CN100547347C - Curvature measurement method based on the little/nano cantilever of micro-interference and finite difference - Google Patents

Abstract

The present invention discloses a kind of curvature measurement method of the little/nano cantilever based on micro-interference and finite difference, is to be finished by following process: the first step, and use the phase shift microscopic interferometry and measure by the surface curvature data of micrometer/nano cantilever; In second step, use method of finite difference and parse by the amount of bow of micrometer/nano cantilever; In the 3rd step, the amount of bow that the surface curvature data that the first step is obtained and second step obtain compares, and measures the curvature value hour of difference between amount of bow and the parsing amount of bow thereby obtain.The present invention adopts the phase shift microscopic interferometry to measure the amount of bow on semi-girder surface, have high precision and high resolving power, to device under test is the nondestructive measurement of non-destruction, using method of finite difference, to parse the realization of semi-girder amount of bow simple, modeling approaches truth more, the result can be used for the measurement of micro-cantilever or nano cantilever (nano cantilever with micron width and nano thickness) curvature accurately and reliably.

Description

Curvature measurement method based on the little/nano cantilever of micro-interference and finite difference

Technical field

The present invention relates to a kind of little/receive the little/nano cantilever curvature measurement method of Mechatronic Systems.It is simple particularly to relate to a kind of realization, and the result has the curvature measurement method of high precision and high-resolution little/nano cantilever based on micro-interference and finite difference accurately and reliably.

Background technology

Little/(Micro/Nano ElectroMechanical System MEMS/NEMS) is the critical size that grows up on the basis of microelectric technique in collection sensing, the information processing of micrometer/nanometer magnitude and is executed in the integrated system of one to receive Mechatronic Systems.The research of MEMS/NEMS relates to multi-door subject, its manufacturing relates to technology such as design, material, technology, measurement, control, the energy and the system integration and encapsulation, and it uses the fields such as automobile, communication, medical treatment, biochemistry, Aero-Space, scientific instrument of having included especially.The MEMS/NEMS device from Design and Machining to the process of drop into using, be a vital step to the mechanical characteristic measurement of its structure.

Since the MEMS/NEMS device often by materials such as silicon and metals by chemical vapor deposition, after method such as sputter and plating forms film, pass through photoetching again, method such as etching and sacrifice layer corrosion is made, therefore this film by the surface micromachined preparation often exists very important residualinternal stress and stress gradient, it greatly affects the yield rate and the usability of device, how to realize to little/micro-nano structure unrelieved stress and stress gradient accurately, on-line measurement is one of vital task of MEMS/NEMS measurement, and when the stress characteristics of film is measured, accurately determine a primary often step of curvature value of membrane structure.Therefore, how easy and realize that accurately curvature measurement has very important significance concerning the analysis of MEMS/NEMS stresses of parts characteristic.

Little/nano cantilever is one of basic building block of MEMS/NEMS device, and it is the key structure in many MEMS/NEMS devices such as resonator, RF switch and biochemical sensor.Little/nano cantilever simple in structure, machining manufacture is easy, and modeling method is relatively ripe, be little/receive one of the most frequently used test structure in the test, can estimate the stress characteristics of membrane structure well by little/nano cantilever test structure.Microscopic interferometry utilizes optical interference to realize the profile measurement on micro devices surface, and the application of phase shift micro-interference has particularly greatly improved vertical resolution and the precision measured, become little/receive one of the most frequently used measurement means in measuring.And method of finite difference is a kind of important numerical algorithm, has obtained using widely in engineering analysis.

Summary of the invention

Technical matters to be solved by this invention is, provides a kind of realization simple, and the result has the curvature measurement method of high precision and high-resolution little/nano cantilever based on micro-interference and finite difference accurately and reliably.

The technical solution adopted in the present invention is: a kind of curvature measurement method of the little/nano cantilever based on micro-interference and finite difference, and measuring process comprises: use the phase shift microscopic interferometry and measure by the surface curvature data of micrometer/nano cantilever; Using method of finite difference parses by the amount of bow of micrometer/nano cantilever; Surface curvature data and amount of bow that preceding two processes are obtained compare, be adopt to intend a kind of in Newton's algorithm and two kinds of algorithms of least square method, measurement amount of bow by semi-girder relatively with resolve amount of bow, thereby obtain making both differences optimal curvatures matching value hour.

Described application phase shift microscopic interferometry is measured by the surface curvature data of micrometer/nano cantilever, its process comprises: use phase shift micro-interference measuring system and gather by the interferogram of micrometer/nano cantilever, by obtaining the original height information on semi-girder surface behind phase extraction and the phase unwrapping; On the original height information that obtains, remove droop error as zero reference level and obtain the vertical curve data on semi-girder surface with the surface of support section.

