CN103823960A - Method for solving elastic deformation of Al/Si3N4/Si three-layer MEMS (micro-electromechanical system) cantilever structure - Google Patents

Abstract

The method for solving the flexible deformation of tri- layers of MEMS cantilever beam structure of Al/Si3N4/Si,The following steps are included: (1) utilizes tri- layers of cantilever beam structure of Al/Si3N4/Si of MEMS technology production,Its each layer Young's modulus and thickness be it is known thatIf Young's modulus ratio γ 2=E3/E2 of 1=E1/E2, γ,Thickness ratio r1=h1/h2,R2=h3/h2,Wherein E1,E2,E3,H1,H2,H3 respectively successively indicate silicon base,Silicon nitride film,The Young's modulus and thickness of aluminium film; (2) under certain process conditions,By processing and manufacturing and last handling process,Residual stress caused by different materials film can be obtained by inquiring pertinent literature,Silicon base,Silicon nitride film,The residual stress of aluminium film is followed successively by σ res, and 1,σ res, 2,σres,3; (3) above-mentioned data are substituted into Stoney and extends formula κ ben is exactly the bending curvature of tri- layers of MEMS cantilever beam structure of Al/Si3N4/Si.

Description

One solves Al/Si 3n 4the method of the elastic deformation of tri-layers of MEMS cantilever beam structure of/Si

Technical field

The present invention relates to a kind of method that solves each layer of unrelieved stress of three layers of MEMS structure and its malformation.

Background technology

The conventional MEMS capacitive temperature sensor of one class adopts the multilayer semi-girder solid variable capacitor structure being made up of conductor (or semiconductor)/dielectric layer/conductor (or semiconductor), asks and has filled insulating medium at two-layer capacitor plate.Electrically conductive film in multilayer beam can be by heavily doped silicon, and doped polycrystalline silicon or metal form, and deielectric-coating can be SiO ₂or Si ₃n ₄or be complex media.Due to the layers of material difference of composition multilayer beam solid variable capacitance, the thermal expansivity of layers of material is just different.In the time of temperature variation, the layers of material of multilayer beam produces thermal stress because of coefficient of thermal expansion mismatch, thereby makes the deformation that bends of multilayer beam.Dielectric substance can produce electrostriction enhancement effect under external force, and specific inductive capacity can produce deformation under external force and change along with material, and this solid electric capacity just changes along with temperature variation.Al/Si ₃n ₄tri-layers of semi-girder of/Si are the structures that this class capacitive temperature sensor is relatively commonly used, golden as electrode, Si ₃n ₄as dielectric layer, Si is another electrode, and the thermal expansivity of layers of material is just different, thereby causes these three layers of semi-girders bendings and change capacitance size.

But due to a variety of causes, conventionally show larger unrelieved stress (strain), the inharmonious measurement performance that can cause the flexural deformation of structure to affect structure of stress (strain) through in each layer of processing and manufacturing and this class multilayer cantilever beam structure of last handling process.Therefore in each tunic, the sign of the relation of unrelieved stress and this malformation just seems most important.Conventionally we adopt Stoney formula to be used as characterizing method, but Stoney formula be suitable for must be based on many harshnesses hypothesis, for example the hypothesis of Stoney formula requires the Young modulus of substrate and film close.Because the Young modulus of each layer differs larger, each layer thickness is more approaching again, the Al/Si in this capacitive temperature sensor ₃n ₄the distortion of/Si three-decker is also not suitable for going to characterize with Stoney formula.

Summary of the invention

In order to solve prior art due to the each layer of inhomogeneous Al/Si causing of residual stress distribution ₃n ₄the problem on deformation of tri-layers of MEMS cantilever beam structure of/Si, the present invention proposes one and solves Al/Si ₃n ₄the method of the elastic deformation of tri-layers of MEMS cantilever beam structure of/Si.

One solves Al/Si ₃n ₄the method of the elastic deformation of tri-layers of MEMS cantilever beam structure of/Si, comprises the following steps:

(1) Al/Si that utilizes MEMS technique to make ₃n ₄tri-layers of cantilever beam structure of/Si, its each layer of Young modulus and thickness are known, establish Young modulus and compare γ ₁=E ₁/ E ₂, γ ₂=E ₃/ E ₂, Thickness Ratio r ₁=h ₁/ h ₂, r ₂=h ₃/ h ₂, wherein E ₁, E ₂, E ₃, h ₁, h ₂, h ₃represent successively respectively Young modulus and the thickness of silicon substrate, silicon nitride film, aluminium film;

(2) under certain process conditions, through processing and manufacturing and last handling process, the unrelieved stress that different materials film produces can obtain by inquiry pertinent literature, and the unrelieved stress of establishing silicon substrate, silicon nitride film, aluminium film is followed successively by σ _{res, 1}, σ _{res, 2}, σ _{res, 3};

