DE102019106510B4 - Method for determining the properties of layers in layer systems - Google Patents

Abstract

Method for determining properties of layers of layer systems from the measured values of the frequency-dependent phase velocity of surface acoustic waves on this layer system, in which the properties sought are determined by fitting the calculated values of the frequency-dependent phase velocity of surface acoustic waves on this layer system to the corresponding measured values with the help of optimization software The layer system is a layer stack on a substrate, characterized in that several directions of propagation of the surface acoustic waves are used, whereby a) the layers form at least one layer stack and the layer stacks are arranged on one and the same solid substrate or on different solid substrates, and b) the The layer properties to be determined are a selection from a list of layer properties which, in addition to density and layer thickness, have at least properties of one isot ropen layer or an anisotropic layer contains according to the real crystal symmetry of the layer, andc) the measured values of the frequency-dependent phase velocity for a solid substrate or several solid substrates and several directions of propagation of the surface acoustic waves in layers or layer stacks on the respective solid substrate are determined experimentally, andd) the object function, which is minimized in the course of the fitting, is calculated from the values of the calculated as well as from the measured values of the frequency-dependent phase velocity of surface acoustic waves of all measured directions of propagation on all measured crystal sections of the substrate, and e) the fitting process is repeated with randomly selected starting parameters of the properties sought until a reliable global minimum of the object function is recognizable in the arrangement of the usually found local minima of the object function or can be determined from this arrangement.

Description

Field of application of the invention

The present invention relates to the fields of measurement technology and acoustics and relates to a method for determining properties of conductive or dielectric layers in layer systems, in particular the elastic properties, the density and the thickness of such layers. The invention is based on the use of surface acoustic waves on piezoelectric and non-piezoelectric substrates.

State of the art

Methods are known for determining properties of at least one layer system on at least one solid substrate from the measured values of the frequency-dependent phase velocity of surface acoustic waves on this layer system, in which the properties sought are obtained by fitting the calculated values of the frequency-dependent phase velocity of surface acoustic waves on this layer system to the corresponding measured values can be determined with the aid of optimization software, whereby a layer system is to be understood as an individual layer on a substrate or a layer stack on a substrate.

A special version, hereinafter referred to as solution [1], is a method for determining properties of layers of a layer system with the help of software from the measured values of the frequency-dependent phase velocity of surface acoustic waves, which are determined with the help of a measurement method that is described in DIN EN 15042-1: 2006-06 (Layer thickness measurement and characterization of surfaces by means of surface waves - Part 1: Guidelines for the determination of elastic constants, density and thickness of layers by means of laser-induced ultrasonic surface waves; German version EN 15042-1: 2006). As a result of the application of the process, examples of a layer system with a tetrahedral amorphous carbon layer (ta-C) on silicon and a layer system ta-C / Cr on a Si are given in the 2016 annual report of the Fraunhofer Institute for Material and Beam Technology (IWS) Dresden -Substrate described. For the ta-C / Si layer system, the layer properties determined with the aid of the method - modulus of elasticity, density and layer thickness - are given, but no second elastic parameter such as the Poisson ratio. Since each amorphous layer is characterized by two elastic parameters that are independent of one another, the lack of the second elastic parameter is disadvantageous. For the layers ta-C / Cr / Si, the modulus of elasticity, density and layer thickness of the ta-C layer and the layer thickness of the chromium intermediate layer are also given. A second elastic parameter of the tetrahedral amorphous carbon layer is also not specified here. It is not mentioned whether the elastic properties and the density were specified, because these parameters are also not specified. In both cases it is not known whether, as in Figure 8 of the DIN EN 15042-1: 2006-06 (Solution [1],), an alternative, experimentally determined correlation between the modulus of elasticity and the density of tetrahedral (diamond-like) amorphous carbon layers was used for the evaluation. In this case the demands on the performance of the optimization are lower.

In another special version (Tsung-Tsong Wu, Yung-Yu Chen and Guo-Tsai Huang, "Evaluation of properties of sub-micrometer thin films using high frequency ultrasonic waves", in Proc.11th International Congress on Fracture (ICF) 2005 , Vol. 5, p.3769, referred to as solution [2]) describes a device and a method for determining the elastic constants c11 and c44 and the density of SiO ₂ layers, based on surface acoustic waves, in which the arrangement of a Delay line on YZ LiNbO ₃ as a piezoelectric substrate, the two interdigital transducers with inclined prongs (SFIT - Slanted Finger Interdigital Transducers) and a SiO ₂ layer between each two interdigital transducers and a reference delay line with the same arrangement of interdigital transducers with inclined prongs, but does not contain a layer, the delay line and the reference delay line forming a delay line composite and wherein di e layer thickness of the SiO ₂ layer is measured separately and in which the method is an optimization method that minimizes the difference between experimental and calculated values of the phase velocity of the surface acoustic waves with the help of the simplex algorithm.

