultaneously analyzing flexural and torsional thermal noise spectra, along with deflection signals during a force-distance curve measurement. The calibration steps remain identical to the conventional thermal noise method, but a computer performs additional calculations to account for mode coupling. We demonstrate the robustness of the calibration method by determining the sensitivity of calibration results to the laser spot position on the cantilever, to the orientation of the cantilever in the cantilever holder, and by repeated measurements. We validated the quantitative force measurements against the known unfolding force of a protein, the I91 domain of titin, which resulted in consistent unfolding force values among six independently calibrated cantilevers.

Wide-frequency band measurement and analysis of electrochemical noise of Li/MnO2 primary battery

The electrochemical noise of a commercially produced Li/MnO2 primary battery was measured during discharge via a constant value resistor. Power spectral density frequency dependences were calculated for different state of charge values across the frequency band from 0.02 to 5 kHz. It was shown that they possess close to linear frequency dependences. For high state of charge values, an intersection with the thermal noise spectrum was observed in a high-frequency band. The electrochemical impedance of the investigated type of battery was measured at different charge values both during discharge and following a relaxation pause. Impedance spectra obtained under load conditions were used to model the electrochemical noise spectra. It was shown that impedance parameters can be used to describe the electrochemical noise spectra in the frequency band above 500 Hz at state of charge values higher than 50%. Dependencies of electrochemical noise amplitude on discharge current value were investigated at different states of charge. It was shown that power spectral density has a linear dependency on discharge current at a power of about 2 for a charged battery and 2.5 for a discharged battery.

Calibration of the oscillation amplitude of electrically excited scanning probe microscopy sensors.

Atomic force microscopy (AFM) is an analytical surface characterization tool which can reveal a sample's topography with high spatial resolution while simultaneously probing tip-sample interactions. Local measurement of chemical properties with high-resolution has gained much popularity in recent years with advances in dynamic AFM methodologies. A calibration factor is required to convert the electrical readout to a mechanical oscillation amplitude in order to extract quantitative information about the surface. We propose a new calibration technique for the oscillation amplitude of electrically driven probes using the principle of energy balance. Our technique relies on the measurement of the energy input to maintain the oscillation amplitude constant. With the measurement of the energy input to the probe, a mechanical oscillation amplitude is calculated and a calibration factor to convert the electrical readout in volts to a mechanical oscillation amplitude in Ångströms is obtained. We demonstrate the application of the new technique with a quartz tuning fork including the qPlus configuration, while the same principle can be applied to other piezoelectric resonators such as length extension resonators or piezoelectric cantilevers. The calibration factor obtained by this technique is found to be in agreement with using the thermal noise spectrum method for capsulated and decapsulated tuning forks and tuning forks in the qPlus configuration.

Cross-correlation limit of a SQUID-based noise thermometer of the pMFFT type

The primary magnetic field fluctuation thermometer (pMFFT) is a SQUID-based noise thermometer for temperatures below 1 K, which complies with metrological requirements. It combines two signal channels in order to apply the cross-correlation technique, but it requires statistically independent noise signals for proper operation. In order to check the limit of the cross-correlation readout, we have performed zero measurements in the millikelvin range in a setup that is identical to the pMFFT, except for the removed temperature sensor. We examined the influence of different parameters such as SQUID working point or flux-lock loop parameters on the minimum cross-correlation signal down to 24 mK and below 100 kHz. Depending on the configuration, typical minimum SQUID-referred cross-power spectral densities of 1.5 × 10−15 /Hz or even smaller values were observed. For the pMFFT, considering its thermal noise spectrum, these flux densities correspond to a device noise temperature of ≤2.5 µK, thereby ensuring a negligible uncertainty contribution at the lower end of the PLTS-2000 (0.9 mK).

Visualization of Au Nanoparticles Buried in a Polymer Matrix by Scanning Thermal Noise Microscopy

Several researchers have recently demonstrated visualization of subsurface features with a nanometer-scale resolution using various imaging schemes based on atomic force microscopy. Since all these subsurface imaging techniques require excitation of the oscillation of the cantilever and/or sample surface, it has been difficult to identify a key imaging mechanism. Here we demonstrate visualization of Au nanoparticles buried 300 nm into a polymer matrix by measurement of the thermal noise spectrum of a microcantilever with a tip in contact to the polymer surface. We show that the subsurface Au nanoparticles are detected as the variation in the contact stiffness and damping reflecting the viscoelastic properties of the polymer surface. The variation in the contact stiffness well agrees with the effective stiffness of a simple one-dimensional model, which is consistent with the fact that the maximum depth range of the technique is far beyond the extent of the contact stress field.

