# Frequency and phase noise of ultra-high Q silicon nitride nanomechanical resonators

**ABSTRACT** We describe the measurement and modeling of amplitude noise and phase noise

in ultra-high Q nanomechanical resonators made from stoichiometric silicon

nitride. With quality factors exceeding 2 million, the resonators' noise

performance is studied with high precision. We find that the amplitude noise

can be well described by the thermomechanical model, however, the resonators

exhibit sizable extra phase noise due to their intrinsic frequency

fluctuations. We develop a method to extract the resonator frequency

fluctuation of a driven resonator and obtain a noise spectrum with dependence,

which could be attributed to defect motion with broadly distributed relaxation

times.

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**ABSTRACT:**We have characterized the mechanical resonance properties (both linear and nonlinear) of various doubly-clamped silicon nitride nanomechanical resonators, each with a different intrinsic tensile stress. The measurements were carried out at 4 K and the magnetomotive technique was used to drive and detect the motion of the beams. The resonant frequencies of the beams are in the megahertz range, with quality factors of the order of 104. We also measure the dynamic range of the beams and their nonlinear (Duffing) behaviour.Journal of Low Temperature Physics 06/2012; 171(5-6). · 1.18 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We demonstrate the room-temperature operation of a silicon nanoelectromechanical resonant-body field effect transistor (RB-FET) embedded into phase-locked loop (PLL). The very-high frequency resonator uses on-chip electrostatic actuation and transistor-based displacement detection. The heterodyne frequency down-conversion based on resistive FET mixing provides a loop feedback signal with high signal-to-noise ratio. We identify key parameters for PLL operation, and analyze the performance of the RB-FET at the system level. Used as resonant mass detector, the experimental frequency stability in the ppm-range translates into sub atto-gram (10−18 g) sensitivity in high vacuum. The feedback and control system are generic and may be extended to other mechanical resonators with transistor properties, such as graphene membranes and carbon nanotubes.Applied Physics Letters 10/2012; 101(15). · 3.52 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**The rapid development of micro- and nanomechanical oscillators in the past decade has led to the emergence of novel devices and sensors that are opening new frontiers in both applied and fundamental science. The potential of these devices is however affected by their increased sensitivity to external perturbations. Here we report a non-perturbative optomechanical stabilization technique and apply the method to stabilize a linear nanomechanical beam at its thermodynamic limit at room temperature. The reported ability to stabilize a nanomechanical oscillator to the thermodynamic limit can be extended to a variety of systems and increases the sensitivity range of nanomechanical sensors in both fundamental and applied studies.Nature Communications 12/2013; 4:2860. · 10.74 Impact Factor

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Frequency and phase noise of ultra-high Q silicon nitride

nanomechanical resonators

King Y. Fong, Wolfram H. P. Pernice*, and Hong X. Tang†

Departments of Electrical Engineering, Yale University,

New Haven, CT 06511, USA

Abstract

We describe the measurement and modeling of amplitude noise and phase noise in ultra-high

Q nanomechanical resonators made from stoichiometric silicon nitride. With quality factors

exceeding 2 million, the resonators’ noise performance is studied with high precision. We find

that the amplitude noise can be well described by the thermomechanical model, however, the

resonators exhibit sizable extra phase noise due to their intrinsic frequency fluctuations. We

develop a method to extract the resonator frequency fluctuation of a driven resonator and obtain

a noise spectrum with /

B k Tf dependence, which could be attributed to defect motion with

broadly distributed relaxation times.

PACS numbers: 85.85.+j, 62.25.Fg, 05.40.-a, 63.50.Lm

* Present address: Institute of Nanotechnology, Karlsruhe Institute of Technology, 76133 Karlsruhe, Germany

† Corresponding author: hong.tang@yale.edu

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Nanoelectromechanical systems (NEMS) have shown great promise for both fundamental

science and applications, by introducing new tools for studying quantum physics [1-3] and

enabling high performance devices for sensing and oscillator applications [4-6]. Recently,

nanomechanical resonators made of high stress stoichiometric silicon nitride have provided a

solution to one of the most challenging quests in NEMS research, i.e., achieving very high

mechanical quality factors (Q factors) [7-9]. For a fixed resonance frequency, high Q is

equivalent to a low dissipation rate, which implies weaker coupling to the thermal bath, higher

sensitivity to external signals, and lower minimum operating power [10]. To enable real world

applications based on such devices, understanding the underlying dissipation and noise

mechanism is crucial. Such an understanding is also a key for implementing quantum

measurements of phonon shot noise as recently proposed [11]. Several models have been

developed to explain the damping mechanism [12-14]. Yet, the noise characteristics such as the

amplitude/phase noise and frequency fluctuation have not been addressed so far.