(3) above-mentioned data substitution Stoney is extended to formula wherein

$\begin{array}{c}{\mathrm{<figure-callout\; id="1"\; label="I"\; filenames="HDA0000406976090000012.png,HDA0000406976090000015.png"\; state="\{\{state\}\}">I</figure-callout>}}_1=2[-{\mathrm{\&gamma;}}_1{r}_1({\mathrm{<figure-callout\; id="1"\; label="r"\; filenames="HDA0000406976090000012.png,HDA0000406976090000015.png"\; state="\{\{state\}\}">r</figure-callout>}}_1+{\mathrm{\&gamma;}}_2{r}_1{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000406976090000012.png"\; state="\{\{state\}\}">r</figure-callout>}}_2+1+2{\mathrm{\&gamma;}}_2{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000406976090000012.png"\; state="\{\{state\}\}">r</figure-callout>}}_2+{\mathrm{\&gamma;}}_2{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000406976090000012.png"\; state="\{\{state\}\}">r</figure-callout>}}_2^2){\mathrm{\&epsiv;}}_{\mathrm{res},1}\\ +({\mathrm{\&gamma;}}_1{\mathrm{<figure-callout\; id="1"\; label="r"\; filenames="HDA0000406976090000012.png,HDA0000406976090000015.png"\; state="\{\{state\}\}">r</figure-callout>}}_1^2+{\mathrm{\&gamma;}}_1{r}_1-{\mathrm{\&gamma;}}_2{r}_2-{\mathrm{\&gamma;}}_2{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000406976090000012.png"\; state="\{\{state\}\}">r</figure-callout>}}_2^2){\mathrm{\&epsiv;}}_{\mathrm{res},2}+{\mathrm{\&gamma;}}_2{r}_2({\mathrm{\&gamma;}}_1{\mathrm{<figure-callout\; id="1"\; label="r"\; filenames="HDA0000406976090000012.png,HDA0000406976090000015.png"\; state="\{\{state\}\}">r</figure-callout>}}_1^2+2{\mathrm{\&gamma;}}_1{\mathrm{<figure-callout\; id="1"\; label="r"\; filenames="HDA0000406976090000012.png,HDA0000406976090000015.png"\; state="\{\{state\}\}">r</figure-callout>}}_1+1+{\mathrm{\&gamma;}}_1{\mathrm{<figure-callout\; id="1"\; label="r"\; filenames="HDA0000406976090000012.png,HDA0000406976090000015.png"\; state="\{\{state\}\}">r</figure-callout>}}_1{\mathrm{\&gamma;}}_2+r_2){\mathrm{\&epsiv;}}_{\mathrm{res},3]}\end{array}$

$\begin{array}{c}{\mathrm{<figure-callout\; id="2"\; label="I"\; filenames="HDA0000406976090000012.png"\; state="\{\{state\}\}">I</figure-callout>}}_2={\mathrm{\&gamma;}}_1^2{\mathrm{<figure-callout\; id="1"\; label="r"\; filenames="HDA0000406976090000012.png,HDA0000406976090000015.png"\; state="\{\{state\}\}">r</figure-callout>}}_1^4+4{\mathrm{\&gamma;}}_1{\mathrm{<figure-callout\; id="1"\; label="r"\; filenames="HDA0000406976090000012.png,HDA0000406976090000015.png"\; state="\{\{state\}\}">r</figure-callout>}}_1^3+4{\mathrm{\&gamma;}}_1{\mathrm{\&gamma;}}_2{\mathrm{<figure-callout\; id="1"\; label="r"\; filenames="HDA0000406976090000012.png,HDA0000406976090000015.png"\; state="\{\{state\}\}">r</figure-callout>}}_1^3{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000406976090000012.png"\; state="\{\{state\}\}">r</figure-callout>}}_2+4{\mathrm{\&gamma;}}_1{\mathrm{<figure-callout\; id="1"\; label="r"\; filenames="HDA0000406976090000012.png,HDA0000406976090000015.png"\; state="\{\{state\}\}">r</figure-callout>}}_1+1+4{\mathrm{\&gamma;}}_2{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000406976090000012.png"\; state="\{\{state\}\}">r</figure-callout>}}_2+4{\mathrm{\&gamma;}}_1{\mathrm{\&gamma;}}_2{r}_1{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000406976090000012.png"\; state="\{\{state\}\}">r</figure-callout>}}_2^3\\ +4{\mathrm{\&gamma;}}_2{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000406976090000012.png"\; state="\{\{state\}\}">r</figure-callout>}}_2^3+{\mathrm{\&gamma;}}_2^2{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000406976090000012.png"\; state="\{\{state\}\}">r</figure-callout>}}_2^4+6{\mathrm{\&gamma;}}_1{\mathrm{<figure-callout\; id="1"\; label="r"\; filenames="HDA0000406976090000012.png,HDA0000406976090000015.png"\; state="\{\{state\}\}">r</figure-callout>}}_1^2+12{\mathrm{\&gamma;}}_1{\mathrm{\&gamma;}}_2{\mathrm{<figure-callout\; id="1"\; label="r"\; filenames="HDA0000406976090000012.png,HDA0000406976090000015.png"\; state="\{\{state\}\}">r</figure-callout>}}_1^2{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000406976090000012.png"\; state="\{\{state\}\}">r</figure-callout>}}_2+6{\mathrm{\&gamma;}}_1{\mathrm{\&gamma;}}_2{\mathrm{<figure-callout\; id="1"\; label="r"\; filenames="HDA0000406976090000012.png,HDA0000406976090000015.png"\; state="\{\{state\}\}">r</figure-callout>}}_1^2{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000406976090000012.png"\; state="\{\{state\}\}">r</figure-callout>}}_2^2\\ +12{\mathrm{\&gamma;}}_1{\mathrm{\&gamma;}}_2{r}_1{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000406976090000012.png"\; state="\{\{state\}\}">r</figure-callout>}}_2+12{\mathrm{\&gamma;}}_1{\mathrm{\&gamma;}}_2{r}_1{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000406976090000012.png"\; state="\{\{state\}\}">r</figure-callout>}}_2^2+6{\mathrm{\&gamma;}}_2{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000406976090000012.png"\; state="\{\{state\}\}">r</figure-callout>}}_2^2\end{array}$

Here ε _{res, i}=σ _{res, i}/ E _i, i=1,2,3.

