Member, IEEE Michiel Postema
email: m.postema@erasmusmc.nl

Ayache Bouakaz

Michel Versluis

Associate Member, IEEE Nico De Jong

Ultrasound-induced Gas Release from Contrast Agent Microbubbles

We investigated gas release from two hard-shelled ultrasound contrast agents, by subjecting them to high-MI ultrasound and simultaneously capturing high-speed photographs. At an insonifying frequency of 1.7 MHz, a larger percentage of contrast bubbles is seen to crack than at 0.5 MHz. Most of the released gas bubbles have equilibrium diameters between 1.25 and 1.75 µm. Their disappearance was observed optically. Free gas bubbles have equilibrium diameters smaller than the bubbles from which they have been released. Coalescence may account for the long dissolution times acoustically observed and published in previous studies. After sonic cracking, the cracked bubbles stay acoustically active.

I. INTRODUCTION

Sonic cracking is the ultrasound-induced release of gas from hard-shelled microbubbles [1]. This phenomenon has been observed in insonified contrast agent with the aid of high-speed photography [START_REF] Dayton | Simultaneous optical and acoustical observations of contrast agents[END_REF], [START_REF] Postema | Optically observed microbubble coalescence and collapse[END_REF], [START_REF] Postema | Ultrasound-induced encapsulated microbubble phenomena[END_REF], [START_REF] Bloch | Optical observation of lipid-and polymer-shelled ultrasound microbubble contrast agents[END_REF]. The following clinical applications of sonic cracking have been suggested [START_REF] Postema | Ultrasound-induced encapsulated microbubble phenomena[END_REF]:

Imaging applications. Release burst imaging makes use of the strong scattering response of released gas microbubbles [START_REF] Frinking | Ultrasound Contrast Agents: acoustic characterization and diagnostic imaging[END_REF]. While the free gas dissolves, the strong acoustic response fades away. The measurement of the reappearance of contrast microbubbles is a quantitative perfusion indicator [START_REF] Wei | Quantification of myocardial blood flow with ultrasound-induced destruction of microbubbles administered as a constant venous infusion[END_REF].

Drug delivery. If the gas content of the encapsulated bubble has therapeutic properties, sonic cracking may find an application in localized drug delivery. We think of nitric oxide and gaseous anaesthetics.

The released gas bubbles might be targeted to the vessel wall owing to primary radiation forces [START_REF] Postema | Ultrasound-induced encapsulated microbubble phenomena[END_REF], [START_REF] Shortencarier | A method for radiation-force localized drug delivery using gas-filled lipospheres[END_REF]. The behavior of contrast agent microbubbles near cells has been recently studied by Van Wamel, Kudo, and Wolfrum [START_REF] Van Wamel | Effects of diagnostic ultrasound parameters on molecular uptake and cell viability[END_REF], [START_REF] Kudo | Optical observation of cell-bubble behavior using a high-speed camera[END_REF], [START_REF] Kudo | Study on mechanism of cell damage caused by microbubbles exposed to ultrasound[END_REF], [START_REF] Wolfrum | Observations of pressure-wave-excited contrast agent bubbles in the vicinity of cells[END_REF].

Noninvasive blood pressure measurements. When a free gas bubble dissolves into a liquid medium, its acoustic response changes with its radius. 1 The changes in oscillating behavior of the dissolving gas bubble lead to changes in scattering cross sections [START_REF] Hilgenfeldt | Sound scattering and localized heat deposition of pulse-driven microbubbles[END_REF], and thus, the scattering behavior of an insonified, diffusing gas bubble can be calculated, dependent of the applied overpressure.

Hence, a noninvasive method for blood pressure assessment would be the measurement of the acoustic response from a dissolving bubble population [START_REF] Bouakaz | Noninvasive measurement of the hydrostatic pressure in a fluid-filled cavity based on the disappearance time of micrometer-sized free gas bubbles[END_REF]. For all applications, a contrast agent consisting of hard-shelled bubbles acts as a vehicle that carries a gas compound to a region of interest (cf. Fig. 1). The gas compound is released by a high-MI ultrasonic burst. The dissolving gas compound is tracked by a low-MI ultrasonic signal. A low-MI signal can also be used to induce microbubble translation [START_REF] Shortencarier | A method for radiation-force localized drug delivery using gas-filled lipospheres[END_REF]. The mechanical index is defined by:

MI = p - ac √ f , ( 1
)
where p - ac is the peak rarefactional acoustic pressure normalized by 1 MPa and f is the center frequency of the ultrasound normalized by 1 MHz.

Especially for the latter two potential applications, the gas release has to be controlled. In this study, we take a step towards controlled gas release from hard-shelled microbubbles, by visualizing and quantifying gas release under different conditions. Furthermore, we discuss if the sizes of the released gas bubbles are in agreement with previous measurements of acoustic decay times.

II. THEORY

The dissolving time of released gas bubbles is related to the hydrostatic pressure as follows [START_REF] Bouakaz | Noninvasive measurement of the hydrostatic pressure in a fluid-filled cavity based on the disappearance time of micrometer-sized free gas bubbles[END_REF]:

dR dt = DL C i C0 -1 -2 σ R p0 -pov p0 1 + 4 σ 3 R p 0 1 R + 1 √ π D t , ( 2
)
1 The resonance frequency fr of a free gas bubble is related to its diameter D according to [START_REF] Minnaert | On musical air bubbles and the sound of running water[END_REF]:

f r ≈ 6.5 m s -1 D .
where C i C0 is the ratio of the dissolved gas concentration to the saturation concentration (saturation ratio), D is the diffusion constant, L is the Ostwald coefficient, p 0 is the ambient pressure, p ov is the applied overpressure, R is the instantaneous bubble radius, t is the time starting (t = 0) when the bubble surface is exposed to the liquid surface, and σ is the surface tension. Equation [START_REF] Dayton | Simultaneous optical and acoustical observations of contrast agents[END_REF] shows that the disappearance of gas bubbles in a liquid medium is highly influenced by gas diffusion parameters and applied overpressure, and that the disappearance time of gas bubbles is shorter when the liquid medium is under pressure.

Simulations of dissolving gas bubbles were presented in [START_REF] Bouakaz | Noninvasive measurement of the hydrostatic pressure in a fluid-filled cavity based on the disappearance time of micrometer-sized free gas bubbles[END_REF], [START_REF] Kabalnov | Dissolution of multicomponent microbubbles in the bloodstream: 1. Theory[END_REF], [START_REF] Chen | The disappearance of ultrasound contrast bubbles: observations of bubble dissolution and cavitation nucleation[END_REF], [START_REF] Postema | Noninvasive microbubble-based pressure measurements: a simulation study[END_REF].

III. EXPERIMENTAL SETUP

Overview

For the observations of gas release, we made use of the Brandaris-128 fast framing camera system [START_REF] Chin | Brandaris 128: a 25 million frames per second digital camera with 128 highly sensitive frames[END_REF]. An overview of this experimental setup is shown in Figure 2. The amplitude of the electrical signal generated by an AWG 520 arbitrary waveform generator (Tektronix, Inc., Beaverton, OR) was adjusted by two variable 355C/D attenuators (Hewlett Packard Company, Palo Alto, CA) in series, and an A-500 60 dB linear power amplifier (ENI technology, Inc., Rochester, NY). The signal was converted to ultrasound by a V389-SU 500 kHz (focal width 3 mm), or by a V397-SU 2.25 MHz single-element transducer (focal width 1 mm) (Panametrics Inc., Waltham, MA), both spherically focused at 7.5 cm. The transducer was mounted in a Perspex container at an angle of 45 • relative to the top of the container (cf. Fig. 3). This container was filled with saturated water. A 200 µm cellulose Cuprophan R capillary tube (Membrana GmbH, Wuppertal, Germany) was fixed in the focal area of the transducer, through which contrast agent was flowing.

Optics

Underneath the capillary tube an optic fiber was mounted. This fiber was connected to an MVS-7010 Fiber Optic Strobe (PerkinElmer Optoelectronics, Salem, MA), and to a KLS-201 continuous fiber light source (Olympus KMI (KeyMed Ltd), Southend-on-Sea, UK).

The container was positioned beneath a customized BXFM microscopic system (Olympus Optical Co., Ltd., Tokyo, Japan) with a U-CA magnification changer (Olympus Optical Co., Ltd.), switched to 2× magnification and a LUMPlanFl 60× water immersion objective lens (Olympus Optical Co., Ltd.). For image control purposes, an LCL-902K CCD camera (Watec Co., Ltd., Yamagata, Japan) was mounted to the top of the microscope.

Camera

The optical observations of the situation during gas release and of the situation 100 ms after insonification were recorded with a Brandaris-128 (cf. Figure 4) fast framing camera system [START_REF] Chin | Brandaris 128: a 25 million frames per second digital camera with 128 highly sensitive frames[END_REF]. The Brandaris-128 captured sequences of 128 image frames at speeds up to 25 million frames per second.

Typical frame sizes correspond to 89 × 68 µm 2 . In all observations, image frames were captured before, during, and after ultrasound insonification.

For the observations of the gas dissolution process itself (cf. Fig. 8), which is a relatively slow process compared to bubble oscillations, we also made use of a data set recorded with a CR 2000 camera (Redlake MASD, LLC, San Diego, CA) that operated at a speed of two thousand frames per second. This camera had been installed on top of the microscope, as shown in Figure 4.

Contrast agents

We investigated the ultrasound contrast agent Quantison TM (Upperton Limited). It consists of human serum albumin-encapsulated air bubbles with a mean diameter of 3.2 µm. Shell thicknesses are between 0.2 and 0.3 µm [START_REF] Frinking | Acoustic modeling of shell-encapsulated gas bubbles[END_REF]. The resonance frequency of the bulk agent is 4 MHz [START_REF] Frinking | Acoustic modeling of shell-encapsulated gas bubbles[END_REF]. The content of a Quantison TM vial was resuspended in 5 ml of Isoton R II (Beckman Coulter, Inc., Fullerton, CA), and shaken gently for 20 seconds before further dilution (approximately 1:50). Further dilution ensures that only a few freely flowing contrast agent microbubbles are in the optical focal plane in the field of view.

We also investigated PB127 (POINT Biomedical Corporation, San Carlos, CA), which is specified as follows: PB127 consists of bilayered microspheres encapsulating nitrogen bubbles with a mean diameter of 4 µm. The outermost layer is albumin and the inner layer is composed of a biodegradable polymer. The resonance frequency of PB127 lies between 6 and 7 MHz. The content of a PB127 vial was resuspended in 2 ml of deionized water, and shaken gently for 20 seconds before further dilution (approximately 1:50).

