DE102013103404A1 - Method for determining a sample-specific size of a piezoelectric thin-film sample - Google Patents

Abstract

A method for determining a sample-specific size of a piezoelectric thin-film sample (4) located on an elastic substrate (2) and arranged between electrodes (3, 4) is presented, which comprises the following steps: a) Measuring a first value for a longitudinal piezoelectric coefficient with a first electrode measuring arrangement, b) measuring at least a second value for the same longitudinal piezoelectric coefficient with a second electrode measuring arrangement, c) calculating the sample-specific size by means of a function dependent on the measured values of the longitudinal piezoelectric coefficient. In this way, the transverse piezoelectric coefficient is preferably determined.

Description

The invention relates to a method for determining a sample-specific size of a piezoelectric thin-film sample located on an elastic substrate and arranged between electrodes.

The piezoelectric effect is the occurrence of electrical stress on the surface of a solid when it is elastically deformed. In the inverse piezoelectric effect, the solid deforms by applying a voltage on its surface. The relationship between the mechanical and electrical properties of a piezoelectric material is described by the piezoelectric coefficients which are components of a tensor. There are two types of coefficients, namely the piezoelectric distortion coefficient d _ij , which describes the relationship between the mechanical distortion and the acting electric field, and the piezoelectric stress coefficient e _ij , which stands for the relationship between the mechanical stress and the electric field. The coefficients d _ij and e _ij are interconvertible by means of elastic constants of the piezoelectric material.

Solids with piezoelectric properties have an anisotropic crystal structure with a direction of polarization, which is why changes in the length of a sample are generally not only parallel to the direction of the electric field involved. In a reference to a piezoelectric sample coordinate system with mutually perpendicular axes 1 to 3, the polarization direction is referred to as axis 3. Essential in connection with the work with piezoelectric materials are the longitudinal piezoelectric coefficients d ₃₃ and e ₃₃ , which describes the relationship between an electric field component parallel to the polarization direction and the mechanical distortion parallel thereto, and the transverse piezoelectric coefficient d ₃₁ or e ₃₁ , which describes the relationship between the electric field component parallel to the polarization direction and the mechanical distortion perpendicular thereto in the direction of the axis 1. Piezoelectric thin-film samples are strongly influenced in their properties by boundary conditions, z. Example, by their geometric shape or by a substrate holding the sample, so that the piezoelectric coefficients are different from those in a corresponding bulk material, so are sample-specific. The sample specific coefficients are referred to as the effective piezoelectric coefficients, e.g. E _{31, f} .

For the production of piezoelectric thin films on substrates, the knowledge and testing of the piezoelectric coefficients is of crucial importance. Typical materials for piezoelectric thin films are, for example, lead zirconate titanate (PZT) or aluminum nitride (AlN). Such layers are used in particular for microelectromechanical systems (MEMS) as actuators. For such actuators in particular the transverse piezoelectric coefficient e ₃₁ and d _{31 is} important, the bending of a structure such. As membrane or bending beam causes.

From the DE 10 2005 006 958 A1 For example, it is known to measure the effective transverse piezoelectric coefficient e _{31, f} by subjecting a sample comprising a piezoelectric material to a bending load and determining the amount of charge generated by the consequent tensile or compressive loading of the piezoelectric material. The necessary measuring structures, z. B. bending beam or membranes are expensive to produce. Alternatively, it can be measured by bending small samples, but this destruction of the substrate z. B. required by sawing.

From the DE 10 2005 006 958 A1 It is further known to determine the longitudinal effective piezoelectric coefficient d _{33, f} by applying a specific electric field in the direction of polarization and determining the change in thickness thereby produced in the direction of polarization. For the determination of the thickness change z. B. the double-beam interferometry can be used.

It is further known from the aforementioned document to apply a lower electrode to a substrate, for example a silicon wafer, and to place the piezoelectric material thereon, which in turn is at least partially covered by an upper electrode.

From Nicolas Zachalas et al. (Effective Piezoelectric Coefficients of Ferroelectric Thin Films on Elastics Substrates; Journal of Intelligent Material Systems and Structures, Vol. 20-2009, page 681 ff) For example, theoretical models are known which allow at least approximate calculation of effective piezoelectric components of a thin-film sample applied to a substrate under various boundary conditions from the known piezoelectric coefficients of the bulk material and elastic constants should. Three models with different boundary conditions for the thin-film sample are presented: a sample on a completely rigid substrate as model I, a sample completely covered by an electrode on an elastic substrate as model II and as model III a sample with a small electrode on the elastic substrate , The disclosure of the above-mentioned Zachalas paper is fully incorporated.