κ _benbe exactly Al/Si ₃n ₄the bending curvature of tri-layers of MEMS cantilever beam structure of/Si.

It is as follows that the present invention Stoney used extends formulation process:

Suppose the three-dimensional structure that has the adhesion of n tunic to form, be subject to the unrelieved stress that through-thickness distributes arbitrarily, as Fig. 1.The existence of initial strain in each layer under Constrained state can cause total to produce distortion after constraint disappears.In this patent, the mechanical analysis of sandwich construction is based on following hypothesis: (i) thickness of every one deck structure is enough little with respect to its length; (ii) material has homogeneity, isotropy, linear elasticity; (iii) edge effect of structure proximal border is negligible; (iv) be parallel to all material property preservation including Young modulus of layers of material at interface constant; (v) line strain and angular strain are infinitely small.And, overstrain ε _resrepresent unrelieved stress σ _resin the potential elastic deformation without causing under retraining.According to hypothesis (ii) and (iii), the relation between overstrain and unrelieved stress can be expressed as ε _res=σ _res/ E, in cantilever beam structure, E is young modulus of material; In plate structure, E represents twin shaft modulus.The overstrain through-thickness distribution character of i tunic can be expressed as a polynomial function:

${\mathrm{\&epsiv;}}_{\mathrm{res},i}\left(z\right)=\underset{k=0}{\overset{\mathrm{\&alpha;}}{\mathrm{\&Sigma;}}}{\mathrm{\&epsiv;}}_{\mathrm{res},i,k}{\left[\frac{z-z_i}{h_i}\right]}^k,z_i\&le;z\&le;z_{i+1}---\left(1\right)$

Wherein, h _iwith

h _jrepresent respectively the thickness of i layer and the position of i layer bottom surface, k represents exponent number.From the physical significance of above formula, i tunic 0 rank overstrain ε _{res, i, 0}may be produced by the imbalance of film-substrate thermal expansivity; And some local effects, as the atom diffusion along film thickness, atom shot-peening effect, change along the crystallite dimension of film thickness, gap or displacement defect, all can cause the generation of overstrain gradient.

Only the unrelieved stress on k rank in formula (1) is discussed below.If each layer is peeled off, every layer of structure has distortion separately, but will show identical displacement due to each layer at interface location, and each layer all can have internal force and moment of resistance to produce at adhesion face place.As shown in Figure 2, in i layer, there is the internal force N acting between end face and i+1 layer _i,kand act on the internal force N between bottom surface and i-1 layer _{i-1, k}.Same, the moment of resistance in i layer and between adjacent layer can be used M _i,kand M _{i-1, k}represent.There is N according to definition _{0, k}=N _n,k=0, M _{0, k}=M _n,k=0.

Use respectively ε _{aix, i, k}and κ _{ben, i, k}be illustrated in axial strain and bending curvature that i layer is caused by the overstrain on k rank in formula (1), the deformation strain ε of through-thickness _{def, i, k}(z) can be expressed as

${\mathrm{\&epsiv;}}_{\mathrm{def},i,k}\left(z\right)={\mathrm{\&epsiv;}}_{\mathrm{axi},i,k}-{\mathrm{\&kappa;}}_{\mathrm{ben},i,k}(z-z_i-\frac{h_i}{2}),z_i\&le;z\&le;z_{i+1}---\left(2\right)$

Shaft distortion (stretching or the compression) ε of each layer _{axi, i, k}by internal force and the average overstrain of this layer

determine,

${\mathrm{\&epsiv;}}_{\mathrm{axi},i,k}=-\stackrel{\&OverBar;}{{\mathrm{\&epsiv;}}_{\mathrm{res},i,k}}-\frac{N_{i,k}-N_{i-1,k}}{E_i{h}_{i}b}---\left(3\right)$

Wherein, b represents the width of sandwich construction, E _irepresent the Young modulus of i tunic, the average overstrain in i tunic k rank for

$\stackrel{\&OverBar;}{{\mathrm{\&epsiv;}}_{\mathrm{res},i,k}}=\frac{1}{h_i}{\&Integral;}_{z_i}^{z_{i+1}}{\mathrm{\&epsiv;}}_{\mathrm{res},i,k}{\left(\frac{z-z_i}{h_i}\right)}^k\mathrm{dz}=\frac{{\mathrm{\&epsiv;}}_{\mathrm{res},i,k}}{k+1}---\left(4\right)$

According to beam theory, i tunic k rank bending curvature κ _{ben, i, k}with bending stiffness E _ii _i, bending moment M _i,krelation as follows:

${\mathrm{\&kappa;}}_{\mathrm{ben},i,k}=\frac{M_{i,k}}{E_i{I}_i}---\left(5\right)$

Wherein

it is cross section rotary inertia.Internal force between adjacent layer, moment of resistance, affect the gradient overstrain that also has this layer of each layer of bending moment,

$M_{i,k}=\frac{(N_{i-1,k}+N_{i,k})h_i}{2}+(M_i-M_{i-1})+\stackrel{\&OverBar;}{M_{i,k}}---\left(6\right)$

Wherein represent the moment of flexure that in gradient overstrain, k rank overstrain causes i tunic to produce

