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Study on the design of anti-sticking structures in MEMS under different forces

Study on the design of anti-sticking structures in MEMS under different forces
Study on the design of anti-sticking structures in MEMS under different forces

V ol. 44 supplement SCIENCE IN CHINA (Series A)June 2001 Study on the design of anti-sticking structures in MEMS under different forces

DING Jianning (丁建宁)*, MENG Yonggang (孟永钢), WEN Shizhu (温诗铸)

(National Tribology Laboratory, Tsinghua University, Beijing, 100084, China, *Corresponding author, Email: dingjn@https://www.wendangku.net/doc/be13987011.html,)

Abstract Mechanical stability and sticking are the troublesome problems in microfabrication and operation processes when separations of components in MEMS are in the sub-micrometer regime. Some mechanical effect, including quantum mechanical effect should be taken into account for solving the problems. The influence of capillary forces on sticking of a surface micromachined microcantilever in ambient environment or the rinse liquid and the influence of quantum mechanical effect such as the Casimir effect on sticking and stability of a micro membrane strip cavity structure in vacuum were investigated. A factual rough model theory about sticking problem under the Casimir effect was suggested for the first time. The study on the design of anti-sticking structures under different forces shows that sticking and stability of microcantilevers and micro membrane strip cavities has something to do with Young’s modulus of materials, surface properties, length of structures, thickness of structures and separation between the fixed surface and the deflecting component. But, it is independent of width of structures. The map of the size design of anti-sticking structures was put forward for the first time. This also provides a way to design a MEMS structure with high resistance to collapse.

keywords Capillary force, Casimir force, MEMS, mechanical stability, sticking

Both size of and separations between the components in some microelectromechanical sys-tems (MEMS) are already in the sub-micrometer regime. There is growing need for better under-standing of interactions between micromachined surfaces at small separations [1]. At separations of the order of a micrometer there has been reported occurrence of the unwanted “sticking” phe-nomenon, wherein a thin micromachined membrane or cantilever has been observed to unexpect-edly latch onto an adjacent parallel surface and remain attached to that surface [2]. Many factors may affect sticking, and all these factors should be investigated. Surface micromachined structures which have been fabricated using the wet sacrificial layer etching technique can be pulled down to the substrate by capillary forces during drying [3]. It has been suggested that etch residues [4], residual stress gradient [5] and condensation of water [6] between the structures may be responsi-ble for sticking during fabrication. It has also been suggested that the van der Waals force [7], the electrostatic force and the acceleration force [3] may be responsible for sticking of released struc-tures during their operational life-time.Capillary forces will play a dominant role on sticking in ambient environment or the rinse liquid. On the other hand, many people have reported that the microdevices would stick to adjacent parallel surface and remain attached to that surface even in

vacuum. In this situation, there is no effect of capillary forces. The proliferation of smaller and lighter components will require MEMS designs to account for some effect that have been ne-glected until now. Some quantum mechanical effect will need to be taken into account, for exam-ple, the Casimir effect [8]. The Casimir force can and do play a significant role in micro- and nanometer-size structure [9]. With continuing reduction in the size and separation of structures, Casimir forces will need to be accounted for in the design and modeling of MEMS and nanoelec-tromechanical systems (NEMS). In this paper, the influence of capillary forces on sticking of a surface micromachined microcantilever in ambient environment or the rinse liquid and the influ-ence of quantum mechanical effect such as the Casimir effect on sticking and stability of a micro membrane strip cavity structure in vacuum were investigated.

1 The capillary forces on sticking of a surface micromachined micrcantilever

A thin liquid layer between two solid plates can work as an adhesive. When the gap between plates is small, the meniscus area is small compared to the bridged area. The energy of the liq-uid-air is neglected. The total surface energy as a function of the bridged area can be written in a general form:

C la b s A C E θγcos 2?= (1)

where γla cos θC is the adhesion tension, and C takes into account the constant term. The total en-ergy of the system consists of the elastic deformation energy and the surface energy, which is a constant minus the adhesion energy. This energy balance is easily made for a cantilever beam. Considering a cantilever beam of length L , width b , thickness h , separation w 0, and Young’s modulus E , as shown in Fig.1. The beam is

sticking to its substrate with a distance l = (L -s )

from its tip. The stored elastic energy of the

beam in the segment 0 ≤ x ≤ s induces a re-

storing force that tends to peel the beam from

the segment s ≤ x ≤ L induces another force

that holds the beam in contact with the sub-

strate. Since there are no external force acting

on the beam for 0 ≤ x ≤ s , its deflection u (x ) is

the solution of 04

4=dx u d EI , 123bh I = (2) The shear deformation is particularly impor-

tant when s approaches L , as shown in Fig.1.