Described application method of finite difference parses by the amount of bow of micrometer/nano cantilever, be in the amount of bow of using method of finite difference parsing semi-girder on the two dimensional model of semi-girder and the relation between the curvature, and considered this imperfect effect of boundary conditions of initial end corner.

The curvature measurement method of the little/nano cantilever based on micro-interference and finite difference of the present invention adopts the phase shift microscopic interferometry to measure the amount of bow on semi-girder surface, has high precision and high resolving power, is the nondestructive measurement of non-destruction to device under test; Consider this imperfect effect of boundary conditions of non-vanishing initial end corner when model is resolved, made modeling approach truth more; Use method of finite difference to carry out approximate treatment, precision is high more more for a long time when node is obtained, and this can reach very high precision in the little/micro-nano structure surface measurement interpretation of result that with the pixel is unit; Provide two kinds of numerical analysis methods to realize measuring and resolving the comparison of amount of bow, the result that two kinds of methods obtain can be compared, checking result's accuracy.The method realizes simple, and the result can be used for the accurate measurement of micro-cantilever or nano cantilever (nano cantilever with micron width and nano thickness) curvature accurately and reliably.

Description of drawings

Fig. 1 is a phase shift micro-interference synoptic diagram of the present invention;

Fig. 2 be of the present invention little/the nano cantilever model;

Fig. 3 be of the present invention little/nano cantilever model node synoptic diagram;

Wherein:

1: measured surface 2: reference mirror

3: spectroscope 4: interferogram

5: light source 6: semi-girder

7: initial end corner 8: the support section of beam

9: node-1,0,1 ... N: the sequence number of node

L: beam length λ: the interval between the neighborhood of nodes

Embodiment

Make a detailed description below in conjunction with the curvature measurement method of drawings and Examples the little/nano cantilever based on micro-interference and finite difference of the present invention.

The curvature measurement method of the little/nano cantilever based on micro-interference and finite difference of the present invention, its measuring process comprises:

1, using the phase shift microscopic interferometry measures by the surface curvature data of micrometer/nano cantilever:

Using phase shift micro-interference measuring system gathers by the interferogram of micrometer/nano cantilever, by obtaining the original height information on semi-girder surface behind phase extraction and the phase unwrapping; Surface with the beam support section on the original height information that obtains is that reference field carries out rotation of coordinate removal droop error, obtains the vertical height information of semi-girder surface with respect to supporting plane.

The concrete practice is: at first will be collected and be preserved by the surperficial interference image 4 by micrometer/nano cantilever that the principle phase shift micro-interference system among Fig. 1 that is shown in obtains; Then, interference image is carried out the initial profile image that phase extraction and phase unwrapping obtain measured surface; At last, with the surface of the support section 8 of Fig. 2 central sill is that reference field carries out rotation of coordinate and eliminates droop error, and with the surface elevation of support section 8 as the zero elevation benchmark, thereby obtain the vertical curve amount on semi-girder 6 surfaces, a column data of getting on the axis of beam is standby.

2, using method of finite difference parses by the amount of bow of micrometer/nano cantilever:

This step is passed through in the relation of using on the two dimensional model of semi-girder between method of finite difference parsing deflection of beam amount and the curvature, and has considered this imperfect effect of boundary conditions of initial end corner.

The concrete practice is: the distribution according to the node among Fig. 39 is divided into N unit with semi-girder, calculates the relation between the initial end corner 7 among the vertical height of each node 9 and curvature and Fig. 2 with method of finite difference.

The distortion of little/nano cantilever meets the small deformation principle, satisfies the Bernoulli-Euler equation:

$\frac{{\∂}^{2}z\left(x\right)}{\∂x^2}=\frac{M\left(x\right)}{\mathrm{EI}}=k\left(x\right)---\left(1\right)$

$\mathrm{\θ}\left(x\right)\≈\tan \mathrm{\θ}\left(x\right)=\frac{\∂z\left(x\right)}{\∂x}---\left(2\right)$

M is a moment of flexure in the formula, and E is a Young modulus, and I is a moment of inertia, and k is a curvature, and θ is a corner.Since semi-girder z to distortion be actually very little with respect to the length of beam, so we are considered as constant with respect to x with the curvature k of each point on the beam.Whole semi-girder is divided into N junior unit with N+1 node, and introducing dummy node-1, the substitution boundary condition: the displacement of beam initial end is zero, and the corner of initial end is often non-vanishing, single order and second order method of finite difference formula (3) and (4) are used in equation (1) and (2) carry out approximate treatment, try to achieve deflection of beam amount (amount of deflection z (x)) and curvature k and beam initial end rotational angle theta ₀Analytic relationship between (corner at node 0 place) as the formula (5).λ in formula (3) and (4) is the interval between the neighborhood of nodes.