$\begin{array}{c}\stackrel{\&OverBar;}{M_{i,k}}={\&Integral;}_{z_i}^{z_{i+1}}{E}_{i}b{\mathrm{\&epsiv;}}_{\mathrm{res},i,k}{\left(\frac{z-z_i}{h_i}\right)}^k(z-z_i-\frac{h_i}{2})\mathrm{dz}\\ =\frac{E_{i}b{h}_i^2}{2}\frac{{\mathrm{k\&epsiv;}}_{\mathrm{res},i,k}}{(k+1)(k+2)}\end{array}---\left(7\right)$

Be sticked together due to each layer, therefore every one deck all has identical bending curvature,

κ _ben,i,k=κ _ben,k (8)

According to obtaining in formula (5) (6) (8)

$\begin{array}{c}{\mathrm{\&kappa;}}_{\mathrm{ben},k}{E}_i{I}_i=\stackrel{\&OverBar;}{M_{i,k}}+\frac{(N_{i-1,k}+N_{i,k})h_i}{2}\\ +(M_i-M_{i-1})\end{array}---\left(9\right)$

Formula (9) can further be derived

${\mathrm{\&kappa;}}_{\mathrm{ben},k}\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{E}_i{I}_i=\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}\stackrel{\&OverBar;}{M_{i,k}}+\frac{1}{2}\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}(N_{i-1,k}+N_{i,k})h_i---\left(10\right)$

The deformation strain of adhering layer must meet continuity,

${\mathrm{\&epsiv;}}_{\mathrm{def},i,k}\left(z_{i+1}\right)={\mathrm{\&epsiv;}}_{\mathrm{def},i+1,k}\left(z_{i+1}\right),1\&le;i\&le;n-1---\left(11\right)$

Composite type (2) (3) (11) can obtain

${\mathrm{\&kappa;}}_{\mathrm{ben},k}\frac{h_{i+1}+h_i}{2}=\stackrel{\&OverBar;}{{\mathrm{\&epsiv;}}_{\mathrm{res},i+1,k}}-\stackrel{\&OverBar;}{{\mathrm{\&epsiv;}}_{\mathrm{res},i,k}}+\frac{N_{i+1,k}-N_{i,k}}{E_{i+1}{h}_{i+1}b}-\frac{N_{i,k}-N_{i-1,k}}{E_i{h}_{i}b}---\left(12\right)$

Simultaneous formula (10) (12) (system of linear equations of n equation composition) can solve bending curvature κ _{ben, k}with internal force N _i,k(1≤i≤n-1).

By every κ _{ben, k}be added the total curvature κ that just obtains this sandwich construction _ben,

${\mathrm{\&kappa;}}_{\mathrm{ben}}=\underset{k=0}{\overset{\mathrm{\&alpha;}}{\mathrm{\&Sigma;}}}{\mathrm{\&kappa;}}_{\mathrm{ben},k}---\left(13\right)$

In tri-layers of MEMS cantilever beam structure of Fig. 3, establish Young modulus and compare γ ₁=E ₁/ E ₂, γ ₂=E ₃/ E ₂, Thickness Ratio r ₁=h ₁/ h ₂, r ₂=h ₃/ h ₂.Can calculate three layers of semi-girder bending curvature according to formula (10) (12) (13):

${\mathrm{\&kappa;}}_{\mathrm{ben},k}=\frac{6}{(k+1)(k+2)h_2{I}_2}\frac{{\mathrm{<figure-callout\; id="1"\; label="I"\; filenames="HDA0000406976090000012.png,HDA0000406976090000015.png"\; state="\{\{state\}\}">I</figure-callout>}}_1^{*}}{I_2}---\left(14a\right)$

Wherein

i ₂be respectively

$\begin{array}{c}{\mathrm{<figure-callout\; id="1"\; label="I"\; filenames="HDA0000406976090000012.png,HDA0000406976090000015.png"\; state="\{\{state\}\}">I</figure-callout>}}_1^{*}=k({\mathrm{\&gamma;}}_1{\mathrm{\&gamma;}}_1^2{\mathrm{\&epsiv;}}_{\mathrm{res},1,k}+{\mathrm{\&epsiv;}}_{\mathrm{res},2,k}+{\mathrm{\&gamma;}}_2{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000406976090000012.png"\; state="\{\{state\}\}">r</figure-callout>}}_2^2{\mathrm{\&epsiv;}}_{\mathrm{res},3,k})({\mathrm{\&gamma;}}_1{\mathrm{<figure-callout\; id="1"\; label="r"\; filenames="HDA0000406976090000012.png,HDA0000406976090000015.png"\; state="\{\{state\}\}">r</figure-callout>}}_1+1+{\mathrm{\&gamma;}}_2{r}_2)\\ +(k+2)[-{\mathrm{\&gamma;}}_1{r}_1({\mathrm{<figure-callout\; id="1"\; label="r"\; filenames="HDA0000406976090000012.png,HDA0000406976090000015.png"\; state="\{\{state\}\}">r</figure-callout>}}_1+{\mathrm{\&gamma;}}_2{r}_1{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000406976090000012.png"\; state="\{\{state\}\}">r</figure-callout>}}_2+1+2{\mathrm{\&gamma;}}_2{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000406976090000012.png"\; state="\{\{state\}\}">r</figure-callout>}}_2+{\mathrm{\&gamma;}}_2{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000406976090000012.png"\; state="\{\{state\}\}">r</figure-callout>}}_2^2){\mathrm{\&epsiv;}}_{\mathrm{res},1,k}\\ +({\mathrm{\&gamma;}}_1{\mathrm{<figure-callout\; id="1"\; label="r"\; filenames="HDA0000406976090000012.png,HDA0000406976090000015.png"\; state="\{\{state\}\}">r</figure-callout>}}_1^2+{\mathrm{\&gamma;}}_1{r}_1-{\mathrm{\&gamma;}}_2{r}_2-{\mathrm{\&gamma;}}_2{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000406976090000012.png"\; state="\{\{state\}\}">r</figure-callout>}}_2^2){\mathrm{\&epsiv;}}_{\mathrm{res},2,k}+{\mathrm{\&gamma;}}_2{r}_2({\mathrm{\&gamma;}}_1{\mathrm{<figure-callout\; id="1"\; label="r"\; filenames="HDA0000406976090000012.png,HDA0000406976090000015.png"\; state="\{\{state\}\}">r</figure-callout>}}_1^2+2{\mathrm{\&gamma;}}_1{\mathrm{<figure-callout\; id="1"\; label="r"\; filenames="HDA0000406976090000012.png,HDA0000406976090000015.png"\; state="\{\{state\}\}">r</figure-callout>}}_1+1+{\mathrm{\&gamma;}}_1{r}_1{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000406976090000012.png"\; state="\{\{state\}\}">r</figure-callout>}}_2+r_2){\mathrm{\&epsiv;}}_{\mathrm{res},3,k}]\end{array}---\left(14b\right)$