The pressure field exerted by an oscillating contrast microbubble depends on the volumetric acceleration of the bubble [START_REF] Landau | Fluid Mechanics[END_REF], [START_REF] Postema | Medical Bubbles[END_REF]. For hard-shelled contrast microbubbles this pressure is very low compared to the ultrasonic field. Hence, for the concentrations we use, the presence of neighboring hard-shelled bubbles is negligible.

Ultrasound

The agents were insonified either by 8 cycles of 500 kHz ultrasound at peak rarefactional pressures p - ac in the range 0.2 < p - ac < 1.3 MPa, 2 or by 8 cycles of 1.7 MHz ultrasound at peak rarefactional pressures in the range 0.4 < p - ac < 2.5 MPa. For both frequencies transmitted, peak rarefactional acoustic pressures were equivalent to mechanical indices in the range 0.3 < MI < 1.9. We refer to ultrasound transmission with mechanical indices greater than 0.8 as high-MI insonification. Acoustic pressures in this regime are high enough to ensure gas release [START_REF] Burns | Looking at and listening to breaking bubbles: a correlative optical acoustic study of some experimental polymer/air agents[END_REF]. This is the regime where other destruction mechanisms have been observed with encapsulated bubbles, such as fragmentation and jetting [START_REF] Postema | Ultrasound-induced encapsulated microbubble phenomena[END_REF]. Sonic cracking is expected to increase with MI. Since it is used in clinical settings, and has to be comparable to other studies, we use the MI as a measure to compare sonic cracking under different frequencies.

Acoustic pressures applied were measured with a calibrated (for both frequencies) MH28-10 PVDF needle hydrophone (FORCE Technology, Brøndby, Denmark) in a separate water tank.

Procedure

We recorded 533 image sequences with the Brandaris-128 system, and 55 sequences with the Kodak CR 2000 camera. For each event, the total number of contrast agent microbubbles visible in the frames were counted, and the number of contrast agent microbubbles from which gas was released. In total, we counted 428 released bubbles. From the frames recorded after ultrasound insonification, the diameters of the released gas bubbles (if still present) or fragments thereof were measured, as well as the diameters of the contrast agent microbubbles from which they had escaped. Fifty eight resting diameters of released PB127 gas bubbles were measured. Bubble diameters and distances were measured manually or by using an image processing method described in [START_REF] Postema | Simulations and measurements of optical images of insonified ultrasound contrast microbubbles[END_REF].

IV. RESULTS

Quantison TM

An example of the sonic cracking of a Quantison TM bubble is shown in Figure 5. Frames numbering is from left to right, then from upper to lower. The camera system operated at a speed of 10 million frames per second. Gas is seen to escape from a 4.3 µm shelled Quantison TM bubble in the third frame, owing to the rarefaction phase of the driving ultrasound. Apparently, the Quantison TM shell is too rigid to be seen expanded. The free gas expands to 12.3 µm in the eighth frame, after which it contracts. In the eleventh frame, the free gas bubble, which has been subjected to motion blur, appears to be detached from the shelled bubble. In the twelfth frame, the gas is hardly visible, owing to the compressive phase of the driving ultrasound.

Another example of gas release is given in Figure 6. The camera system operated at a speed of 3 million frames per second. In the second frame, gas escapes from a 3.7 µm bubble. The response of the released gas to ultrasound can be clearly appreciated. After expanding to a 8.0 µm maximum in frame three, it is seen detached and contracted in frames four and five. The encapsulated bubble clearly looks different than before cracking, as if there is no gas left inside the shell. A few microseconds after insonification, the left frame of Figure 7 was captured. The released gas bubble is visible to the upper left of the bubble from which we saw it escape. Its diameter is estimated to be less than 1 µm. One hundred ms later, the right frame was captured. Here, the released gas bubble has disappeared.

The dissolving process itself of the released gas bubbles was captured at 2000 frames per second. The upper images of Figure 8 show 8-bit photographs of three Quantison TM bubbles before ultrasound arrival.

The lower images show the same photographs after gray-level window-slicing. After insonification, the Quantison TM bubbles have translated, and two fragments are visible in the second frame. The diameters of the fragments are less than 1 µm. These fragments slowly disappear, and are hardly visible in the last frame.

PB127

An example of the sonic cracking of PB127 bubbles is shown in Figure 9, at MI=0.9 and a driving frequency of 1.7 MHz. Frames numbering is from left to right, then from upper to lower. The camera system operated at a speed of 9.4 million frames per second. Upon ultrasound arrival in the second frame, two 7 µm PB127 bubbles start to contract. From frame four on, gas is released from both bubbles. In frame eight, both gas bubbles have been detached from the shell. Starting with frame ten, gas release is again observed from the lower PB127 bubble. The released gas bubbles interact, and both fragmentation and coalescence occur several times between frames 21 and 47. After insonification had finished, one resulting free gas bubble remained, which was still seen to pulsate. The left frame of Figure 10 was captured after insonification. The free gas bubble is the lower bubble in the diamond-shaped bubble group. Its resting diameter is 4 µm. One hundred ms later, the right frame was captured. Here, the free gas bubble has disappeared.

Our results show that the sizes of the PB127 bubbles from which gas was released, are normally distributed. The mean diameter of these bubbles is 3.6 µm, with a standard deviation of 1.5 µm.

The size distribution of the released gas bubbles from PB127 is shown in Figure 11. For both frequencies, most released gas bubbles have equilibrium diameters between 1.25 and 1.75 µm. We did not find a correlation between the sizes of the released gas bubbles and the peak acoustic pressures.

Overview

A quantitative overview of gas release is demonstrated in Figure 12. Typically, 10-15 contrast agent bubbles were visible in the field of view. For both agents, the percentage of released gas bubbles is greater at 1.7 MHz than at 0.5 MHz insonification.

Released gas bubbles have been observed to translate, to fragment, and to coalesce with other released bubbles. The contrast agent bubbles did not demonstrate these phenomena. Our measurements of resting sizes of released gas bubbles include fragments of released gas bubbles and coalesced bubbles. Bubbles that did not show gas release upon insonification, have been observed to crack during a subsequent ultrasonic burst.

After sonic cracking, contrast bubbles of both agents can stay acoustically active. Gas remainder has been observed inside the shells from sonically cracked bubbles, which was released by a subsequent ultrasonic burst.

V. DISCUSSION AND CONCLUSIONS

We can only speculate on why certain bubbles crack while others stay intact. Tiny flaws in the shells may account for this observation. Such flaws are apparently formed or widened during ultrasound insonification, causing the bubble to crack during a subsequent ultrasonic burst. The partly emptied shell must still be traceable after sonic cracking, and might have different acoustic properties. This is the first study where subsequent gas release was witnessed from the same contrast agent microbubble. This is also the first study where ulltrasound-induced coalescence of released gas bubbles has been observed.

The difference in average cracking percentage observed at 0.5 MHz and 1.7 MHz, may lie in the proximity of the resonance frequency of the agent to the insonifying frequency. The resonance frequency of Quantison TM is closest to 1.7 MHz, indeed. So is the resonance frequency of PB127: f r > 1.7 MHz.

This would imply that at a driving frequency closer to the resonance frequency (given the same MI), sonic cracking would be observed more often.

The phenomenon of bubbles translating during an ultrasonic cycle has only been observed in our results with free gas bubbles, and not with the hard-shelled bubbles. We attribute this behavior to the small size of the free gas bubbles during contraction, and the lack thereof with the hard-shelled contrast agent bubbles. When in contraction phase, free gas bubbles have a very small translating mass, as it is equivalent to half the mass of the displaced fluid [START_REF] Leighton | The Acoustic Bubble[END_REF]. When subjected to a radiation force, such light bubbles can accelerate much faster than the relatively heavy encapsulated bubbles.

The mean diameter of the PB127 bubbles from which gas was released, agrees with the published mean diameter of PB127 bubbles, but the free gas bubbles have final resting diameters smaller than the bubbles from which they have been released. The observations of gas left inside shell after cracking, may account for this difference.

The acoustic response from an ensemble of Quantison TM microbubbles disappears between 15 and 30 ms after transmission of a high-MI ultrasonic burst [START_REF] Bouakaz | Noninvasive measurement of the hydrostatic pressure in a fluid-filled cavity based on the disappearance time of micrometer-sized free gas bubbles[END_REF], whereas the disappearance time of the fundamental acoustic response of an ensemble of PB127 microbubbles is shorter than 100 ms after transmission of a high-amplitude ultrasonic burst [START_REF] Burns | Looking at and listening to breaking bubbles: a correlative optical acoustic study of some experimental polymer/air agents[END_REF]. Figure 13 demonstrates simulations of the dissolution of nitrogen microbubbles with diameters between 0.5 and 4.5 µm into saturated water at ambient pressure. For the environment of a blood vessel, the saturation ratio of the gas will be different from the (saturated) in vitro situation. However, this difference in dissolution time for nitrogen is low (a few ms) compared to the dissolution time itself (10-100 ms). To overcome this difference, it has been suggested by us that one might use noble gases for released microbubble-based pressure measurements, so that the saturation ratio is zero [START_REF] Postema | Noninvasive microbubble-based pressure measurements: a simulation study[END_REF]. Since most released gas bubbles have equilibrium diameters between 1.25 and 1.75 µm, they will disappear within 15 ms. The longer decay times measured in Quantison TM [START_REF] Bouakaz | Noninvasive measurement of the hydrostatic pressure in a fluid-filled cavity based on the disappearance time of micrometer-sized free gas bubbles[END_REF] and PB127 [START_REF] Burns | Looking at and listening to breaking bubbles: a correlative optical acoustic study of some experimental polymer/air agents[END_REF] must be attributed to the relatively small number of large released gas bubbles around 4 µm (dissolution in 90 ms). These large bubbles might be formed by coalescence of released gas bubbles or fragments. The mechanism of ultrasound-induced lipid-encapsulated microbubble coalescence was recently studied [START_REF] Postema | Ultrasound-induced microbubble coalescence[END_REF]. Ultrasound-induced coalescence of released gas microbubbles is currently under investigation [START_REF] Postema | Ultrasound-induced coalescence of free gas microbubbles[END_REF].

For noninvasive pressure measurements, the influence of applying hydrostatic overpressures on the occurrence of sonic cracking will have to be investigated. Furthermore, the influence of pulse length and pulse repetition are of great practical interest.

LIST OF FIGURES -0.5 ms 0.0m s 0 .5 ms 1.0m s 1 .5 ms

Region of interest

High-MI ultrasound

Low-MI ultrasound

Fig. 1 .Fig. 2 .Fig. 3 .