${\left(\frac{\∂f}{\∂x}\right)}_i=\frac{f_{i+1}-f_{i-1}}{2\mathrm{\λ}}---\left(3\right)$

${\left(\frac{{\∂}^{2}f}{{\∂x}^2}\right)}_i=\frac{f_{i-1}+f_{i+1}-f_i}{{\mathrm{\λ}}^2}---\left(4\right)$

$z\left(x\right)={\mathrm{\θ}}_{0}x+k\frac{x^2}{2}---\left(5\right)$

3, calculating optimum curvature matching value:

With the measurement amount of bow and parsing amount of bow that obtains previously, by comparing with intending Newton's algorithm or least square method, obtain and measure the curvature value hour of difference between amount of bow and the parsing amount of bow, as the optimal curvatures matching value, this step has two kinds of concrete practices:

(1) adopt the plan Newton's algorithm:

Intending Newton's algorithm is a kind of numerical optimization method, can be used for the minimum problems of solved function.With the amount of bow of the axis of beam and the amount of bow pointwise contrast of formula simulation, get and make the k and the θ of root mean square difference minimum ₀Value is for finally separating.The expression formula of root mean square difference as the formula (6), wherein f is the measured value of amount of bow and the root mean square difference between the analytic simulation value, z (x _i) be semi-girder x _iThe flexural measurement value at place.

$f=\sqrt{\frac{{[{\mathrm{\&theta;}}_0{x}_1+k\frac{{x_1}^2}{2}-z\left(x_1\right)]}^2+{[{\mathrm{\&theta;}}_0{x}_2+k\frac{{x_2}^2}{2}-z\left(x_2\right)]}^2+\&CenterDot;\&CenterDot;\&CenterDot;+{[{\mathrm{\&theta;}}_0{x}_N+k\frac{{x_{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="C20071005712400081.png"\; state="\{\{state\}\}">N</figure-callout>}}}^2}{2}-z\left(x_N\right)]}^2}{N}}---\left(6\right)$

As long as from certain initial value, intend Newton's algorithm and just can obtain making value hour k and the θ of objective function f ₀Optimum solution.

(2) adopt least square method:

Least square method is if formula (6) is configured to an overdetermined equation group, and as the formula (7), the equation left side is and k and θ ₀Relevant parsing amount of bow, equation the right is the flexural measurement value, k and θ in the time of can using least square method to obtain thus to make two amount of bow approximately equals ₀Separate.

$\left[\begin{array}{cc}{x}_1& \frac{{x_1}^2}{2}\\ x_2& \frac{{x_2}^2}{2}\\ \&CenterDot;& \&CenterDot;\\ \&CenterDot;& \&CenterDot;\\ \&CenterDot;& \&CenterDot;\\ x_N& \frac{{x_{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="C20071005712400081.png"\; state="\{\{state\}\}">N</figure-callout>}}}^2}{2}\end{array}\right]\left[\begin{array}{c}{\mathrm{\&theta;}}_0\\ k\end{array}\right]=\left[\begin{array}{c}z\left(x_1\right)\\ z\left(x_2\right)\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ z\left(x_N\right)\end{array}\right]---\left(7\right)$

The present invention is open and that disclose, and method can be used for reference this paper disclosure, although method of the present invention is described by preferred embodiment, but those skilled in the art obviously can be spliced method as herein described in not breaking away from content of the present invention, spirit and scope or change, or increase and decrease some links, more particularly, the replacement that all are similar and change apparent to those skilled in the artly, they all are regarded as being included in spirit of the present invention, scope and the content.

Claims (3)

1. the curvature measurement method based on the little/nano cantilever of micro-interference and finite difference is characterized in that measuring process comprises: use the phase shift microscopic interferometry and measure by the surface curvature data of micrometer/nano cantilever; Using method of finite difference parses by the amount of bow of micrometer/nano cantilever; Surface curvature data and amount of bow that preceding two processes are obtained compare, specifically be adopt to intend a kind of in Newton's algorithm and two kinds of algorithms of least square method, measurement amount of bow by semi-girder relatively with resolve amount of bow, thereby obtain making both differences optimal curvatures matching value hour.

2. the curvature measurement method of the little/nano cantilever based on micro-interference and finite difference according to claim 1, it is characterized in that, described application phase shift microscopic interferometry is measured by the surface curvature data of micrometer/nano cantilever, its process comprises: use phase shift micro-interference measuring system and gather by the interferogram of micrometer/nano cantilever, by obtaining the original height information on semi-girder surface behind phase extraction and the phase unwrapping; On the original height information that obtains, remove droop error as zero reference level and obtain the vertical curve data on semi-girder surface with the surface of support section.

3. the curvature measurement method of the little/nano cantilever based on micro-interference and finite difference according to claim 1, it is characterized in that, described application method of finite difference parses by the amount of bow of micrometer/nano cantilever, be in the amount of bow of using method of finite difference parsing semi-girder on the two dimensional model of semi-girder and the relation between the curvature, and considered this imperfect effect of boundary conditions of initial end corner.

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