Quantum optomechanics with silicon nanostructures

Mechanical resonators are the most basic and ubiquitous physical systems known. In on-chip form, they are used to process high frequency signals in every cell phone, television, and laptop. They have also been in the last few decades in different shapes and forms, a critical part of progress in quantum information sciences with kilogram-scale mirrors for gravitational wave detection measuring motion at its quantum limits, and the motion of single ions being used to link qubits for quantum computation.

Optomechanics is a field primarily concerned with coupling light to the motion of mechanical structures. This thesis contains descriptions of recent work with mechanical systems in the megahertz to gigahertz frequency range, formed by nanofabricating novel photonic/phononic structures on a silicon chip. These structures are designed to have both optical and mechanical resonances, and laser light is used to address and manipulate their motional degrees of freedom through radiation pressure forces. We laser cool these mechanical resonators to their ground states, and observe for the first time the quantum zero-point motion of a nanomechanical resonator. Conversely, we show that engineered mechanical resonances drastically modify the optical response of our structures, creating large effective optical nonlinearities not present in bulk silicon. We experimentally demonstrate aspects of these nonlinearities by proposing and observing ``electromagnetically induced transparency'' and light slowed down to 6 m/s, as well as wavelength conversion, and generation of nonclassical optical radiation. Finally, the application of optomechanics to longstanding problems in quantum and classical communications are proposed and investigated.

Nonlinear optics in planar silica-on-silicon disk resonators

Optical frequency combs (OFCs) provide direct phase-coherent link between optical and RF frequencies, and enable precision measurement of optical frequencies. In recent years, a new class of frequency combs (microcombs) have emerged based on parametric frequency conversions in dielectric microresonators. Micocombs have large line spacing from 10's to 100's GHz, allowing easy access to individual comb lines for arbitrary waveform synthesis. They also provide broadband parametric gain bandwidth, not limited by specific atomic or molecular transitions in conventional OFCs. The emerging applications of microcombs include low noise microwave generation, astronomical spectrograph calibration, direct comb spectroscopy, and high capacity telecommunications.

In this thesis, research is presented starting with the introduction of a new type of chemically etched, planar silica-on-silicon disk resonator. A record Q factor of 875 million is achieved for on-chip devices. A simple and accurate approach to characterize the FSR and dispersion of microcavities is demonstrated. Microresonator-based frequency combs (microcombs) are demonstrated with microwave repetition rate less than 80 GHz on a chip for the first time. Overall low threshold power (as low as 1 mW) of microcombs across a wide range of resonator FSRs from 2.6 to 220 GHz in surface-loss-limited disk resonators is demonstrated. The rich and complex dynamics of microcomb RF noise are studied. High-coherence, RF phase-locking of microcombs is demonstrated where injection locking of the subcomb offset frequencies are observed by pump-detuning-alignment. Moreover, temporal mode locking, featuring subpicosecond pulses from a parametric 22 GHz microcomb, is observed. We further demonstrated a shot-noise-limited white phase noise of microcomb for the first time. Finally, stabilization of the microcomb repetition rate is realized by phase lock loop control.

For another major nonlinear optical application of disk resonators, highly coherent, simulated Brillouin lasers (SBL) on silicon are also demonstrated, with record low Schawlow-Townes noise less than 0.1 Hz^2/Hz for any chip-based lasers and low technical noise comparable to commercial narrow-linewidth fiber lasers. The SBL devices are efficient, featuring more than 90% quantum efficiency and threshold as low as 60 microwatts. Moreover, novel properties of the SBL are studied, including cascaded operation, threshold tuning, and mode-pulling phenomena. Furthermore, high performance microwave generation using on-chip cascaded Brillouin oscillation is demonstrated. It is also robust enough to enable incorporation as the optical voltage-controlled-oscillator in the first demonstration of a photonic-based, microwave frequency synthesizer. Finally, applications of microresonators as frequency reference cavities and low-phase-noise optomechanical oscillators are presented.