 123 Fig. 1. Schematic overview of proposed clinical setup.

Fig. 4 .Fig. 5 .

 45 Fig. 4. Line drawings of the housing of the Brandaris-128 (left) and the CR 2000 camera (right).

Fig. 6 .

 6 Fig. 6. Gas release from a Quantison TM contrast microbubble at 0.5 MHz and MI = 0.9. Each frame corresponds to a 13 × 13 µm 2 area. Interframe times are 0.33 µs. The frames cover the second ultrasonic cycle.

Fig. 7 .

 7 Fig. 7. Dissolution of released gas. Each frame corresponds to a 13 × 13 µm 2 area. Time between recordings is 100 ms. The free gas bubble is visible to the upper left of the PB127 bubble in the left frame.

Fig. 8 .Fig. 9 .

 89 Fig. 8. Dissolution of released gas, captured at 2 kHz. Upper frames show raw CCD images, lower frames show segmented images. Each frame corresponds to a 10 × 10 µm 2 area. The released gas fragments (arrow) are less than 1 µm.

Fig. 10 .

 10 Fig.10. Three PB127 bubbles and one released gas bubble (left). After 100 ms the released gas bubble has disappeared (right). Each frame corresponds to a 23 × 23 µm 2 area. Time between recordings is 100 ms.

Fig. 11 .Fig. 12 .Fig. 13 .

 111213 Fig. 11. Size distribution of released gas from PB127 for MI> 0.3.

1

  Schematic overview of proposed clinical setup. . . . . . . . . . . . . . . . . . . . . . . . . . 2 Basic setup for taking sonic cracking pictures. . . . . . . . . . . . . . . . . . . . . . . . . . Quantison TM contrast microbubble at 0.5 MHz and MI = 1.1. Each frame corresponds to a 19×19 µm 2 area. Interframe times are 0.1 µs. The frames cover the second ultrasonic cycle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Gas release from a Quantison TM contrast microbubble at 0.5 MHz and MI = 0.9. Each frame corresponds to a 13 × 13 µm 2 area. Interframe times are 0.33 µs. The frames cover the second ultrasonic cycle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Dissolution of released gas. Each frame corresponds to a 13 × 13 µm 2 area. Time between recordings is 100 ms. The free gas bubble is visible to the upper left of the PB127 bubble in the left frame. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23×23 µm 2 area. Time between recordings is 100 ms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Size distribution of released gas from PB127 for MI> 0.3. . . . . . . . . . . . . . . . . . . 12 Average number of cracked contrast agent bubbles for MI> 0.8. . . . . . . . . . . . . . . . . 13 Dissolution of nitrogen microbubbles with diameters between 0.5 and 4.5 µm into saturated water at ambient pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

	3	Overview of the optical/ultrasound part of the experimental bubble-system. . . . . . . . . . .
	4	Line drawings of the housing of the Brandaris-128 (left) and the CR 2000 camera (right). .
	5	Gas release from a

8 Dissolution of released gas, captured at 2 kHz. Upper frames show raw CCD images, lower frames show segmented images. Each frame corresponds to a 10×10 µm 2 area. The released gas fragments (arrow) are less than 1 µm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Gas release from two PB127 contrast microbubbles (arrows) at 1.7 MHz and MI = 0.9. Each frame corresponds to a 46 × 30 µm 2 area. Interframe times are 0.1 µs. After release (first three rows), the free gas bubbles interact, until one free gas bubble remains (last row). . . .

10

Three PB127 bubbles and one released gas bubble (left). After 100 ms the released gas bubble has disappeared (right). Each frame corresponds to a

[START_REF] Landau | Fluid Mechanics[END_REF] DRAFT

or by 6 cycles in the recordings with the CR 2000 camera. An example of such a recording is given in Fig.8.[START_REF] Landau | Fluid Mechanics[END_REF] DRAFT

September 25, DRAFT

ACKNOWLEDGMENTS

We thank POINT Biomedical Corporation for supplying PB127, Upperton Limited for supplying Quantison TM , Leo Bekkering and Jan Honkoop for technical assistance, and Cees Pakvis for drawing Figure 4.

This work has been supported by the Technology Foundation STW (RKG.5104).


T Meyer

A Girard

G Le Dain

A Rhallabi

E Baudet

V Nazabal

P Němec

C Cardinaud

Surface composition and micromasking effect during the etching of amorphous Ge-Sb-Se thin films in SF 6 and SF 6 /Ar plasmas

Keywords: Ge-Sb-Se, Etching, ICP, SF 6, Micromasking

A functional waveguide for photonic applications must fulfil some specific requirements in terms of dimension, shape, etch rate and roughness. In this study, Ge-Sb-Se thin films were etched using an Inductively Coupled Plasma reactor via fluorine-based chemistry. In a SF 6 plasma, etch rate and roughness highlight a micro masking effect which originates from the formation of SbF 3 , (Se)-Sb-F x and (Sb)-Se-F environments. The latter have been identified with in situ XPS.

Systematically, a SF 6 plasma is associated with a quasi-isotropic profile and a rough surface.

In a SF 6 /Ar plasma, the impact of pressure and the argon content has been investigated. The addition of argon affects directly the fluorine atom flux and the argon atom flux which were calculated using a global model. It was found that there is a strong coherence between the fluorine atom flux, the proportion of fluorine at the surface and the RMS roughness. A synergistic effect, between ion bombardment and reactive neutral species, is observed when varying both parameters. Surface is free of fluorinated products for a high percentage of argon (95%) and low-pressures (< 4mTorr). A smooth surface and a quasi-vertical profile were achieved in a SF 6 /Ar with a gas mixture ratio 5/95 and at a pressure of 1.5 mTorr.

Introduction

Chalcogenide glasses (ChGs) contain at least one chalcogen element (S, Se, Te) and one or more elements from the 14 th (typically Ge) and/or 15 th (As, Sb) columns of the periodic table.

These materials are known for their good optical transmittance in the infrared region and their high linear and nonlinear refractive index [START_REF] Adam | Chalcogenide Glasses: Preparation, Properties and Applications[END_REF]. Consequently, ChGs such as As-S [START_REF] Madden | [END_REF], Ge-Sb-S [3], Ge-Sb-Se [4] or Ge-As-Se [5] are used as waveguides dedicated for infrared spectroscopy, all-optical signal processing etc. However, the use of the arsenic based-ChGs could be an issue for environmental applications because of its elemental form toxicity. A way to get around that is to substitute antimony to arsenic.

Many efforts are invested in the manufacturing of the Ge-Sb-Se waveguides with a promising application in monitoring of molecule concentrations, exploiting the molecule signatures in the mid-infrared region [6][7][8]. Nevertheless, the performance of a waveguide relies on its optical losses.

Key parameters as the composition, the dimension and the sidewall roughness of the waveguide affect its optical properties [9]. Therefore, there is a vast room of improvement to limit the optical losses by means of the deposition method and the patterning process. Plasma etching is a well-known technique providing anisotropic profile and smooth surfaces. Etching of chalcogenide thin films has been systematically studied using halogen-based plasmas such as CF 4 [10,11], CHF 3 [12], Cl 2 [13] etc. The etching of Ge 20 Sb 15 Se 65 and Ge 28.1 Sb 6.3 Se 65.6 has been performed using CHF 3 and CF 4 plasmas [14][15][16]. An etch rate of 1.4 µm/min and a sidewall angle of 85 • have been reported during the etching of Ge 20 Sb 5 Se 75 in CHF 3 [17]. To the best of our knowledge, the interaction between fluorine atoms and the Ge-Sb-Se system was not investigated in SF 6 plasma yet. Furthermore, using a different chemistry extends the fundamental knowledge of ChGs, and we can discuss about the viability for industrial application.

In this paper, we studied the etching of amorphous Ge-Sb-Se thin films using SF 6 and SF 6 /Ar plasmas. The former chemistry, though inadequate, brings out the limitation of fluorine-based plasmas, and the latter introduces an unused and effective way to achieve a quasi-vertical sidewall profile. The surface composition, the etch rate and the roughness are investigated when the pressure and the argon content are varied.

Methods

Ge 19 Sb 21 Se 60 thin films with a thickness of ~1.5 µm were deposited by pulsed laser deposition, by ablating a Ge 19.5 Sb 17.8 Se 62.7 bulk target with a KrF excimer laser (Compex 102, Lambda Physik) [18]. The composition of the thin films (± 1%) and the bulk target (± 0.5%) were determined by Energy Dispersive X-ray Spectroscopy (JSM 6400-OXFORD Link INCA).

In addition, vitreous Se (4N pellet; Codex International), germanium (5N 4" wafer; Goodfellow), Sb (1*2 cm ingot; HiChem SPOL S.R.O) and Sb 2 Se 3 (5 cm diameter target; HiChem SPOL S.R.O) were used. Sb and Sb 2 Se 3 powders were prepared from the related bulk materials.

All thin film samples were kept under inert atmosphere until etching or analysis in order to avoid oxidation and surface contamination.

Etching was performed in an Inductively Coupled Plasma reactor (ICP). The plasma source is located in the upper part of the reactor and consisted of an alumina tube of 18 cm in diameter and 26 cm in height, with an external loop antenna operating at 13.56 MHz. The diffusion chamber is 35 cm in diameter and 47 cm in height. Samples are placed at the centre of the reactor on a 100 mm substrate. The latter is mechanically clamped to the substrate holder and located 17 cm below the bottom of the alumina tube. The substrate temperature was maintained at 20 • C thanks to a He cooling system. Mass flow controllers adjust the process gas flow rate inside the diffusion chamber. Before the etching sequence, the chamber conditioning consists in a C 4 F 8 plasma for 30 minutes at 10 mTorr and 400 W to deposit a teflon-like layer on the reactor wall. Between processes, a cleaning procedure using H 2 /Ar plasma removes the etching products (GeF x , SbF x and SeF x ) from the reactor wall and the substrate holder.

For patterned samples, a standard photolithography process was applied using a S1818 photoresist with a thickness of ~1.5 µm. Etch depths were measured using a surface profilometer (DEKTAK 8 VEECO). Atomic force microscopy (NanoWizard 3 JPK Instruments) was performed to measure the RMS roughness with a scanning size of 3×3 µm 2 on blank samples.

The surface morphology was examined using a scanning electron microscope (SEM, JEOL JSM 7600F).