In NEMS, resonator frequency fluctuations can arise due to various processes, such as

temperature fluctuations [15], adsorption-desorption of molecules on the device surface [16,17],

molecule diffusion along the resonator [18]. These frequency fluctuations are usually

accompanied by thermal motion of the resonator, which has to be separated to reveal true

resonator frequency fluctuations as recently suggested [19]. Here, we experimentally study the

frequency fluctuation of ultra high Q nanomechanical resonators made of stoichiometric silicon

nitride. We derive and validate a mathematical model to account for the observed frequency

fluctuations. Our results show that high Q devices are actually more susceptible to frequency

flickering. By measuring the phase noise of the device, a resonator frequency noise spectrum

with /

B k Tf dependence is extracted. We propose that this fluctuation is due to defect motion

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with a broad spectrum of thermally-activated relaxation times. Our results give insight into the

noise mechanism of high Q silicon nitride resonators and are important for NEMS based

applications, such as sensors and low phase noise optomechanical oscillators.

We fabricate nanomechanical resonators from LPCVD stoichiometric silicon nitride. Fig. 1(a)

shows the SEM image of a fabricated device, which contains a doubly-clamped beam resonator

with dimensions 380μm 0.33μm 0.2μm

, an optical readout circuit with grating coupler

input/output ports as well as an on-chip interferometer. (Fabrication details can be found in Refs.

[9,20]). The devices are mounted in a helium cryostat and the temperature of the sample stage is

regulated from 5K to room temperature with milli-Kelvin precision. The pressure is kept below

10-6 Torr for all the measurements carried out. The sample stage is equipped with a piezodisk

actuator for mechanical actuation. The mechanical quality factor is measured using a ring-down

method [9] and is plotted against temperature in Fig. 1(b). At room-temperature we measure a

quality factor of the fundamental in-plane flexural mode of roughly 400,000, which increases up

to 2.2 million when the device is cooled down to 5K. The observed high Q of tensile stressed

silicon nitride nanostring can be attributed to the increased elastic energy stored in the tension

[12,14]. For devices with cross-sectional dimensions similar to the device used here, it was

suggested that the Q is limited by internal material damping [12]. It has also been shown that the

quality factor of silicon nitride nanostring resonators is not sensitive to pressure change once the

pressure is reduced below milli-Torr levels [7,9]. Therefore, we can safely neglect the effect of

pressure change on the Q during the cool down.

Under tensile stress, a beam with length of L has a mechanical fundamental mode profile of

( , ) z t ( )sin(x t / ) z L

, where the displacement ( ) x t is defined as the displacement of the

middle of the beam. Under such normalization condition, we define the effective mass m as half

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of the total mass so that

2

( ) / 2mx t

correctly expresses the total kinetic energy of the resonator.

(Details of definition of effective mass can be found, for example, in section 3.1 of Ref [3].)

Because of the sensitive optical interferometric read-out, thermomechanical noise is resolved at

all temperatures concerned in this work and thus allows us to calibrate the displacement of the

resonator. Meanwhile, the optical power is kept minimal at around 10μW in order not to perturb

the thermal equilibrium or induce any optomechanical backaction. Device displacement at

different driving intensity is plotted in Fig. 1(c). At large drive the device response displays

Duffing nonlinearity due to stiffening effects in the displaced beam. If the nonlinear equation of

motion

2

r

2

r

3

3

2cos( )/

d

x

x

x K xFtm

and harmonic motion cos()

d

xAt

are

assumed, where ωr is the resonance frequency, / 2

r

Q

is the damping rate, F is the external

driving force, ωd is the driving frequency and A is the amplitude of the displacement, the

nonlinear coefficient K3 can be found by fitting a cubic polynomial to the graph

2

F vs

2

A . For

the present case, we obtain

92

3

6.9 10 nm

K

, which agrees with the theoretical value

22

3

/ 4 KEL

expected for a tensile stressed beam experiencing additional stress due to large

strain [21]. Here, E and are the Young’s modulus and the stress, respectively.