$\begin{array}{c}{\mathrm{<figure-callout\; id="2"\; label="I"\; filenames="HDA0000406976090000012.png"\; state="\{\{state\}\}">I</figure-callout>}}_2={\mathrm{\&gamma;}}_1^2{\mathrm{<figure-callout\; id="1"\; label="r"\; filenames="HDA0000406976090000012.png,HDA0000406976090000015.png"\; state="\{\{state\}\}">r</figure-callout>}}_1^4+4{\mathrm{\&gamma;}}_1{\mathrm{<figure-callout\; id="1"\; label="r"\; filenames="HDA0000406976090000012.png,HDA0000406976090000015.png"\; state="\{\{state\}\}">r</figure-callout>}}_1^3+4{\mathrm{\&gamma;}}_1{\mathrm{\&gamma;}}_2{\mathrm{<figure-callout\; id="1"\; label="r"\; filenames="HDA0000406976090000012.png,HDA0000406976090000015.png"\; state="\{\{state\}\}">r</figure-callout>}}_1^3{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000406976090000012.png"\; state="\{\{state\}\}">r</figure-callout>}}_2+4{\mathrm{\&gamma;}}_1{\mathrm{<figure-callout\; id="1"\; label="r"\; filenames="HDA0000406976090000012.png,HDA0000406976090000015.png"\; state="\{\{state\}\}">r</figure-callout>}}_1+1+4{\mathrm{\&gamma;}}_2{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000406976090000012.png"\; state="\{\{state\}\}">r</figure-callout>}}_2+4{\mathrm{\&gamma;}}_1{\mathrm{\&gamma;}}_2{r}_1{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000406976090000012.png"\; state="\{\{state\}\}">r</figure-callout>}}_2^3\\ +4{\mathrm{\&gamma;}}_2{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000406976090000012.png"\; state="\{\{state\}\}">r</figure-callout>}}_2^3+{\mathrm{\&gamma;}}_2^2{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000406976090000012.png"\; state="\{\{state\}\}">r</figure-callout>}}_2^4+6{\mathrm{\&gamma;}}_1{\mathrm{<figure-callout\; id="1"\; label="r"\; filenames="HDA0000406976090000012.png,HDA0000406976090000015.png"\; state="\{\{state\}\}">r</figure-callout>}}_1^2+12{\mathrm{\&gamma;}}_1{\mathrm{\&gamma;}}_2{\mathrm{<figure-callout\; id="1"\; label="r"\; filenames="HDA0000406976090000012.png,HDA0000406976090000015.png"\; state="\{\{state\}\}">r</figure-callout>}}_1^2{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000406976090000012.png"\; state="\{\{state\}\}">r</figure-callout>}}_2+6{\mathrm{\&gamma;}}_1{\mathrm{\&gamma;}}_2{\mathrm{<figure-callout\; id="1"\; label="r"\; filenames="HDA0000406976090000012.png,HDA0000406976090000015.png"\; state="\{\{state\}\}">r</figure-callout>}}_1^2{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000406976090000012.png"\; state="\{\{state\}\}">r</figure-callout>}}_2^2\\ +12{\mathrm{\&gamma;}}_1{\mathrm{\&gamma;}}_2{r}_1{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000406976090000012.png"\; state="\{\{state\}\}">r</figure-callout>}}_2+12{\mathrm{\&gamma;}}_1{\mathrm{\&gamma;}}_2{r}_1{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000406976090000012.png"\; state="\{\{state\}\}">r</figure-callout>}}_2^2+6{\mathrm{\&gamma;}}_2{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000406976090000012.png"\; state="\{\{state\}\}">r</figure-callout>}}_2^2\end{array}---\left(14c\right)$

Suppose that in each tunic, unrelieved stress through-thickness is uniformly distributed, in formula (1), only comprise 0 rank item, three layers of semi-girder bending curvature:

${\mathrm{\&kappa;}}_{\mathrm{ben}}={\mathrm{\&kappa;}}_{\mathrm{ben},0}=\frac{3}{h_2}\frac{I_1}{I_2}---\left(15a\right)$