The composition of the surface and chemical bonding were examined by in situ X-Ray Photoelectron Spectroscopy (XPS) on blank samples. Samples are transferred under ultra-high vacuum from the etching chamber to the analysis chamber, via an intermediate buffer chamber. This procedure avoids any post etching contamination of the sample surface and preserves the XPS cham-ber from contamination from the etching vessel. The device is composed of a monochromatic Al Kα X-ray source (hν = 1486.6 eV, SPECS XR 50 M and FOCUS 500) and a hemispherical analyser (PHOIBOS HR 150). Wide spectra were recorded with a pass energy of 30 eV with an energy step of 1 eV. XPS narrow spectra were obtained using a pass energy of 14 eV with an energy step of 0.1 eV. Sample charging was neutralized with an electron flood gun. Data were processed using CASA-XPS software where Ge 2p 3/2 , F 1s, Sb 3d and Se 3d core levels were decomposed using a Shirley background [19] and a Lorentzian function convoluted with a Gaussian. Sensitivity factors were retrieved from GeSe 2 and Sb 2 Se 3 powders and Ge 19 Sb 21 Se 60 thin film. To probe variation of the relative atomic proportions of Ge, Sb and Se within the analysed depth and versus the etching conditions, atomic percentages were calculated using the Ge 2p 3/2 , Sb 3d and Se 3d core levels without taking into account the F 1s area. However, the relative area of the F 1s region was calculated by taking into account all investigated core levels. It should be pointed out that the inelastic mean free path of Ge 2p 3/2 , Sb 3d and Se 3d photoelectrons are different. Thus, the atomic percentages are calculated over different analysed depths. Energy calibration of Ge, Sb and Se XPS spectra was performed using the Ge-Ge (Binding Energy, BE = 29.0 eV), Sb-Sb (BE = 528.0 eV) and Se-Se (BE = 55.3 eV) chemical states. Sb 2 Se 3 and Ge-Sb-Se XPS spectra were calibrated using the [SbSe 3/2 ] motif (BE = 529.0 eV) as the reference value. The uncertainty of the binding energy values is ± 0.2 eV. The d 5/2 and d 3/2 components of individual doublets were constrained with an equal Full Width Half Maximum. Moreover, the peak separation and the peak area ratio were set at 9.35 eV -0.667 for Sb 3d and 0.85 eV -0.72 for Se 3d. The mentioned fit parameters were retrieved from nonetched Ge, Sb and Se samples.

A kinetic model was used to provide a basic understanding of the plasma chemistry. It was developed for a low-pressure, RF (13.56 MHz), ICP SF 6 /Ar mixture. The comparison between modeling and experimental data can be found elsewhere [20]. The model takes into account kinetic reactions related to SF x , F and Ar as the attachment, excitation, dissociation and ionization by electronic impact. Rate coefficients are calculated from cross-section data considering a Maxwellian electron energy distribution function [21][22][23][24]. Rate coefficients for neutral recombination and ion recombination reactions are included in the model [25,26]. Diffusion of reactive neutrals and positive ions is considered [27]. Onto surface, adsorption and desorption can occur, regarding the atom or molecule concerned. On the basis of experimental data made in SF 6 [20], the sticking coefficient was set at 0.02 for SF x (x = 0, 1, 3, 5) and F species. The model solves plasma physics equations system: the mass-balance equations, the charge neutrality equation and the power balance equation. Reaction rate coefficients were injected in the mass-balance equations to determine the density of each species considered. Charge neutrality equation and power balance equation evaluated the electron density and the electronic temperature, respectively. Input parameters are the dimension of the simplified reactor (cylindrical geometry), the source power, the pressure, the total flow rate and gas temperature. The latter was assumed equal to the temperature of the wall (300 K). The model provides information about the plasma parameters and the particles characteristics (density, flux). In the present paper, the model was used for the SF 6 /Ar chemistry to calculate the radical and ion fluxes.

Results and discussion

Etching in pure SF 6

To understand the etching mechanisms, it is crucial to get information about the volatility of the etching products. Without pertinent data on the species, only a minimalist approach can be made with the use of the boiling temperatures as for GeF 2 and GeF 4 . In other cases, the vapour pressures can be calculated from Antoine coefficients as for SbF 3 , SbF 5 , SeF 2 and SeF 4 [START_REF] Addison | Inorganic Chemistry of the Main-Group Elements[END_REF][START_REF] Abney | [END_REF][START_REF] Yaws | The Yaws Handbook of Vapor Pressure (Second Edition)[END_REF]. As displayed in Fig. 1, it is expected that the SbF 5 , SeF 6 , SeF 4 , GeF 4 and GeF 2 species are volatile and the SbF 3 molecule is nonvolatile, and so, the latter should be detected at the surface. The vapour pressures data are reliable for the etching of pure elements. For binary or ternary materials, mixed nonvolatile products may occur depending on the structure of the etched material and the plasma chemistry.

In situ XPS analysis

Wide XPS spectra are presented in Fig. 2 after etching. Literature concerning the XPS analysis of Ge-Sb-Se system shows that SbSe 3/2 pyramids and GeSe 4/2 tetrahedra are the dominant motifs [18,[START_REF] Wei | [END_REF]32]. The quantification of relative atomic percentages shows a Ge-depletion considering Ge, Sb and Se elements. In the absence of bias, a weak Al contamination is present at the surface, coming from the alumina tube. In this specific case, the Al 2s and Al 2p are distinct from the Ge 3p core level (BE = 123 eV) and from the Se 3d energy loss (BE ≈ 70/80 eV), respectively. In addition, the F 1s area represents 55% of the summed areas (Ge 2p 3/2 + F 1s+ Sb 3d + Se 3d), implying the presence of fluorinated compounds at the surface. Note that sulfur is not detected due to an overlapping of its contributions with the Se 3s and 3p core levels. The relative atomic percentage of sulfur at the surface is considered to be negligible (at. < 1%) compared to that of Se. Se 3d XPS spectra are presented in Fig. 5. A pure vitreous Se and a Sb 2 Se 3 powder provide information about the Se-F and the (Sb)-Se-F species, although the Sb 2 Se 3 spectrum is not decomposed because of large uncertainty. For the selenium sample, no other contribution than the homopolar Se-Se bond is detected, although two peaks at 55.3 eV and 55.7 eV are needed since vitreous Se can form Se 8 rings and chains [36,37]. The absence of fluorinated species confirms the volatility of the SeF 4 and SeF 6 products. Then, the bonding of Sb to Se, Sb 2 Se 3 , induces a chemical shift to a lower binding energy since the electronegativity of antimony (2.05) is lower than that of selenium (2.55). The broad contribution from 57 to 61 eV corresponds to the mixed (Sb)-Se-F products which is obviously independent of a Ge depletion. It should be noticed that the neighbouring atoms of a Se atoms is different than that of Sb atoms. [18,[START_REF] Wei | [END_REF]32,38].

Turning to the etched Ge-Sb-Se thin film, the large contribution centred at 57.2 eV is assigned to be the (Sb)-Se-F environment. As shown by comparing with the pure selenium sample, the doublet cannot be solely the Se-F species because of their volatility. Besides, no energy shift is reported for the Se-Se-(Ge, Sb) (BE = 53.8 eV) and Se-(Ge, Sb) 2 (BE = 54.6 eV) environments.

A micromasking effect

Table 1 presents the etch rate, the RMS roughness and simulation data regarding the etching conditions. It is expected that the interaction, between fluorine atoms and the elements within the amorphous material, forms nonvolatile products regardless of the conditions. Thus, the process could be apprehended as an outcome between the formation rates of these products, the etching of these products and the etching of the thin film. At lower bias potential (-7 V), nonvolatile products accumulate at the surface forming a rough surface and reducing the interaction between fluorine atoms and the underlying material. As the bias increases, the energetic ions break the Sb-F bonds and eject the atoms from the surface, giving also a smoothing effect. The etch rate is closely related to the sputtering yields. Therefore, it is coherent to observe an increase of the etch rate with the bias potential.

Fig. 6 shows the simulation data as function of the chamber pressure. The high dissociation rate of SF 6 (Fig. 6a) and the high electronic density (Table 1) lead to higher fluorine atom flux and fluorine ion flux (Fig. 6b), which are likely to react with the Ge-Sb-Se. Note there is discrepancy in literature concerning the variation of the fluorine atom density which must be discussed. Yang densities increase [41]. Haidar et al. reported an increase of the fluorine flux with the decreasing pressure, in which the variation is thought to be related to the dissociation of SF 6 [42]. The major difference between the different simulation studies is mainly due to the electronic density which can strongly increase with the decreasing pressure, or at the opposite, exhibits a low variation.

In the former case, the fluorine atom flux is expected to increase with the dissociation degree of SF 6 , compensating the loss of SF 6 total pressure, whereas in the latter case, the increasing feed gas pressure enhances the production of fluorine atom. As the pressure increases, the total positive ionic flux decreases faster than the fluorine atom flux, resulting in a slight increase of the Γ F Γ I+ ratio (Table 1). Therefore, the etching of nonvolatile species is less effective and the etch rate drops with the working pressure. 1.

Regardless of the conditions used in SF 6 pure plasma, the surface can be assimilated as nanoislands as shown in Fig. 7. The extreme situations arise at lower pressure and lower bias potential with an important accumulation of nonvolatile species.

As can be seen in Fig. 8a, the etch rate is sharply reduced and the RMS roughness is increased until the nanostructured islands reach a maximum height (Fig. 8b), which is typical of a micromasking effect. The mechanism is known to arise from metal contamination (hardmask, wall) or nonvolatile etch products formation. Although an Al-F environment is detected with an unbiased substrate holder, the environment is undetected when applying a bias (V bias < -50 V) to the substrate holder. In view of this information, the origin of the micromasking, discussed in Fig. 8 and further in this paper, is solely attributed to nonvolatile etch products. An etch rate of approximately 3.5 µm/min has been reported for the etching of the Ge-Se related material in a 100% SF 6 plasma with an unbiased substrate, confirming that the [GeSe 4/2 ] motif is chemically etched [43]. At the surface of the Ge-Sb-Se thin film, most of the Ge atoms are chemically etched during the first minute, producing GeF 2 and GeF 4 volatile etch products, whereas the (Se)-Sb-F x , SbF 3 and (Sb)-Se-F environments accumulate at the surface. During that short etching period, both chemical etching and physical etching are effective, resulting in a higher etch rate. As the surface becomes Ge-depleted, the process relies mainly on the ion sputtering of nonvolatile products, and the synergistic phenomenon between ion bombardment and chemical etching is significantly lessened, which leads to a stabilization of the etch rate.

