CN101769898A - On-line measuring method of Poisson ratio of micro-electromechanical system (MEMS) film based on resonance frequency method - Google Patents

Abstract

The invention discloses an on-line measuring method of the Poisson ratio of a micro-electromechanical system (MEMS) film based on a resonance frequency method. The method comprises the following steps of: making an annular MEMS film as a testing structure, fixedly supporting the inner boundary of the MEMS film on an anchor zone and enabling the outer boundary of the MEMS film to be free; fixing the anchor zone on a substrate; measuring the resonance frequency of the radial vibration and the transverse vibration of the annular MEMS film; and calculating the Poisson ratio of the MEMS film by using the resonance frequency of the radial vibration and the transverse vibration. In the method, the concrete values of Young modulus, material density and the like of the film are not needed to be known in advance, thereby errors caused by the Young modulus, material density and the like can be avoided, and the precision is improved. Due to the symmetrical structure of the annular film, the fixed supporting in the anchor zone more approaches to the ideal state. The measuring method belongs to non-contact measurement so that the measuring process can not influence the testing structure, and the repeatability of the measurement can be ensured. The measuring method is suitable for both a conductor and a non-conductor, and the testing structure is simple to make.

Description

MEMS film Poisson ratio On-line Measuring Method based on resonance frequency method

Technical field:

The invention provides a kind of method that can on-line measurement MEMS (micro electro mechanical system) (MEMS) film Poisson ratio, belong to MEMS material parameter field of measuring technique.

Technical background:

The MEMS film is widely used in MEMS (micro electro mechanical system).The quality of thin film mechanical performance is very big to device performance influence, the performance of structural sheet film of especially making movable member fine or not most important to the success or failure of MEMS element manufacturing and performance.The mechanics parameter of MEMS film mainly comprises Poisson ratio, elastic modulus, unrelieved stress and along stress gradient of thickness direction etc.The performance of mechanical properties in films and macroscopical large volume same material has very big-difference, can not be equal to the MEMS mechanical properties in films to the mechanical property of macroscopical large volume material.And the mechanical property of material also can change along with the change of fabrication process condition (for example temperature, reaction gas pressure etc.), thereby need be on processing line often measure (on-line measurement), with the mechanical properties in films that obtains to process under a certain specific process conditions or the stability of characterization processes.Measurement on processing line (on-line measurement) must be easy, accurate, chip occupying area is little, do not have destructiveness, can't adopt the method for measuring macroscopical large volume material mechanical performance to measure.The numerical value of Poisson ratio is all smaller usually, and on-line measurement is difficulty more.Therefore research is applicable to that the On-line Measuring Method of the film Poisson ratio of the actual process of MEMS just becomes of this field and presses for.The present invention has studied the On-line Measuring Method of MEMS film Poisson ratio.

The method of on-line measurement MEMS film Poisson ratio mainly contains at present: torsional technique, pulling method etc.

Torsional technique: make two continuous annulus (Fig. 1) on structural sheet, the anchor district, exerts pressure with the nano impress meter at two annulus connecting places after the release at the right and left, and two ring connecting places produce downward displacement.Derive the theoretical relationship between the displacement, pressure, Poisson ratio, Young modulus of annulus connecting place, the displacement of the two ring connecting places that measure from nano impress and the relation between institute's plus-pressure can be calculated the Poisson ratio of material.The limitation of this method is to measure the Young modulus that Poisson ratio also must be known material in advance, will influence the measuring accuracy of Poisson ratio like this

Pulling method: make one group of cantilever beam structure, at first the two ends of beam are directly applied pulling force by PZT (piezoelectric ceramics) device, speckle image after measuring beam and stretch with atomic force microscope then, just can draw external force change cause laterally reach linear deformation, thereby the Poisson ratio of obtaining.This method is based on the direct measurement of Poisson ratio definition, and precision is very high, but complicated operation, and needs expensive device such as atomic force microscope, is not suitable for on-line measurement.