I. A recirculating charged-couple device. II. The mercury selenide on N-silicon Schottky barrier

A recirculating charge-coupled device structure has been devised. Entrance and exit gates allow a signal to be admitted, recirculated a given number of times, and then examined. In this way a small device permits simulation of a very long shift register without passing the signal through input and output diffusions. An oscilloscope motion picture demonstrating degradation of an actual circulating signal has been made. The performance of the device in simulating degradation of a signal by a very long shift register is well fit by a simple model based on transfer inefficiency.

Electrical properties of the mercury selenide on n-type chemically-cleaned silicon Schottky barrier have been studied. Barrier heights measured were 0.96 volts for the photoresponse technique and 0.90 volts for the current-voltage technique. These are the highest barriers yet reported on n-type silicon.

Hall coefficient and resistivity of an amorphous palladium-silicon alloy

The Hall coefficient and resistance in several specimens of an amorphous metallic alloy containing 80 at.% palladium and 20 at.% silicon have been investigated at temperatures between 4.2°K and room temperature. An ideal limiting behavior of these transport coefficients was analyzed on the basis of the nearly free electron model to yield a carrier density of 9 x 10²² cm.^-3, or about 1.7 electrons per palladium atom, and a mean free path of about 9Å which is almost constant with temperature. The deviations of the individual specimens from this ideal behavior, which were small but noticeable in the relative resistivity and much greater in the Hall coefficient, can be explained by invoking disk-shaped crystalline regions with low resistivity and a positive Hall coefficient. A detailed calculation shows how a volume fraction of such crystalline material too small to be noticed in X-ray diffraction could have a significant effect on the resistivity and a much greater effect on the Hall coefficient.

Part I. Kinetics of coarsening of the gamma-prime precipitate in nickel-silicon alloys. Part II. Rate of crystallization of an amorphous Fe-P-C alloy

The coarsening kinetics of Ni₃ Si(γ') precipitate in a binary Ni-Si alloy containing 6.5 wt. % silicon was studied by magnetic techniques and transmission electronmicroscopy. A calibration curve was established to determine the concentration of silicon in the matrix. The variation of the Si content of the Ni-rich matrix as a function of time follows Lifshitz and Wagner theory for diffusion controlled coarsening phenomena. The estimated values of equilibrium solubility of silicon in the matrix represent the true coherent equilibrium solubilities.

The experimental particle-size distributions and average particle size were determined from dark field electron micrographs. The average particle size varies linearly with t^-1/3 as suggested by Lifshitz and Wagner. The experimental distributions of particle sizes differ slightly from the theoretical curve at the early stages of aging, but the agreement is satisfactory at the later stages. The values of diffusion coefficient of silicon, interfacial free energy and activation energy were calculated from the results of coarsening kinetics. The experimental value of effective diffusion coefficient is in satisfactory agreement with the value predicted by the application of irreversible the rmodynamics to the process of volume constrained growth of coherent precipitate during coarsening. The coherent γ' particles in Ni-Sialloy unlike those in Ni-Al and Ni-Ti seem to lose coherency at high temperature. A mechanism for the formation of semi-coherent precipitate is suggested.

Conduction-electron localized-moment interaction in palladium-silicon base amorphous alloys containing transition metals

The electrical and magnetic properties of amorphous alloys obtained by rapid quenching from the liquid state have been studied. The composition of these alloys corresponds to the general formula M_xPd_80-xSi₂₀, in which M stands for a metal of the first transition series between chromium and nickel and x is its atomic concentration. The concentration ranges within which an amorphous structure could be obtained were: from 0 to 7 for Cr, Mn and Fe, from 0 to 11 for Co and from 0 to 15 for Ni. A well-defined minimum in the resistivity vs temperature curve was observed for all alloys except those containing nickel. The alloys for which a resistivity minimum was observed had a negative magnetoresistivity approximately proportional to the square of the magnetization and their susceptibility obeyed the Curie-Weiss law in a wide temperature range. For concentrated Fe and Co alloys the resistivity minimum was found to coexist with ferromagnetism. These observations lead to the conclusion that the present results are due to a s-d exchange interaction. The unusually high resistivity minimum temperature observed in the Cr alloys is interpreted as a result of a high Kondo temperature and a large s-d exchange integral. A low Fermi energy of the amorphous alloys (3.5 eV) is also responsible for the anomalies due to the s-d exchange interaction.