A 100% SF 6 etching process is known to induce a chemical etching process. It is notably efficient for the elements of the 14 th column (Si, Ge) since the process forms volatile products (SiF 4 , GeF 4 ). However, the efficiency of a process is mostly affected by its limit. For the ternary material, we identify the constraint as the formation of (Se)-Sb-F x , SbF 3 and (Sb)-Se-F entities. Fig. 9 shows a SEM oblique image of a trench-patterned in the Ge-Sb-Se thin film, after the removal of the resist mask. Region (1) corresponds to the trench top, region (2) to the trench sidewall and region (3) to the trench bottom. As seen on the figure, the trench sidewall is very rough as compared to the trench bottom. Our interpretation is that this roughness results from the accumulation of nonvolatile etch products. Two main origins can be proposed: the deposition of non-volatile species coming from the etching of the trench bottom and the less efficient etching of the sidewall in relation to the weaker ion flux received. Fig. 9: Oblique view SEM images of a trench-patterned Ge-Sb-Se thin film after etching with a SF 6 plasma at 10 mTorr, 1200 W, -100 V and 20 sccm (Condition C 5 in Table 1 during 5 minutes). ( 1) Top ( 2) Sidewall (3) Bottom.

Thereafter, we propose to slow down the formation rate of the (Se)-Sb-F x , SbF 3 and (Sb)-Se-F entities by adding argon to SF 6 . The idea is to increase the ionic flux and to decrease the fluorine atom flux. The same approach was experimented on silica glass [44] and on the Ge-Sb-Te material [45].

Etching in SF 6 /Ar

Effects of argon addition

Plasma modeling data are displayed in Fig. 10a, Fig. 10b and Table 2. The increase of the electronic density (Table 2) with the argon percentage in the SF 6 /Ar mixture is consistent with the experimental data [20] and the literature [39,[START_REF] Hebner | Characterization of SF6/Argon Plasmas for Microelectronics Applications[END_REF][START_REF] Oh | [END_REF]. As the plasma density increases, the dissociation of the precursor is enhanced as well as the ionization of reactive neutral species. It results in a slight decrease of the fluorine atom flux, and a near-constant variation of the F + and SF +

x fluxes. Above 40% of argon, the dominant contribution of the neutral flux is the fluorine atom flux with values one to three orders of magnitude higher than those of the SF x radicals.

It was experienced that adding argon to fluorine-based plasmas increases the ionic flux for a high content of argon [26,44,48]. The same behaviour is observed with plasma modeling.

For the investigated argon percentage (40 to 95%), Ar + is the dominant contribution of the ionic flux. Even if the flux of F + is of the same order of magnitude, one must consider the sputtering yield which is higher for Ar + sputtering than F + sputtering. Between 80 to 95% of argon, a noticeable variation of the Γ F Γ Ar + ratio (Table 2) provides a compelling evidence that the etching moves towards an important ion bombardment regime. 2.

due to the Sb-F bonding, which is responsible of the micromasking effect. As can be seen in Table 2, we note a nonmonotonic variation of the etch rate at 60% of argon, resulting of a synergistic effect because of the increasing Ar ion flux (see Fig. 10b). As evidenced in Fig. 11a, there is a noticeable drop of the relative area of the fluorinated environments (Sb-F x and Se-F) from 40 to 60% of argon. With the increasing Ar ion flux, the removal rate of nonvolatile species such as (Se)-Sb-F x and SbF 3 is increased, leading to an increase of the Sb-Se bonds at the surface. Concerning the Sb 3d core level, the [SbSe 3/2 ] motif is dominant at 60% of argon, and the sputtering of atoms (Sb and Se) is likely to be enhanced with the increasing ionic flux.

Nevertheless, it is worth noting that the relative area of fluorinated species remains significant. In that regard, the surface contains two distinct chemical bonds for Sb atoms: Sb-F and Sb-Se. As the bond energy for SbF 3 is higher (440 kJ/mol) [START_REF] King | Encyclopedia of inorganic chemistry[END_REF] than that of the pyramidal [SbSe 3/2 ] (179 to 184 kJ/mol) [START_REF] Dahshan | [END_REF][START_REF] Sharda | Defect Diffus[END_REF][52], it is expected that both species reacts differently to the increasing ion bombardment. Such disparity could be responsible of the higher RMS roughness (30.5 nm). At 80% of argon, the surface contains less fluorine (10.9% of total area) and is G e-enriched. Since the quantification is performed using the Ge 2 p 3/2 core level and not the Ge 3d core level, the Ge relative percentage is overestimated, and so, Se and Sb relative percentages are underestimated.

At 95% of argon, SbF 3 , (Se)-Sb-F x and (Sb)-Se-F environments are undetected (at. < 1%).

Simultaneously, the AFM measurement presents the lowest value of RMS roughness (1.4 nm) and the lowest etch rate (53 nm/min), meaning that the rate limiting step of the SF 6 /Ar (5%/95%) etching is the fluorine atom flux. 2.

Impact of varying the plasma mixture on pattern etching was also investigated. SEM images of the Ge-Sb-Se patterned samples (etch depth ~1.2 µm) are shown in Fig. 12. As can be seen at 95% of argon in Fig. 12a, there is a very smooth surface with a quasi-vertical profile. It is also the condition where the fluorine atom concentration is the lowest whether it is in the plasma or at the surface. At 80% of argon (Fig. 12b), the patterned Ge-Sb-Se contains some residues on the surface and a large sidewall angle. At argon percentage lower than 60% of total pressure (Fig. 12c and Fig. 12d), the micromasking effect is apparent and it is related to the important concentration of nonvolatile species at the surface. It seems that there is a strong coherence with blank (Table 2) and patterned samples regarding the roughness.

Effects of pressure

Along with the argon content and the bias, the pressure is usually a key parameter to optimize an etching process. Through the variation of pressure, the plasma chemistry can be modified drastically. In this section, we investigate the etching of the Ge-Sb-Se thin films using a gas mixing ratio SF 6 /Ar=5/95.

Plasma modeling data are presented in Fig. 13 and Table 2. Total radical fluxes and total ionic for Ar content greater than 30%, attributed to the competition between the ionization processes (i.e. electron production) and the dissociative attachment (i.e. electron loss) [39]. It should be noticed that the mass-balance equation takes into account the loss rates of neutral species to the wall due to the diffusion [20]. The gain of fluorine through dissociation of SF 6 is sharply offset by the surface loss at lower pressure. Therefore, the fluorine atom flux drops as going from 8 to 1 mTorr. As the pressure decreases, the energy to produce a positive ion is reduced because of the high dissociation rate of the SF x fragments. It accentuates the ionization processes at lower pressure.

The area ratios extracted from Sb 3d and Se 3d XPS spectra are plotted in Fig. 14a. The relative area of F 1s and the relative atomic percentages are displayed in Fig. 14b. At 2 mTorr, the formation rate of the fluorinated species is offset by the significant ion bombardment (Fig. 13). In other terms, the newly formed Sb-F bonds are constantly broken, leading to a very smooth surface from 0.5 to 2 mTorr with RMS roughness values between 0.9 and 1.4 nm (Table 2). Beyond 2 mTorr, the argon ion flux drops and the chemical etching of Ge become efficient. In addition, the SbF 3 and (Se)-Sb-F x species are detected at the surface and, as mentionned in Sec 3. 2 with a bias set at -150 V.

Fig. 15 shows the SEM images of etched profiles for low and high Γ F Γ Ar + conditions (see Fig. 13 and Table 2). As can been seen in Fig. 15a and Fig. 15b, the lower pressure case (1.5 mTorr) exhibits a very smooth surface and a quasi-vertical profile. The condition at 16 mTorr (Fig. 15c and Fig. 15d) suggests, as with a 100% SF 6 etching, that the nonvolatile species induces a micromasking effect and a tapered profile due to a redeposit on the sidewall.

Conclusion

Ge-Sb-Se thin films were studied in pure a SF 6 plasma. We identified the origin of roughness as the formation of SbF 3 , (Se)-Sb-F x and (Sb)-Se-F environments. We presented a way to reduce the formation rate of the nonvolatile products by adding argon to SF 6 plasma. XPS analyses showed explicitly that the Sb-F or Se-F bonds were not detected for a high argon percentage (95%) and for low-pressure processes (< 4 mTorr). The surface composition and the etching profile were improved by adding argon, despite the fact that the etch rate decreases with argon content. By optimizing the argon content and the pressure, it was possible to obtain a very smooth surface and a vertical profile. These are promising results for the SF 6 /Ar mixture. Further optimizations should be focused to improve the sidewall roughness.

It should be pointed out that this study is of interest for the etching of antimony-based materials using fluorine-based plasmas at ambient temperature.

Fig. 1 :

 1 Fig. 1: Vapour pressures and boiling temperatures of fluorinated compounds [28-30].

Fig. 2 :

 2 Fig. 2: Wide XPS spectra before and after SF 6 etching. (Etching conditions: 10 mTorr, 20 sccm, 700 W and unbiased substrate)

Fig. 3 :

 3 Fig. 3: Ge 2p 3/2 spectra after SF 6 etching. (Etching conditions: 10 mTorr, 20 sccm, 700 W and unbiased substrate)

Fig. 4

 4 Fig. 4 shows the Sb 3d spectra before and after SF 6 etching. For comparison purposes, the analysis was also performed on Sb and Sb 2 Se 3 powders. On the former, the Sb-Sb homopolar bond is positioned at 528.1 eV. The doublets at 529.6 eV (CS = 1.5 eV), 531.1 eV (CS = 3.0 eV) and 532.1 eV (CS = 4.0 eV) are assigned to the (Sb)-Sb-F x species owing to the shift to higher values of binding energy. Important shifts have been reported for SbF 3 (BE = 531.9 eV; CS = 3.4 eV) [33], SbF 4 (BE = 531.0/532.0 eV) [34], KSbF 5 (BE = 532.7 eV; CS = 4.7 eV) [35], CSbF 5 (BE = 532.3). On the nonetched Ge-Sb-Se sample, the dominant contribution is attributed to the [SbSe 3/2 ] motif at 529.0 eV (CS = 0.9 eV). The Sb-O and O 1s contributions originate from the air contamination. As displayed on the Sb 2 Se 3 spectra, the contribution at lower energy (BE = 529.0 eV) is assigned to the [SbSe 3/2 ] motif. The superposition of the selenide component with the (Sb)-Sb-F x species is not considered because it would imply the detection of the Sb-Sb homopolar bond. The adsorption of fluorine on the Sb-Se system generates the (Se) 2 -Sb-F entity at 531.2 eV (CS = 3.1 eV) and the SbF 3 species at 532.1 eV (CS = 4.0 eV). 7

Fig. 4 :

 4 Fig. 4: Sb 3d spectra after SF 6 etching. (Etching conditions: 10 mTorr, 20 sccm, 700 W and unbiased substrate)

Fig. 5 :

 5 Fig. 5: Se 3d spectra after SF 6 etching. (Etching conditions: 10 mTorr, 20 sccm, 700 W and unbiased substrate)

Fig. 6 :

 6 Fig. 6: a) Radical fluxes and b) positive ion fluxes according to the pressure at 1200 W and 20 sccm (modeling).

Table 1 :

 1 Etch rate, roughness and simulation data (Γ F : fluorine atom flux; Γ I+ : positive ionic flux; n e : electronic density) according to the plasma conditions with a etch time of 5 minutes and a SF 6 flow rate set at 20 sccm. The margin error of the etch rate is ± 2 nm/min.

et

  et al. reported an increase of the fluorine atomic density[39], whereas Kokoris et al. observed a nonmonotonic variation[40]. Tinck et al. justified the drop of SF x densities with the chamber pressure due to the recombination reactions within the plasma, while showing the F and SF 6

Fig. 7 :

 7 Fig. 7: AFM scans of etched Ge-Sb-Se thin films. Corresponding conditions are shown in Table1.

Fig. 8 :

 8 Fig. 8: a) Etch rate and RMS roughness; b) AFM trace after SF 6 etching. (Etching conditions: 10 mTorr, 20 sccm, 700 W and -100 V)

Fig. 10 :

 10 Fig. 10: a) Radical fluxes and b) positive ion fluxes according to argon content at 900 W, 2 mTorr and 40 sccm (modeling).