In another special version (FS Hickernell and TS Hickernell, "Surface Acoustic Wave Characterization of PECVD Films on Gallium Arsenide" in IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, Vol. 42, No. 3, May 1995, referred to as solution [3]) the elastic constants C ₁₁ and C ₄₄ of layers are measured by fitting the calculated dependence of the phase velocity of the surface acoustic wave on the normalized layer thickness to the corresponding one Dependency determined. In order to have a wide frequency range available, higher harmonics of the surface acoustic wave are used in a delay line. The density of the layers is determined from the difference between the masses of the delay line with and without the layer and the layer thickness, which must be measured separately. There is no reference delay line.

Another special version ( M. Knapp, AM Lomonossov, P.Warkentin, PM Jäger, W. Ruile, H.-P. Kirschner, M. Honal, I. Bleyl, AP Mayer, and L. M Reindl, "Accurate Characterization of SiO2 Thin Films Using Surface Acoustic Waves", IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, Vol. 62, no. 4th , April 2015, referred to as solution [4]) also describes a delay line network in which the delay line and reference delay line differ only in the spaces between the interdigital transducers, and the entire delay line network is coated _{with SiO 2.} Two different sections of LiNbO ₃ are used as piezoelectric substrates to determine the elastic constants c11 and c44 and the density of the layer, namely 128 ° YXLiNbO ₃ and 64 ° YXLiNbO ₃ . An optimization method is used that can adapt the calculated frequency-dependent phase to the experimentally determined frequency dependency of the phase, the frequency-dependent phase being calculated with the aid of an analysis method from the layer parameters specified by the optimization method. However, it is not mentioned which optimization method is used to determine the elastic constants c11 and c44 and the density of the layer.

The solution [1] comes closest to its own solution, but has the disadvantages that firstly, the method can only process the data that has been obtained by measuring in only one direction of propagation of the surface acoustic waves, secondly, the layer thickness can only be processed the fitting process can be determined, and must be measured separately with the help of an alternative method and entered as a constant in the start parameters if at least one other layer parameter is not determined using the method described.Third, only such sections and directions of propagation of the substrate crystal can be used that are approximately isotropic in their immediate vicinity. Fourth, isotropic layers are assumed, although anisotropic layers are also of technical importance. And fifth, the elastic properties such as the modulus of elasticity and Poisson's ratio and the density can only be determined from a single layer.

The common disadvantage of the solutions [2, 3, 4] is the need for the substrate carrying the layer to have piezoelectric properties.

In addition, US 2014/0 007 692 A1 discloses a system for monitoring the deposition of ultra-thin layers and nanomaterials. This system uses embodiments employing surface acoustic waves that incorporate differential delay lines. These in turn contain integral reference devices which are arranged on the same substrate as the sensor device or on a separate device and are in thermal contact with the monitoring system in order to enable temperature-compensated measurements. This monitoring system allows an inherent temperature compensation, a higher sensitivity to surface interactions than quartz crystal microbalances (QCM) and is able to work at extreme temperatures.

Another theoretical study is known in which the dispersive characteristics of surface acoustic waves are analyzed, which propagate on layered structures of the type low-k / SiO2 / Si substrate and low-k / Cu / Si substrate ( Xia Xiao et al. in: Microelectr. J., Vol. 37, 2006, pp. 1052-1055 ). The mechanical properties of these low dielectric constant layers can be determined by fitting the theoretical and experimental dispersion curves. However, the fitting process is not described, and no fitting results are given. The numerical results for given values of the modulus of elasticity of the layers with a low dielectric constant show that the surface wave technology can be used in most cases below a frequency limit of 250 MHz.

Subject of the invention

The present invention is based on the object of developing a method based on the use of surface acoustic waves for determining properties of layers of a layer system, with which the scope of the layer parameters to be determined is expanded compared to the prior art and in which no conditions apply the isotropy of the substrate crystal in the vicinity of the cut used and the direction of propagation of the substrate crystal used. The task includes that the layer thickness is also determined by a fitting process and that the elastic properties should also be determinable from anisotropic layers and that the elastic properties such as elastic modulus and Poisson's ratio and the density should be determinable both from a single layer and from a stack of layers.

The object is achieved with the features contained in the patent claims, the invention also including combinations of the individual dependent patent claims in the sense of an AND link, as long as they are not mutually exclusive.

The method according to the invention for determining properties of layers of layer systems is based on a method in which the properties sought are determined from the measured values of the frequency-dependent phase velocity of surface acoustic waves on this layer system by fitting calculated values of the phase velocity to the corresponding measured values with the help of optimization software .