Wherein

$\begin{array}{c}{\mathrm{<figure-callout\; id="1"\; label="I"\; filenames="HDA0000406976090000012.png,HDA0000406976090000015.png"\; state="\{\{state\}\}">I</figure-callout>}}_1=2[-{\mathrm{\&gamma;}}_1{r}_1({\mathrm{<figure-callout\; id="1"\; label="r"\; filenames="HDA0000406976090000012.png,HDA0000406976090000015.png"\; state="\{\{state\}\}">r</figure-callout>}}_1+{\mathrm{\&gamma;}}_2{r}_1{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000406976090000012.png"\; state="\{\{state\}\}">r</figure-callout>}}_2+1+2{\mathrm{\&gamma;}}_2{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000406976090000012.png"\; state="\{\{state\}\}">r</figure-callout>}}_2+{\mathrm{\&gamma;}}_2{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000406976090000012.png"\; state="\{\{state\}\}">r</figure-callout>}}_2^2){\mathrm{\&epsiv;}}_{\mathrm{res},1}\\ +({\mathrm{\&gamma;}}_1{\mathrm{<figure-callout\; id="1"\; label="r"\; filenames="HDA0000406976090000012.png,HDA0000406976090000015.png"\; state="\{\{state\}\}">r</figure-callout>}}_1^2+{\mathrm{\&gamma;}}_1{r}_1-{\mathrm{\&gamma;}}_2{r}_2-{\mathrm{\&gamma;}}_2{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000406976090000012.png"\; state="\{\{state\}\}">r</figure-callout>}}_2^2){\mathrm{\&epsiv;}}_{\mathrm{res},2}+{\mathrm{\&gamma;}}_2{r}_2({\mathrm{\&gamma;}}_1{\mathrm{<figure-callout\; id="1"\; label="r"\; filenames="HDA0000406976090000012.png,HDA0000406976090000015.png"\; state="\{\{state\}\}">r</figure-callout>}}_1^2+2{\mathrm{\&gamma;}}_1{\mathrm{<figure-callout\; id="1"\; label="r"\; filenames="HDA0000406976090000012.png,HDA0000406976090000015.png"\; state="\{\{state\}\}">r</figure-callout>}}_1+1+{\mathrm{\&gamma;}}_1{\mathrm{<figure-callout\; id="1"\; label="r"\; filenames="HDA0000406976090000012.png,HDA0000406976090000015.png"\; state="\{\{state\}\}">r</figure-callout>}}_1{\mathrm{\&gamma;}}_2+r_2){\mathrm{\&epsiv;}}_{\mathrm{res},3}]\end{array}---\left(15b\right)$

ε in formula _{res, 1}, ε _{res, 2}, ε _{res, 3}represent respectively film 1,2,3 overstrain.

Advantage of the present invention is: each layer of Young modulus in trilamellar membrane structure be there is no to strict demand than with Thickness Ratio, guaranteed higher accuracy simultaneously, can be applicable to be similar to Al/Si in capacitive temperature sensor ₃n ₄the such each layer of Young modulus of/Si three-decker differs large and the approaching situation of thickness.

Accompanying drawing explanation

Fig. 1 is multi-layer film structure of the present invention and residual stress distribution sketch

Fig. 2 is each tunic force analysis of the present invention

Fig. 3 is three layers of MEMS cantilever beam structure schematic diagram of the present invention

Fig. 4 is Al/Si of the present invention ₃n ₄tri-layers of MEMS cantilever beam structure schematic diagram of/Si

Fig. 5 is Al/Si of the present invention ₃n ₄the ANSYS emulation (Y-axis displacement) of tri-layers of cantilever beam structure distortion of/Si

Embodiment:

With reference to accompanying drawing, further illustrate the present invention:

Solve Al/Si ₃n ₄the method of the elastic deformation of tri-layers of MEMS cantilever beam structure of/Si, comprises the following steps:

(1) Al/Si that utilizes MEMS technique to make ₃n ₄tri-layers of cantilever beam structure of/Si, its each layer of Young modulus and thickness are known, establish Young modulus and compare γ ₁=E ₁/ E ₂, γ ₂=E ₃/ E ₂, Thickness Ratio r ₁=h ₁/ h ₂, r ₂=h ₃/ h ₂, wherein E ₁, E ₂, E ₃, h ₁, h ₂, h ₃represent successively respectively Young modulus and the thickness of silicon substrate, silicon nitride film, aluminium film.

(2) under certain process conditions, through processing and manufacturing and last handling process, the unrelieved stress that different materials film produces can obtain by inquiry document, and the unrelieved stress of establishing silicon substrate, silicon nitride film, aluminium film is followed successively by σ _{res, 1}, σ _{res, 2}, σ _{res, 3}.