It is an object of the present invention to provide a novel method of the type mentioned above, with which a sample-specific size can be carried out with less effort and without destroying the substrate.

This object is solved by the features of claim 1.

Advantageous embodiments of the method according to the invention will become apparent from the dependent claims.

In the following, the theoretical background and an exemplary preferred embodiment of the method according to the invention are illustrated by means of figures.

It shows schematically:

1 FIG. 2 is a side cross-sectional view of a prior art structure of a piezoelectric thin film sample on a substrate.

2 : a diagram with measured values for d ₃₃ as a function of the electrode size,

3a : a portion of a thin film sample on a substrate without an applied electric field,

3b : the structure according to 3a with applied electric field and

4 in plan view, a thin film sample with a plurality of upper electrodes.

1 schematically shows a known from the prior art layer structure of a measuring structure 1 with a substrate 2 , one on the substrate 2 applied lower electrode 3 , a thin-film sample 4 and an upper electrode 5 , To measure the longitudinal piezoelectric distortion coefficient d ₃₃ , a voltage is applied between the electrodes and the thickness change of the thin-film sample produced thereby 4 measured in polarization direction. The change in thickness is preferred on both sides of the measuring structure 1 provided measuring points, one of which is located on the upper electrode 5 located.

The invention now uses the knowledge not shown in the prior art that different values for the same effective longitudinal piezoelectric coefficient are measured for the same sample with different electrode arrangements. 2 shows an example of a thin-film sample 4 made of PZT on a silicon wafer as a substrate 2 the measured values for the longitudinal piezoelectric coefficient d ₃₃ as a function of the edge length of a square electrode 5 together with a theoretical value d _33theor for the longitudinal coefficient represented by the horizontal line. The finding was surprising that the measured value is greater than the theoretical value of the expected longitudinal coefficient for a large electrode structure compared to the substrate thickness, but lower for small electrode structures.

This effect is probably due to the influence of the thin-film sample 4 firmly bonded substrate 2 due. 3a (without applied electrical voltage) and 3b (with applied electrical voltage) show a simplified principle of the overall deformation of substrate 2 and part 6 the thin-film sample 4 , The electrodes 3 and 5 (please refer 1 ) are not shown here for the sake of clarity. The illustrated part 6 the thin-film sample 4 is from the not reproduced upper electrode 5 covered. The unillustrated part of the thin-film sample 4 that feel like in 1 to the edge of the substrate 2 is not from an active upper electrode 5 covered. Because of the transverse piezo coefficient, when the measuring voltage is applied, it contracts below that in 3a and 3b not shown upper electrode 5 the part shown 6 the thin-film sample 4 perpendicular to the polarization direction, ie parallel to the layer surface. As a result, the substrate experiences 2 , as in 3b shown in the area below the part 6 the thin-film sample 4 also a cross-compression. But this affects that Substrate in a hitherto disregarded in the art, the measurement of the longitudinal coefficient d _33rd

It was found that the transverse compression of the substrate 2 an additional thickening in the area below the thin-film sample 4 only with such upper electrodes 5 causes their longitudinal extent large against the thickness of the substrate 2 is. For smaller upper electrodes 5 however, cross-compression probably can not cover the entire substrate 2 penetrate, whereby the effect is reduced or absent.

The method according to the invention exploits the above-described effect of measuring different values for the longitudinal piezoelectric coefficient in order to determine indirectly a further sample-specific variable, preferably the transverse piezoelectric coefficient. For the representation of the mathematical relationship of the effect with the transverse coefficient, the theoretical model III mentioned in the introduction to the description of FIG Zachalas et al. (Effective Piezoelectric Coefficients of Ferroelectric Thin Films on Elastics Substrates, supra) based on.