1. a) die Schichten mindestens einen Schichtstapel bilden und die Schichtstapel auf ein und demselben Festkörpersubstrat oder auf verschiedenen Festkörpersubstraten angeordnet werden, und
2. b) die zu bestimmenden Schichteigenschaften eine Auswahl aus einer Liste von Schichteigenschaften sind, die neben Dichte und Schichtdicke mindestens Eigenschaften einer isotropen Schicht oder einer anisotropen Schicht enthält entsprechend der realen Kristallsymmetrie der Schicht, und
3. c) die Messwerte der frequenzabhängigen Phasengeschwindigkeit für ein Festkörpersubstrat oder mehrere Festkörpersubstrate und mehrere Ausbreitungsrichtungen der akustischen Oberflächenwellen in Schichten oder Schichtstapeln auf dem jeweiligen Festkörpersubstrat experimentell bestimmt werden, und
4. d) die Objektfunktion, die im Verlauf des Fittens minimiert wird, sowohl aus den Werten der berechneten als auch aus den gemessenen Werten der frequenzabhängigen Phasengeschwindigkeit von akustischen Oberflächenwellen aller gemessenen Ausbreitungsrichtungen auf allen gemessenen Kristallschnitten des Substrats berechnet wird, und
5. e) der Fitprozess mit zufällig gewählten Startparametern der gesuchten Eigenschaften solange wiederholt wird, bis in der Anordnung der gewöhnlich gefundenen lokalen Minima der Objektfunktion ein zuverlässiges globales Minimum der Objektfunktion erkennbar ist oder aus dieser Anordnung bestimmt werden kann.

According to the invention, several directions of propagation of the surface acoustic waves are used, with

1. a) the layers form at least one layer stack and the layer stacks are arranged on one and the same solid-state substrate or on different solid-state substrates, and
2. b) the layer properties to be determined are a selection from a list of layer properties which, in addition to density and layer thickness, contain at least properties of an isotropic layer or an anisotropic layer according to the real crystal symmetry of the layer, and
3. c) the measured values of the frequency-dependent phase velocity for a solid substrate or several solid substrates and several directions of propagation of the surface acoustic waves in layers or layer stacks are determined experimentally on the respective solid substrate, and
4. d) the object function, which is minimized in the course of the fitting, is calculated both from the values of the calculated and from the measured values of the frequency-dependent phase velocity of surface acoustic waves of all measured directions of propagation on all measured crystal sections of the substrate, and
5. e) the fitting process is repeated with randomly selected starting parameters of the sought properties until a reliable global minimum of the object function is recognizable in the arrangement of the local minima of the object function usually found or can be determined from this arrangement.

The object function describes the deviation of the calculated phase velocity from the measured phase velocity, both as a function of the frequency. The measured phase speed depends on the real layer properties, while the calculated phase speed depends on theoretical layer properties, which are selected step by step by the optimization software according to the optimization algorithm so that the calculated and measured phase speeds approximate each other. A device according to DIN 15042-1: 2006-06, for example, can be used to determine the measured values. The commercial software “Optimization Toolbox” from Matlab, for example, is suitable as optimization software.

In the method according to the invention, as a result of the use of several directions of propagation of the surface acoustic waves, the scope of the layer parameters to be determined is significantly expanded compared to the prior art. In addition, in the method according to the invention, advantageously no conditions or requirements are placed on the isotropy of the substrate crystal in the vicinity of the cut used and the direction of propagation of the substrate crystal used. It is also advantageous that the layer thickness can also be determined by a fitting process and that the elastic properties can also be determined of anisotropic layers. Another advantage is that the elastic properties, such as the modulus of elasticity and the Poisson's ratio, as well as the density can be determined both from a single layer and from a stack of layers.

• Für die Festkörpersubstrate können vorteilhaft Kristallschnitte ein und derselben Kristallart verwendet werden.

Appropriate or advantageous embodiments of the method according to the invention are:

• Crystal cuts of one and the same type of crystal can advantageously be used for the solid substrates.

A search area is defined for each layer property to be determined, in that a lower and an upper limit of the respective search area are specified for the optimization software for each layer property to be determined.

The greatest possible number of layer properties to be determined is advantageously selected.

Advantageously, all material-specific properties of the layers are included in the determination, in particular also the elastic stiffnesses c11 and c55 as well as the density and the thickness of the layer under consideration.

It is also advantageous if the frequency range from which the values of the parameters originate from which the object function is calculated is selected as large as possible.

The parabolic component of the measured curves of the speed of acoustic surface waves as a function of the frequency is expediently as large as possible and other non-linear components, however, are very small.

As a measure of the parabolic components of the measured curves of the speed of acoustic surface waves as a function of the frequency and as the mean curvature of these curves in the frequency range used for optimization, the mean 2nd derivative of the speed according to the frequency is used if the mean curvature of these curves in the frequency range used for optimization, the absolute value is greater than 0.0015 m / (s * MHz ² ).

At least one parameter from the list of layer properties to be determined is determined by an alternative method if the mean 2nd derivative of the speed according to the frequency as the mean curvature of these curves in the frequency range used for the optimization is not greater than 0.0015 in absolute terms m / (s * MHz ² ) is.

