JP2020118513A - Method and device for measuring layer thickness and viscoelastic coefficient - Google Patents

Abstract

To measure the thickness of a layer formed on a piezoelectric element with high accuracy from the initial stage.SOLUTION: The method for measuring a layer thickness and a viscoelastic coefficient, in which a sensor including a piezoelectric element having a resonance frequency is immersed into a solution, comprises: acquiring a variation ΔFby which the fundamental frequency Fof a resonance frequency generated by forming a layer on the piezoelectric element in the solution shifts, a variation ΔFby which the high order wave frequency Fof the resonance frequency shifts, a variation ΔFby which the half-value half-width Fof a peak waveform having the fundamental frequency as a peak shifts and a variation ΔFby which the half-value half-width Fof a peak waveform having a high order wave as a peak shifts; and calculating the thickness h of the layer, the storage modulus G' of the layer, a loss modulus G'' on the basis of the fundamental frequency Fof the layer and a loss modulus G'' on the basis of the high order wave frequency Fof the layer from the variation ΔF, the variation ΔFand the variation ΔFand the variation ΔFusing an evaluation function.SELECTED DRAWING: Figure 1

Description

The present invention relates to a method for measuring a layer thickness and a viscoelastic coefficient, and a device for measuring a layer thickness and a viscoelastic coefficient.

In film thickness measurement using a piezoelectric element such as a crystal oscillator, when a substance adheres to the electrode surface of the piezoelectric element, the resonance frequency of the piezoelectric element changes according to the amount of the substance. According to such film thickness measurement, the thickness of the layer formed in the vacuum container can be accurately measured.

In particular, in recent years, not only for vacuum applications, but also methods for immersing a piezoelectric element in a solution and measuring the thickness and viscoelastic coefficient of a layer formed on the piezoelectric element have been provided. Since the film in the solution often fits the Voight model, for example, a method of applying the Voight model from three frequency data and measuring the layer thickness and the viscoelastic coefficient is provided (for example, Patent Document 1). reference).

Japanese Patent No. 5372263

However, when the Voight model is applied to the film in the solution, the layer thickness may not be measured at the initial stage when the layer starts to form on the piezoelectric element in the solution and the layer thickness cannot be measured at the initial stage.

In view of the above circumstances, an object of the present invention is to measure the thickness of a layer formed on a piezoelectric element with high accuracy from an initial stage, and further to measure the viscoelastic coefficient of the layer at that time, the layer thickness Another object of the present invention is to provide a viscoelastic coefficient measuring method and a measuring device therefor.

In order to achieve the above object, in the method for measuring the layer thickness and viscoelastic coefficient according to one embodiment of the present invention, a sensor having a piezoelectric element having a resonance frequency is immersed in a solution.
A change amount ΔF _{s1 in} which the fundamental frequency F _{s1 of the} resonance frequency shifts and a change amount in which the higher-order wave frequency F _sn of the resonance frequency shifts caused by forming a layer on the piezoelectric element in the solution. ΔF _sn , the amount of change ΔF _{w1 in} which the full width at half maximum F _w1 of the peak waveform having the apex at the fundamental frequency shifts, and the amount of change ΔF _{wn in} which the full width at half maximum F _wn of the peak waveform having the apex at the higher order wave shifts Is obtained.
Using the evaluation function from the variation ΔF _s1 , the variation ΔF _sn , the variation ΔF _w1 and the variation ΔF _wn , the thickness h of the layer, the storage elastic modulus G′ of the layer, and The loss elastic modulus G ₁ ″ based on the fundamental frequency F _s1 of the layer and the loss elastic modulus G _n ″ based on the frequency F _{sn of the} higher order wave of the layer are calculated.

According to such a method of measuring the layer thickness and the viscoelastic coefficient, the layer thickness can be accurately measured from the initial stage when the layer is formed on the piezoelectric element.

In the above method of measuring the viscoelastic coefficient, the following frequency change amounts ΔF _s1 ′, ΔF _w1 ′, ΔF _sn ′, and ΔF _wn ′, which are determined from the change ΔZ in the impedance of the piezoelectric element, are used:

・・・（Ａ１）式

...(A1) formula

・・・（Ａ２）式

...(A2) formula

・・・（Ａ３）式

...(A3) formula

・・・（Ａ４）式

...(A4) formula

In addition, the complex elastic moduli of the fundamental frequency and the higher-order wave (n-th harmonic) are expressed by G ₁ =G ₁ ′+iG ₁ ″... (A5) Formula G _n =G _n ′+iG _n ″...( A6),
Equation serving as the evaluation function when the said fundamental frequency and the higher order wave storage elastic modulus of the (n _{harmonic) G '= G 1' =} G n '··· (A7) equation,
S ₁ =(ΔF _s1 −ΔF _s1 ′) ² +(ΔF _w1 −ΔF _w1 ′) ² (A8) Equation S _n =(ΔF _sn −ΔF _sn ′) ² +(ΔF _wn −ΔF _wn ′) the sum of ² ··· (A9) type or _{S 1 = (ΔF s1 -ΔF s1} ') of the absolute value + (ΔF _w1 -ΔF _w1' absolute value ··· (A8) 'formula) S _n, = ( ΔF _sn −ΔF _sn ′) absolute value+(ΔF _wn −ΔF _wn ′) absolute value (A9)′, S ₁ +S _n (A10) is evaluated as the minimum. However, a set of h, G′, G ₁ ″, and G _n ″ may be calculated.
(In the formula, Z _q : shear mode acoustic impedance of quartz (gm/sec/cm ² ), f ₀ : fundamental frequency (Hz), ω: angular frequency, ρ ₁ : density of layer (g/cm ³ ), h: Layer thickness (nm), G: Complex elastic modulus (MPa), G′: Storage elastic modulus (MPa), G ₁ ″: Loss elastic modulus (MPa) based on fundamental frequency, G _n ″: High Loss elastic modulus (MPa) based on frequency of next wave, ρ ₁ : density of layer (g/cm ³ ), ρ ₂ : density of solution (g/cm ³ ), η ₂ : viscosity of solution (Pa·s) )