Optical properties of excited silicon and germanium at low temperatures

Part I of this thesis deals with 3 topics concerning the luminescence from bound multi-exciton complexes in Si. Part II presents a model for the decay of electron-hole droplets in pure and doped Ge.

Part I.

We present high resolution photoluminescence data for Si doped With Al, Ga, and In. We observe emission lines due to recombination of electron-hole pairs in bound excitons and satellite lines which have been interpreted in terms of complexes of several excitons bound to an impurity. The bound exciton luminescence in Si:Ga and Si:Al consists of three emission lines due to transitions from the ground state and two low lying excited states. In Si:Ga, we observe a second triplet of emission lines which precisely mirror the triplet due to the bound exciton. This second triplet is interpreted as due to decay of a two exciton complex into the bound exciton. The observation of the second complete triplet in Si:Ga conclusively demonstrates that more than one exciton will bind to an impurity. Similar results are found for Si:Al. The energy of the lines show that the second exciton is less tightly bound than the first in Si:Ga. Other lines are observed at lower energies. The assumption of ground state to ground-state transitions for the lower energy lines is shown to produce a complicated dependence of binding energy of the last exciton on the number of excitons in a complex. No line attributable to the decay of a two exciton complex is observed in Si:In.

We present measurements of the bound exciton lifetimes for the four common acceptors in Si and for the first two bound multi-exciton complexes in Si:Ga and Si:Al. These results are shown to be in agreement with a calculation by Osbourn and Smith of Auger transition rates for acceptor bound excitons in Si. Kinetics determine the relative populations of complexes of various sizes and work functions, at temperatures which do not allow them to thermalize with respect to one another. It is shown that kinetic limitations may make it impossible to form two-exciton complexes in Si:In from a gas of free excitons.

We present direct thermodynamic measurements of the work functions of bound multi-exciton complexes in Al, B, P and Li doped Si. We find that in general the work functions are smaller than previously believed. These data remove one obstacle to the bound multi-exciton complex picture which has been the need to explain the very large apparent work functions for the larger complexes obtained by assuming that some of the observed lines are ground-state to ground-state transitions. None of the measured work functions exceed that of the electron-hole liquid.

Part II.

A new model for the decay of electron-hole-droplets in Ge is presented. The model is based on the existence of a cloud of droplets within the crystal and incorporates exciton flow among the drops in the cloud and the diffusion of excitons away from the cloud. It is able to fit the experimental luminescence decays for pure Ge at different temperatures and pump powers while retaining physically reasonable parameters for the drops. It predicts the shrinkage of the cloud at higher temperatures which has been verified by spatially and temporally resolved infrared absorption experiments. The model also accounts for the nearly exponential decay of electron-hole-droplets in lightly doped Ge at higher temperatures.

Magnetic moments in amorpous palladium-cobalt-silicon alloys

The magnetic moments of amorphous ternary alloys containing Pd, Co and Si in atomic concentrations corresponding to Pd_(80-x)Co_xSi_(20) in which x is 3, 5, 7, 9, 10 and 11, have been measured between 1.8 and 300°K and in magnetic fields up to 8.35 kOe. The alloys were obtained by rapid quenching of a liquid droplet and their structures were analyzed by X-ray diffraction. The measurements were made in a null-coil pendulum magnetometer in which the temperature could be varied continuously without immersing the sample in a cryogenic liquid. The alloys containing 9 at.% Co or less obeyed Curie's Law over certain temperature ranges, and had negligible permanent moments at room temperature. Those containing 10 and 11 at.% Co followed Curie's Law only above approximately 200°K and had significant permanent moments at room temperature. For all alloys, the moments calculated from Curie's Law were too high to be accounted for by the moments of individual Co atoms. To explain these findings, a model based on the existence of superparamagnetic clustering is proposed. The cluster sizes calculated from the model are consistent with the rapid onset of ferromagnetism in the alloys containing 10 and 11 at.% Co and with the magnetic moments in an alloy containing 7 at.% Co heat treated in such a manner as to contain a small amount of a crystalline phase. In alloys containing 7 at.% Co or less, a maximum in the magnetization vs temperature curve was observed around 10°K. This maximum was eliminated by cooling the alloy in a magnetic field, and an explanation for this observation is suggested.