The selection of properties is determined separately for each layer of each layer system.

A layer property or several layer properties can also be selected from the list of the layer properties to be determined if these have been measured separately using other methods.

Non-piezoelectric substrates, in particular sections of a silicon crystal, can be used as solid substrates. However, piezoelectric substrates can also be used.

The method is advantageously used in layer systems with one or more electrically conductive layers or with one or more electrically non-conductive layers.

In the case of non-conductive layers, the list of the layer properties to be determined also contains the dielectric constant.

The method according to the invention includes that the layer properties of several layer systems arranged next to one another on one and the same solid substrate are determined, which consist of layers with different layer thicknesses made of the same material, the same selection of layer properties being determined for all layers in order to avoid the change in the To investigate layer properties with increasing layer thickness and this selection also includes the property layer thickness.

According to the invention, the object function is calculated from a sum of the squares of the differences from the calculated and the measured phase velocity of surface acoustic waves.

An expedient embodiment of the method according to the invention is that the properties of the layers are determined independently of one another without using known relationships between two or more properties.

The fitting process, or called optimization, is carried out according to the invention with the aid of optimization software. This consists of two components: the actual optimization software and the Analysis software. With the help of its algorithm, the actual optimization software calculates the current values of the layer properties to be determined, taking into account the search areas, and transfers these values to the analysis software. When starting the optimization, it uses the specified start parameters. The analysis software uses the current values of the layer properties to be determined and the specified substrate parameters to calculate the frequency-dependent phase velocity of the surface acoustic waves, and from this and from the measured values of the frequency-dependent phase velocity, calculates the value of the object function that belongs to the current values of the layer properties to be determined and outputs it Value of the object function back to the actual optimization software. Using randomly selected starting parameters, it uses its algorithm to calculate the current values of the layer properties to be determined for the next step in the optimization. The optimization ends when a minimum of the object function is reached. That can be a local or a global minimum. The goal is to get the global minimum, which is characterized by the smallest possible minimum of the object function. That can be found if the optimization is repeated several times. There are also cases in which the minimum object function values are arranged as a function of each layer property to be determined in such a way that they can be fitted by a polynomial function which contains a minimum within the search areas. In this case, the independent variable of the polynomial at the location of the minimum is the value sought for the respective layer property to be determined.

Figure list

• 1: Ergebnisse der Messung der Geschwindigkeit akustischer Oberflächenwellen in Abhängigkeit von der Frequenz auf dem Schichtsystem Hydrogel/(001)Si für die Abweichungen von der Ausbreitungsrichtung [110] um 0° und 30°, zugehörig zum Beispiel 1,
• 2: Ergebnisse der Messung der Geschwindigkeit akustischer Oberflächenwellen in Abhängigkeit von der Frequenz auf dem Schichtsystem SiO2/(001)Si für die Abweichungen von der Ausbreitungsrichtung [110] um 0°, 30° und 45°, zugehörig zum Beispiel 2,
• 3: Ergebnisse der Messung der Geschwindigkeit akustischer Oberflächenwellen in Abhängigkeit von der Frequenz auf dem Schichtsystem RuAI/(001)Si für die Ausbreitungsrichtung [110], zugehörig zum Beispiel 3,
• 4: Ergebnisse der Messung der Geschwindigkeit akustischer Oberflächenwellen in Abhängigkeit von der Frequenz auf dem Schichtsystem AIN/(001)Si für die Abweichungen von der Ausbreitungsrichtung [110] um 0°, 30° und 45°, zugehörig zum Beispiel 4.

The invention is explained in more detail below with reference to four exemplary embodiments and associated diagrams for the presentation of measurement results and 5 tables for the presentation of fitting results, also referred to as optimization results. The "Optimization Toolbox" from Matlab was used as the optimization software in all examples. The diagrams are called figures and show:

• 1 : Results of the measurement of the velocity of surface acoustic waves as a function of the frequency on the layer system hydrogel / (001) Si for the deviations from the direction of propagation [110] by 0 ° and 30 °, belonging to example 1,
• 2 : Results of the measurement of the speed of acoustic surface waves as a function of the frequency on the layer system SiO2 / (001) Si for the deviations from the direction of propagation [110] by 0 °, 30 ° and 45 °, belonging to example 2,
• 3 : Results of the measurement of the velocity of surface acoustic waves as a function of the frequency on the layer system RuAI / (001) Si for the direction of propagation [110], belonging to example 3,
• 4th : Results of the measurement of the velocity of surface acoustic waves as a function of the frequency on the layer system AIN / (001) Si for the deviations from the direction of propagation [110] by 0 °, 30 ° and 45 °, belonging to example 4.

example 1

This embodiment relates to 1 and Tables 1.1 and 1.2.