According to such a method for measuring the layer thickness and the viscoelastic coefficient, the layer thickness can be accurately measured from the initial stage when the layer starts to be formed on the piezoelectric element, and the measured ΔF _s1 , ΔF _sn , ΔF _w1 , and ΔF _{w1 In} order to obtain the same number of parameters h, G′, G ₁ ″, and G _n ″ as _wn , h, G′ constituting ΔF _s1 ′, ΔF _sn ′, ΔF _w1 ′, and ΔF _wn ′, The error does not concentrate on any of G ₁ ″ and G _n ″.

In the measurement method of the viscoelastic coefficient, h by the evaluation _{function, G ', G 1''},' in the calculation for obtaining a set of, using an iterative _{method, h, G 'G n'} , G 1 ' The set of “, G _n ”may be optimized.

According to such a method of measuring the layer thickness and the viscoelastic coefficient, the iterative method is adopted, so that the set of h, G′, G ₁ ″, and G _n ″ can be accurately obtained in a short time.

In the above-mentioned viscoelastic coefficient measuring method, in the above iterative method, an optimization method of any one of the nonlinear least squares method of the steepest descent method, the Newton method, the Levenberg-Markt method, the Gauss-Newton method, and the symplectic method is used. Then, the set of h, G′, G ₁ ″, and G _n ″ may be optimized.

According to the method for measuring the layer thickness and the viscoelastic coefficient, the steepest descent method, the Newton method, the Levenberg-Markt method, the Gauss-Newton method, the simplex method, etc. are incorporated in the iterative calculation. The set of', G ₁ ″ and G _n ″ can be accurately obtained in a short time.

In the above method of measuring the viscoelastic coefficient, a triple wave (n=3) may be used as the higher-order wave.

According to such a method for measuring the layer thickness and the viscoelastic coefficient, h, G′, G ₁ ″, G can be calculated by ΔF _s1 , ΔF _s3 , ΔF _w1 and ΔF _w3 based on the fundamental frequency and the third harmonic. _{The set of n} ″ is calculated accurately.

In order to achieve the above object, a layer thickness and viscoelastic coefficient measuring device according to an embodiment of the present invention includes a sensor, a measuring unit, and a computing unit.
The sensor has a piezoelectric element having a resonance frequency.
The measuring unit changes the amount ΔF _s1 of the fundamental frequency F _{s1 of the} resonance frequency, which is caused by forming a layer on the piezoelectric element in the solution, and the frequency F _sn of the higher-order wave of the resonance frequency. a change amount [Delta] F _sn, half width at half maximum F _w1 of the peak waveform whose vertices the fundamental frequency and the change amount [Delta] F _w1 to shift, the half width at half maximum F _wn peak waveform whose vertices the higher order wave shift but to shift The change amount ΔF _wn is acquired.
The calculation unit uses an evaluation function from the change amount ΔF _s1 , the change amount ΔF _sn , the change amount ΔF _w1 , and the change amount ΔF _wn to calculate the thickness h of the layer and the storage elastic modulus of the layer. G′, the loss elastic modulus G ₁ ″ of the layer based on the fundamental frequency F _s1 , and the loss elastic modulus G _n ″ of the layer based on the frequency F _{sn of the} higher-order wave are calculated.

According to such a layer thickness and viscoelastic coefficient measuring device, the layer thickness can be accurately measured from the initial stage when the layer is formed on the piezoelectric element.

As described above, according to the present invention, the thickness of the layer formed on the piezoelectric element is accurately measured from the initial stage, and the viscoelastic coefficient of the layer at that time is measured. An elastic modulus measuring method and a measuring apparatus therefor are provided.

FIG. 6A is a diagram showing a peak waveform having a resonance frequency F _s of the crystal unit in the admittance method as an apex. FIG. 6B shows the fundamental frequency F _s1 of the resonance frequency F _s and the higher-order waves F _s3 and F _{s5 of} the resonance frequency. It is a block block diagram which shows the measuring apparatus of the layer thickness and viscoelastic coefficient which concerns on this embodiment. It is an example of a flowchart explaining a measuring method of a layer thickness and a viscoelastic coefficient of the present embodiment. It is an example of a flowchart explaining a measuring method of a layer thickness and a viscoelastic coefficient of the present embodiment. It is an example of a flowchart explaining a measuring method of a layer thickness and a viscoelastic coefficient of the present embodiment. It is a graph which shows the relationship between the time when a layer is formed in a crystal oscillator, and the measured value of thickness.