Space-charge-limited current in fast neutron irradiated silicon

DC and transient measurements of space-charge-limited currents through alloyed and symmetrical n^+ν n^+ structures made of nominally 75 kΩcm ν-type silicon are studied before and after the introduction of defects by 14 MeV neutron radiation. In the transient measurements, the current response to a large turn-on voltage step is analyzed. Right after the voltage step is applied, the current transient reaches a value which we shall call "initial current" value. At longer times, the transient current decays from the initial current value if traps are present.

Before the irradiation, the initial current density-voltage characteristics J(V) agree quantitatively with the theory of trap-free space-charge-limited current in solids. We obtain for the electron mobility a temperature dependence which indicates that scattering due to impurities is weak. This is expected for the high purity silicon used. The drift velocity-field relationships for electrons at room temperature and 77°K, derived from the initial current density-voltage characteristics, are shown to fit the relationships obtained with other methods by other workers. The transient current response for t > 0 remains practically constant at the initial value, thus indicating negligible trapping.

Measurement of the initial (trap-free) current density-voltage characteristics after the irradiation indicates that the drift velocity-field relationship of electrons in silicon is affected by the radiation only at low temperature in the low field range. The effect is not sufficiently pronounced to be readily analyzed and no formal description of it is offered. In the transient response after irradiation for t > 0, the current decays from its initial value, thus revealing the presence of traps. To study these traps, in addition to transient measurements, the DC current characteristics were measured and shown to follow the theory of trap-dominated space-charge-limited current in solids. This theory was applied to a model consisting of two discrete levels in the forbidden band gap. Calculations and experiments agreed and the capture cross-sections of the trapping levels were obtained. This is the first experimental case known to us through which the flow of space-charge-limited current is so simply representable.

These results demonstrate the sensitivity of space-charge-limited current flow as a tool to detect traps and changes in the drift velocity-field relationship of carriers caused by radiation. They also establish that devices based on the mode of space-charge-limited current flow will be affected considerably by any type of radiation capable of introducing traps. This point has generally been overlooked so far, but is obviously quite significant.

Energy loss of light ions channeling in silicon

Experimental measurements of rate of energy loss were made for protons of energy .5 to 1.6 MeV channeling through 1 μm thick silicon targets along the <110>, <111>, and <211> axial directions, and the {100}, {110}, {111}, and {211} planar directions. A .05% resolution automatically controlled magnetic spectrometer was used. The data are presented graphically along with an extensive summary of data in the literature. The data taken cover a wider range of channels than has previously been examined, and are in agreement with the data of F. Eisen, et al., Radd. Eff. 13, 93 (1972).

The theory in the literature for channeling energy loss due to interaction with local electrons, core electrons, and distant valence electrons of the crystal atoms is summarized. Straggling is analyzed, and a computer program which calculates energy loss and straggling using this theory and the Moliere approximation to the Thomas Fermi potential, V_TF, and the detailed silicon crystal structure is described. Values for the local electron density Z_loc in each of the channels listed above are extracted from the data by graphical matching of the experimental and computer results.

Zeroth and second order contributions to Z_loc as a function of distance from the center of the channel were computed from ∇²V_TF = 4πρ for various channels in silicon. For data taken in this work and data of F. Eisen, et al., Rad. Eff. 13, 93 (1972), the calculated zeroth order contribution to Z_loc lies between the experimentally extracted Z_loc values obtained by using the peak and the leading edge of the transmission spectra, suggesting that the observed straggling is due both to statistical fluctuations and to path variation.