Since l = (L -s ) is very small, the shear deforma-

tion at the tip will occur before complete de-

tachment and the beam will touch the substrate under an angle. Thus, equation is solved subject to Figure 1. Model used for sticking of a micro-cantilever

the boundary conditions

s

m w dx du s 0==θ, 0)0(=u , 0)(w s u = (3) where θ is the shear angle of the tip, and m is a non-dimensional number. In equilibrium, the total energy is minimal, which means dU /ds =0. Thus, the deformation energy at the point snap back is about four times lower, and for the critical length can be written

4203cos 163C

la w Eh L θγ= (4) From equation (4), we can find that the sticking and stability of microcantilevers under capillary forces has something to do with Young’s modulus of materials, surface properties, length of struc-tures, thickness of structures and separation between the fixed surface and the deflecting compo-nent. For a typical polysilicon material, Young's modulus is 164 GPa [10]. When the water liquid bridge is considered, equation (4) can be calculated with an adhesion energy of about 100 mJ/m 2 based on this sticking model. When the separation between the microcantilever and the surface S is 0.1 μm, 0.2 μm, 0.5 μm, 1 μm, and 2 μm respectively, the size of microcantilever beam struc-tures under capillary forces can be designed according to Fig.2. Thus, the microcantilever can be prevented from collapsing into the substrate surface. As the sticking of microcantilever under cap-illary forces has something to do with the surface energy, the method to reducing surface energy by coated with SAMs (Self-assembled monolayers) can be used. In this paper, OTS (octadecyltri-chlorosilane) films were used to reduce the surface energy of hydrophilic polysilicon films from 100 mJ/m 2 to 23 μJ/m 2. The size of microcantilever beam structures with SAMs films coated un-der capillary forces can be designed according to Fig.3. Fig.3 shows the high aspect ratio (L /h )

structure that is not easily to collapse into the surface can be designed.

Figure 3. The size of microcantilever beam structures with SAMs films coated

Figure 2. The map of the size design of

anti-sticking microcantilever structures

2 The casimir force on sticking and stability of a micromembrane strip cavity structure The Casimir force per unit area between two perfectly parallel plates with infinite conductivity and a distance d apart is given by [11]

4201240)(d

c d U d F h π?=???= , 2402c h π≡? (5) in which ?=h /2π, h being Planck's constant, c the velocity of light, and d the surface separation measured from contact of the two smooth surfaces. The equation is valid for d ≥100 nm. The force constant ? is equal to 1.3×10-27 Nm 2 when the distance d is given in meters. The minus sign indi-cates an attraction. As the surfaces are expected to form a boundary to the electromagnetic waves, there is a correction due to the finite conductivity of the plates, the surface roughness and the fi-nite temperature. When corrected, the Casimir force can be calculated by [12]

)()(0d F d F T r c ηηη= (6)

where ηc , ηr ,and ηT are the correction coefficients of conductivity, surface roughness, and tem-perature respectively.

We consider a micromembrane strip cavity structure under the Casimir force with the plate strip subject to the fixed-fixed boundary conditions as shown in Fig.4 [13]. If bending is insignifi-cant and can be neglected, external load may be assumed to be carried almost entirely by the membrane force N . The differential equation then

reduces to the second order in x :

)()(22x q dx

x w d N ?= (7) Thus, the differential equations and the bound-

ary conditions describing the static equilibrium of

the membrane strip are for L x ≤≤0:

)()(//d F x Nw = (8)

)(0x w w d ?= (9)

0/=N (10)

