NOISE MEASUREMENTS IN FERROELECTRICS

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NOISE MEASUREMENTS IN FERROELECTRICS

L. Godefroy

To cite this version:

L. Godefroy. NOISE MEASUREMENTS IN FERROELECTRICS. Journal de Physique Colloques,

1972, 33 (C2), pp.C2-44-C2-48. �10.1051/jphyscol:1972210�. �jpa-00214946�

L. GODEFROY

NOISE MEASUREMENTS IN FERROELECTRICS

Abstract. - Due to the strong non-linearity in the vicinity of the Curie temperature the classical bridge measurements of permittivity are irrelevant for ferroelectric materials. In order to avoid this difficulty several authors have suggested deducing the permittivity from noise measurements. The theoretical aspect of the problem and that of the experimental design are described

classically the crystal is loaded with a variable resistor

more recently, an auxiliary noise generator has been used.

This new method seems to allow a more accurate solution of the problem.

I. Introduction.

In ionic crystals the thermal motion of the ions results in polarization fluctuations which produce voltage fluctuations across the terminals of the electrodes. Thermodynamics informs us that this noise voltage is related to the temperature T and to the capacitance C by

In a more detailed way the spectrum of the fluctuat- ing voltage may be calculated by Nyquist's formula

Even if the method is useful near the Curie tempe- rature, it may be noticed that it is based on Nyquist's formula which cannot be used for non-linear systems.

This important point has never been discussed in full details.

In this review we will explain and discuss the theore- tical and experimental attempts made to develop the noise measurement method and we will give the chief results which have been obtained in this field. We limit ourselves to the case of thermal equilibrium excluding experiments with variable temperature or field.

Su(o)

4 k ~ % ( Z ( o ) ) (2) 11. Theoretical approachs.

The first theoretical where Z(o) is the impedance of the dipole and

is the investigation of polarization fluctuations in ferro- electric materials was presented by Burgess [B

real part of the complex number

As a model of ferroelectric crystal he used a system If the electrodes are short-circuited a fluctuating of N indentical dipoles on equivalent sites, each of current appears. The spectrum of its fluctuations is them having a moment equal to + ^rn. Coupling

given by between the dipoles is taken into account by a Lorentz

si(w)

4 ~ T % ( A ( u ) ) (3) factor fl

E,,,

E + ^PP

where A ( o ) is the admittance of the dipole. A and Z are

related by where P is the polarization

P

M/V ( M net dipole

The general idea for the determination of impedance by noise measurements is the following

if the simul- taneous measurements of Su(o) and Si(o) are per- formed, the real parts of Z and A may be deduced from (2) and (3) and by using the relation (4) both real and imaginary parts of Z and A may be calculated.

Such a method, which requires difficult measure- ments of noise voltage and current, it not useful for normal dielectrics for which classical bridge measure- ments are much more easy. But the method becomes useful for the ferroelectrics crystals in the vicinity of Curie temperautre : indeed in this case the non-linearity is very strong and the a. c. field of the bridge may pertrub the crystal in such a way that the results become irrelevant. Moreower near the transition the fluctuations are expected to be enhanced by critical phenomena and to be more easy to measure.

moment, V volume). In this model M is related to the field E by an implicit equation

There is a critical or Curie temperature To given by kT,

,!7Nm2/V such that, for E

0, M

0 for T > To and M

for T < T,.

Moment fluctuations can be evaluated using a Taylor's expansion of the free energy :

The partial derivatives of F are generally different

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972210

NOISE MEASUREMENTS IN FERROELECTRICS C2-45

from zero, except at the Curie temperature where both F" and F" vanish. Near to but nqt at the Curie tempe- rature the first term of the expansion gives a gaussian distribution for M from which we deduce for E

0

At Tc the distribution is no longer gaussian and is governed by the fourth derivative of F, giving

M2

1 . 1 7 ~ ~ ' ~ r n ' a t T = Tc.

The range of validity of the formulas (8) may be evaluated by equating the values of %' given by (8) and (9). Eq. ( 8 ) fails in a range of temperature defined by

'P

According to the high value of N this range is very narrow.

To discuss the validity of this derivation we have to point out that in this model the long range interaction is taken into account by the Lorentz factor p only.

Near Tc this is inadequate because of the need to include a distance-dependent correlation between the dipoles. Dipoles may group together, acting as a whole, and thereby decreasing the total number N of fluctuat- ing elements. The influence of this modification of N and of the correlated modification of rn does not act in the same way for T

T, and for T

Tc (formulas ( 8 ) and (9)). This should be kept in mind when discussing the phenomena which occur near the transition when the crystal in heated through the Curie temperature.