Summary of the invention:

The objective of the invention is to solve the problems referred to above that prior art exists, proposed a kind of MEMS film Poisson ratio On-line Measuring Method based on resonance frequency method.Its basic ideas are:

As test structure, interior boundary is propped up admittedly in the anchor district, the outer boundary freedom to make an annular MEMS film (being called for short the annular film); The outer boundary radius is a, and the inner boundary radius is b, and thickness is h.The anchor district is fixed on the substrate.After utilizing MSA-500 micro laser vialog to measure the resonance frequency of the radial vibration of this structure and laterally (vertical thin face) vibration, the theoretical formula of the calculating Poisson ratio that substitution is derived just can obtain the Poisson ratio of material.Principle of the present invention is as follows:

1, the radial vibration of annular film:

Make an inner boundary and prop up admittedly in anchor district, outer boundary annular film freely, the inside radius of annular film is b, and external radius is a, and thickness is h, and is that initial point is set up three-dimensional polar coordinate system (Fig. 2) with the circle ring center.The axisymmetric radial vibration of research annular film.Because be film, thickness is much smaller than radius, so belong to the plane stress situation, at this moment, the stress of z direction is zero.For axisymmetric radial vibration, can take up an official post at annulus and get an infinitesimal, and it is carried out force analysis (Fig. 3).Can get the diaphragm oscillatory differential equation by (Fig. 3) is:

Wherein: u is radial displacement, σ _rBe radial stress,

Be tangential stress, ρ is a density, and t is the time.Behind the abbreviation:

Because be the radial axle symmetric vibration, so there is following relationship to set up:

${\mathrm{\ϵ}}_r=\frac{\∂u}{\∂r},$

Wherein, ε _rBe the radial line strain, Be tangential line strain,

Be shear strain.

Physical equation is:

(3) formula substitution (4) (5) two formulas, bring (2) formula again into, behind abbreviation, can get:

$\frac{{\∂}^{2}u}{\∂r^2}+\frac{\∂u}{r\∂r}-\frac{u}{r^2}=\frac{{\∂}^{2}u}{k\∂t^2}---\left(6\right)$

Wherein: E is a Young modulus, and v is a Poisson ratio.

If (6) formula separate for: u (r, t)=δ (r) e ^{J ω t}, behind substitution (6) the formula abbreviation:

$r^2\frac{d^2\mathrm{\δ}}{{\mathrm{dr}}^2}+r\frac{\mathrm{d\δ}}{\mathrm{dr}}+({\mathrm{\β}}^2{r}^2-1)\mathrm{\δ}=0---\left(7\right)$

Wherein: ${\mathrm{\β}}^2=\frac{{\mathrm{\ω}}^2}{k}=\frac{\mathrm{\ρ}(1-v^2){\mathrm{\ω}}^2}{E}---\left(8\right)$

Obviously, (7) formula is that parameter is β ²1 rank Bessel equation.Its general solution is:

δ(r)＝AJ ₁(βr)+BY ₁(βr) (9)

A, B are arbitrary constant, J ₁(β r) is real argument the 1st class 1 rank Bessel function, Y ₁(β r) is real argument the 2nd class 1 rank Bessel function.So (6) general solution of formula is:

u(r，t)＝δ(r)e ^jωt＝[AJ ₁(βr)+BY ₁(βr)]e ^jωt (10)

Because annular film inner boundary is fixed, outer boundary is freely, so, be zero in the radial displacement of inner boundary place, the radial stress of boundary diaphragm is zero outside, promptly

Can obtain (10) formula substitution (11) formula:

$\left\{\begin{array}{c}[\left(\mathrm{\βa}\right)J_0\left(\mathrm{\βa}\right)-(1-v)J_1\left(\mathrm{\βa}\right)]A+[\left(\mathrm{\βa}\right)Y_0\left(\mathrm{\βa}\right)-(1-v)Y_1\left(\mathrm{\βa}\right)]B=0\\ J_1\left(\mathrm{\βb}\right)A+Y_1\left(\mathrm{\βb}\right)B=0\end{array}\right.---\left(12\right)$