Hereinafter, embodiments of the present invention will be described with reference to the drawings. XYZ axis coordinates may be introduced into each drawing. Further, the same members or members having the same function are denoted by the same reference numerals, and the description thereof may be appropriately omitted after the description of the members. As the piezoelectric element, for example, a crystal oscillator will be described as an example to describe the embodiment.

In each formula in the present embodiment, ω: angular frequency, ρ ₁ : layer density (g/cm ³ ), h: layer thickness (nm), G: complex elastic modulus (MPa), G′: storage Elastic modulus (MPa), G ₁ ″: Loss elastic modulus (MPa) based on fundamental frequency, G _n ″: Loss elastic modulus (MPa) based on higher-order frequency, ρ ₁ : Density of layer (g/ cm ³ ), ρ ₂ : density of solution (g/cm ³ ), η ₂ : viscosity of solution (Pa·s), Z _q : shear mode acoustic impedance of crystal (gm/sec/cm ² ), f ₀ : Represents the fundamental frequency (Hz) of a crystal resonator without layers attached.

The basic principle of the method for measuring the layer thickness and the viscoelastic coefficient according to this embodiment will be described.

FIG. 1A is a diagram showing a peak waveform having a resonance frequency F _s of a crystal resonator as an apex in the admittance method. In FIG. 1A, the half-width at half maximum F _w is shown in addition to the resonance frequency F _s . The full width at half maximum F _w is half the full width at half maximum, and means, for example, half the full width at half maximum of the peak waveform. FIG. 1B shows the fundamental frequency F _s1 of the resonance frequency F _s and the higher-order waves F _s3 and F _{s5 of} the resonance frequency.

According to the transmission theory of Martin et al. (VE Granstaff, SJ.Martin, J.Appl.Phys. 1994,75,1319), when a viscoelastic layer is adsorbed on a crystal oscillator in a solution, the viscoelastic layer is in solution. The difference between the impedance Z of the crystal unit before being adsorbed to the crystal unit and the impedance Z of the crystal unit to which the viscoelastic layer is attached, that is, the amount of change in impedance ΔZ, is expressed by the following equation (1). To be done.

・・・（１）式

...(1) formula

From the equation (1), the difference between the resonance frequency F _s1 of the crystal unit before the viscoelastic layer is adsorbed to the crystal unit in the solution and the resonance frequency F _s1 of the crystal unit to which the viscoelastic layer is attached, that is, The resonance frequency variation ΔF _s1 is expressed by the following equation (2). Furthermore, the amount of change ΔF _w1 of the half-width at half maximum F _w1 of the peak waveform having the resonance frequency F _s1 as its apex is expressed by the following equation (3). In _addition, the _{F s1,} subscript is included in the _{F w1 "s1", "w1"} refers to the basic frequency of the resonance frequency.

・・・（２）式

...(2) formula

・・・（３）式

...(3) formula

Here, in this embodiment, the condition is that the viscous load of the solution does not occur. For example, a change in the viscous load of a solution occurs when the viscosity of a buffer solution and a sample solution differ greatly, and when a sample containing a high-concentration glycerol solution is added to a buffer solution for measurement, in most cases This is because the sample solution and the buffer solution have almost equal viscosities, and no viscosity change occurs. Therefore, the first terms of the expressions (2) and (3) can be ignored, and the expressions (2) and (3) can be rewritten as the following expressions (4) and (5). ..

・・・（４）式

...(4) formula

・・・（５）式

・・・Equation (5)

In addition to the fundamental frequency F _s1 , the resonance frequency includes frequencies of higher-order waves. The frequency of the higher-order wave is, for example, a third harmonic wave, a fifth harmonic wave, a seventh harmonic wave, which is an odd multiple of the fundamental frequency F _s1 .

Regarding the frequency of such a higher-order wave, the resonance frequency F _sn of the higher-order wave in the crystal unit before the viscoelastic layer is adsorbed to the crystal unit in the solution and the crystal frequency of the crystal unit to which the viscoelastic layer adheres The difference between the higher-order wave and the resonance frequency F _sn , that is, the change amount ΔF _sn of the resonance frequency is expressed by the following equation (6). Further, the amount of change ΔF _wn of the half-width at half maximum F _wn of the peak waveform having the resonance frequency F _sn as the apex is expressed by the following equation (7). Here, n is an odd number.

・・・（６）式

・・・Equation (6)

・・・（７）式

...(7) formula

For example, when the higher-order wave is the third harmonic, the order n is “3”, and equations (6) and (7) can be rewritten as the following equations (8) and (9).

・・・（８）式

・・・Equation (8)

・・・（９）式

・・・Equation (9)

Further, in the complex elastic modulus G, since it can be written as G=G′ (storage elastic modulus)+iG ₁ ″ (loss elastic modulus), each of G ₁ and G _n is
_{G 1 = G 1 '+ iG} 1''··· (10) formula _{G n = G n' + iG} n '' ··· (11) equation Kakiarawaseru.

ΔF _s1 , ΔF _w1 , ΔF _sn , and ΔF _wn represented by these mathematical expressions can be directly measured using a measuring device. These frequencies are measured by a method using an oscillator circuit, a method obtained by frequency sweeping from an external device, such as an impedance analyzer or a network analyzer. As long as the resonant frequency F _s, the frequency F _w of the half width at half maximum with half of the conductance value of the resonance frequency F _s can be measured, the measurement method is not particularly limited.