Another theoretical approach is to develop general considerations without reference to any model. The basic theorems are Nyquist's relation and Kramer- Kronig's formula.

A synthesis of thie approach has been presented by Petersson [P

P 70b]

If P is the reciprocal permitti- vity

Nyquist's relation gives for the spectrum of voltage fliuctuations

where L is the thickness and A the area of the sample.

On the other hand Kramers Kronig's relation gives

Integrating (1 1) over the whole range of frequencies and using (12) we obtain for the total voltage fluctua- tions

By a similar derivation the following dual formula may be obtained

(1 3 b

This implies that the total voltage fluctuation is directly related to the difference between high and low frequency permittivity, that is t o the ionic contribution to the permittivity.

This rather surprising result must be handled care- fully, for the validity of Nyquist's formula (1 1) requires the linearity of the systems and does not apply up to the highest frequencies. On the other hand, the finite band width of the apparatus does not allow a true evaluation of p. Nevertheless the experimental testing of eq. (13) should be of interest in pointing out the influence of non-linearity on Nyquist's formula.

111. Standard Experimental studies.

Standard measurements are performed by amplifying and recording the noise voltage which appears accross the electrodes of a condenser built with the material as dielectric. A known adjustable loading resistor R, is connected parallel to the condensor in order to obtain data sufficient to calculate both capacitance C and resistance R of the sample. Network analysis gives the spectrum of voltage fluctuations

(14)

The values of R and C are obtained from the simultaneous solution of relations corresponding to eq. (14) for different RL _values.

Brophy and Webb [B 621 have studied triglycene sulfate. In figure 1 the noise voltage density is plotted versus temperature for different values of

These results point out the difficulty of the method. For low values of RL (RL - lo4 0) the loading resistor acts as a short-circuit for the sample. For high values of RL

FIG. 1. - Variation of noise voltage across the load resistor as a function of temperature of a triglycene sulfate crystal near the Curie temperature for several values of load resistor and at a

frequency of 1 000 cps (after Brophy and Webb).

C2-46 L. _GODEFROY

(RL - ^lo7

the noise of the loading resistor is

preponderant and the sample cannot be detected.

Therefore only a narrow range of values of RL may be used. Even in this range the calculations are tedious because the values which are to be inserted on the left hand side of the eq. (14) are very close to one another.

Nevertheless Brophy and Webb have performed the measurements for R and C and they give the permitti- vity

versus the frequency at the Curie temperature (cf. Fig. 2 and ref. [B 621).

Similar measurements on T. G. S. were performed by Ishibashi et al. [I 691 and by Fuhrmans [F 681.

Unfortunately their results do not agree with those of Brophy and Webb (Fig. 2). In particular, the decrease of

for frequencies greater than 10' Hz has not been found by Fuhrmans.

Sekido-Mitsui

FIG. 2. - Comparison of the results of different authors for the permittivity of T. G . S.

Rochelle Salt (R. S.) was investigated by noise method by Bittel et al. [B 651. Figure 3 shows the noise spectrum for different temperature (Re, is defined by Re,

Su( f)/4 kT). For low frequencies the load resistance RL

10 kS2 is predominant whereas for high frequencies a l /

noise is observed. It was observed that between 15 Hz and 15 kHz the noise may

FIG. 3. - Noise voltage of a Rochelle salt crystal at different temperatures expressed by the equivalent resistance Req. Load resistance I0 kQ, X experimental values, calculated with the half-frequency value of the capacitance, 0 calculated with the

bridge-values of R and C (after H. Bittel et al.).

be described using Nyquist's formula, even at Curie temperature. Non-linearity is investigated

the mean value of the fluctuating field is shown to be about 2 pV/cm whereas a 3 V/cm field is required to lower the permittivity by 1 %. Some experimental difficulties are pointed out

a very long time must elapse before the crystal reaches thermal equilibrium

a lack of thermal homogeneity moreower, perturbs the results.

The preceding results were obtained with crystals in thermal equilibrium without applied d. c. field. They were completed by Micheron et al. [M 661 and Jan- nin [J 711 who applied a d. c. field. Their results for T. G. S. are summarized in figure 4 which gives the total polarization fluctuations versus temperature for different values of the d. c. field. The agreement

FIG. 4. - Total polarization fluctuation for a triglycene sulfate crystal as a function of temperature for different d. c. applied

fields (after M. Jannin).

between bridge measurements (solid line) and noise measurements (dots) is good. There is complete concordance with Devonshire's laws for displacement of the maximum. The first step was to analyse the results, assuming that the three parameters R, C and 2

were independant

this procedure makes it possible to

verify the validity of the proportionality of R and 3

no discrepancy was found. Secondly, the number of

independant parameters is reduced to two (R and C)

and the agreement between noise and bridge measure-

ment is better than 2 % for C and 10 % for R provided

that the a.

field in bridge measurement is less than

0.5 V/cm.

NOISE MEASUREMENTS IN FERROELECTRICS

IV. A new method.

It has been pointed out

(Section 3) that due to the great difficulty in measuring Si(w) it was necessary t o use load resistors in order to get sufficient data to solve the equations. Unfortunately these loading resistors often hide the true phenomena.