To separate and quantify the amplitude and phase noise, we used a lock-in amplifier (Zurich

Instruments HF2LI [22]) to measure the two quadrature components of the device in response to

a sinusoidal drive. The lock-in bandwidth is chosen such that within the bandwidth

LI

the

noise power is dominated by thermomechanical noise, while /

LIrQ

so that most of the

thermal noise power is captured. Fig. 2(a) shows the measured two quadrature components in the

complex plane in the presence (lower datasets) and absence (upper datasets) of the drive. The

data acquisition time for each dataset is kept at 100s, which is much longer than the device

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ringdown time. The black solid line traces out the polar plot of the driven response when the

driving frequency is swept across the resonance. To ensure linear operation, i.e.,

2

3

1/ K AQ

,

the device is driven to an amplitude of A ~ 4.5nm. In the absence of the drive, the fluctuation of

the two quadrature components is due to thermomechanical noise. As expected, the noise is

random in phase and therefore has a circularly symmetric distribution [23], as illustrated for

clarity in a zoom-in in Fig. 2(b). Fig. 2(c) shows that the fluctuation is Gaussian distributed and

has a standard deviation agrees with the expected value

22

r

/xkTm

. Here

2

x

is the

ensemble average of the squared displacement fluctuation. In the driven case we would expect to

see the same fluctuation profile if thermomechanical noise was the dominant noise source.

However, this is apparently not the case. While the amount of amplitude noise is similar to that

of the thermomechanical noise, there are large fluctuations along the resonance circle. Fig. 2(d)

shows the measured two quadrature components under different driving intensity. It is clear that

the phase angle enclosed by the extra fluctuation remains constant when the drive is increased.

Therefore, this extra phase noise is independent of the driving intensity. To confirm that the two

quadrature components move indeed along the resonance circle, we show in Fig. 2(e) a dataset

with a longer acquisition time of 2000s. The results therefore suggest that it is the resonator

frequency that is fluctuating. Here, we point out that frequency fluctuation induced by amplitude

noise via mechanical nonlinear effects cannot account for the observed phenomenon. Such effect

can be estimated by

3r

K A A

, which for our case has a value of

4

2 10 Hz

at 5K.

This value is orders of magnitude smaller than the observed fluctuation. Besides, the

corresponding phase fluctuation is expected to be proportional to the amplitude, which is

contrary to our observation.

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To understand the observed phenomenon, we theoretically analyze a driven harmonic

resonator whose resonance frequency ( )

rt

fluctuates in time. Its displacement ( )x t follows a

stochastic differential equation

2

( ) x t

2( ) x t

( ) t x t( )cos( )/( )/t

rd th

Ftmfm

, (1)

where cos()

d

Ft

is the external driving force, and is the damping rate. )(tfth

is the

corresponding thermal fluctuation force due to coupling to the external bath. It is a white force

noise assumed to have a correlation time much shorter than any time scale concerned, i.e.,

( )t f()( )

thth

ft

. The Duffing nonlinear term is assumed to be negligible, as justified

from above discussions. The displacement ( )

x t can be expressed in terms of amplitude and

phase as

( ) x t( )cos A t( ) t

d

t

. Here, is defined as the phase relative to the drive.

Before we proceed, we would like to emphasize that there are two distinct frequency concepts,

namely the resonator frequency

r

and the instantaneous frequency

i . The former is the

natural frequency at which the resonator would oscillate in the absence of drive while the latter is

the apparent oscillation frequency which is defined as /

id

d dt

. Note that a driven

passive resonator should always follow the driving frequency

d

, with a shift of /ddt

. A

change of resonator frequency

r

indirectly affects the instantaneous frequency

i by altering

the phase (, )

rt

.