Fig. 11a presents

  Fig. 11a presents the area ratios calculated from decomposed Sb 3d and Se 3d XPS spectra; Fig. 11b shows the relative atomic proportion of Ge, Sb and Se; and the relative area of the F 1s region; and Table 2 lists the etch rate and roughness according to the Ar percentage in the SF 6 /Ar mixture. Drastic change of surface composition are evidenced with the reduction of the fluorinated species compared with the related selenide species. The analysis of the Ge 2p 3/2 is not reliable because of the overlapping between the (Se)-Ge-F x species and the [GeSe 4/2 ] motif.

Fig. 12 :

 12 Fig. 12: SEM images of trench-patterned of Ge-Sb-Se thin films after etching with SF 6 /Ar at a) 95% Ar (etch time: 23 min) b) 80% Ar (etch time: 6 min) c) 60% Ar (etch time: 2 min 30) and d) 40% Ar (etch time: 6 min). Corresponding conditions are shown in Table2.

Fig. 13 :

 13 Fig. 13: Fluxes according to the pressure at 900 W, 95% of Ar and 40 sccm (modeling). (Γ n : total radical flux; Γ i+ : total ionic flux)

2 . 1 ,

 21 Fig. 14: a) Area ratio b) Ge, Sb and Se relative atomic percentages and F 1s relative area as a function of the pressure.The dash lines correspond to the atomic percentages of Ge, Sb and Se of the nonetched Ge-Sb-Se thin film. Corresponding conditions are shown in Table2with a bias set at -150 V.

Fig. 15 :

 15 Fig. 15: SEM images of patterned trench of Ge-Sb-Se thin films and blank Ge-Sb-Se thin films after dry etching with SF 6 /Ar (5%/95%) at 900 W, -200 V, 40.5 sccm at a) 1.5 mTorr (oblique view; etch time: 20 min) b) 1.5 mTorr (top view; etch time: 5 min) c) 16 mTorr (oblique view; etch time: 20 min) d) 16 mTorr (top view; etch time: 5 min).

Table 2 :

 2 Etch rate, RMS roughness and simulation data (Γ F : fluorine atom flux; Γ Ar + : argon ion flux; n e : electronic density) according to plasma conditions with the source power set at 900 W. The margin error of the etch rate is ± 2 nm/min.

	Ar percentage	Pressure Bias potential Etch rate RMS roughness	Γ F	n e
	(of total pressure) (mTorr)	(-V)	(nm/min)	(nm)	Γ Ar +	(10 11 cm -3 )
	95	2	150	53	1.4 ± 0.1	7.1	7.76
	90	2	150	89	1.5 ± 0.1	30.0	9.17
	80	2	150	197	7.2 ± 0.2	93.5	8.14
	60	2	150	467	30.5 ± 4.0	441.8	4.11
	40	2	150	203	8.5 ± 0.6	1050.5	2.58
	95	0.5	200	78	0.9 ± 0.1
	95	1	200	94	0.9 ± 0.1	2.4	5.97
	95	2	200	94	1.4 ± 0.2	7.1	7.76
	95	4	200	129	4.6 ± 0.2	19.3	10.4
	95	10	200	68	4 ± 0.1	61.2	5.12
	95	16	200	63	6.6 ± 0.7	82.3	2.24

Acknowledgements

Czech Science Foundation (project no. 19-24516S) and the Barrande exchange program between France and the Czech Republic are greatly acknowledged for supporting this work.


M Goryachev
email: maxim.goryachev@femto-st.fr

S Galliou

Philippe Abbé

V Komine

Oscillator Frequency Stability Improvement by Means of Negative Feedback

Maxim Goryachev, Serge Galliou, Philippe Abbé, and Vadim Komine Abstract-A novel, simple method is proposed to increase the frequency stability of an oscillator. An additional negative feedback is used in combination with the positive loop of the harmonic oscillator to decrease the phase sensitivity to fluctuations of parameters other than the resonator. The main advantage of the proposed correction approach is that it does not require expensive external elements such as mixers or resonators. The validity of the method is theoretically demonstrated on a Colpitts oscillator using the control system theory approach and numerical simulations, and is experimentally verified with phase noise measurements of an actual oscillatormockup. It is shown that the medium-term frequency stability can be easily improved by a factor of ten.

I. I

I  the oscillator frequency stability [1]- [3] is an important issue for many modern systems dealing with communication, navigation, radar applications, etc. Traditionally, frequency stability is achieved by minimizing the environmental influence, maximizing the loaded quality-factor Q of the resonator, minimizing the noise of the sustaining amplifier, and optimizing the oscillator elements. Even if the oscillator phase noise performance is optimized by these means, it can be further improved by other noise reduction techniques. A summary of the most common amplifier and oscillator feedforward and feedback noise reduction techniques is given in [START_REF] Mcneilage | Review of feedback and feedforward noise reduction techniques[END_REF]. Some of these techniques are VCO-based locking involving one of the following principles: high-Q resonance element as an external discriminator [START_REF] Walls | High spectral purity x-band source[END_REF], Pound discriminator [START_REF] Pound | Electronic frequency stabilization of microwave oscillators[END_REF]- [START_REF] Santiago | Improved performance of a temperature compensated LN2 cooled sapphire oscillator[END_REF], or carrier suppression using a high-Q cavity [START_REF] Santiago | Microwave frequency discriminator with a cryogenic sapphire resonator for ultra-low phase noise[END_REF]. The main disadvantage of such techniques is that an additional resonant element is required, which is usually the most critical device and adds its own noise. To avoid this, at least two other techniques may be considered: interferometric carrier suppression [START_REF] Ivanov | Tunable microwave oscillator for low phase noise applications[END_REF]- [START_REF] Ivanov | Ultra-low noise microwave oscillator with advanced phase noise suppression system[END_REF], and the high-Q cavity as an external discriminator [START_REF] Tsarapkin | Stable microwave oscillator with combined stabilization[END_REF], [START_REF] Galani | Analysis and design of a single-resonator GaAsFET oscillator with noise degeneration[END_REF]. These techniques use the information extracted from the oscillating loop for its correction with the voltage-controlled phase shifter. When analyzing all of these approaches, one common idea can be retained: all of the techniques try to transform the oscillator phase noise into a dc voltage by a phase detec-tion for further feedback correction. In contrast with these traditional phase-noise reduction approaches [START_REF] Mcneilage | Review of feedback and feedforward noise reduction techniques[END_REF], the idea developed in this paper does not require a signal-phaseto-voltage conversion. The proposed structure decreases the dependency of the oscillation frequency on fluctuations of oscillator components other than the resonator. This improvement is achieved because of the high-level improvement of the oscillator architecture (structural optimization) and not by a parametric optimization as is proposed, for example, in [START_REF] Filicori | Frequency stability in resonator-stabilized oscillators[END_REF].

A basic harmonic oscillator consists of a resonator and a sustaining amplifier interconnected in the form of a positive-feedback closed loop [START_REF] Rubiola | Phase Noise and Frequency Stability in Oscillators[END_REF]. To sustain harmonic oscillations, the loop must be designed with respect to the well-known Barkhausen conditions. In actual applications, both the amplifier and the resonator are subject to parametric fluctuations. Because the oscillator feedback is positive, these fluctuations reproduce themselves after passing through the loop, according to the Leeson effect [START_REF] Leeson | A simple model of feedback oscillator noise spectrum[END_REF]. This results in a multiplication of the close-to-carrier open loop phase-noise power spectral density (PSD) by f -2 within the resonator bandwidth. To avoid this drawback, the sign of the feedback must be negative as it is in a phase-locked loop (PLL) [START_REF] Klapper | Phase-Locked and Frequency-Feedback Systems[END_REF]. Indeed, the resulting PSD of a PLL depends on the open-loop PSD of its voltagecontrolled oscillator (VCO). Unfortunately, the PLL requires an external frequency reference source because a single negative loop is not able to sustain oscillations. As a result, the output close-to-carrier noise depends on the noise of this external frequency source. In addition, at least two resonators are used in this case to implement the system: one is in the VCO and another in the external reference. Is it possible to combine the ability of the usual positive feedback to sustain the oscillation and the property of the negative feedback to reduce the close-to-carrier phase noise by means of their structural integration using just one resonator? In this paper we try to answer this question by combining both in one oscillator.

It must be noted that the proposed method of phasenoise reduction is not an automatic gain control (AGC). The proposed solution deals with the reduction of oscillator frequency, but not amplitude fluctuations. Thus, the amplitude detector and a traditional variable gain amplifier are not used (even though detectorless AGC systems can also be of great interest). Instead, the oscillating signal is fed back into a sustaining amplifier with no averaging or detection of any kind.

The system under investigation is based on a common Colpitts quartz-crystal oscillator. The corresponding analytical description [START_REF] Goryachev | Oscillator phase noise optimization and correction[END_REF] differs from traditional approaches [START_REF] Rhea | Oscillator Design and Computer Simulation[END_REF]- [START_REF] Quere | Low phase noise oscillator modeling in a system environment[END_REF]. Also, for simplicity, some elements are assumed to be ideal. The analytical results are further confirmed with numerical models using a commercial circuit simulator. This is an intermediate stage toward a conceptual verification. Though an actual implementation of the analyzed circuit is obviously slightly more complicated, both analytical and simulation investigations are in good qualitative agreement with experimental results. The improvement of the oscillator long-term frequency stability is demonstrated.