For example, FIG. 2 is a block configuration diagram showing a layer thickness and viscoelastic coefficient measuring device according to the present embodiment.

As shown in FIG. 1, the measuring apparatus 100 includes a sensor 1 having a piezoelectric element 1a, a network analyzer 2 as a measuring unit, a control unit 3, a calculation unit 6, a calculation unit 7, and a display unit 8. And a storage unit 9.

The sensor 1 and the network analyzer 2 are controlled by the control unit 3. The calculation unit 6 calculates the frequency component of the frequency measured by the network analyzer 2. The calculation unit 7 calculates the film thickness and viscoelasticity using the result obtained by the calculation unit 6. The display unit 8 displays the values of the frequency, viscoelasticity and the like obtained by the calculation units 6 and 7. The control unit 3, the calculation units 6 and 7, and the display unit 8 may be composed of a normal personal computer and a monitor.

Further, in order to adjust the temperature of the sensor 1, a temperature control unit 4 such as a Peltier element may be provided below the sensor 1. The temperature adjustment unit 5 for adjusting the temperature control unit 4 is controlled by the control unit 3.

The piezoelectric element 1a having a resonance frequency may be an APM (ACOUSTIC PLATE MODE SENSOR), an FPW (FLEXURAL PLATE-WAVE SENSOR), a SAW (SOURFACE ACOUSTIC-WAVE SENSOR), or the like, in addition to the crystal oscillator.

For example, in this embodiment, caused by the layer in the crystal oscillator is formed by the network analyzer 2 is in solution, the amount of change [Delta] F _s1 of the fundamental frequency F _s1, the change amount [Delta] F _sn frequency F _sn, the half width at half maximum a change amount [Delta] F _w1 of F _w1, and acquires the change amount [Delta] F _wn of half width at half maximum _{F wn.}

The calculation units 6 and 7 establish an evaluation function including the change amount ΔF _s1 , the change amount ΔF _sn , the change amount ΔF _w1 , and the change amount ΔF _wn to ΔF _s1 , ΔF _sn , ΔF _w1 and the change amount ΔF _wn , and then the layers The thickness h of the layer, the storage elastic modulus G′ of the layer, the loss elastic modulus G ₁ ″ based on the fundamental frequency F _s1 of the layer, and the loss elastic modulus G _n ″ based on the frequency F _sn of the higher-order wave of the layer. calculate. A computer program stored in the storage unit 9 is used for the calculation in the calculation units 6 and 7.

The storage unit 9 has a recording medium such as a hard disk, an optical disk, or a flash memory, and stores a computer program that automatically executes the algorithm of the method for measuring the layer thickness and the viscoelastic coefficient according to the present embodiment, or the acquired data. , Save calculated data, etc.

Next, a method of measuring the layer thickness and the viscoelastic coefficient according to this embodiment will be described.

In the present embodiment, at least two frequencies are selected from the frequency changes of the n-th harmonic (n=2n+1, n is an integer) (for example, the fundamental frequency and the third-harmonic), and the layers formed on the crystal resonator are selected. The thickness of the layer and the viscoelastic coefficient of the layer are calculated without the limitation of the Voight model.

Here, in the solution, as the layer becomes thicker, water is contained and approaches the Voight model. However, in the present embodiment that does not use the solution method by the simultaneous equations based on the Voight model, the frequency dependence is present in G′ included in G. If not, the frequency is constant. That is,
G and _{'= G 1' = G n} '··· (12) formula.

Further, G ₁ ″ based on the fundamental frequency and G _n ″ based on the higher-order wave (nth harmonic) are calculated independently. Then, each of G′, G ₁ ″, G _n ″, and h is obtained by using the evaluation function.

As the measuring method, first, the sensor 1 of the measuring device 100 is immersed in the solution. The layer formed in the piezoelectric element 1a is dissolved in the solution.

Next, the measuring apparatus 100, caused by the layer to the piezoelectric element 1a is formed in solution, and the change amount [Delta] F _s1 of the fundamental frequency _{F s1,} the change amount [Delta] F _sn frequency _{F sn,} the half width at half maximum _{F w1} The change amount ΔF _w1 and the change amount ΔF _wn of the half-width half-width F _wn are acquired.

Next, in the measuring apparatus 100, the thickness h of the layer and the storage of the layer are stored using the evaluation function from the measured variation ΔF _s1 , variation ΔF _sn , variation ΔF _w1 and variation ΔF _wn. The elastic modulus G′, the loss elastic modulus G ₁ ″ based on the fundamental frequency F _s1 of the layer, and the loss elastic modulus G _n ″ based on the frequency F _sn of the higher-order wave of the layer are calculated.

Specifically, in order to obtain the thickness h, the storage elastic modulus G′, the loss elastic modulus G ₁ ″, and the loss elastic modulus G _n ″, the thickness h, the storage elastic modulus G′ are arranged along with the measured values. , The loss elastic modulus G ₁ ″, the loss elastic modulus G _n ″, and the like, the same formulas as the formulas (4) to (7) are introduced.