If A, B has untrivialo solution, and then the determinant of (12) formula is zero.That is:

$\left|\begin{array}{cc}\left(\mathrm{\βa}\right)J_0\left(\mathrm{\βa}\right)-(1-v)J_1\left(\mathrm{\βa}\right)& \left(\mathrm{\βa}\right)Y_0\left(\mathrm{\βa}\right)-(1-v)Y_1\left(\mathrm{\βa}\right)\\ J_1\left(\mathrm{\βb}\right)& Y_1\left(\mathrm{\βb}\right)\end{array}\right|=0---\left(13\right)$

Abbreviation (13) formula obtains:

[J ₀(βa)Y ₁(βb)-Y ₀(βa)J ₁(βb)](βa)+

(14)

[Y ₁(βa)J ₁(βb)-J ₁(βa)Y ₁(βb)](1-v)＝0

(14) formula is a transcendental equation, can't obtain the analytic solution of equation, can only obtain numerical solution with numerical method, obtains its root then with fitting of a polynomial, and when a=2b, it first resolved positive root and be (corresponding radial symmetry vibration pitch circle number is zero):

β a=-0.1405v ²+ 0.7444v+2.9728 (quadratic fit)

Or: β a=0.6742v+2.9786 (once fitting)

Relatively the seeing of the result of once fitting and quadratic fit and (14) formula numerical solution (Fig. 4).As seen quadratic fit precision is very high, is analytic solution so get the quadratic fit result, that is:

βa＝-0.1405v ²+0.7444v+2.9728 (15)

By (8) Shi Kede: $f_1=\frac{\mathrm{\βa}}{2\mathrm{\πa}}\sqrt{\frac{E}{\mathrm{\ρ}(1-v^2)}}$

(15) formula substitution (16) formula:

$f_1=\frac{-0.1405v^2+0.7444v+2.9728}{2\mathrm{\πa}}\sqrt{\frac{E}{\mathrm{\ρ}(1-v^2)}}---\left(17\right)$

2, the transverse vibration of annulus:

According to theory of elastic mechanics, this annular film at 0 pitch diameter rotational symmetry transverse vibration differential equation of appearance is:

$(\frac{{\&PartialD;}^2}{\&PartialD;r^2}+\frac{1}{r}\frac{\&PartialD;}{\&PartialD;r})(\frac{{\&PartialD;}^2}{\&PartialD;r^2}+\frac{1}{r}\frac{\&PartialD;}{\&PartialD;r})w+\frac{\mathrm{\&rho;<figure-callout\; id="0"\; label="h\; D"\; filenames="HSA00000021846900013.png"\; state="\{\{state\}\}">h</figure-callout>}}{D_{}}\frac{{\&PartialD;}^{2}w}{\&PartialD;t^2}=0---\left(18\right)$

In the formula:

Be the thin plate bendind rigidity, w is a transversal displacement, and h is a thickness, and ρ is a density, and E is a Young modulus, and v is a Poisson ratio.

Formula (18) separate for:

Wherein: z (r)=[C ₁J ₀(β r)+C ₂I ₀(β r)+C ₃Y ₀(β r)+C ₄K ₀(β r)] (20)

${\mathrm{\&beta;}}^4=\frac{\mathrm{\&rho;<figure-callout\; id="0"\; label="h\; D"\; filenames="HSA00000021846900013.png"\; state="\{\{state\}\}">h</figure-callout>}}{D_{}}\&CenterDot;{\mathrm{\&omega;}}^2---\left(21\right)$

J ₀(β r) is the real argument first kind 0 rank bessel function, Y ₀(β r) is real argument second class 0 rank bessel function, I ₀(β r) is the empty argument first kind 0 rank bessel function, K ₀(β r) is empty argument second class 0 rank bessel function.ω is an angular frequency.

