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THE NONLINEAR SUBSTRUCTURE OF l/f NOISE IN QUARTZ CRYSTAL RESONATORS
M. Planat
To cite this version:
M. Planat. THE NONLINEAR SUBSTRUCTURE OF l/f NOISE IN QUARTZ CRYS- TAL RESONATORS. Journal de Physique Colloques, 1989, 50 (C3), pp.C3-65-C3-69.
�10.1051/jphyscol:1989309�. �jpa-00229448�
THE NONLINEAR SUBSTRUCTURE OF l/f NOISE IN QUARTZ CRYSTAL RESONATORS
M. PLANAT
Laboratoire de Physique et M6trologie des Oscillateurs du CNRS associ6 a l'Universit6 de Franche-ComtB-Besanpn, 32, Avenue de I'Observatoire, F-25000 BesanCon, France
Resume
-Les resonateurs acoustiques a quartz font apparaitre aux basses frequences des fluctuations anormales avec un spectre de puissance hyperbolique (d'oh le nom de bruit en 110. Les caractCristiques dispersives et non lineaires du reseau cristallin sont prises en compte pour etudier ces fluctuations.
Abstract
-A t low frequencies quartz crystal resonators m a n i f e s t a n o r m a l o u s fluctuations with an hyperbolic power spectrum (they are called l/f noise).Dispersive and nonlinear characteristics of the crystal lattice are used to study these fluctuations.
1. I N T R O D U C T I O N
Fluctuations in a physical system a r e usually well explained by assuming Brownian motion This was shown by Einstein (1905) and Langevin (1908) Using the fluctuation-dissipation theorem a full description of fluctuations can be obtained from knowledge of the macroscopic friction constant In this approach the system is assumed to be in stationnary thermal equilibrium and a n y deviation relaxes in a finite time a The corresponding power spectrum shows a lorentzien dependence, i e i t is uniform for frequencies lower than the cut-off frequency llr. and decreases like lIf2 for frequencies higher than 111 [ I ]
However, most natural systems exhibit anomalous low frequency fluctuations with a n hyperbolic power spectrum down to t h e lower observed frequency. This so-called llf noise manifests in the voltage, current or resistance of any solid s t a t e device, in t h e frequency of quartz crystal resonators and even in economic, geographic o r biologic d a t a 121.
A fundamental approach for llf noise yet does not exist. The infrared divergence theory by Handel [31 i s strongly criticized because it violates the rules of quantum mechanics 141. On the other hand theories based on the thermal activation of traps [5] could explain the very long time correlations required by llf noise but a r e in poor agreement with experiments.
I recently proposed a new approach for llf noise from the concept of dispersive wave 161. This approach is used here for t h e case of acoustic waves. In part 1 t h e low frequency dispersion for a one dimensional lattice is identified, t h e wave amplitude is computed a t the geometrical approximation and the corresponding llf spectrum is obtained. At very low frequencies t h e h-ont of the disturbance is approached, the amplitude of the wave increases and the linear ray approximation fails (together with the standard concept of group velocity. The approach has to be modified in two ways :
- introducing t h e concept of caustic to compute the displacement a t higher order
-
introducing nonlinearitics.In p a r t 2 a realistic nonlinear dispersive equation is established. This equation is shown to support stationary nonlinear excitations (kinks). A threshold condition for such excitations to occur is given together with consequences for t h e low frequency spectrum.
2
-
LOW FREQUENCY D I S P E R S I V E WAVES O N t i LATTICE A N D I R N O I S E The crystal i s first considered one-dimensional, purely elastic, dispersive and linear.For a discrete chain of identical spheres of mass m, linked together by springs of force constant p the dispersion relation for a wave of wavenumber k and frequency w i s given by
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1989309
JOURNAL DE PHYSIQUE
where a is the separation between spheres in equilibrium and w, = 2(p/m)1/2 is the cut-off frequency of the first Brillouin zone.
Since we a r e interested with low frequencies we adopt the continuum approximation of long waves. The dispersion relation (1) can be written
where c, = 112 aw, is the phase velocity for the non dispersive case and = 1/48 a 3 0 , a parameter characterizing the discrete nature of the lattice.
We here restrict for the case of waves propagating towards the right (k ) 0) ; the mechanical displacement is
where the phase factor y(k) is given by
Integral (3) can be computed asymptotically for large times t and fixed values of x/t with the method of stationary phase. For large t the phase equal a big number of times 2n and corresponds to the "geometrical" approximation of zero wave length. The "ray" equation is obtained when the first order derivative of the phase vanishes
yl(ks) = 0 =.c - 30 k z
-
xlt (5)Expanding the phase around the stationary phase point k, equation (3) can be converted into a Fresnel integral [7, p.
3711 with the result
By using eqs. (3)-(5) and for a n observer travelling a t the velocity dt (c,, the wave has the following form
For a n observer travelling a t the velocity x/t ) c,, the stationary wavenumber k, becomes imaginary and the wave (6) shows a n exponential decay. in the transition region, near xit = c,, the approximation (6) predicts a n infinite amplitude. From (5) we see that this case corresponds to the wavenumber k, = 0 where the second order derivative y9'(k,) = - 6pk, vanishes.
The ray x = cot is called a caustic and to obtain a valid approximation in the transition region we have to add higher order forms in the Taylor expansion of the phase [7, p. 441 1. The result is
1 x - c t u(x,t) = F(0)
-
where Ai is the standard Airy integral. From (8) we observe that behind the wave front (x/t ( c,) the argument of the Airy integral is negative and the wave is oscillatory ; on the other hand the wave is decaying exponentially ahead ( d t ) c,). We can check that away from the transition region the asymptotic expansion of Airy integral recovers the previous expression (7).