II. M D

The analyzed crystal oscillator is shown in Fig. 1. This modified Colpitts design has its usual positive feedback loop including the crystal filter and a sustaining amplifier. An additional negative feedback includes an idealized transfer function H(p) (where p ≡ d/dt is the differentiation operator) which drives a voltage-controlled voltage source U v . The first feedback loop is intended to sustain oscillations whose frequency is mainly defined by the crystal (X1) resonance frequency, slightly shifted by other oscillator components such as C 1 and C 2 . The second feedback loop uses the voltage across the resonator to reinject the oscillating signal into the sustaining amplifier, but with an open-loop phase shift close to ±π.

The main idea of the proposed analysis is to simplify the design through successive equivalent transformations (∆ -Y and Y -∆ transforms, series and parallel connections, transforms of dependent signal sources) and to investigate the influence of the second negative feedback loop [different solutions for H(p)]. The oscillator is analyzed like a feedback system, as is usual in the control system theory.

At the first step, the oscillator must be properly transformed. In addition to equivalent transformations of the linear elements, the transistor is represented with its wellknown Ebers-Moll model. The result is shown in Fig. 2. This oscillator representation has only one nonlinear element: the base-emitter junction modeled with I e (V be ). Special attention should be paid to the fact that elements Z x , Z l , and Z e are functions of the parameters C 1 , R e , C 2 , and X 1 , or more precisely,

Zp L pC RZ pC Z pC R xx x xl ee =+ += = 11 1 21 ,, || , (1)
where L x , C x , and R x are the motional parameters of the crystal resonator. Other elements of the system in Fig.

++ -                         () , () =1 , II Z IU I V u exp           (2)
where I is a current through elements E u , U a (I e ) = -Z n (p) I e . I s and u t are transistor parameters.

System (2) corresponds to the idealized physical system shown in Fig. 2. The same system may be represented from the point of view of control system theory, i.e., using blocks of transfer functions as follows (see Fig. 3): In the same way as an original circuit, the system shown in Fig. 3 represents a two-loop nonlinear system with positive and negative feedbacks. The nonlinearity is represented with a nonlinear relationship I e (U be ). The common positive loop is represented with the linear transfer function (or physically the impedance) Z fb1 (p) and the nonlinear transconductance G = I e (U be ). The negative loop consists of the linear transfer functions K a (p), H(p), K s (p), Z a (p), K u (p), and the nonlinear transconductance G = I e (U be ).

K Z Z K ZZ ZZ ZZ ZZ ZZ ZZ ZZ ZZ Z
It must be noted that if H(p) ≡ 0, the second loop is broken and the system dynamics are characterized only by the linear transfer function Z fb1 (p), and the corresponding amplitude is determined by nonlinear transconductance G and impedance Z fb1 (p). This case corresponds to the classical Colpitts oscillator. If H(p) ≠ 0, the two-loop system may be equivalently transformed into a system with one loop and another linear transfer function Z fb2 (p).

Further, for simplicity, it is assumed that K u (p) = 1 without lack of generality. This is possible when the capacitance C f is large enough.

III. M A

The nonlinear dynamic system shown in Fig. 3 can be represented by nonlinear and linear parts in one closed loop. Different methods to find oscillation amplitude and frequency are known for this type of nonlinear systems: for example, the method of describing functions [START_REF] Levine | Control System Fundamentals[END_REF], [START_REF] Atherton | Stability of Nonlinear Systems[END_REF], also called the harmonic balance method in control theory. This method is very useful for high-order systems and helps to find the oscillation conditions from the open-loop transfer function of the whole system.

In the absence of the additional negative feedback, the linearized open-loop transfer function of the oscillator can be expressed as

H GZ ol1 fb1 = 0 - , (4)
where G 0 is a linearized transconductance of the nonlinear element I e (U be ) in the vicinity of the operating point. As previously mentioned, if the second loop exists, it may be represented as a modification of the first positive loop. In this case, one can also calculate the open-loop transfer function of the oscillating loop as

H GZ ZZ ZZ ZZ ZH p ZZ ZZ Hp ol2l an xa ee x al ex = () (1 () ) () (1 () ) 0 - -+ +- ++ +- Z ZZ Z xl e () , + (5)
Thus, for example, in the ideal case of H(p) = 1, the transfer function ( 5) is considerably simplified:

H GZ ZZ ZZ Z Z Z ol2l xe lex n a = 1 0 - - ( ) + ++ . ( 6
)
The existence of oscillations is found from Barkhausen conditions, which may be written as ℑ{} H i ol = 0 and |H oli | > 0, where i ∈ {1, 2} and ℑ{} denotes an imaginary part. From the first equation of the Barkhausen conditions, one may write the equation necessary for calculating the oscillation frequency. Thus, in the absence of the second loop, the oscillating frequency may be found from the sixth-order equation

ω ωω ω ωω ττ ωω ω ττ 64 0 2 0 2 2 1 22 1 2 0 4 0 2 2 1 2 1 -- ++-         ++     - Q Q Σ Σ Σ 2 2 2 =0 0 2 1 0 22 0 2 1 2 1 22 1 0 4 1 0 ω ττ ωω ωω ωω ω τ ω ττ ω Σ Σ Σ Σ Σ ++ ++     + Q , (7)
where

ω ω ω ω ττ 0 2 0 1 2 1 2 12 12 11 = 1 ,= ,= 1 , =, =, CL Q L CK CL CC LCC KRC xx x xm x x mm Σ + Σ Σ = 12 . 12 R CC CC m + (8)
For simplicity, it is assumed that R m = Z 2 and K m = 1 -Z n /Z a are real. This is relevant when the presence of L c may be neglected. In this work all of the parameters defined in (8) are considered fluctuating with time.

For the case of a system having a second loop with a transfer function H(p) = 1, the oscillating frequency is determined by the fourth-order equation

ω ωω ω ωω ωω ωω ωω 42 0 2 0 2 2 1 22 0 4 0 2 1 22 2 1 2 2( ) =0 -- ++ ++ + +         Q ΣΣ Σ , (9)
with two physically based solutions in the case of a high quality factor Q: ω 2 ≈ ω ω

0 2 1 2 + and ω 2 ≈ ω ω 0 22
+ Σ . For the further analysis, this case of high-Q resonators is considered, i.e., that of crystal resonators whose Q • f 0 product is greater than 1 • 10 13 (Q is typically close to 1 • 10 6 at f 0 = ω 0 /(2π) = 10 MHz). Then, all the terms inversely proportional to Q can be neglected in ( 7) and [START_REF] Santiago | Microwave frequency discriminator with a cryogenic sapphire resonator for ultra-low phase noise[END_REF]. In fact, [START_REF] Luiten | Cryogenic sapphire resonator oscillator with exceptional frequency stability: An update[END_REF] and ( 9) are of the same type, and can be represented in the following form:

ω ωα βω αβ β 64 11 2 22 3 [] [] =0 , -+ ++ + qq q ( 10
)
where q is a generic parameter, such that if q = 0, (10) reduces to [START_REF] Santiago | Microwave frequency discriminator with a cryogenic sapphire resonator for ultra-low phase noise[END_REF], and if q = 1, this equation represents the system without correction, i.e., described by [START_REF] Luiten | Cryogenic sapphire resonator oscillator with exceptional frequency stability: An update[END_REF]:

β 1 = - (1/(τ 1 τ Σ )), β 2 = -() () , 2
To compare the frequency sensitivity in both cases, small changes of parameters around their mean values must be considered. Thus, the oscillating frequency is a sum of a constant value ω c and a small change ∆ω. The same representation can be made for equation parameters, i.e., α 1 = α 1c + ∆α 1 , α 2 = α 2c + ∆α 2 , β 1 = β 1c + ∆β 1 , β 2 = β 2c + ∆β 2 , β 3 = β 3c + ∆β 3 . Next, substituting these expressions into (10) and neglecting the higher-order terms of small variations, an expression for the frequency change can be deduced:

() 2(

) . ∆∆ qq [START_REF] Ivanov | Ultra-low noise microwave oscillator with advanced phase noise suppression system[END_REF] Eq. ( 12) clearly shows that in the case of q = 0, the last term does not exist. It can also be easily shown that the denominator of both terms slightly depends on q in the case ω ωω c 2 0 2 0 2 -≪ , which is also relevant. It can be noted that the second term could vanish even for q = 1 in the marginal case ω c = ω 0 . In this case, the oscillating frequency depends only on ω 0 and no improvement is achieved with the second feedback. However, ω c never actually equals ω 0 (although the condition ω ωω c 2 0 2 0 2 -≪ holds true), simply because some other reactive components exist in the circuit. Moreover, resonators are intentionally fabricated with a resonant frequency less than the desired oscillation frequency. The difference is usually tuned with a series capacitance (not shown in our schematics, for simplicity) in the course of oscillator design. This is done to cover resonator parameter dispersion. So, in reality, the second term in (12) always exists for q = 1. Thus, the circuit sensitivity to parameter fluctuations can be decreased. The improvement effect exists even for parameters L x and C x of the resonator itself (again only in the case ω c ≠ ω 0 ). Thus, this analysis demonstrates that the oscillator with the second feedback is less sensitive to parametric fluctuations, because its related frequency change ∆ω has fewer contributing terms of instability.

In general, the resonator should always be considered as a BAW device in its fluctuating environment (electrical, thermal, mechanical). Therefore, the proposed method can be considered as an improvement of the electrical environment in terms of the oscillating frequency's sensitivity to circuit component fluctuations.

The sign of the transfer function H(p) = 1 is justified by the fact that the second feedback is inherently negative because of the sign introduced by the system configuration. In the same way, the first loop is positive in spite of the sign in ( 4)-( 6). In the case of a practical circuit, in which the sustaining amplifier of the oscillating loop is more complicated, the negative sign of the second feedback is introduced by an inverting amplifier.

IV. S S

To verify the results of the previous section, the phase noise of the considered oscillator has been simulated with Advanced Design System (ADS, Agilent Technologies Inc., Santa Clara, CA) (for phase noise simulation principles see, for example, [START_REF] Demir | Phase noise in oscillators: A unifying theory and numerical methods for characterization[END_REF], [START_REF] Djurhuus | Oscillator phase noise: A geometrical approach[END_REF]). An ADS design of the system is shown in Fig. 4. This design represents a more realistic variant of the circuit shown in Fig. 1, with an additional common-emitter amplifier in the oscillator negative feedback. The operational concept remains the same. The voltage across the crystal resonator is amplified and injected back into the sustaining amplifier.