wobei ν_Su die Querkontraktionszahl, Y_Su der Elastizitätsmodul des Substrats 2, I_Su die Dicke des Substrats 2 und T_su die mechanische Spannung im Substrat 2 ist. ist. Es wird angenommen, dass diese Dickenänderung bei Normierung auf die bei der Messung angelegte elektrische Spannung der Differenz zwischen dem festgestellten Messwert und dem erwarteten theoretischen Wert für den longitudinalen Koeffizienten entspricht.The mathematical relationship can be plausibly derived as follows:
First, generally, a thickness change D of the substrate will be made 2 from a mechanical transverse contraction of the substrate 2 considered. For this applies

where ν _{Su is} the transverse contraction number, Y _{Su is} the modulus of elasticity of the substrate 2 , I _Su the thickness of the substrate 2 and T _su the stress in the substrate 2 is. is. It is assumed that this normalized normalization to the voltage applied during the measurement corresponds to the difference between the observed value and the expected theoretical value for the longitudinal coefficient.

wobei I_PP die Dicke der piezoelektrischen Schicht, I_Su die Dicke des Substrats, T_PP die mechanische Spannung in der Dünnschichtprobe und T_Su die mechanische Spannung im Substrat ist (siehe Gleichung (14) aus Zachalas et al.; a. a. O., S. 683 ).According to the theoretical considerations of Zachalas et al , For a thin film sample deposited on an elastic substrate, which is completely covered by an electrode, the following relationship is assumed:

where I _{PP is} the thickness of the piezoelectric layer, I _{Su is} the thickness of the substrate, T _{PP is} the stress in the thin film sample, and T _{Su is} the stress in the substrate (see Equation (14)) Zachalas et al .; cit., p. 683 ).

(Vergleiche Zalachas a. a. O., S. 688, die dortige Gleichung 18 ).If the piezoelectric layer only partially from the acting electrode 5 covered, the upper equation with a z. B. empirically or by simulation calculations to be determined geometric factor a, so it reads

(Compare Zalachas loc. Cit., P. 688, the equation 18 there ).

If, in equation (1), the quantity T _{Su is replaced} by equation (3), the result is:

wobei Y_PP der Elastizitätsmodul, ν_PP die Querkontraktion der Dünnschichtprobe 4, sowie V_PP die angelegte elektrische Spannung ist.From known basic equations of the piezoelectricity T _{PP is given} in first approximation by

where Y _{PP is} the modulus of elasticity, ν _{PP is} the transverse contraction of the thin-film sample 4 , and V _{PP is} the applied voltage.

wobei d₃₁ der transversale piezoelektrische Verzerrungskoeffizient und e_31,f der gesuchte effektive transversale piezoelektrische Spannungskoeffizient ist. Die geometrieabhängige Konstante a hängt von der Elektrodenmessanordnung ab, die sich aus der Ausdehnung der Elektrode 5, insbesondere parallel zur Schicht der Dünnschichtprobe 4, und/oder aus der Platzierung der Messstelle für die Dickenmessung auf der Elektrode 5 ergibt. Die geometrieabhängige Konstante kann durch Referenzmessungen und/oder durch Simulationen genau und spezifisch bestimmt werden. Dies kann z. B. durch die im dargestellten Stand der Technik bekannten Verfahren zur Messung des transversalen Koeffizienten an gleichen Proben erfolgen.Equation (5) inserted into equation (4)

where d _{31 is} the transverse piezoelectric distortion coefficient and e _{31, f is} the sought effective transverse piezoelectric stress coefficient. The geometry-dependent constant a depends on the electrode measuring arrangement resulting from the expansion of the electrode 5 , in particular parallel to the layer of the thin-film sample 4 , and / or from the placement of the measuring point for the thickness measurement on the electrode 5 results. The geometry-dependent constant can be determined precisely and specifically by reference measurements and / or by simulations. This can be z. Example, by the known in the prior art method for measuring the transverse coefficient of the same samples.

D / V _PP = d _{33, meas} - d _{33, f} , from which the following dependence of the measured value for the transverse coefficient on the geometric constant a results:

In two measurements with two different geometries and correspondingly different known geometric constants a ₁ and a ₂ , different measured values d _{33, meas 1} and d _{33, meas 2} result for the transverse coefficient. By simple subtraction results:

Thus, the transverse piezoelectric coefficient e _{31, f can be determined} from the measured values as well as the known geometric constants a ₁ and a ₂ as well as the well-known quantities Y _Su and ν _Su .

