Comités especiales

Table 2.1 – Effective piezoelectric coefficients for standard PZT ceramic grades.

PZT grade e31(C/m2) e33(C/m2) e31, f(C/m2) e33, if(C/m2)

PZT-4 _−5.2 15.1 ₋₁₅ 17.9

PZT-5H _−6.55 23.3 _−23.3 27.7

piezoelectric coefficients and is given by [35]:

e33, if= e33−

c₁₃E

c₁₁E e31 (2.11)

As will be seen in chapter 3, it is more accurate to write E3= V /(a + ∆a) where a + ∆a is an

effective gap for electrical properties. In this case we have:

e33, if= − w (x2) 3x1(2x2− x1)(1 − η) (a + ∆a)Ests2 V cftf (2.12)

Table 2.1 shows the values of the effective piezoelectric coefficients for two common com- mercial PZT ceramic grades. We have e33, if> e33and |e31, f| > |e31|. This means that the film

clamping improves the piezoelectric response in the IDE case, which is also true in the PPE case as was shown by Muralt [36], although the improvement is less pronounced for IDE. We expect a larger piezoelectric response in the IDE configuration, since e_{33, if> |e31, f|.}

2.3 Piezoelectric coefficient in direct mode

2.3.1 Setup description and cantilever bending geometry

For measuring the transverse piezoelectric coefficient through the direct piezoelectric effect, a setup first proposed by Dubois and Muralt [39] is used. In this setup, a cantilever beam is clamped at one end and a piezoelectric actuator applies an alternating displacement on the beam other end. The setup is pictured in Figure 2.5. Since the piezoelectric film is much thinner than the substrate, it is assumed to experience the same strain as that induced at the surface of the bent substrate. The electrodes are connected either to a charge amplifier to collect the piezoelectric charges generated by the oscillating strain — the piezoelectric film is then in closed-circuit condition, since the charge amplifier input behaves as a virtual ground; or to a voltage amplifier to measure the voltage difference induced by the oscillating strain — the open-circuit condition is then realized in the film. An oscilloscope is used to record the peak-to-peak variation of the charge (resp. the voltage). This allows to obtain the effective e and h coefficients, respectively.

We set x = 0 at the clamping position and x = l > 0 at the point where the actuator is in contact with the substrate at zero excursion. The actuator peak-to-peak displacement is written as

charge amplifier or voltage amplifier clamping stage substrate film piezoelectric actuator x y

Figure 2.5 – Configuration of the clamping stage for direct transverse piezoelectric effect measurements. The dimensions are not to scale. The cantilever excursiony is exaggerated

for clarity. The piezoelectric actuator moves back and forth as indicated by arrows, causing a small excursion oscillation around an average excursion value.

∆yact, imposed on the cantilever beam at x = l . The active zone starts at x = 0. The full

derivation of the coefficients is given in appendix B; here, only the final results are presented.

2.3.2 Parallel plate electrodes

This case is treated by Dubois and Muralt [39]. Let d be the length of the top electrode along thex direction. As before we name w_elthe (rectangular) top electrode width. e_{31, f}is given as function of the sample geometry, peak-to-peak actuator displacement and recorded peak-to-peak charge signal∆Q as follows:

e31, f= − 2l3∆Q 3∆yactts(1 − νs)weld ³ l −d₂´ (2.13)

And h31, fis obtained from the measured peak-to-peak voltage∆V by:

h_{31, f=} 2l 3_∆V 3∆yactts(1 − νs)tf ³ l −d₂ ´ (2.14)

2.3. Piezoelectric coefficient in direct mode

2.3.3 Interdigitated electrodes

This case is treated by Chidambaram et al. [43]. We must define, unlike for the PPE case, a new effective piezoelectric coefficient eIDE:

eIDE= e31 Ã νsc12E − c13E c₁₁E − νs ! + e33 (2.15)

Note that eIDEis not a material parameter but depends on the Poisson’s ratio of the substrate.

We show in appendix B that it can be approximated by e33, ifwith good accuracy.

With the same notations as before, eIDEis given by [43]:

eIDE= − 2l3∆Q 3∆yacttsweltfNg ³ l −d₂´ (2.16)

where Ngis the number of IDE gaps and d = Ng(a + b) is the length of the interdigitation zone.