For example, the following equations (13) to (16) are introduced for calculation. Here, the formulas used to calculate the thickness h, the storage elastic modulus G′, the loss elastic modulus G ₁ ″, and the loss elastic modulus G _n ″ are the measured values ΔF _s1 , ΔF _w1 , ΔF _sn , In order to distinguish from ΔF _wn , “′” is added in the upper right.

・・・（１３）式

...(13) formula

・・・（１４）式

...(14) formula

・・・（１５）式

・・・Equation (15)

・・・（１６）式

...(16) formula

Then, the following two equations,
S ₁ =(ΔF _s1 −ΔF _s1 ′) ² +(ΔF _w1 −ΔF _w1 ′) ² (17) Formula S _n =(ΔF _sn −ΔF _sn ′) ² +(ΔF _wn −ΔF _wn ′) ² ... (18), which is the sum of S ₁ +S _n (19)
Then, a set of h, G′, G ₁ ″, and G _n ″ at which is minimized is calculated. In this embodiment, the expression (19) is used as the evaluation function 1.

In the above, the sum of S ₁ and S _n (evaluation function 1) is minimized by the so-called least squares method. That is, the sum (S ₁ +S _n ) is evaluated by this set of equations so that the sum (S ₁ +S _n ) is minimized. Therefore, the sum (S ₁ +S _n ) is used as the evaluation function.

Also, another method may be used for evaluation, for example,
S ₁ =absolute value of (ΔF _s1 −ΔF _s1 ′)+absolute value of (ΔF _w1 −ΔF _w1 ′) (17)′ Expression S _n =(ΔF _sn −ΔF _sn ′) absolute value+( Absolute value of ΔF _wn −ΔF _wn ')... Sum of equation (18)', S ₁ +S _n ... (19)' when h, G', G ₁ '' when the equation is minimum , G _n ″ may be calculated. In this embodiment, the expression (19)′ is the evaluation function 2.

The evaluation method (that is, the evaluation functions 1 and 2) disclosed in the present application is based on the measurement results (ΔFs1, ΔFw1, ΔFsn, ΔFwn) and the calculation results (G′, G ₁ ″, G _n ″, h) ( ΔFs1′, ΔFw1′, ΔFsn′, ΔFwn′) are technical thoughts that should be the same. In the present embodiment, since it is ideal that the measurement result and the calculation result are the same, the evaluation function is used that takes the difference and approaches zero. In other words, another calculation method, that is, an evaluation function may be selected as long as it can be evaluated that the values approach the same.

That is, after the sensor 1 of the measuring device 100 is immersed in the solution, the total value (S ₁ +S) in which the respective variation amounts (ΔF _s1 , ΔF _w1 , ΔF _sn , ΔF _wn ) acquired and obtained by the evaluation device 100 are incorporated. _{Using the} evaluation that minimizes _{n 2} ) as the evaluation criterion, the unknown parameters are shaken to determine the unknown parameters.

Further, in the above, when obtaining ΔF _s1 ′, ΔF _w1 ′, ΔF _sn ′, and ΔF _wn ′, the expression “shake parameters” is used. In the calculation of the present embodiment, a so-called iterative method is used in which an algorithm for obtaining a root by slightly increasing (shaking) an arbitrary initial value (parameter) and performing a plurality of calculations is used. Here, the "root" is a root that converges to the value indicated by the evaluation function (for example, the film thickness h).

That is, when calculating a set of h, G′, G ₁ ″, and G _n ″, iterative calculation is performed. At this time, h, G′, G ₁ ″, and G _n ″ are obtained by using a nonlinear least-squares optimization method such as the steepest descent method, the Newton method, the Levenberg-Marquardt method, the Gauss-Newton method, or the simplex method. May be optimized.

If the load on the arithmetic units 6 and 7 increases, the time required for the arithmetic processing increases, so that the time resolution of the measuring apparatus 100 decreases and the performance decreases. For this reason, it is preferable to adopt an optimal method for the arithmetic units 6 and 7 among the above-mentioned iterative methods.

3 to 5 are examples of flowcharts illustrating the method for measuring the layer thickness and the viscoelastic coefficient of the present embodiment.

The flowchart shows a method of selecting the third harmonic (n=3) as the higher-order wave and measuring the layer thickness and the viscoelastic coefficient. When the third harmonic (n=3) is selected as the higher-order wave, equations (17) to (19) are
_{S 1 = (ΔF s1 -ΔF s1} ') 2 + (ΔF w1 -ΔF w1') 2 ··· (17) formula _{S 3 = (ΔF s3 -ΔF s3} ') 2 + (ΔF w3 -ΔF w3') ² ... (18) Formula S ₁ +S ₃ ... (19) Formula

Or
S ₁ =absolute value of (ΔF _s1 −ΔF _s1 ′)+absolute value of (ΔF _w1 −ΔF _w1 ′) (17)′ Expression S ₃ =(ΔF _s3 −ΔF _s3 ′) absolute value+( The absolute value of ΔF _w3 −ΔF _w3 ′ is the sum of the equations (18)′, S ₁ +S ₃ ... (19)′.

In particular, the measuring method of the present embodiment works effectively when the thickness of the layer formed on the piezoelectric element 1a is an extremely thin thickness of several nm or less.