To free outer boundary: r=a place, moment M _rWith shearing Q _rAll be zero, that is: M _r=0, Q _r=0, to fixing inner boundary, r=b place, amount of deflection z (r) and corner

All be zero, that is:

Can get following formula:

$\left\{\begin{array}{c}\frac{d^{2}z}{{\mathrm{dr}}^2}+\frac{v}{r}\frac{\mathrm{dz}}{\mathrm{dr}}{|}_{r=a}=0\\ \frac{d}{\mathrm{dr}}(\frac{d^{2}z}{{\mathrm{dr}}^2}+\frac{1}{r}\frac{\mathrm{dz}}{\mathrm{dr}}){|}_{r=a}=0\\ z\left(r\right){|}_{r=b}=0\\ \frac{\mathrm{dz}}{\mathrm{dr}}{|}_{r=b}=0\end{array}\right.---\left(22\right)$

Is (20) substitution (22) Shi Kede:

$\left\{\begin{array}{c}[J_0\left(\mathrm{\&beta;a}\right)-\frac{(1-v)}{\mathrm{\&beta;a}}{J}_1\left(\mathrm{\&beta;a}\right)]C_1-[I_0\left(\mathrm{\&beta;a}\right)-\frac{(1-v)}{\mathrm{\&beta;a}}{I}_1\left(\mathrm{\&beta;a}\right)]C_2\\ +[Y_0\left(\mathrm{\&beta;a}\right)-\frac{(1-v)}{\mathrm{\&beta;a}}{Y}_1\left(\mathrm{\&beta;a}\right)]C_3-\\ [K_0\left(\mathrm{\&beta;a}\right)+\frac{(1-v)}{\mathrm{\&beta;a}}{K}_1\left(\mathrm{\&beta;a}\right)]C_4=0\\ J_1\left(\mathrm{\&beta;a}\right)C_1+I_1\left(\mathrm{\&beta;a}\right)C_2+Y_1\left(\mathrm{\&beta;a}\right)C_3-K_1\left(\mathrm{\&beta;a}\right)C_4=0\\ J_0\left(\mathrm{\&beta;b}\right)C_1+I_0\left(\mathrm{\&beta;b}\right)C_2+Y_0\left(\mathrm{\&beta;b}\right){\mathrm{<figure-callout\; id="3"\; label="C"\; filenames="HSA00000021846900031.png"\; state="\{\{state\}\}">C</figure-callout>}}_3+K_0\left(\mathrm{\&beta;b}\right)C_4=0\\ -J_1\left(\mathrm{\&beta;b}\right)C_1+I_1\left(\mathrm{\&beta;b}\right)C_2-Y_1\left(\mathrm{\&beta;b}\right)C_3-K_1\left(\mathrm{\&beta;b}\right)C_4=0\end{array}\right.---\left(23\right)$

In the formula, C ₁, C ₂, C ₃, C ₄Untrivialo solution is arranged, and then the determinant of coefficient of system of equations is necessary for zero, that is:

$\left|\begin{array}{cccc}[J_0\left(\mathrm{\&beta;a}\right)-\frac{(1-v)}{\mathrm{\&beta;a}}{J}_1\left(\mathrm{\&beta;a}\right)]& -[I_0\left(\mathrm{\&beta;a}\right)-\frac{(1-v)}{\mathrm{\&beta;a}}{I}_1\left(\mathrm{\&beta;a}\right)]& [Y_0\left(\mathrm{\&beta;a}\right)-\frac{(1-v)}{\mathrm{\&beta;a}}{Y}_1\left(\mathrm{\&beta;a}\right)]& -[K_0\left(\mathrm{\&beta;a}\right)+\frac{(1-v)}{\mathrm{\&beta;a}}{K}_1\left(\mathrm{\&beta;a}\right)]\\ J_1\left(\mathrm{\&beta;a}\right)& I_1\left(\mathrm{\&beta;a}\right)& Y_1\left(\mathrm{\&beta;a}\right)& -K_1\left(\mathrm{\&beta;a}\right)\\ J_0\left(\mathrm{\&beta;b}\right)& I_0\left(\mathrm{\&beta;b}\right)& Y_0\left(\mathrm{\&beta;b}\right)& K_0\left(\mathrm{\&beta;b}\right)\\ J_1\left(\mathrm{\&beta;b}\right)& -I_1\left(\mathrm{\&beta;b}\right)& Y_1\left(\mathrm{\&beta;b}\right)& K_1\left(\mathrm{\&beta;b}\right)\end{array}\right|=0---\left(24\right)$