The layer system under consideration is a hydrogel layer on (001) silicon. The aim is to determine the elastic stiffnesses c11 and c55 as well as the density and thickness of the layer with the help of an optimization process that adapts the calculated values of the phase velocity of surface acoustic waves as a function of the frequency to the corresponding measured values. For a first optimization, a value of the layer thickness was used, which was measured by an alternative method with 1.73 µm. The phase velocity of surface acoustic waves as a function of frequency, referred to as the dispersion curve, was measured with the aid of a device using the method mentioned in the prior art as solution [1]. The device works without the piezoelectric properties of the layer system to be examined, because the surface acoustic waves are excited by laser pulses and received by a piezoelectric film that is in contact with the cutting edge of the detector wedge of the device used on the layer system.

1 shows two measured dispersion curves for different directions of propagation (0 degrees corresponds to the direction of propagation [110], 30 degrees correspond to a direction of propagation that is rotated by 30 ° compared to [110]). Due to very small wave amplitudes, strong disturbances occur at the limits of the measured frequency range which cannot be used for optimization because they too would lead to correspondingly large errors in the parameters to be determined. Therefore, these curve areas are hidden.

The optimization results with a given layer thickness of 1.73 µm are summarized in Table 1.1 for 5 optimizations. The selected frequency range is 30 to 110 MHz, shown in 1 as the frequency range used. Although randomly selected starting parameters were used for the optimization for each result line in Table 1.1, the values for c11, c55 and density of the layer only scatter slightly.

Table 1.2 shows 14 optimization results for the same hydrogel layer on (001) silicon, whereby in addition to table 1.1 the layer thickness together with the other parameters was also determined by optimization. Random starting parameters were chosen for all optimizations in Table 1.2. Nevertheless, the parameters only scatter slightly. The layer thickness is determined to be approximately 1.795 µm. The property values in Tables 1.1 and 1.2 are quite close together. The differences are in the single-digit percentage range. Regardless of which of the two optimizations produces the smaller errors, it is found that the described method for determining the properties of at least one layer system on a solid substrate is, under certain conditions, capable of determining the layer thickness in addition to the layer properties c11, c55 and density . The prerequisite for this is that the parabolic component in the measured curves of the speed of acoustic surface waves as a function of the frequency is as large as possible, other disruptive non-linear components as in 1 the curve sections above 30 MHz and below 110 MHz, in 2 the curve sections above 60 MHz and below 270 MHz and in 3 however, the curve sections above 70 MHz and below 250 MHz are very small. This relationship is described quantitatively in a comparison of the exemplary embodiments described here. Table 1.1 # c11 / GPa c55 / GPa Density / (g / cm ³ ) Thickness / µm E / GPa Poisson's ratio 1 12.85630 1.52039 1.32870 1.73000 4.35725 0.43294 2 12.85570 1.52039 1.32869 1.73000 4.35724 0.43294 3 12.85550 1.52040 1.32869 1.73000 4.35727 0.43293 4th 12.85550 1.52040 1.32869 1.73000 4.35727 0.43293 5 12.85560 1.52039 1.32869 1.73000 4.35724 0.43294

Layer system hydrogel / (001) Si; the layer thickness was determined alternatively (E = modulus of elasticity).
Variable: c11, c55, density
Constant: layer thickness = 1.73 µm
Frequency range used: 30 ... 110 MHz
40 times the point spacing compared to the measurement
Results of 5 optimizations with randomly chosen starting parameters Table 1.2 # c11 / GPa c55 / GPa Density / (g / cm ³ ) Thickness / µm E / GPa Poisson's ratio 1 12.26900 1.56324 1.28073 1.79517 4.46146 0.42699 2 12.26980 1.56321 1.28077 1.79512 4.46139 0.42700 3 12.27010 1.56321 1.28074 1.79517 4,46140 0.42700 4th 12.27730 1.56299 1.28091 1.79496 4,46096 0.42706 5 12.26910 1.56323 1.28073 1.79517 4.46143 0.42699 6th 12.29140 1.56263 1.28115 1.79468 4,46030 0.42718 7th 12.27500 1.56307 1,28085 1.79504 4.46113 0.42704 8th 12.27260 1.56311 1,28084 1.79504 4.46119 0.42702 9 12.26650 1.56327 1.28073 1.79516 4.46149 0.42697 10 12.26960 1.56322 1.28074 1.79517 4.46142 0.42700 11 12.26750 1.56331 1.28066 1.79527 4.46161 0.42698 12 12.26960 1.56321 1.28075 1.79516 4.46139 0.42700 13 12.26940 1.56322 1.28075 1.79515 4.46141 0.42699 14th 12.26550 1.56339 1.28059 1.79536 4.46179 0.42696

Layer system hydrogel / (001) Si; Layer thickness together with the other parameters determined by optimization (E = modulus of elasticity).
Variables: c11, c55, density, layer thickness
Constant: none
Frequency range: 30 ... 110 MHz
40 times the point spacing compared to the measurement
Results of 14 optimizations with randomly chosen starting parameters

Example 2

This embodiment relates to 2 and Tables 2.1 and 2.2.