Cryogenic Silicon Optical Reference Cavities

Thermodynamical fluctuations in temperature and position exist in every physical system, and show up as a fundamental noise limit whenever we choose to measure some quantity in a laboratory environment. Thermodynamical fluctuations in the position of the atoms in the dielectric coatings on the mirrors for optical cavities at the forefront of precision metrology (e.g., LIGO, the cavities which probe atomic transitions to define the second) are a current limiting noise source for these experiments, and anything which involves locking a laser to an optical cavity. These thermodynamic noise sources scale physical geometry of experiment, material properties (such as mechanical loss in our dielectric coatings), and temperature. The temperature scaling provides a natural motivation to move to lower temperatures, with a potential huge benefit for redesigning a room temperature experiment which is limited by thermal noise for cryogenic operation.

We design, build, and characterize a pair of linear Fabry-Perot cavities to explore limitations to ultra low noise laser stabilization experiments at cryogenic temperatures. We use silicon as the primary material for the cavity and mirrors, due to a zero crossing in its linear coefficient of thermal expansion (CTE) at 123 K, and other desirable material properties. We use silica tantala coatings, which are currently the best for making high finesse low noise cavities at room temperature. The material properties of these coating materials (which set the thermal noise levels) are relatively unknown at cryogenic temperatures, which motivates us to study them at these temperatures. We were not able to measure any thermal noise source with our experiment due to excess noise. In this work we analyze the design and performance of the cavities, and recommend a design shift from mid length cavities to short cavities in order to facilitate a direct measurement of cryogenic coating noise.

In addition, we measure the cavities (frequency dependent) photo-thermal response. This can help characterize thermooptic noise in the coatings, which is poorly understood at cryogenic temperatures. We also explore the feasibility of using the cavity to do macroscopic quantum optomechanics such as ground state cooling.

I. Stokes flow past a thin screen. II. Viscous flows past porous bodies of finite size

Part I

The slow, viscous flow past a thin screen is analyzed based on Stokes equations. The problem is reduced to an associated electric potential problem as introduced by Roscoe. Alternatively, the problem is formulated in terms of a Stokeslet distribution, which turns out to be equivalent to the first approach.

Special interest is directed towards the solution of the Stokes flow past a circular annulus. A "Stokeslet" formulation is used in this analysis. The problem is finally reduced to solving a Fredholm integral equation of the second kind. Numerical data for the drag coefficient and the mean velocity through the hole of the annulus are obtained.

Stokes flow past a circular screen with numerous holes is also attempted by assuming a set of approximate boundary conditions. An "electric potential" formulation is used, and the problem is also reduced to solving a Fredholm integral equation of the second kind. Drag coefficient and mean velocity through the screen are computed.

Part II

The purpose of this investigation is to formulate correctly a set of boundary conditions to be prescribed at the interface between a viscous flow region and a porous medium so that the problem of a viscous flow past a porous body can be solved.

General macroscopic equations of motion for flow through porous media are first derived by averaging Stokes equations over a volume element of the medium. These equations, including viscous stresses for the description, are more general than Darcy's law. They reduce to Darcy's law when the Darcy number becomes extremely small.

The interface boundary conditions of the first kind are then formulated with respect to the general macroscopic equations applied within the porous region. An application of such equations and boundary conditions to a Poiseuille shear flow problem demonstrates that there usually exists a thin interface layer immediately inside the porous medium in which the tangential velocity varies exponentially and Darcy's law does not apply.

With Darcy's law assumed within the porous region, interface boundary conditions of the second kind are established which relate the flow variables across the interface layer. The primary feature is a jump condition on the tangential velocity, which is found to be directly proportional to the normal gradient of the tangential velocity immediately outside the porous medium. This is in agreement with the experimental results of Beavers, et al.

The derived boundary conditions are applied in the solutions of two other problems: (1) Viscous flow between a rotating solid cylinder and a stationary porous cylinder, and (2) Stokes flow past a porous sphere.