In the measurement method, a top-down approach is used in which a functional block of calculation is defined as a subroutine, and the subroutine is stepwise detailed and calculated. For example, the process 1 (FIG. 3) includes the subroutine of the process 2 (FIG. 4), and the process 2 includes the subroutine of the process 3 (FIG. 5). The flowcharts shown in FIGS. 3 to 5 can be executed by a computer program stored in the storage unit 9, for example.

The Levenberg-Markt method is an iterative method for finding the minimum of a function expressed in the form of sum of squares of nonlinear functions. For example, the Levenberg-Marquardt method works in the same way as the steepest descent method that converges to the correct solution when the current solution (latest solution) is far from the correct solution, and executes the Newton method when the current solution is close to the correct solution. To do.

As shown in FIG. 3, in the process 1, first, the layer thickness h is set to the minimum value (h _min ) as the initial value of the layer thickness (step S11).

Next, processing 2 which is a subroutine is executed (step S20). Details of the process 2 will be described later.

After the process 2 is executed, a minute amount of the layer thickness Δh is added to the layer thickness h, and the layer thickness h is increased by the minute amount Δh (step S12).

Next, when it is determined that the layer thickness h is equal to or less than the maximum value (h _max ), the process returns to the process 2 again, and the steps S20 and S12 are executed. When it is determined that the layer thickness h is larger than the maximum value (h _max ), the steepest descent method and the Newton method are used with h, G′, G ₁ ″, and G ₃ ″ obtained by the process 1 as initial values. , And the Levenberg-Markt method, the Gauss-Newton method, the simplex method, etc., the set of h, G′, G ₁ ″, and G ₃ ″ that minimizes S ₁ +S ₂ is calculated.

The process 2 included in the process 1 will be described.

As shown in FIG. 4, in the process 2, first, the storage elastic modulus G′ is set to the minimum value (G′ _min ) as the initial value of the storage elastic modulus (step S21).

Next, processing 3 which is a subroutine is executed (step S30). Details of the process 3 will be described later.

After the process 3 is executed, a minute amount of the storage elastic modulus ΔG′ is added to the storage elastic modulus G′, and the storage elastic modulus G′ is increased by the minute amount ΔG′ (step S22).

Next, when it is determined that the storage elastic modulus G′ is the maximum value (G′ _max ) or less, the process returns to the process 3 again, and steps S30 and S22 are executed. When it is determined that the storage elastic modulus G′ is larger than the maximum value (G′ _max ), the process 2 ends, and the process proceeds to step S12 of the process 1.

The process 3 included in the process 2 will be described.

As shown in FIG. 5, in the process 3, the flow of steps S31 to S34 and the flow of steps 35 to S38 are individually executed. These two flows may be made to proceed in parallel, or one of them may be made to go before and after the other.

Steps S31 to S34 will be described.

First, the loss elastic modulus G ₁ ″ is set to the lowest value (G ₁ ″ _min ) as the initial value of the loss elastic modulus (step S 31 ).

Next, the set of G′, G ₁ ″, and h that minimizes S ₁ is stored (step S 32 ).

Then, the loss modulus G _{1 '',} the small loss modulus .DELTA.G 1 amounts _'are subject to', loss modulus G _{1 ''} is a small amount .DELTA.G 1 _'increases' minute (step S33).

Then, if the loss modulus G _{1 ''} is the maximum value (loss modulus _{G 1} 'is _{determined' max)} below, returns again to the step S32, step S33, step S34 is executed. When it is determined that the loss elastic modulus G ₁ ″ is larger than the maximum value (G ₁ ″ _max ), the process proceeds to the next step S39.

Steps S35 to S38 will be described.

First, the loss elastic modulus G ₃ ″ is set to the minimum value (G ₃ ″ _min ) as the initial value of the loss elastic modulus (step S35).

Next, the set of G′, G ₃ ″, and h that minimizes S ₂ is stored (step S36).

Next, 'the, small amounts loss modulus .DELTA.G 3 _of' the loss modulus G _{3 'are} subject to', loss modulus G _{3 ''} is a small amount .DELTA.G 3 _'increases' minute (step S37).

Then, if the loss modulus G _{3 ''} is the maximum value (loss modulus _{G 3} 'is _{determined' max)} below, returns again to the step S36, step S37, step S38 is executed. When it is determined that the loss elastic modulus G ₃ ″ is larger than the maximum value (G ₃ ″ _max ), the process proceeds to the next step S39.

Next, in step S39, a set of h, G′, G ₁ ″, and G ₃ ″ that minimizes S ₁ +S ₂ is stored. This saved set becomes the initial values of h, G′, G ₁ ″, and G ₃ ″ processed by the process 1. Then, in process 2, step S22 is executed.

In the present embodiment, the higher order wave is not limited to the third harmonic wave, and if at least two frequencies of the fundamental frequency (n=1) and the higher order wave shown in FIG. Can be performed. For example, in addition to the combination of the fundamental frequency and the third harmonic, the combination of the fundamental frequency and the fifth harmonic, and the like. In the above description, the layer thickness and the viscoelastic coefficient are measured by using the fundamental frequency and the third harmonic as an example.

Here, in the method applying the Voight model (hereinafter, the method of the comparative example), when h, G′, G ₁ ″, and G ₃ ″ of the layers formed in the crystal unit are obtained, the same as in the present embodiment. Then, the change amount ΔF _s1 of the resonance frequency of the fundamental frequency, the change amount ΔF _{w1 of} the half width at half maximum, the change amount ΔF _{s3 of the} triple wave of the resonance frequency, and the half width ΔF _w3 at half thereof are measured.