(3) above-mentioned data substitution Stoney is extended to formula

wherein

$\begin{array}{c}{I}_1=2[-{\mathrm{\&gamma;}}_1{r}_1({\mathrm{<figure-callout\; id="1"\; label="r"\; filenames="HDA0000406976090000012.png,HDA0000406976090000015.png"\; state="\{\{state\}\}">r</figure-callout>}}_1+{\mathrm{\&gamma;}}_2{r}_1{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000406976090000012.png"\; state="\{\{state\}\}">r</figure-callout>}}_2+1+2{\mathrm{\&gamma;}}_2{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000406976090000012.png"\; state="\{\{state\}\}">r</figure-callout>}}_2+{\mathrm{\&gamma;}}_2{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000406976090000012.png"\; state="\{\{state\}\}">r</figure-callout>}}_2^2){\mathrm{\&epsiv;}}_{\mathrm{res},1}\\ +({\mathrm{\&gamma;}}_1{\mathrm{<figure-callout\; id="1"\; label="r"\; filenames="HDA0000406976090000012.png,HDA0000406976090000015.png"\; state="\{\{state\}\}">r</figure-callout>}}_1^2+{\mathrm{\&gamma;}}_1{r}_1-{\mathrm{\&gamma;}}_2{r}_2-{\mathrm{\&gamma;}}_2{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000406976090000012.png"\; state="\{\{state\}\}">r</figure-callout>}}_2^2){\mathrm{\&epsiv;}}_{\mathrm{res},2}+{\mathrm{\&gamma;}}_2{r}_2({\mathrm{\&gamma;}}_1{\mathrm{<figure-callout\; id="1"\; label="r"\; filenames="HDA0000406976090000012.png,HDA0000406976090000015.png"\; state="\{\{state\}\}">r</figure-callout>}}_1^2+2{\mathrm{\&gamma;}}_1{\mathrm{<figure-callout\; id="1"\; label="r"\; filenames="HDA0000406976090000012.png,HDA0000406976090000015.png"\; state="\{\{state\}\}">r</figure-callout>}}_1+1+{\mathrm{\&gamma;}}_1{\mathrm{<figure-callout\; id="1"\; label="r"\; filenames="HDA0000406976090000012.png,HDA0000406976090000015.png"\; state="\{\{state\}\}">r</figure-callout>}}_1{\mathrm{\&gamma;}}_2+r_2){\mathrm{\&epsiv;}}_{\mathrm{res},3}]\end{array}$

$\begin{array}{c}{I}_2={\mathrm{\&gamma;}}_1^2{\mathrm{<figure-callout\; id="1"\; label="r"\; filenames="HDA0000406976090000012.png,HDA0000406976090000015.png"\; state="\{\{state\}\}">r</figure-callout>}}_1^4+4{\mathrm{\&gamma;}}_1{\mathrm{<figure-callout\; id="1"\; label="r"\; filenames="HDA0000406976090000012.png,HDA0000406976090000015.png"\; state="\{\{state\}\}">r</figure-callout>}}_1^3+4{\mathrm{\&gamma;}}_1{\mathrm{\&gamma;}}_2{\mathrm{<figure-callout\; id="1"\; label="r"\; filenames="HDA0000406976090000012.png,HDA0000406976090000015.png"\; state="\{\{state\}\}">r</figure-callout>}}_1^3{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000406976090000012.png"\; state="\{\{state\}\}">r</figure-callout>}}_2+4{\mathrm{\&gamma;}}_1{\mathrm{<figure-callout\; id="1"\; label="r"\; filenames="HDA0000406976090000012.png,HDA0000406976090000015.png"\; state="\{\{state\}\}">r</figure-callout>}}_1+1+4{\mathrm{\&gamma;}}_2{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000406976090000012.png"\; state="\{\{state\}\}">r</figure-callout>}}_2+4{\mathrm{\&gamma;}}_1{\mathrm{\&gamma;}}_2{r}_1{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000406976090000012.png"\; state="\{\{state\}\}">r</figure-callout>}}_2^3\\ +4{\mathrm{\&gamma;}}_2{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000406976090000012.png"\; state="\{\{state\}\}">r</figure-callout>}}_2^3+{\mathrm{\&gamma;}}_2^2{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000406976090000012.png"\; state="\{\{state\}\}">r</figure-callout>}}_2^4+6{\mathrm{\&gamma;}}_1{\mathrm{<figure-callout\; id="1"\; label="r"\; filenames="HDA0000406976090000012.png,HDA0000406976090000015.png"\; state="\{\{state\}\}">r</figure-callout>}}_1^2+12{\mathrm{\&gamma;}}_1{\mathrm{\&gamma;}}_2{\mathrm{<figure-callout\; id="1"\; label="r"\; filenames="HDA0000406976090000012.png,HDA0000406976090000015.png"\; state="\{\{state\}\}">r</figure-callout>}}_1^2{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000406976090000012.png"\; state="\{\{state\}\}">r</figure-callout>}}_2+6{\mathrm{\&gamma;}}_1{\mathrm{\&gamma;}}_2{\mathrm{<figure-callout\; id="1"\; label="r"\; filenames="HDA0000406976090000012.png,HDA0000406976090000015.png"\; state="\{\{state\}\}">r</figure-callout>}}_1^2{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000406976090000012.png"\; state="\{\{state\}\}">r</figure-callout>}}_2^2\\ +12{\mathrm{\&gamma;}}_1{\mathrm{\&gamma;}}_2{r}_1{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000406976090000012.png"\; state="\{\{state\}\}">r</figure-callout>}}_2+12{\mathrm{\&gamma;}}_1{\mathrm{\&gamma;}}_2{r}_1{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000406976090000012.png"\; state="\{\{state\}\}">r</figure-callout>}}_2^2+6{\mathrm{\&gamma;}}_2{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000406976090000012.png"\; state="\{\{state\}\}">r</figure-callout>}}_2^2\end{array}$

Here ε _{res, i}represent the overstrain of i layer structure, ε _{res, i}=σ _{res, i}/ E _i, i=1,2,3.

κ _benbe exactly Al/Si ₃n ₄the bending curvature of tri-layers of MEMS cantilever beam structure of/Si.