The layer system considered is an SiO ₂ layer on (001) silicon. The aim is to determine the elastic stiffnesses c11 and c55 as well as the density and thickness of the layer with the help of an optimization process that adapts the calculated values of the phase velocity of surface acoustic waves as a function of the frequency to the corresponding measured values. For a first optimization, a value of the layer thickness was used, which was measured by an alternative method with 0.785 µm. The phase velocity of surface acoustic waves as a function of the frequency, referred to as the dispersion curve, was measured using a device in accordance with DIN 15042-1: 2006-06. The device was briefly described in Example 1.

2 shows three measured dispersion curves for different directions of propagation (0 degrees corresponds to the direction of propagation [110], 30 and 45 degrees correspond to a direction of propagation that is rotated by 30 ° or 45 ° compared to [110]). At the limits of the frequency range, as a result of very small wave amplitudes, strong interferences occur which cannot be used for the optimization because they would lead to correspondingly large errors in the parameters to be determined. Therefore, these curve sections are hidden. The frequency range used is in 2 shown.

The optimization results with a given layer thickness of 0.785 µm are summarized in Table 2.1 for 5 optimizations. The frequency range is selected to be 60 to 270 MHz. Although randomly selected starting parameters were used for the optimization for each result line in Table 2.1, the values for c11, c55 and density of the layer only scatter slightly.

In Table 2.2, 10 optimization _{results of the same SiO 2} layer on (001) silicon are presented, with the layer thickness and the other parameters also being determined by optimization in addition to Table 2.1. For all optimizations in Tables 2.1 and 2.2, random starting parameters were chosen. Nevertheless, the parameters only scatter slightly. However, the values in both tables differ significantly. The values in table 2.1 are obviously more reliable than those in table 2.2 because the values given in table 2.1 for the density of the SiO ₂ layer with approximately 2.18 come very close to the known value for amorphous SiO ₂ (quartz glass) of 2.20 . In the present case, the described method for determining properties of at least one layer system on a solid substrate does not offer the prerequisites for determining the layer thickness in addition to the layer properties c11, c55 and density. The prerequisite for this is that the parabolic component of the measured curves of the speed of acoustic surface waves as a function of the frequency is as large as possible, while other non-linear components are very small. This relationship is described quantitatively in a comparison of the exemplary embodiments described here. Table 2.1 # c11 / GPa c55 / GPa Density / (g / cm ³ ) Thickness / µm 1 75.4565 32.1027 2.18055 0.785 2 75.4593 32.109 2.1806 0.785 3 75.4574 32.1055 2.18057 0.785 4th 75.457 32.1044 2.18056 0.785 5 75.4582 32.1073 2.18059 0.785

Layer system SiO ₂ / (001) Si, layer thickness determined alternatively
Variable: c11, c55 density
Constant: layer thickness = 0.785 µm
Frequency range used: 60 ... 270 MHz
40 times the point spacing compared to the measurement
Results of 5 optimizations with randomly chosen starting parameters Table 2.2 # c11 / GPa c55 / GPa Density / (g / cm ³ ) Thickness / µm 1 121.314 36.2439 2.88296 0.627455 2 113.531 35.5908 2.77546 0.650785 3 122.722 36.3628 2.90171 0.623566 4th 121.947 36.2952 2.89136 0.625704 5 119.624 36.1065 2.86033 0.632222 6th 119.356 36.0895 2.85684 0.63297 7th 120.546 36.1877 2.8729 0.629571 8th 122.287 36.3224 2.89583 0.624779 9 119.024 36.0635 2.85238 0.63392 10 112.715 35.51 2.76345 0.653495

Layer thickness together with the other parameters determined by optimization,
Variables: c11, c55, density, layer thickness
Constant: none
Frequency range: 60 ... 270 MHz
40 times the point spacing compared to the measurement
Results of 10 optimizations with randomly chosen starting parameters

Example 3

This embodiment relates to 3 and Table 3.

The layer system considered is a RuAI layer on (001) silicon. This layer was produced as a layer system of several aluminum and ruthenium layers and then homogenized by a thermal treatment, for example tempering.

3 shows only a measured dispersion curve for the direction of propagation [110] because measurements with alternative directions of propagation could not be carried out. No parabolic component can be seen on this curve. Therefore, no attempt was made to determine the layer thickness in addition to the elastic stiffnesses c11 and c55 as well as the density with the help of an optimization process that adapts the calculated values of the phase velocity of surface acoustic waves as a function of the frequency to the corresponding measured values. For the optimization, a value of the layer thickness was used that was determined by an alternative method to be 0.22 µm. The phase velocity of surface acoustic waves as a function of the frequency, referred to as the dispersion curve, was measured using a device in accordance with DIN 15042-1: 2006-06. The device was briefly described in Example 1. At the limits of the In the frequency range, very small wave amplitudes result in strong interferences which cannot be used for optimization because they would lead to correspondingly large errors in the parameters to be determined. Therefore these are hidden. The frequency range used is in 3 shown.