For a resonator with low damping rate ( )

rt

driven near resonance, i.e.,

( )

t

( )

t

rd

and ( )

d

t

, it can be shown that the slowly varying complex displacement

( )( )expA t( )u ti t

satisfies the equation

( )( )( )( )t2

thd

u t

i t u t

Ffim

, where

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( )t2 ( )exp(t)

ththd

f

fit

[24]. Using an appropriate integrating factor, ( )u t can be expressed in

integral form [19] as

'

( )'exp( ')" ( ")

( ')t2

tt

thd

t

u tdttti dttFf

im

. (2)

We assume that in the time span concerned the phase accumulation due to the resonator

frequency fluctuation is small, i.e.,

'

" ( ")

1

t

tdt

t

. This condition can always be satisfied if the

chosen time span is short enough. Upon integration by parts, the amplitude and phase of the

displacement, given by

( ) Im{ ( )}u tA t

and ( ) Re{ ( )} (u t / 2)/ 2

d

tFm

, can be re-

written as

( ')t

'exp(')( ') ( ')2

t

RId

A tdtttFf t

g t

m

(3a)

( ')

F

( ')

F

( )t'exp(')( ')t ( ')t

2

t

IR

f t

g t

dttt

, (3b)

where

Rf

,

If

and

R

g

,

Ig

are the real and imaginary parts of

thf

and

'

( ')t" exp

( 't")( ")t

t

thth

g

dttf

, respectively. The first term of Eq. (3a) gives the steady

state driven amplitude

0

/ 2

d

AFm

. In both expressions for ( )A t and ( ) t

, the second terms

represent the contribution from thermal fluctuation and the third terms represent the mixing

between thermal and resonator frequency fluctuation.

The first term of Eq. (3b) is the phase noise due to resonator frequency fluctuations. Note that

this is the only term that is independent of the driving force while all the other terms are

inversely proportional to the drive. Therefore this term will eventually dominate if the drive is

sufficiently large. Our experimental data shown in Fig. 2(d) confirms that the first term

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dominates in the present case. Thus, the expression for the phase can be simplified as

( )t exp( ')( ')t dt

' / 2

t

tt

. It is then straightforward to show that the transfer-

function between the spectral densities of ( ) t

and ( ) t

is given by

22

( )

( )

SS

. (4)

Note that here we did not assume any physical model of the frequency noise nor make any

assumption about the correlation time scale of ( ) t

. Our result is thus generally valid as long as

the condition

'

" ( ")

1

t

tdt

t

is satisfied. This condition can always be guaranteed by

considering a short enough measurement time.

One important implication of Eq. (4) is that the area under the transfer function

22

0

0

/()/dQ

is proportional to Q, which means that the amount of resonator

frequency noise transferred into phase noise is larger for devices with higher Q. It can be

intuitively understood as the result of steeper slope in the phase-frequency curve. Another way of

looking at this is to consider the instantaneous frequency spectral density

2222

( )

( )

() ( )

f SSS

. It can be seen that the transfer function is a high pass

filter: the higher the Q, the lower the roll-off frequency and more noise is allowed to pass. This is

essentially the reason why we are able to quantify the frequency noise in our ultrahigh Q devices.

It is generally believed that high Q always has positive impact on device performance. Here we

show however that an increase in device performance based on higher Q is accompanied by

higher susceptibility to the resonator’s inherent frequency noise that exists.

This intrinsic resonator frequency noise imposes a detection limit for applications that rely on

the measurement of resonance shift, such as mass and force gradient sensing. The minimum

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resolvable frequency change

min

can be expressed as the change of frequency that produces

the same amount of phase shift as the total phase noise within the measurement bandwidth

bw

,

or

1/2

22

min

0

/ 2 ( )

bw

r

Qd S

. Contrary to the frequency detection limit

imposed by thermomechanical noise [25],

min

does not depend on the driven oscillation

amplitude. Therefore, the dynamic range of the device is no longer a determinant factor for the

frequency sensitivity when the phase noise of the device is dominated by the intrinsic resonator

frequency noise.