)](2

1)([2//x w x u Eh N += (11) 0)()0(==L u u , 0)()0(==L w w (12) Eq.(8) is the same as Eq.(7), but with the load q (x ) replaced by the Casimir pressure (load) at a distance x from the left edge of the membrane strip. Substitute Eq.(6) into Eq.(8), multiply by w /(x )dx and integrating indefinitely. In the mean time, we defined the normalized deflection and its value at x =L /2: w (x )/w 0 ≡ δ(x ), δ(L /2) ≡ δL /2, and w (L /2) ≡ w 0δL /2. Then, we solve Eq.(11) for u /(x ), multiplying through by dx

, and then integrating over the entire length of the strip. We next

Figure 4. The cross section of a mi-cromembrane strip cavity structure

solve for δ(x ), performing a separation of the variables, and integrating over half the length of the membrane (i.e., 0≤x ≤L /2, 0≤δ≤δL/2). Eliminating N , we arrive at an expression for the dimen-sionless system characteristic constant, which are defined as K :

??????+?=≡?∫2/02/12/0030704)(11ln ])([{4L x x w w K Ehw L L L δδδθδδθ???????+∑∑==?+?+1111112/12/0101)(})]1([))](1([i i i L i i i x d w i x w i δδθδθ ??????+?×∫2/02/12/00)(11ln ])([{L x x w L L δδδθδδθ∑=?+?+11101))](1([i i i x w i δθ31112/12/01)(})]1([??????∑=??+i i L i x d w i δδθ (13) Performed a numerical integration for the value of δL/2 from zero to unity, which corresponds to the case of the membrane strip in contact with the surface S . The result for the value of w 0 (w 0=0.1 μm) is shown in Fig.5. Figure 5 shows that, for a given value of K <2.55×10-4, there are

two solutions for δL/2. The smaller value of δL/2 (min 2/L δ) defines a stable static equilibrium state.

This state is the state of minimum potential energy for the membrane strip, subject to the Casimir

force. A larger value, max 2/L δ, corresponds to an unstable static equilibrium state. Thus, with noth-

ing other than the Casimir force loading the strip, the strip will collapse if K is larger than the critical value of K C , such as 2.55×10-4. This provides a way to check if a system with given di-

mensions and material properties will have a stable equilibrium position under Casimir forces. The value of K is independent of the width of the membrane strip, same as the results in the analy-sis of sticking in MEMS. For different w 0, there is different critical value of K C [14]. The critical value of K C increases with the increase in the value of w o , which means the stability of the mem-brane strip has something to do with the structure. The smaller w o

will make the membrane strip

Figure 5. A stable and an unstable equilib-

rium state for w 0=0.1 μm. Figure 6. Typical dimensions of anti-sticking micromembrane strips

collapse into the surface S easily. K changes when the strip deflects under the attractive Casimir force, which means that the strip is becoming unstable easily when the value of w0 is small. This also provides a way to design a membrane strip with high aspect ratio (L/h) that is not easily to collapse into the surface S. For a typical polysilicon material, Young's modulus is 164 GPa [10]. When the separation between the membrane strip and the surface S is 0.1 μm, 0.2 μm, 0.5 μm, 1 μm, and 2 μm respectively, the size of membrane strips can be designed according to Fig.6. Thus, the strip can be prevented from collapsing into the substrate surface S.

3 Conclusions

As stated earlier, capillary forces will play a dominant role on sticking in ambient environment or the rinse liquid. On the other hand, the magnitude of Casimir force is significant when the membranes work in vacuum without the effect of capillary forces.The Casimir effect may play an important role if the separations are small enough. The corrections of roughness, conductivity, and temperature to the Casimir force must be considered for accurate account of their effects. The study on the design of anti-sticking structures under different forces shows that sticking and sta-bility of microcantilevers and micro membrane strip cavities has something to do with Young’s modulus of materials, surface properties, length of structures, thickness of structures and separa-tion between the fixed surface and the deflecting component. But, it is independent of width of structures. The map of the size design of anti-sticking structures was put forward for the first time. This provides a way to check if a system with given dimensions and material properties will be in a stable equilibrium. This also provides a way to design a MEMS structure with high resistance to collapse.

Acknowledgements: The authors would like to acknowledge the support of the Doctoral Science Fundation of China (Contract number: 2000000338)

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