Topology optimization of silicon anode structures for lithium-ion battery applications

This thesis presents a topology optimization methodology for the systematic design of optimal multifunctional silicon anode structures in lithium-ion batteries. In order to develop next generation high performance lithium-ion batteries, key design challenges relating to the silicon anode structure must be addressed, namely the lithiation-induced mechanical degradation and the low intrinsic electrical conductivity of silicon. As such, this work considers two design objectives of minimum compliance under design dependent volume expansion, and maximum electrical conduction through the structure, both of which are subject to a constraint on material volume. Density-based topology optimization methods are employed in conjunction with regularization techniques, a continuation scheme, and mathematical programming methods. The objectives are first considered individually, during which the iteration history, mesh independence, and influence of prescribed volume fraction and minimum length scale are investigated. The methodology is subsequently extended to a bi-objective formulation to simultaneously address both the compliance and conduction design criteria. A weighting method is used to derive the Pareto fronts, which demonstrate a clear trade-off between the competing design objectives. Furthermore, a systematic parameter study is undertaken to determine the influence of the prescribed volume fraction and minimum length scale on the optimal combined topologies. The developments presented in this work provide a foundation for the informed design and development of silicon anode structures for high performance lithium-ion batteries.

Chemical and Electrochemical Behavior of Graphene-Covered Silicon Photoanodes

This dissertation describes efforts over the last five years to develop protective layers for semiconductor photoelectrodes based on monolayer or few-layer graphene sheets. Graphene is an attractive candidate for a protective layer because of its known chemical inertness, transparency, ease of deposition, and limited number of electronic states. Monolayer graphene was found to effectively inhibit loss of photocurrent over 1000 seconds at n-Si/aqueous electrolyte interfaces that exhibit total loss over photocurrent over 100 seconds. Further, the presence of graphene was found to effect only partial Fermi level pinning at the Si/graphene interface with respect to a range of nonaqueous electrolytes. Fluorination of graphene was found to extend the stability imparted on n-Si by the monolayer sheet in aqueous Fe(CN)₆^3-/4- electrolyte to over 100,000 seconds. It was demonstrated that the stability of the photocurrent of n-Si/fluorinated graphene/aqueous electrolyte interfaces relative to n-Si/aqueous electrolyte interfaces is likely attributable to the inhibition of oxidation of the silicon surface.

This dissertation also relates efforts to describe and define terminology relevant to the field of photoelectrochemistry and solar fuels production. Terminology describing varying interfaces employed in electrochemical solar fuels devices are defined, and the research challenges associated with each are discussed. Methods for determining the efficiency of varying photoelectrochemical and solar-fuel-producing cells from the current-voltage behavior of the individual components of such a device without requiring the device be constructed are described, and a range of commonly employed performance metrics are explored.

Paths Towards High Efficiency Silicon Photovoltaics

While photovoltaics hold much promise as a sustainable electricity source, continued cost reduction is necessary to continue the current growth in deployment. A promising path to continuing to reduce total system cost is by increasing device efficiency. This thesis explores several silicon-based photovoltaic technologies with the potential to reach high power conversion efficiencies. Silicon microwire arrays, formed by joining millions of micron diameter wires together, were developed as a low cost, low efficiency solar technology. The feasibility of transitioning this to a high efficiency technology was explored. In order to achieve high efficiency, high quality silicon material must be used. Lifetimes and diffusion lengths in these wires were measured and the action of various surface passivation treatments studied. While long lifetimes were not achieved, strong inversion at the silicon / hydrofluoric acid interface was measured, which is important for understanding a common measurement used in solar materials characterization.

Cryogenic deep reactive ion etching was then explored as a method for fabricating high quality wires and improved lifetimes were measured. As another way to reach high efficiency, growth of silicon-germanium alloy wires was explored as a substrate for a III-V on Si tandem device. Patterned arrays of wires with up to 12% germanium incorporation were grown. This alloy is more closely lattice matched to GaP than silicon and allows for improvements in III-V integration on silicon.

Heterojunctions of silicon are another promising path towards achieving high efficiency devices. The GaP/Si heterointerface and properties of GaP grown on silicon were studied. Additionally, a substrate removal process was developed which allows the formation of high quality free standing GaP films and has wide applications in the field of optics.

Finally, the effect of defects at the interface of the amorphous silicon heterojuction cell was studied. Excellent voltages, and thus efficiencies, are achievable with this system, but the voltage is very sensitive to growth conditions. We directly measured lateral transport lengths at the heterointerface on the order of tens to hundreds of microns, which allows carriers to travel towards any defects that are present and recombine. This measurement adds to the understanding of these types of high efficiency devices and may aid in future device design.