To verify and apply the results obtained above, we measure the two quadrature components

and compute the amplitude and phase spectral densities by periodogram using the Hanning

window function. The measurement times for 5K, 78K and 296K are chosen to be 125s, 25s, and

8s, respectively, such that the condition

'

" ( ")

1

t

tdt

t

is meet. The obtained spectral densities

for the amplitude and phase are plotted in Fig. 3(a) and 3(b), respectively. The amplitude and

phase noise background of the drive source of our lock-in amplifier [22] is well below the device

noise and therefore plays negligible role in the measured spectrum. The black solid lines show

the noise contribution due to thermal fluctuation (the second terms in Eq. (3a) and (3b)). The

amplitude noise matches well with the thermal term but the phase noise is significantly higher

than that, which agrees with the observation from Fig. 2(a). Supported by the above analysis, we

attribute this extra phase noise to resonator frequency fluctuation, whose spectral density is

computed by Eq. (4) and plotted in Fig. 3(c). A 1/f noise clearly dominates from the lowest

frequency of 0.008Hz up to 5Hz. By denoting

0

( )f ( )/S S Tf

, we can extract

0( )S T from the

resonator frequency noise spectrum. In Fig. 4 we plot

0( ) S T against temperature in a log-log

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scale. A power law fit aTn gives an exponent of 0.94 ± 0.10, which strongly suggests that

0( ) S T

is linearly proportional to temperature T. Therefore, the resonator frequency noise spectrum has

an overall /

B k Tf dependence.

Previously, several processes have been identified to be the sources of resonator frequency

fluctuation in nanomechanical systems, such as temperature fluctuation [15], adsorption-

desorption of molecules on the device surface [16,17], and molecule diffusion along the

resonator [18]. However, these known processes do not give rise to the /

B k Tf noise spectrum

we observed. We propose that this /

B k Tf dependent noise spectrum can be understood within

the Dutta-Dimon-Horn model [26], which applies generally to fluctuations with a broad

spectrum of thermally activated relaxation times. We suggest that in the present case the

fluctuations originate from defect motions within the material [15]. For crystalline solids, local

elastic distortion due to a point defect can be considered as an elastic dipole [27]. Depending on

its orientation, the dipole produces a distortion of the local strain under a stress field and hence

locally modifies the Young’s modulus. The idea of elastic dipoles can also be applied to model

the inherent structural disorder in amorphous solids [28]. Following Ref. [15], we consider a

defect state with two possible dipole orientations separated by an energy barrier E and a

thermally activated reorientation time

0exp( /

)

B

E k T

. The idea is conceptually similar to the

well known two-level states (TLS) model, which has successfully explained many thermal

properties of amorphous solids at low-temperature [29] as well as damping of nanomechanical

resonators [30]. While the TLS model describes the situation in which atoms with two energy

minima are allowed to tunnel through the energy barrier quantum mechanically, here we ignore

such tunneling effects and assume that the elastic dipoles reorient through thermal activation. For

a tensile-stressed beam, it can be shown that such fluctuation in elasticity leads to a resonator

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frequency noise spectrum of

222

( ) (1)S

, where

2

is the variance of the

frequency fluctuation. Following the Dutta-Dimon-Horn model [26] and assuming the

distribution of activation energy ( )D E to be relatively flat, i.e, ()( )/( )

nnn

B k T d D EdE D E

,

the spectra for different activation energy can be integrated and give

( , )

B k T

ST D E

, (5)

where

0

ln()

B

E

k T

. The result reproduces the /

B k Tf dependence we observe

experimentally. If we assume

0 to be 10-12s (on the order of the inverse phonon frequency [26]),

the activation energy can be estimated to have a span in range of 0.01-0.8eV, which is consistent

with the typical energy scale of TLS and defect motions [26,29]. Here we would like to

emphasize that the defect states model is phenomenological. Its microscopic basis remains to be

verified by separate experimental approaches, for example material characterization.

In conclusion, we have studied the noise characteristics of a high Q nanomechanical resonator

made of stoichiometric silicon nitride. The amplitude noise can be explained by the thermal

motion of mechanical resonator, while an extra phase noise is observed. We develop a method to

extract the resonator frequency fluctuation and obtain a noise spectrum with /

B k Tf dependence.

We propose that this frequency fluctuation is due to defect states with a broad spectrum of

thermally-activated elastic dipole reorientation time. On the other hand, our theory also suggests

that high Q devices are more susceptible to the resonator’s inherent frequency fluctuation. Our

result is important for understanding the noise mechanism of the high Q silicon nitride

nanomechanical resonators, as well as for applying this high performance device to sensing and

oscillator applications.

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Acknowledgements. We acknowledge funding from DARPA/MTO ORCHID program

through a grant from the Air Force Office of Scientific Research (AFOSR). H.X.T acknowledges

support from a Packard Fellowship and a CAREER award from the National Science Foundation.

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