Accompanying drawing 4 represents an Al/Si ₃n ₄tri-layers of MEMS cantilever beam structure of/Si, the thickness of substrate silicon, middle layer silicon nitride, upper strata aluminium film is followed successively by h ₁=1 μ m, h ₂=0.5 μ m, h ₃=1 μ m.The Young modulus of silicon, silicon nitride, aluminium is followed successively by E ₁=179GPa, E ₂=290GPa, E ₃=70GPa.Under certain process conditions, can know the unrelieved stress of layers of material.If under a certain process conditions, each layer of unrelieved stress is respectively σ _{res, 1}=-50MPa, σ _{res, 2}=80MPa, σ _{res, 3}=60MPa.By above-mentioned data substitution formula (15) and (14c), can calculate this Al/Si ₃n ₄the bending curvature κ of tri-layers of MEMS cantilever beam structure of/Si _ben=6.18 × 10 ^-4(μ m) ^-1.Fig. 5 is this Al/Si ₃n ₄the diastrophic simulation result of tri-layers of cantilever beam structure of/Si under each layer of unrelieved stress effect, length of cantilever 100 μ m, maximum defluxion simulation result is 3.16 μ m.Consider the boundary condition ω at x=0 place | _x=0=0 and d ω/dx| _x=0=0, semi-girder displacement along its length can be expressed as ω (x)=κ _benx ²/ 2, x represents along the coordinate of semi-girder length direction, the simulation result κ of this semi-girder bending curvature _{ben, sim}=6.32 × 10 ^-4(μ m) ^-1.Visible, the analytic solution that calculate with above-mentioned characterizing method and simulation result quite coincide.

Because by changing process for making, can control roughly Al/Si in theory ₃n ₄unrelieved stress in each layer of/Si trilamellar membrane structure, so under the guidance of this characterizing method, can be by make the distortion of this structure reach minimum to the control of process for making.Above-mentioned characterizing method is to the Al/Si in capacitive temperature sensor ₃n ₄the process lowest optimization of tri-layers of MEMS cantilever beam structure of/Si provides certain theory support.

Claims (1)

1. one kind solves Al/Si ₃n ₄the method of the elastic deformation of tri-layers of MEMS cantilever beam structure of/Si, comprises the following steps:

(1) Al/Si that utilizes MEMS technique to make ₃n ₄tri-layers of cantilever beam structure of/Si, its each layer of Young modulus and thickness are known, establish Young modulus and compare γ ₁=E ₁/ E ₂, γ ₂=E ₃/ E ₂, Thickness Ratio r ₁=h ₁/ h ₂, r ₂=h ₃/ h ₂, wherein E ₁, E ₂, E ₃, h ₁, h ₂, h ₃represent successively respectively Young modulus and the thickness of silicon substrate, silicon nitride film, aluminium film;

(2) under certain process conditions, through processing and manufacturing and last handling process, the unrelieved stress that different materials film produces can obtain by inquiry pertinent literature, and the unrelieved stress of establishing silicon substrate, silicon nitride film, aluminium film is followed successively by σ _{res, 1}, σ _{res, 2}, σ _{res, 3};

(3) above-mentioned data substitution Stoney is extended to formula

wherein

$\begin{array}{c}{I}_1=2[-{\mathrm{\&gamma;}}_1{r}_1(r_1+{\mathrm{\&gamma;}}_2{r}_1{r}_2+1+2{\mathrm{\&gamma;}}_2{r}_2+{\mathrm{\&gamma;}}_2{r}_2^2){\mathrm{\&epsiv;}}_{\mathrm{res},1}\\ +({\mathrm{\&gamma;}}_1{r}_1^2+{\mathrm{\&gamma;}}_1{r}_1-{\mathrm{\&gamma;}}_2{r}_2-{\mathrm{\&gamma;}}_2{r}_2^2){\mathrm{\&epsiv;}}_{\mathrm{res},2}+{\mathrm{\&gamma;}}_2{r}_2({\mathrm{\&gamma;}}_1{r}_1^2+2{\mathrm{\&gamma;}}_1{r}_1+1+{\mathrm{\&gamma;}}_1{r}_1{\mathrm{\&gamma;}}_2+r_2){\mathrm{\&epsiv;}}_{\mathrm{res},3]}\end{array}$

$\begin{array}{c}{I}_2={\mathrm{\&gamma;}}_1^2{r}_1^4+4{\mathrm{\&gamma;}}_1{r}_1^3+4{\mathrm{\&gamma;}}_1{\mathrm{\&gamma;}}_2{r}_1^3{r}_2+4{\mathrm{\&gamma;}}_1{r}_1+1+4{\mathrm{\&gamma;}}_2{r}_2+4{\mathrm{\&gamma;}}_1{\mathrm{\&gamma;}}_2{r}_1{r}_2^3\\ +4{\mathrm{\&gamma;}}_2{r}_2^3+{\mathrm{\&gamma;}}_2^2{r}_2^4+6{\mathrm{\&gamma;}}_1{r}_1^2+12{\mathrm{\&gamma;}}_1{\mathrm{\&gamma;}}_2{r}_1^2{r}_2+6{\mathrm{\&gamma;}}_1{\mathrm{\&gamma;}}_2{r}_1^2{r}_2^2\\ +12{\mathrm{\&gamma;}}_1{\mathrm{\&gamma;}}_2{r}_1{r}_2+12{\mathrm{\&gamma;}}_1{\mathrm{\&gamma;}}_2{r}_1{r}_2^2+6{\mathrm{\&gamma;}}_2{r}_2^2\end{array}$

Here ε _{res, i}represent the overstrain of i layer structure, ε _{res, i}=σ _{res, i}/ E _i, i=1,2,3.

κ _benbe exactly Al/Si ₃n ₄the bending curvature of tri-layers of MEMS cantilever beam structure of/Si.

Images