(24) formula is that 0 pitch diameter inner boundary of a correspondence is fixed, outer boundary is the transcendental equation of annulus transverse vibration freely, can't obtain the analytic solution of equation, can only obtain numerical solution with numerical method, obtain its root again with fitting of a polynomial, when a=2b, the positive root of it first (corresponding 0 pitch diameter, 0 pitch circle symmetric vibration) square be:

(β a) ²=-0.5894v ²+ 2.3321v+12.3780 (quadratic fit)

Or: (β is a) ²=2.0374v+12.4025 (once fitting)

Relatively the seeing of the result of once fitting and quadratic fit and (24) formula numerical solution (Fig. 5), visible quadratic fit precision is very high, is analytic solution so get the quadratic fit result, that is:

(βa) ²＝-0.5894v ²+2.3321v+12.3780 (25)

By (21) formula, can get:

$f_2=\frac{{\left(\mathrm{\&beta;a}\right)}^2\&CenterDot;h}{4\mathrm{\&pi;}{a}^2}\sqrt{\frac{\mathrm{<figure-callout\; id="3"\; label="E"\; filenames="HSA00000021846900031.png"\; state="\{\{state\}\}">E</figure-callout>}}{3\mathrm{\&rho;}(1-v^2)}}---\left(26\right)$

(25) formula substitution (26) formula, can get:

$f_2=\frac{(-0.5894v^2+2.3321v+12.3780)\&CenterDot;h}{4\mathrm{\&pi;}{a}^2}\sqrt{\frac{\mathrm{<figure-callout\; id="3"\; label="E"\; filenames="HSA00000021846900031.png"\; state="\{\{state\}\}">E</figure-callout>}}{3\mathrm{\&rho;}(1-v^2)}}---\left(27\right)$

3. calculate the theoretical formula of Poisson ratio

Remove the theoretical formula that (27) can calculate Poisson ratio by formula (17):

$(-0.5894*v^2+2.3321*v+12.3780)*h*f_1-$

$2*\sqrt{3}*a*(-0.1405*v^2+0.7444*v+2.9728)*f_2=0---\left(28\right)$

Wherein: * is a multiplication sign, all represents multiplication sign in following formula.

4. simulating, verifying:

(28) correctness of formula is by the CoventorWare software verification, and concrete proof procedure and data are as follows:

In CoventorWare software, the Poisson ratio of setting polysilicon is v=0.22, Young modulus E=160GPa, density p=2.23E-15kg/ μ m ³With CoventorWare software according to sacrifice layer process simulation generate that one group of inner boundary props up admittedly, outer boundary freely the three-dimensional stereo model of annular film as test structure (Fig. 7).Wherein, the external radius a of annular film is respectively 300 microns, 350 microns, 400 microns, 450 microns, and the thickness h of film is 2 microns, a=2b.The radial vibration resonance frequency f that obtains with CoventorWare emulation ₁With transverse vibration resonance frequency f ₂, and the v ' as a result of the Poisson ratio that calculates with (28) formula sees (table 1).

The result of calculation of table 1:CoventorWare emulated data and Poisson ratio

Annotate: the Poisson ratio v=0.22 that sets in the CoventorWare emulation.

The present invention is based on the MEMS film Poisson ratio On-line Measuring Method of resonance frequency method, the steps include:

Make an annular MEMS film as test structure, interior boundary is propped up admittedly in the anchor district, the outer boundary freedom; The anchor district is fixed on the substrate; Measure the radial vibration of annular MEMS film and the resonance frequency of transverse vibration, calculate MEMS film Poisson ratio with following formula then:

$(-0.5894*v^2+2.3321*v+12.3780)*h*f_1-$

$2*\sqrt{3}*a*(-0.1405*v^2+0.7444*v+2.9728)*f_2=0$

Advantage of the present invention:

1. need not to know in advance the occurrence of film Young modulus and density of material etc., avoided the error that causes because of Young modulus and density of material etc., help the raising of precision.