However, in the case of the comparative example, G′ has no frequency dependency and is a constant value, and G″ is proportional to the frequency. Therefore, _three parameters to be obtained are h, G′, and G″ (G″=mG ₁ ″=G ₃ ″, m is a proportional coefficient), and the measured value is, for example, ΔF. Three of ΔF _s1 , ΔF _w1 , and ΔF _w3 among _s1 , ΔF _w1 , ΔF _s3 , and ΔF _w3 are sufficient. That is, h, G′, and G″ can be calculated directly from the three measured values.

However, from the three measurements, direct, h, G ', G' when calculating the 'calculated h, G', G 'and backward from', for example, when calculating the [Delta] F _s3, the [Delta] F _s3 In some cases, the calculation and the measured value of ΔF _s3 may not match. That is, the measured value that was not included in the calculation from among ΔF _s1 , ΔF _w1 , ΔF _s3 , and ΔF _w3 and the calculated value obtained by back-calculating from h, G′, and G″ did not match and were included in the calculation. Errors concentrate on the missing measurement values.

Further, in the case of the comparative example, in the initial stage when the layer starts to be formed on the piezoelectric element in the solution, the layer thickness may not be accurately measured, for example, because it does not fit the Voight model.

On the other hand, according to the present embodiment, from the four measured values of ΔF _s1 , ΔF _w1 , ΔF _sn , and ΔF _wn , four solutions of the same number as the four measured values, that is, h, G′, G ₁ ″ and G _n ″ are calculated. Therefore, there is no measured value that is not included in the calculation, and errors do not concentrate on the measured value that is not included in the calculation.

Further, according to the present embodiment, the thickness of the layer can be accurately measured from the initial stage when the layer starts to be formed on the piezoelectric element 1a in the solution. In particular, it is effective for measuring the layer thickness of several nm or less.

FIGS. 6A and 6B are graphs showing the relationship between the time when the layer is formed on the crystal unit and the measured value of the thickness. Here, FIG. 6A shows an example of the case of measurement by the method of the comparative example, and FIG. 6B shows an example of case of measurement by the method of the present embodiment. A crystal oscillator is used as the piezoelectric element 1a. Then, superoxide dismutase 1 (SOD1) is formed on this crystal oscillator.

In the comparative example shown in FIG. 6A, the thickness of SOD1 is not accurately measured at the initial stage (within 10 minutes) of immersing the crystal resonator in the SOD1 solution. For example, in the method of the comparative example, it can be seen that the thickness of SOD1 is not measured up to a thickness of 3 nm, and the thickness starts to be measured after the thickness of SOD1 approaches 3 nm.

On the other hand, in the present embodiment shown in FIG. 6B, it can be seen that the thickness of SOD1 is measured from the initial stage of immersing the crystal resonator in the SOD1 solution.

Table 1 shows measured and calculated values of ΔF _s1 , ΔF _w1 , ΔF _s3 , and ΔF _w3 . Example 1 and Comparative Example 1 are examples in which a myosin layer is formed on a crystal resonator. Example 2 and Comparative Example 2 are examples in which a crystal layer has a collagen layer formed thereon. Example 3 and Comparative Example 3 are examples in which a crystal unit is provided with an avidin layer.

From the results of Table 1, it is understood that in Comparative Examples 1 to 3, among ΔF _s1 , ΔF _w1 , ΔF _s3 , and ΔF _w3 , the error of ΔF _s3 is remarkable. That is, the errors are concentrated on ΔF _s3 among ΔF _s1 , ΔF _w1 , ΔF _s3 , and ΔF _w3 .

On the other hand, in Examples 1 to 3, it can be seen that the measured value and the calculated value are approximately the same in each of ΔF _s1 , ΔF _w1 , ΔF _s3 , and ΔF _w3 .

Total error%=((calculated value ΔF _s1 −measured value ΔF _s1 )/calculated value ΔF _s1 +(calculated value ΔF _w1 −measured value ΔF _w1 )/calculated value ΔF _w1 +(calculated value ΔF _s3 −measured value ΔF _s3 ). /Calculated value ΔF _s3 +(calculated value ΔF _w3 −measured value ΔF _w3 )/calculated value ΔF _w3 )×100 (20) Equation

When the total error is defined by the above formula (20), the total error is 10% or more and 50% or less in Comparative Examples 1 to 3, whereas the total error is less than 10% in Examples 1 to 3. I found out.

Although the embodiments of the present invention have been described above, the present invention is not limited to the above-described embodiments and various modifications can be made. For example, the fundamental frequency, the third harmonic, and the fifth harmonic may be used, and G′, G ₁ ″, G ₃ ″, G ₅ ″, and h may be obtained using an evaluation function. Further, the relational expression between the change ΔZ of the impedance Z of the piezoelectric element before and after the film is attached and the frequency of the calculated value is not limited to the expressions given here, and the method of this time can be similarly applied to these. Each embodiment is not limited to an independent form, and can be combined as technically possible.