The complementarity and similarity of magnetorelaxometry and thermal magnetic noise spectroscopy for magnetic nanoparticle characterization

Magnetorelaxometry and thermal magnetic noise spectroscopy are two magnetic characterization techniques enabling one to estimate the magnetic nanoparticle hydrodynamic size distribution. Both techniques are based on the same physical principle, i.e. the thermal fluctuations of the magnetic moment. In the case of magnetorelaxometry these fluctuations give rise to a relaxing magnetic moment after an externally applied magnetic field is switched off, whereas thermal magnetic noise spectra are measured in the absence of any external excitation. Hence, thermal magnetic noise spectroscopy is an equilibrium measurement technique. Here, we compare the similarity and complementarity of both methods and conclude that, for particles within both methods’ sensitivity range, they give the same estimate for the size distribution. For small particles (or samples with low viscosities), the used setup is not sufficiently sensitive to accurately estimate the size distribution from the relaxometry signal whereas this is still possible with thermal magnetic noise spectroscopy. For larger particles, however, magnetorelaxometry is the preferred method because of its higher signal to noise ratio and faster measurement time.

Low-energy theory of transport in Majorana wire junctions

We formulate and apply a low-energy transport theory for hybrid quantum devices containing junctions of topological superconductor (TS) wires and conventional normal (N) or superconducting (S) leads. We model TS wires as spinless $p$-wave superconductors and derive their boundary Keldysh Green's function, capturing both the Majorana end state and continuum quasiparticle excitations in a unified manner. We also specify this Green's function for a finite-length TS wire. Junctions connecting different parts of the device are described by the standard tunneling Hamiltonian. Using this Hamiltonian approach, one also has the option to include many-body interactions in a systematic manner. For N-TS junctions, we provide the current-voltage ($I$-$V$) characteristics at arbitrary junction transparency and give exact results for the shot noise power and the excess current. For TS-TS junctions, analytical results for the thermal noise spectrum and for the $I$-$V$ curve in the high-transparency low-bias regime are presented. For S-TS junctions, we compute the entire $I$-$V$ curve and clarify the conditions for having a finite Josephson current.

Contact-free experimental determination of the static flexural spring constant of cantilever sensors using a microfluidic force tool.

Micro- and nanocantilevers are employed in atomic force microscopy (AFM) and in micro- and nanoelectromechanical systems (MEMS and NEMS) as sensing elements. They enable nanomechanical measurements, are essential for the characterization of nanomaterials, and form an integral part of many nanoscale devices. Despite the fact that numerous methods described in the literature can be applied to determine the static flexural spring constant of micro- and nanocantilever sensors, experimental techniques that do not require contact between the sensor and a surface at some point during the calibration process are still the exception rather than the rule. We describe a noncontact method using a microfluidic force tool that produces accurate forces and demonstrate that this, in combination with a thermal noise spectrum, can provide the static flexural spring constant for cantilever sensors of different geometric shapes over a wide range of spring constant values (≈0.8–160 N/m).

Colored spectrum characteristics of thermal noise on the molecular scale.

Thermal noise is of fundamental importance to many processes. Traditionally, thermal noise has been treated as white noise on the macroscopic scale. Using molecular dynamics simulations and power spectrum analysis, we show that the thermal noise of solute molecules in water is non-white on the molecular scale, which is in contrast to the conventional theory. In the frequency domain from 2 × 1011 Hz to 1013 Hz, the power spectrum of thermal noise for polar solute molecules resembles the spectrum of 1/f noise. The power spectrum of thermal noise for non-polar solute molecules deviates only slightly from the spectrum of white noise. The key to this phenomenon is the existence of hydrogen bonds between polar solute molecules and solvent water molecules. Furthermore, for polar solute molecules, the degree of power spectrum deviation from that of white noise is associated with the average lifetime of the hydrogen bonds between the solute and the solvent molecules.