The optimization results with a given layer thickness of 0.22 µm are summarized in Table 3 for 2 optimizations. The frequency range is selected to be 70 to 250 MHz. Table 3 # c11 / GPa c55 / GPa Density / (g / cm ³ ) Thickness / µm 1 139.244 38.8679 3.81595 0.22 2 134.783 36.8539 3.74777 0.22

RuAI / (001) [110] Si, annealed at 900 ° C, thickness: sum of all individual layer thicknesses, variables: c11, c55, density
Constant: layer thickness = 0.22 µm
Frequency range used: 70 ... 250 MHz
40 times the point spacing compared to the measurement
Results of 2 optimizations with randomly chosen starting parameters

Example 4

This embodiment relates to 4th and Table 4.

The layer system under consideration is an AlN layer on (001) silicon. The aim is to determine the elastic stiffnesses c11 and c55 as well as the density and thickness of the layer with the help of an optimization process that adapts the calculated values of the phase velocity of surface acoustic waves as a function of the frequency to the corresponding measured values. The phase velocity of surface acoustic waves as a function of the frequency, referred to as the dispersion curve, was measured using a device in accordance with DIN 15042-1: 2006-06. The device has been briefly described in Example 1.

4th shows three measured dispersion curves for different directions of propagation (0 degrees corresponds to the direction of propagation [110], 30 and 45 degrees correspond to a direction of propagation that is rotated by 30 ° or 45 ° compared to [110]). At the limits, especially at the upper limit, of the frequency range, very small wave amplitudes result in strong disturbances which cannot be used for the optimization because they would lead to correspondingly large errors in the parameters to be determined. Therefore, these curve sections are hidden. The frequency range used (50 to 500 MHz) is in 4th shown.

Table 4 shows 10 optimization results of the AlN layer on (001) silicon, the layer thickness also being determined by optimization together with the other parameters. For all optimizations in Table 4 random starting parameters were chosen. Nevertheless, the parameters only scatter slightly. In the present case, the described method for determining properties of at least one layer system on a solid substrate offers the prerequisites for determining the layer thickness in addition to the layer properties c11, c55 and density. The prerequisite for this is that the parabolic component of the measured curves of the speed of acoustic surface waves as a function of the frequency is as large as possible, while other non-linear components are very small. The curves in 4th show significant parabola-like deviations from linearity. In a comparison of the exemplary embodiments described here, this relationship is described quantitatively in Table 5. Table 4 # c11 / GPa c55 / GPa Density / (g / cm ³ ) Thickness / µm 1 435.984 160.895 3.51592 0.911678 2 435.975 160.904 3.51594 0.911659 3 435.99 160.888 3.5159 0.911691 4th 435.985 160.894 3.51592 0.911681 5 436,033 160.839 3.51578 0.911795 6th 436.03 160.844 3.51579 0.911784 7th 435.986 160.891 3.51591 0.911686 8th 435.979 160.899 3.51593 0.911669 9 435.98 160.899 3.51593 0.91167 10 435.968 160.913 3.51597 0.911639

AIN / (001) Si
Variable: c11, c55, density, thickness
Constant: none
Frequency range used: 50 ... 500 MHz
30 times the point spacing compared to the measurement
Results of 10 optimizations with randomly chosen starting parameters

Comparison of the embodiments 1 to 3 with regard to parabolic components in the dispersion curves

As a measure of the parabolic components of the dispersion curves, the mean 2nd derivative of the speed according to the frequency is used as the mean curvature of these curves in the frequency range used for the optimization. The corresponding values for the layer systems under consideration are given in Table 5. It can be seen from this that the hydrogel / (001) [110] Si layer system is most strongly curved in terms of amount and concave from below, which is also true 1 emerges. AIN // (001) [110] Si follows with an order of magnitude distance, nevertheless the mean 2nd derivative is larger in magnitude than that of the subsequent layer systems. In contrast, RuAI / (001) [110] Si is the least curved. From this the conclusion is drawn that hydrogel / (001) [110] Si is sufficiently curved for the determination of the layer thickness as a result of optimization, which is the case for SiO ₂ / (001) [110] Si and especially for RuAI / (001 ) [110] Si does not apply. For AIN // (001) [110] Si, the curvature is sufficiently large to ensure reliable values of the parameters to be determined, as the small fluctuations in table 4 show. Although three directions of propagation are available for the layer system SiO ₂ / (001) [110] Si ( 2 ), the optimization does not provide any reliable values for the layer thickness, probably a consequence of the very small value of the mean 2nd derivative. However, in comparison with RuAl / (001) [110] Si ( 3 ), where only one direction of propagation could be measured, _{parameters were better reproducible for SiO 2} / (001) [110] Si than for RuAl / (001) [110] Si.