2. owing to the symmetry of annulus membrane structure, the more approaching ideal in anchor district is propped up admittedly.

3. this method belongs to non-contact measurement, and measuring process can not exert an influence to test structure, can guarantee the repeatability of measuring.

4. all be suitable for for conductor and nonconductor this method.

5. test structure is made simple.Therefore, this method is fit to the on-line measurement of MEMS film Poisson ratio.

Description of drawings:

Fig. 1 is the structural drawing of torsional technique on-line measurement Poisson ratio, and arrow is represented pressure.

Fig. 2 is after sacrifice layer discharges, the test structure diagrammatic cross-section.

Fig. 3 is an infinitesimal force analysis figure who gets arbitrarily on the annular film.

Fig. 4 is the matched curve computer interface sectional drawing of radial vibration β a value and Poisson ratio relation, and horizontal ordinate is represented Poisson ratio, and ordinate is represented β a.Dot-and-dash line is the once fitting curve, and solid line is the quadratic fit curve, and symbol ※ is a numerical solution.

Fig. 5 is that (β a) for transverse vibration ²The matched curve computer interface sectional drawing of value and Poisson ratio relation, horizontal ordinate is represented Poisson ratio among the figure, and (β is a) in the ordinate representative ²Dot-and-dash line is the once fitting curve, and solid line is the quadratic fit curve, and symbol ※ is a numerical solution.

Fig. 6 is that inner boundary props up admittedly in anchor district, outer boundary annular films test structure fabrication process synoptic diagram freely.Wherein: 6-1～6-6 has represented manufacturing process successively.

Embodiment

Below in conjunction with embodiment, the present invention is elaborated.

Embodiment:

Shown in (Fig. 6-1), in sacrificial silicon layer technology, with LPCVD (low pressure chemical gaseous phase deposition) method thick silicon dioxide (sio of deposit one deck 0.3 μ m on monocrystalline substrate 1 ₂) layer 2, the silicon nitride layer 3 (Fig. 6-1) that deposit one deck 0.2 μ m is thick again, and then the thick phosphorosilicate glass of deposit one deck 2 μ m is made sacrifice layer 4 (Fig. 6-2), through etching a radius with HF solution on sacrifice layer 4 after gluing, exposure, the development is 200 microns circular pothole 5 (Fig. 6-3), the thick polysilicon membrane of deposit one deck 2 μ m is made structural sheet 6 (6-4) again, on structural sheet, be the center with the anchor district, etching external radius is the annular disk 7 (Fig. 6-5) of 400 μ m, remove sacrifice layer with HF solution, can obtain desired structure (Fig. 6-6).Measure the resonance frequency f of radial vibration with the micro-formula laser vibration measurer of MSA-500 ₁The resonance frequency f of=1.0650E07 hertz and transverse vibration ₂=6.3160E04 hertz is f ₁=1.0650E07 hertz, f ₂=6.3160E04 hertz, the a=400 micron, h-2 micron substitution (28) formula can be tried to achieve Poisson ratio v=0.2253.

Claims (1)

1. based on the MEMS film Poisson ratio On-line Measuring Method of resonance frequency method, the steps include:

Make an annular MEMS film as test structure, interior boundary is propped up admittedly in the anchor district, the outer boundary freedom; The anchor district is fixed on the substrate; Measure the radial vibration of annular MEMS film and the resonance frequency of transverse vibration, calculate MEMS film Poisson ratio with following formula then:

$(-0.5894*V^2+2.3321*v+12.3780)*h*f_1-$

$2*\sqrt{3}*a*(-0.1405*v^2+0.7444*v+2.9728)*{f_2=0.}_{}$

Images