To avoid their use Micheron [M 711 has proposed a new method (Fig. 5). An ajustable auxiliary noise

Sample

4

FIG. 5. - Diagram of the noise measurement operating process as designed by Micheron.

generator is placed parallel to the sample

in general the mean square voltage across the terminals is given by

When I

I

I Z I this reduces to

- -

v 2

e2 +

^{(15 bis)}

The measurement process is divided into three stages. In the first place switch I, is on, switch I, is off, and Nyquist's, formula gives the real part of Z.

Secondly, both switches are on and the noise generator is adjusted in such a way that equals twice the value it had during the first stage. This means that

2 e2 from which we deduce by (15 bis)

Finally, without modifying 2 I, is turned off while

is hept on, then

2 can be measured. Having obtained 2 during the first stage, 2 during the final stage and 1

l2 by construction, we can deduce 1 2 l2

from

From

1 and R both real and imaginary parts of the impedance are available.

Strictly speaking the sample is not in thermal equili- brium during the second stage because of the noise generator in parallel with it. But we must note that this perturbation is weak

its effect is only to double the value of the mean square voltage, and this should be compared with the much stronger effect of the a. c.

measuring field in a bridge.

This new method was applied to baryum titanate single crystals. Figure 6 shows the inverse permittivity versus temperature as it is deduced from bridge measurements and from noise measurements. In the paraelectric range the agreement is good. But in the ferroelectric range the discrepancy is significant

the

FIG. 6. - Results for a baryum titanate single crystal (after Micheron).

transition seems to be much more drastic where noise measurements are used. The smoothness of the transi- tion in bridge measurements is imputed to the a. c.

measuring field which however was reduced to 1 V/cm in this experiment.

Due to the smoothness of one plot a discrepancy of about 2

is found between the Curie temperature deduced from one or the other method.

On figure 6 are also plotted the conductance G of the crystal deduced from bridge measurements at 1 kHz and the square of the spectrum component of the fluctuating polarization, at the same frequency. The conductance shows a small anomaly at T, whereas p2 is much more anomalous. Moreover, a few degrees below T,, p2 shows anomaly which is completely absent for G.

No explanation is given but work is still proceeding.

V. Conclusion.

The validity of classical bridge measurements of permittivity in ferroelectric crystals may be questioned because of the presence of an a. c.

field which is strong enough to pertub the crystal in such a way that the results become irrelevant. That is why several authors have tried to measure the impedance of ferroelectric crystals by means of the voltage fluctua- tions which appear at the terminals of the electrodes.

As is the case with all noise measurements, this process is rather delicate but the greatest difficulty arises from the impossibility of finding the spectrum of the fluctuating short-circuit current. As long as noise voltage measurements are the only means available, expedients must be used to get sufficient data to solve the problem of measuring both real and imaginary parts of the impedance.

In the classical methods loading resistors are used.

By working with different values of the load, the authors can collect enough data to calculate the impedance. This is useful but very tedious and rather tortuous.

The new method described above works with an

auxiliary noise generator, providingresults in a simpler

way. Though it is not yet worked out in its complete

form, this method seems to be a very promising one.

L. _GODEFROY

References

@

581 BURGESS (R. E.), Can. Jour.

1958, 36, 1569. [J 701 JANNIN (M.), ThBse de

Cycle Dijon, 1970, a

@

621 BROPHY

WEBB

L.),

Rev., 1962, paraitre dans le Journal

Physique.

128,

584. [M 661 MICHERON (F.), BAUMBERGER

GODEFROY (L.),

@ 651 BITTEL

(H.), HELLMISS (G.), UNRUH (H.

Proceedings of the 1st International Meeting on

Phys.,

1965,

1. Ferroelectricity, Prague, 1966.

[M 711 MICHERON (F.), Thbse

Doctorat d'Etat, Dijon, [F 681 FUHRMANS (M.), Diplom. Miinster, 1968. 1971.

[I 691 ISHIBASHI (Y.), SAWADA (A.), TAKAGI

[P 70aJ PETERSSON (J.), Zeit. f:

1970,

148.

SOC. Japan.,

1969,

705. [P 70b] PETERSSON (J.), Zeit. f.

1970,

261.