The open circuit coefficient is also a new coefficient, defined as:

hIDE= h31 Ã νsc₁₂D− c₁₃D c₁₁D − νs ! + h33 (2.17)

With the same notations as before, we have:

hIDE= 2l3∆V 3∆yacttsa ³ l −d₂´ (2.18)

As will be seen in chapter 3, it is more accurate to use an effective gap value of a + ∆a. The formula then becomes :

hIDE=

2l3∆V 3∆yactts(a + ∆a)

³

l −d₂´

3

Physical behavior of ferroelectric thin

films with IDE and characterization

methodology

3.1 Introduction

The interdigitated electrode (IDE) configuration allows for a large voltage response [32], which is important when using ferroelectric thin films with high dielectric constants such as PbZr_xTi_1–xO₃(PZT). It offers a number of further advantages : better polarization stability at small fields due to less polarization back-switching [45], a lower hysteresis in the unipolar mode [35], and lower dielectric losses [45]. There is therefore a large interest to improve characterization of such devices. The main difficulty to characterize ferroelectric properties in ferroelectric films with IDE lies in the inhomogeneous and curved electric field, resulting in much more complex patterns than in the PPE case. When the structures are biased, neither the direction nor the intensity of the electric field is uniform inside the film [35]. For this reason, a number of corrections are required if one wants to extract the material properties from standard measurements [35, 45, 77]. There are particularly questions about the interpretation of a ferroelectric loop. How is the polarization derived from the charges captured by the ID electrodes? And how should one correct the electric field to obtain the real coercive field?

In this chapter, we first review the previous approaches to this problem. There is the simple model assuming that the electrodes traverse the ferroelectric thin film, approximating the IDE with parallel plate capacitors that are connected in parallel. This model underestimates the dielectric constant [45]. Much better agreement with expected dielectric constants is obtained by applying a conformal mapping transformation approach following Gevorgian and coworkers [78] or Igreja and Dias [79]. However, this approach does not give an answer on how one has to transform electric fields, and to the question of how polarization charges are transmitted to the interdigitated electrodes. We will show that for typical geometries — with a film thickness smaller than the electrode gap — one can simplify the expressions of Gevorgian’s and Igreja’s methods to extract the dielectric constant, and that the electric field between the fingers can be considered as homogeneous, depending on the gap distance and the ferroelectric film thickness only. We also calculate the effective area that should be

Si

(a)

t

a

b

+V

-V

+V

W

(b)

Figure 3.1 – (a) : Stack design of the samples used in this study. (b) : Geometry of the interdigi- tation zone and definition of the geometrical quantities. The electric field lines, as well as the local polarization vector, which will attempt to align with the local direction of the electric field, are shown to illustrate the expected behavior of the ferroelectric film.

used to obtain the polarization from switching current measurements; and we show that the Si substrate — though separated through 2µm of SiO₂and only weakly doped — adds a parasitic capacitance that should be removed. Our corrections result in polarization versus field (PV) and permittivity versus field (CV) loops that overlap for a large value range of IDE gaps, showing that they must be close to intrinsic material values.

Figure 3.1 (b) defines the geometrical quantities used to describe the IDE structure. a is the finger distance or gap, b is the finger width, W is the finger length (only considering the distance over which fingers of opposite polarity are facing each other), N is the number of fingers for each electrode (thus 2N is the total number of fingers), and tf is the ferroelectric

film thickness. V is the potential difference applied across the IDE capacitor. Our PZT films are grown on a 100-nm-thick MgO layer on Si substrate with 2µm wet oxide to provide electrical insulation, as pictured in Figure 3.1 (a). The detailed fabrication route can be found in a previous publication [35].

This chapter is adapted from the following publications :

A) R Nigon, TM Raeder, and P Muralt. Characterization methodology for lead zirconate titanate thin films with interdigitated electrode structures. Journal of Applied Physics, 121(20):204101, 2017

B) CH Nguyen, R Nigon, TM Ræder, U Hanke, E Halvorsen, and P Muralt. Extraction of

properties of ferroelectric thin films in interdigitated electrode systems. In preparation