As observed from eq. 2 a n observer travelling a t fixed velocity x/t will see a quasiperiodic signal with a n envelope decaying slowly as a function of time like t-l/2.From this we can explain the shape of the power spectrum.
Assuming elementary excitations like (7) are produced a t random starting times, with a mean frequency u , the power spectrum takes the form (61
In this expression u(x.0 is the time Fourier transform ofdisplacement u(x,t) and is defined by
where the carrier frequency is
The power spectrum has the 1/F shape where F = W(ks)/2n - f is the interval1 with respect to the carrier frequency.
For a n observer approaching the velocity c, of the wavefront the amplitude of the spectrum becomes larger and larger ; a t the same time the carrier frequency becomes lower and lower. In the transition region eq. (8) should be used to compute the power spectrum.
For a n observer a t rest (the measurement set up), the carrier frequency is maximum equal to 2c03f213d@.
Correspondingly the spectrum has the lower amplitude.
NONLINEAR DlSPEKSlVE WAVES ON A LATTICE
In the previous section we have shown that, in the transition region, lower is the wave frequency higher a r e its amplitude and its associated 1/73 power spectrum. It is an interesting question to analyze anharmonic effects for this case.
To establish the model we first neglect dispersion. The nonlinear propagation equation for a wave in the direction x is given by [8]
utt = c0 uxxg(ux) 2 with g(w) = 1
+
y w+
6 w 2 (13)and y and 6 a r e parameters taking into account anharmonic effects in the crystal.
Introducing the particle velocity v = ut and the strain w = u,, eq. (16) can be integrated with respect to v and w by using the Riemann's method of characteristics.They a r e two families of characteristics called C + and C- with slopes d d d t =
f
d g ( w ) . A pair of Riemann's invariants r, and so can be defined byW -
v - c
1
"ig(a) d a = r o oiongc+0 (14)
This system can be studied for the case where the medium initially a t rest is excited a t time t = 0 with an initial state
For this case the solution is a simple wave since it describes a continuous transition from a constant initial state. In the present case, this is the region of the (t,x) plane covered by negative-sloping characteristics that lead back to the zone of constant initial conditions. More precisely we have
s = O and u = 1 / 2 r (16)
This implies that v, w and are constants along characteristics of positive slope and these characteristics are straight lines.
If one characteristic C + interesects the axis t = 0 a t x =
5,
then w = w,(<) on the whole of that curve. The corresponding equation of characteristics C + is thenAllowing
5
to vary we determine equations for strain w and particle velocity vJOURNAL DE PHYSIQUE
woC<)
w(x,t) = wo(Q with o(x,t) = -c0
1
d a d oOn the other hand, mechanical displacement can be easily obtained since
is constant along C +
Integrating this equation, we obtain
u(x,t) = u,(U
+
[V+
w d g ( w )-
ItFrom eqs. (17)-(20) the simple wave solution of eq. (13) can be drawn easily. This solution exhibits a breakdown (shock) after a finite time, since it becomes multivalued ; v and w shows the Riemann-Hugoniot singularity and u the so-called swallow-tail singularity.This is not surprising since the simple wave mechanical displacement u can be rewritten in the form of the Hamilton-Jacobi equation
* -
(211u = u with Mw) =
-
co1
d g ( o ) do' I
Beyond the breakdown time, the solution (17)-(20) is no longer valid. Equation (13) has to be supplemented by new conditions characterizing the shock. In our case dispersion effects prevent the shock from forming ; this will be studied now.
From the two basic ingredients : dispersion and nonlinearity we can establish the equation for waves along the one- dimensional lattice. The propagation equation for the simple dispersive wave is
The corresponding equation for the strain w = u, is
w, + c o d g ( w ) w x
+ fl
w x n = 0 (23)This is a Korteweg's type equation of Hamiltonian form [lo]. Its unique integral of motion correspond to the so-called solitons.
We consider the typical case where y # 0 and 8 is a small parameter (see eq. (13))
Using the coordinate transform z = x - cot and r; = +c,yt, the equation for the displacement takes the form u + - u t 1
+
b uZZZ = 0 ; b=
2P/yco (24)T 2
and the equation for the strain takes the standard Korteweg de Vries form.
The parameter b introduced in (24) characterizes the relative degree of dispersion with respect to nonlinearity.
I t is interesting to derive the necessary conditions for solitons to form. We consider the Korteweg de Vries equation (24) for the strain
wZ
+
w wZ+
b wZzz = 0 (25)Looking for stationary solutions in the form
w = w <(x - ct) we obtain from standard considerations [71, [I01
w = wo"(;) where the amplitude wo and width t of solitons are given by
For longitudinal vibrations in X-cut quartz, y = 1 and (34) can be rewrittcn a s w, ) (a/e)z
Some comments can be given on the basis of these preliminary estimates.At high enough frequencies t h e mean free path of phonons does not exceed 1 nm, the width t of the initial pulse will not exceed these value a n d then t h e threshold s t r a i n 10-2 - 10-3 cilnnot be reached by thermal activation.
At low frequencies, the mean free path can reach the size of the sample, say 1 m m and t h e threshold s t r a i n becomcs considerably lower, about 10~13. There exist a possibility t h a t solitons can form. The corresponding excitations for t h e displacement a r e kinks (domain walls).
I n a f u t u r e work, we will examine the statistical mechanics associated to these excitations. In p a r t i c u l a r t h e displacement kink corresponding to eq. (27) h a s just t h e form of t h e kink of the +4 model. It h a s been shown in ref.
[I21 t h a t systems ruled by the a4 equation can undergo a structural phase transition with soft modes. Moreover a central peak appear which could be related to our llf noise problem.
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