Alternatively, further sample-specific quantities can be determined, for example ν _PP or Y _PP with a corresponding equivalent transformation of equation (8), if z. B. already the transverse piezoelectric coefficient is known.

The above derivation assumes scalar sizes and thus an isotropic view. The scalar quantities can be seen as an average over the usually given anisotropic case. More accurate anisotropic viewing is possible in a similar manner and also allows the determination of piezoelectric coefficients related to other axes than those used in the above derivation.

The above considerations are not limited to the illustrated example. So can the upper electrode 5 take other than square forms. It is also possible to use the lower electrode 3 or both electrodes 3 and 5 to change in their geometry. Additionally or alternatively, another measuring point for thickness measurement can be provided. It would also be conceivable to have several on the same side of the thin-film sample 4 located upper electrodes 3 and / or lower electrodes 5 , the spatially separated from each other or preferably adjacent to each other simultaneously to apply voltage. The thin-film sample 4 Nor does the complete, the thin-film sample 4 facing surface of the substrate 2 cover. Ultimately, the only decisive factor is to change the geometric factor a in a suitable manner in order to be able to carry out at least two measurements.

4 shows a thin-film sample 7 in supervision. That the thin-film sample 7 supporting substrate is not shown here. The thin-film sample 7 has a plurality of different upper electrodes 8th different size and geometry. For a plurality of measurements according to the invention, individual ones of the electrodes 8th and / or groups of electrodes 8th be energized successively. It can be provided a single or a plurality of not visible here lower electrodes.

LIST OF REFERENCE NUMBERS

1

measurement structure

2

substratum

3

lower electrode

4

thin film sample

5

upper electrode

6

Part of the thin-film sample

7

thin film sample

8th

upper electrodes

QUOTES INCLUDE IN THE DESCRIPTION

This list of the documents listed by the applicant has been generated automatically and is included solely for the better information of the reader. The list is not part of the German patent or utility model application. The DPMA assumes no liability for any errors or omissions.

Cited patent literature

• DE 102005006958 A1 [0005, 0006]

Cited non-patent literature

• Nicolas Zachalas et al. (Effective Piezoelectric Coefficients of Ferroelectric Thin Films on Elastics Substrates; Journal of Intelligent Material Systems and Structures, Vol. 20-2009, page 681 ff) [0008]
• Zachalas et al. (Effective Piezoelectric Coefficients of Ferroelectric Thin Films on Elastics Substrate, supra). [0023]
• Zachalas et al. [0025]
• Zachalas et al .; cit., p. 683 [0025]
• Zalachas loc. Cit., P. 688, the equation 18 there [0026]

Claims (6)

Method for determining a sample-specific size of a material on an elastic substrate ( 2 ) and between electrodes ( 3 . 4 ) piezoelectric thin-film sample ( 4 comprising: a) measuring a first value for a longitudinal piezoelectric coefficient with a first electrode measuring arrangement, b) measuring at least a second value for the same longitudinal piezoelectric coefficient with a second electrode measuring arrangement, c) calculating the sample-specific size by means of one of the measured values of the longitudinal piezoelectric coefficient dependent function. A method according to claim 1, characterized in that the sample-specific size is a transverse piezoelectric coefficient. A method according to claim 1 or 2, characterized in that the first and second electrode measuring arrangement by different sizes of the sample-side contact surfaces of at least one of the electrodes ( 3 . 4 ) differ from each other. Method according to one of the preceding claims, characterized in that the first and second electrode measuring arrangement differ from each other by a different placement of measuring points for a thickness measurement on at least one of the electrodes. Method according to one of the preceding claims, characterized in that on at least one of the sample sides at least two electrodes ( 3 . 4 ). Method according to one of the preceding claims, characterized in that the function dependent on the two values of the longitudinal piezoelectric coefficient

or an equivalent function is used, where e _{31, f is} the effective transverse piezo coefficient d _{33, meas1 is} the first measured value of the longitudinal piezoelectric coefficient d _{33, meas2 is} the second measured value of the longitudinal piezoelectric coefficient Y _Su the modulus of elasticity of the substrate ( 2 ) ν _{Su is} the transverse contraction number of the substrate ( 2 ) a _{1 is} a given during the first electrode measuring arrangement geometric constant a ₂ given in the second electrode measuring arrangement geometric constant.

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