In summary, it can be stated that both a higher parabolic portion of the dispersion curves of the velocity and the inclusion of more than one direction of propagation in the measurement of the dispersion curves are beneficial for the more precise determination and / or a larger number of independently determinable layer properties. Table 5 Shift system Frequency range used (MHz) ^{Mean 2nd} derivative d 2 v / df ² m / (s * MHz ² ) Hydrogel / (001) [110] Si 30 ... 110 -0.0311 AIN // (001) [110] Si 50 ... 500 -0.00333 SiO ₂ / (001) [110] Si 60 ... 270 0.00151 RuAl / (001) [110] Si 70 ... 250 0.000460

Comparison of the mean 2nd derivative of the speed according to the frequency as a measure of the mean curvature of the curves for the layer systems hydrogel / (001) [110] Si, AlN // (001) [110] Si, SiO ₂ / (001) [ 110] Si and RuAI / (001) [110] Si 1 , 4th , 2 or. 3 .

Claims (17)

Procedure for determining the properties of layers of layer systems from the measured values of the frequency-dependent Characterized phase velocity of surface acoustic waves on this layer system in which the sought properties are determined by fitting the calculated values of the frequency-dependent phase velocity of surface acoustic waves on this layer system to the respective measured values with the aid of an optimization software, wherein the layer system is a layer stack on a substrate, characterized that several directions of propagation of the surface acoustic waves are used, whereby a) the layers form at least one layer stack and the layer stacks are arranged on one and the same solid substrate or on different solid substrates, and b) the layer properties to be determined are a selection from a list of layer properties, which, in addition to density and layer thickness, contains at least properties of an isotropic layer or an anisotropic layer in accordance with the real crystal symmetry of the layer, and c) d The measured values of the frequency-dependent phase velocity for a solid substrate or several solid substrates and several directions of propagation of the surface acoustic waves in layers or layer stacks on the respective solid substrate are determined experimentally, and d) the object function, which is minimized in the course of the fitting, both from the values of the calculated and is also calculated from the measured values of the frequency-dependent phase velocity of acoustic surface waves of all measured directions of propagation on all measured crystal sections of the substrate, and e) the fitting process is repeated with randomly selected starting parameters of the properties sought until the order of the usually found local minima of the object function a reliable global minimum of the object function is recognizable or can be determined from this arrangement. Procedure according to Claim 1 , characterized in that all solid substrates are crystal sections of one and the same type of crystal. Procedure according to Claim 1 , characterized in that a search area is defined for each layer property to be determined, in that a lower and an upper limit of the respective search area are specified for the optimization software for each layer property to be determined. Procedure according to Claim 1 , characterized in that the elastic stiffnesses c11 and c55, the density and the thickness of the layer under consideration are also included in the determination of the layer properties. Procedure according to Claim 1 , characterized in that the mean 2nd derivative of the speed according to the frequency is used as a measure for the parabolic components of the measured curves of the speed of acoustic surface waves as a function of the frequency and as the mean curvature of these curves in the frequency range used for optimization, and the mean curvature of these curves in the frequency range used for the optimization is greater than 0.0015 m / (s * MHz ² ) in absolute terms. Procedure according to the Claims 1 and 5 , characterized in that at least one parameter from the list of layer properties to be determined is determined by a conventional alternative method if the mean 2nd derivative of the speed according to the frequency as the mean curvature of these curves in the frequency range used for the optimization according to the absolute amount is not greater than 0.0015 m / (s * MHz ² ). Procedure according to Claim 1 , characterized in that the selection of properties is determined separately for each layer of each layer system. Procedure according to Claim 1 , characterized in that a layer property or several layer properties are selected from the list of layer properties to be determined, which have been measured separately. Procedure according to Claim 1 , characterized in that only non-piezoelectric substrates are used as solid substrates. Procedure according to Claim 1 , characterized in that cuts of a silicon crystal are used as solid substrates. Procedure according to Claim 1 , characterized in that only piezoelectric are used as solid substrates. Procedure according to Claim 1 , characterized in that the properties of one or more electrically conductive layers are determined. Procedure according to Claim 1 , characterized in that the properties of one or more electrically non-conductive layers are determined. Procedure according to Claim 13 , characterized in that the dielectric constant is determined when determining the properties. Procedure according to Claim 1 , characterized in that the layer properties are determined by several layer systems arranged next to one another on one and the same solid substrate, consisting of layers with different layer thicknesses made of the same material, the same selection of layer properties being determined for all layers in order to reflect the change in layer properties to investigate increasing layer thickness and this selection also includes the property layer thickness. Procedure according to Claim 1 , characterized in that the object function is calculated from a sum of the squares of the differences from the calculated and the measured phase velocity of surface acoustic waves. Procedure according to Claim 1 , characterized in that the properties of the layers are determined independently of one another without using known relationships between two or more properties.

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