DESCRIPTION OF SYMBOLS 1... Sensor 1a... Piezoelectric element 2... Network analyzer 3... Control part 4... Temperature control part 5... Temperature adjustment part 6, 7... Calculation part 8... Display part 9... Storage part 100... Measuring device

Claims (6)

Immersing a sensor with a piezoelectric element with a resonant frequency in a solution,
A change amount ΔF _s1 that shifts the fundamental frequency F _s1 of the resonance frequency and a change amount that shifts the frequency F _sn of a higher-order wave of the resonance frequency, which are caused by forming a layer on the piezoelectric element in the solution. ΔF _sn , the amount of change ΔF _{w1 in} which the half-width at half maximum F _w1 of the peak waveform having the apex at the fundamental frequency is shifted, and the amount of change ΔF _{wn in} which the half-width at half maximum F _wn of the peak waveform having the apex at the higher order wave is shifted. To get
Using the evaluation function from the variation ΔF _s1 , the variation ΔF _sn , the variation ΔF _w1 , and the variation ΔF _wn , the layer thickness h, the storage elastic modulus G′ of the layer, and Calculation of loss elastic modulus G ₁ ″ based on the fundamental frequency F _s1 of the layer and loss elastic modulus G _n ″ based on the frequency F _{sn of the} higher-order wave of the layer Measurement of layer thickness and viscoelastic coefficient Method. The method for measuring the layer thickness and viscoelastic coefficient according to claim 1, wherein the frequency changes ΔF _s1 ′, ΔF _w1 ′, ΔF _sn ′, and ΔF _wn ′ are determined from the change ΔZ of the impedance of the piezoelectric element. Using the formula, the complex viscoelastic moduli of the fundamental frequency and the higher-order waves are G ₁ =G ₁ ′+iG ₁ ″... (A5) Formula G _n =G _n ′+iG _n ″... Formula (A6)
An equation that is the evaluation function when the storage frequencies of the fundamental frequency and the higher-order waves are G′=G ₁ ′=G _n ′ (A7)
S ₁ =(ΔF _s1 −ΔF _s1 ′) ² +(ΔF _w1 −ΔF _w1 ′) ² (A8) Equation S _n =(ΔF _sn −ΔF _sn ′) ² +(ΔF _wn −ΔF _wn ′) ² ... Sum of formula (A9), or
S ₁ =absolute value of (ΔF _s1 −ΔF _s1 ′)+absolute value of (ΔF _w1 −ΔF _w1 ′) (A8)′ Expression S _n =(ΔF _sn −ΔF _sn ′) absolute value+( ΔF _wn −ΔF _wn ′) absolute value... (A9)′ Equation S ₁ +S _n ... (A10) equation is evaluated as the minimum, and h, G′, G ₁ ′ are evaluated. A method of measuring a layer thickness and a viscoelastic coefficient for calculating a set of “, G _n ”.
(In the formula, Z _q : shear mode acoustic impedance of quartz (gm/sec/cm ² ), f ₀ : fundamental frequency (Hz), ω: angular frequency, ρ ₁ : density of layer (g/cm ³ ), h: Layer thickness (nm), G: Complex elastic modulus (MPa), G′: Storage elastic modulus (MPa), G ₁ ″: Loss elastic modulus (MPa) based on fundamental frequency, G _n ″: High Loss elastic modulus (MPa) based on frequency of next wave, ρ ₁ : density of layer (g/cm ³ ), ρ ₂ : density of solution (g/cm ³ ), η ₂ : viscosity of solution (Pa·s) ) A method for measuring a layer thickness and a viscoelastic coefficient according to claim 1 or 2, wherein
In the calculation for obtaining the set of h, G′, G ₁ ″, and G _n ″ by the evaluation function, an iterative method is used to optimize the set of h, G′, G ₁ ″, and G _n ″. Measuring method for layer thickness and viscoelastic coefficient. A method for measuring a layer thickness and a viscoelastic coefficient according to claim 3,
In the above iterative method, a method of measuring a layer thickness and a viscoelastic coefficient using an optimization method of a nonlinear least squares method, such as the steepest descent method, the Newton method, the Levenberg-Markt method, the Gauss-Newton method, or the simplex method. A method for measuring a layer thickness and a viscoelastic coefficient according to any one of claims 1 to 4,
A third-order wave (n=3) is used as the higher-order wave. A method for measuring a layer thickness and a viscoelastic coefficient. A sensor having a piezoelectric element having a resonance frequency,
A change amount ΔF _s1 that shifts the fundamental frequency F _s1 of the resonance frequency and a change amount that shifts the frequency F _sn of a higher-order wave of the resonance frequency, which are caused by forming a layer on the piezoelectric element in the solution. ΔF _sn , the amount of change ΔF _{w1 in} which the half-width at half maximum F _w1 of the peak waveform having the apex at the fundamental frequency is shifted, and the amount of change ΔF _{wn in} which the half-width at half maximum F _wn of the peak waveform having the apex at the higher order wave is shifted. And a measurement unit that acquires
Using the evaluation function from the variation ΔF _s1 , the variation ΔF _sn , the variation ΔF _w1 , and the variation ΔF _wn , the layer thickness h, the storage elastic modulus G′ of the layer, and A layer thickness comprising: a calculation unit for calculating a loss elastic modulus G ₁ ″ based on the fundamental frequency F _s1 of the layer and a loss elastic modulus G _n ″ based on the frequency F _{sn of the} higher-order wave of the layer. And a viscoelastic coefficient measuring device.

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