Thermal magnetic noise spectra of nanoparticle ensembles

Typically, the dynamic behaviour of magnetic nanoparticles is investigated by measuring their response to externally applied magnetic fields. In contrast, we present a study of the magnetic fluctuations in an ensemble of magnetic nanoparticles recorded in the absence of any external excitation. Several samples of magnetic nanoparticles with varying particle size, composition, and environment were investigated. We interpret the thermal magnetic noise spectrum to estimate particle size distributions and compare these to the distributions derived from magnetorelaxometry measurements of the same samples.

Quantitative assessment of contact and non-contact lateral force calibration methods for atomic force microscopy

Atomic Force Microscopy (AFM) has been widely used for measuring friction force at the nano-scale. However, one of the key challenges faced by AFM researchers is to calibrate an AFM system to interpret a lateral force signal as a quantifiable force. In this study, five rectangular cantilevers were used to quantitatively compare three different lateral force calibration methods to demonstrate the legitimacy and to establish confidence in the quantitative integrity of the proposed methods. The Flat-Wedge method is based on a variation of the lateral output on a surface with flat and changing slopes, the Multi-Load Pivot method is based on taking pivot measurements at several locations along the cantilever length, and the Lateral AFM Thermal-Sader method is based on determining the optical lever sensitivity from the thermal noise spectrum of the first torsional mode with a known torsional spring constant from the Sader method. The results of the calibration using the Flat-Wedge and Multi-Load Pivot methods were found to be consistent within experimental uncertainties, and the experimental uncertainties of the two methods were found to be less than 15%. However, the lateral force sensitivity determined by the Lateral AFM Thermal-Sader method was found to be 8–29% smaller than those obtained from the other two methods. This discrepancy decreased to 3–19% when the torsional mode correction factor for an ideal cantilever was used, which suggests that the torsional mode correction should be taken into account to establish confidence in Lateral AFM Thermal-Sader method.

Development of a detachable high speed miniature scanning probe microscope for large area substrates inspection.

We have developed a high speed, miniature scanning probe microscope (MSPM) integrated with a Positioning Unit (PU) for accurately positioning the MSPM on a large substrate. This combination enables simultaneous, parallel operation of many units on a large sample for high throughput measurements. The size of the MSPM is 19 × 45 × 70 mm(3). It contains a one-dimensional flexure stage with counter-balanced actuation for vertical scanning with a bandwidth of 50 kHz and a z-travel range of more than 2 μm. This stage is mechanically decoupled from the rest of the MSPM by suspending it on specific dynamically determined points. The motion of the probe, which is mounted on top of the flexure stage is measured by a very compact optical beam deflection (OBD). Thermal noise spectrum measurements of short cantilevers show a bandwidth of 2 MHz and a noise of less than 15 fm/Hz(1/2). A fast approach and engagement of the probe to the substrate surface have been achieved by integrating a small stepper actuator and direct monitoring of the cantilever response to the approaching surface. The PU has the same width as the MSPM, 45 mm and can position the MSPM to a pre-chosen position within an area of 275×30 mm(2) to within 100 nm accuracy within a few seconds. During scanning, the MSPM is detached from the PU which is essential to eliminate mechanical vibration and drift from the relatively low-resonance frequency and low-stiffness structure of the PU. Although the specific implementation of the MSPM we describe here has been developed as an atomic force microscope, the general architecture is applicable to any form of SPM. This high speed MSPM is now being used in a parallel SPM architecture for inspection and metrology of large samples such as semiconductor wafers and masks.

Plasmonic mass and Johnson–Nyquist noise

The fluctuation-dissipation theorem relates the thermal noise spectrum of a conductor to its linear response properties, with the ohmic resistance arising from the electron scattering being the most notable linear response property. But the linear response also includes the collective inertial acceleration of electrons, which should in principle influence the thermal noise spectrum as well. In practice, this effect would be largely masked by the Planck quantization for traditional conductors with short electron scattering times. But recent advances in nanotechnology have enabled the fabrication of conductors with greatly increased electron scattering times, with which the collective inertial effect can critically affect the thermal noise spectrum. In this paper we highlight this collective inertial effect—that is, the plasmonic effect—on the thermal noise spectrum under the framework of semiclassical electron dynamics, from both fundamental microscopic and practical modeling points of view. In graphene, where non-zero collective inertia arises from zero single-electron effective mass and where both electron and hole bands exist together, the thermal noise spectrum shows rich temperature and frequency dependencies, unseen in traditional conductors.