During the oscillator phase-noise simulation, special attention is devoted to the parametric noise of circuit components, because these component instabilities are supposed to be suppressed by the second loop. Because the usual commercial circuit simulators do not contain any parametric noise models, we solve this problem with the approach that has been successfully used for precision close-to-carrier phase-noise simulations [START_REF] Goryachev | Precision close-to-carrier phase noise simulation of BAW oscillators[END_REF]. To introduce parametric phase fluctuations into the system, a parametric noise source is implemented in the circuit (see Fig. 4). This source includes a voltage-controlled capacitor SRC7, whose value depends on the voltage of two noise sources SRC1 and SRC2. Indeed, voltage noise is converted into parametric capacitance change, and further into oscillation frequency fluctuations related to oscillator phase noise. Thus, the first source SRC1 must have a PSD proportional to f -1/2 to produce an open-loop f -1 noise for medium and high Fourier frequencies, because the created noise on SRC7 has a spectral density proportional to the squared voltage of the noise source SRC1. In other words, SRC1 is a source of parametric flicker noise. Similarly, the second source SRC2, with a PSD proportional to f -1 , gives an f -2 slope of the open-loop noise for very low frequencies (which is mainly caused by temperature randomwalk fluctuations, physically). In other words, SRC2 is a source of parametric random walk. These noise sources are calibrated to be dominant for low Fourier frequencies.

Fig. 5 shows the simulation results in terms of L() f , the phase-noise PSD, of the oscillator with and without the second feedback. The PSD of the oscillator without the second feedback [see curve [START_REF] Rutman | Characterization of phase and frequency instabilities in precision frequency sources: Fifteen years of progress[END_REF], Fig. 5] for Fourier frequencies lower than 5 Hz, i.e., inside the resonator bandwidth, is dominated by the parametric noise introduced as described previously. This conclusion is made by comparing the results with and without the parametric noise source. The slopes of these curves close to the carrier are due to the Leeson effect [START_REF] Leeson | A simple model of feedback oscillator noise spectrum[END_REF] applied to the parametric phase noise. Usually f -2 (because parametric random walk is dominant for f < 5 • 10 -2 Hz) and f -1 (because parametric flicker is dominant for 5 • 10 -2 < f < 2 Hz) are observed here. For the higher frequencies, the oscillator phase noise is mainly due to the additive noise of its components. The second feedback considerably reduces f -3 parametric noise for low Fourier frequencies by about 9 dBc/Hz and almost eliminates f -4 instabilities for the very low Fourier frequencies (curve (2), Fig. 5). These effects strongly correlate with the amplification and phase shift in the second system feedback. Also it must be noted that in the absence of the parametric noise source, the effect of the additional feedback is negligible in the close-tocarrier region. Therefore, when comparing both curves, it can be seen that the frequency stabilization effect exists for parametric instabilities in the region of low Fourier frequencies, where parametric fluctuations are dominant. For higher offset frequencies, the second feedback increases instabilities because of the additional noise of the amplifier in the second feedback. The level of this additive noise is mostly determined by the signal-to-noise ratio in the second feedback.

Thus, the simulation results confirm the analytical analysis made in the previous section. The second negative feedback reduces the oscillator frequency sensitivity to fluctuations of its parameters inside the resonator bandwidth despite an increase outside this bandwidth. Using the same simulation techniques, the effectiveness of this approach has also been confirmed for another crystal oscillator topology known as a Pierce oscillator.

V. E V

A 10-MHz quartz crystal oscillator with two feedback loops has been designed (see Fig. 6) to verify the actual efficiency of the expected phase-noise correction. The implemented mockup-oscillator is based on a cascode amplifier stage with two independent inputs. One input (base of transistor VT1) is connected directly with the resonator and locks the oscillating loop. The other input (base of transistor VT2) is connected to the same resonator, but through the common-emitter amplifier based on transistor VT3, which shifts the phase (ideally) by π to introduce the appropriate correction to the oscillating loop. Though this circuit is much more complicated than the one described in the previous sections, it works on the same principles. The actual circuit parameters are chosen to meet the assumptions made in the previous sections. Particularly, the input current of the inverting amplifier is much less than that of the sustaining one to meet the assumption of a high input impedance in H(p).

The circuit operation has been tested in two configurations to find out the effect of the additional negative feedback. In the first operation mode [see Fig. 7, curves (1) and (3)], this second feedback is unlocked. Thus, for example, resistor R9 is substituted with an open circuit. So, the feedback inverting amplifier is present in the circuit, but it is not locked to the sustaining amplifier. After that, the same circuit is tested with small values of this resistor, i.e., in operation with both loops [see Fig. 7, curves (2) and ( 4)]. The experiment is repeated for several values of the supply voltage VCC.

The goal of this design is not to built an ultra-stable oscillator (that necessarily requires a complex and expensive thermo-mechanical structure), but to demonstrate the efficiency of the phase-correction loop. The whole circuit is simply placed inside a Dewar flask with just a single temperature control of the resonator case. Moreover, the circuit lacks output matching amplifiers and is not parametrically optimized.

Measurements have been made with a TSC 5120A phase noise test set (Timing Solutions Corp., Boulder, CO) using a hydrogen maser as a reference.

The results in Fig. 7 show that for all values of the supply voltage, the long-term frequency stability of the oscillator is improved. At the same time, its short-term stability is slightly degraded because of additional noise of the transistor VT3 in the second feedback. Even though the transistor VT1 is saturated in the 18-V configuration, the effect of stability improvement exists. In the 5-V experiment, this transistor is in the linear operating mode, and in this case the oscillator exhibits better results than for 18 V. Thus, when nonlinearities are very limited, the negative feedback loop is still efficient. This fact confirms the linear nature of the experimentally observed effect. The effect of the negative feedback is also seen on the corresponding phase-noise PSDs which are shown in Fig. 8. This effect is qualitatively the same as predicted by the simulation (see Fig. 5).

The experimental results are in good agreement with simulation data. Parts of the plots on the left of the minima are determined by the white phase noise (see Fig. 5) and mostly depend on the signal-to-noise ratio. In the case of 18-V power supply, better performances are achieved in this region. Here also, the second feedback degrades the system frequency stability. This fact corresponds to the increase of white phase noise for high Fourier frequencies in Fig. 5. Next, the minima of the Allan deviation are determined by the oscillator flicker frequency noise, i.e., the f -3 phase noise in Fig. 5. Here, the second feedback results in stability improvement with a simultaneous shift of the minima toward the higher τ. Frequency stability improvement is clearly observed for the intermediate values of τ (up to about 200 sec) where the Allan deviation corresponds to f -3 and f -4 phase noise (τ 0 and τ +1/2 slopes, respectively). This effect has been predicted by the simulation. In addition, the effect of the second feedback decreases for very long averaging times τ in the 5-V case. Indeed, the very-long-term frequency stability of such oscillators mainly originates from resonator parametric instabilities and aging (all of the sources of oscillator frequency instabilities are still discussed [START_REF] Walls | The quest to understand and reduce 1/f noise in amplifiers and BAW quartz oscillators[END_REF]- [START_REF] Rubiola | On the 1/f frequency noise in ultrastable quartz oscillators[END_REF]) and, thus, according to the suggested analytical analysis, it can hardly be improved. These effects were not simulated previously. In the 18-V case, the improvement still exists, meaning that these instabilities originate from other sources not previously considered here.

VI. C

In this work, investigations of the oscillator architecture are presented. The introduction of a negative feedback loop was suggested in addition to the positive one that sustains the oscillation. The second additional feedback reinjects the inverted signal from the crystal resonator into the sustaining amplifier. It is shown analytically, using a simple model, that this second feedback is able to make the oscillating frequency less dependent on the circuit parameters, and therefore less sensitive to the parametric fluctuations of the circuit. The same idea is further demonstrated with numerical simulations. Additionally, the simulation analysis states that for high Fourier frequencies, the oscillator phase noise could be increased. These conclusions are also confirmed with actual frequency stability measurements of a mock-up oscillator. A non-negligible frequency stability improvement is observed in the area of middle and high averaging times, τ. On the other hand, the short-term frequency stability is slightly degraded because of additive noise of the additional electronic components. These results are achievable for medium-quality oscillators with no additional costs on high-quality resonators, mixers, or temperature control units, which are the most expensive components. The phase-correction loop could be an alternative solution for a crystal oscillator positioned between the temperature-compensated crystal oscillator (TCXO) type and the oven-controlled crystal oscillator (OCXO) type [START_REF] Vig | Quartz crystal resonators and oscillators: For frequency control and timing applications-A tutorial[END_REF]. In further work, the same principle of frequency stability improvement should be investigated for state-ofthe-art oscillators.

A

The authors thank the anonymous reviewers for their invaluable comments and suggestions.

  2 do not depend on C 1 , R e , C 2 , and X 1 . The voltagedependent element K u (p)U v is the result of an equivalent transformation of resistor R f , the ideal voltage-controlled voltage source U v = H(p)U x and the dc voltage source E parallel to the filtering capacitor C f . Thus, K u (p) is a transfer function depending on the resistor and the capacitor.Using Kirchoff laws, the following system of equations may be written to describe the oscillator shown in Fig.2, taking into account that the ideal block H(p) has no input current:

Fig. 1 .

 1 Fig. 1. Modified Colpitts oscillator with the second negative feedback. Fig. 2. Transformed two-loop Colpitts oscillator.

Fig. 3 .

 3 Fig. 3. Block-diagram representation of the Colpitts oscillator with the additional feedback [a path with the transfer functions Z e (p), K a (p), and H(p)].

Fig. 4 .

 4 Fig. 4. Advanced Design System (ADS) design of the two-loop system: (1) sustaining amplifier, (2) crystal resonator, (3) buffer amplifier, (4) parametric noise source. Signal paths of both loops are shown in bold.

Fig. 5 .

 5 Fig. 5. Simulated effect of the oscillator negative feedback on the oscillator phase-noise power spectral density [single-sideband noise spectrum L() f ]: (1) one-loop oscillator, (2) two-loop oscillator.

Fig. 6 .

 6 Fig. 6. Practical realization of the two-loop oscillator. L2 and C8 are added to reject resonator spurious modes.

Fig. 7 .Fig. 8 .

 78 Fig. 7. Effect of the oscillator negative feedback on the Allan deviation: (1) 18-V supply voltage, one-loop oscillator, (2) 18-V supply voltage, two-loop oscillator, (3) 5-V supply voltage, one-loop oscillator, (4) 5-V supply voltage, two-loop oscillator.