Microsystem Technologies 9 (2003) 420426 Springer-Verlag 2003 DOI 10.1007/s00542-002-0250-2
Electromechanical model of RF MEMS switches

L. X. Zhang, Y.-P. Zhao

420 Abstract With the recent rapid growth of Radio Frequency Micro-Electro-Mechanical Systems (RF MEMS) switches, there has developed an emergent requirement for more accurate theoretical models to predict their electromechanical behaviors. Many parameters exist in the analysis of the behavior of the switch, and it is inconvenient for further study. In this paper, an improved model is introduced, considering simultaneously axial stress, residual stress, and fringing-eld effect of the xed-xed bridge structure. To avoid any unnecessary repetitive model tests and numerical simulation for RF MEMS switches, some dimensionless numbers are derived by making governing equation dimensionless. The electromechanical behavior of the xed-xed bridge structure of RF MEMS switches is totally determined by these dimensionless numbers. mainly three types of beams: cantilever, xed-xed beam, and torsion beam [7, 8], and xed-xed beam are under extensive research at present. The present paper goes on study of RF MEMS switches based on simple xed-xed beam. Theoretical model can offer proper and convenient approach for numerical calculations, and promote the design of devices. At present, some theories and models of RF MEMS switches have been established gradually based on conventional theories. Because of miniaturization and electromechanical coupling, some new phenomena and factors, such as axial stress due to beam large deection, residual stress in thin lms and so on, become comparatively more important. To design and optimize RF MEMS switches further, it is essential to set up more accurate models. The dimensionless numbers are useful for scaling purposes and for organizing experimental model tests and numerical calculations to avoid any unnecessary repetition of the results in dimensionless space [9, 10]. In the analysis of improved models of RF switches, taking more physical quantities into account, the dimensionless numbers can play an important role in making the analysis more convenient. In the present paper, a brief review of the existing models is presented in Sect. 3, then an improved model is put forward in Sect. 4 by considering axial stress caused by beam large deection, residual stress as well as fringingeld effect, and four dimensionless numbers are obtained by making governing equation dimensionless. Some existing models of RF switches and data of pull-in voltage are reformulated using the dimensionless numbers and analyzed further. The following begins with brief introduction on structure and working principle of RF MEMS switches.
1 Introduction In recent years, Micro-Electro-Mechanical Systems (MEMS) technology has grown rapidly and entered into many communication and defense applications [13]. At present, as the development in MEMS technology [4], Radio Frequency (RF) MEMS is one of the fastest growing areas in commercial MEMS technology. As a novel switch, RF MEMS switches have a myriad application in radar system and wireless communications [5, 6]. Comparing to semiconductor switches widely used in millimeter wave integrated circuits and microwave circuits, the novel device has a low insertion loss, good isolation, low return loss, high frequency, good Q-factor, and a low cost and power consumption. RF MEMS switches consist of a thin metal membrane, entitled the bridge, suspended over a center conductor, and one beam xed on the ground conductor. There are
Received: 4 February 2002/Accepted: 19 May 2002
2 Structure and working principle of RF MEMS switches L. X. Zhang, Y.-P. Zhao (&) The schematic view of RF MEMS switches as analyzed in State Key Laboratory of Nonlinear Mechanics, this paper is shown in Fig. 1. The device consists of Institute of Mechanics, Chinese Academy of Sciences, transmission line, hinge, movable plate formed by using Beijing 100080, PR China electroplated Au or Cu, and substrate. One SEM photo of E-mail: yzhao@lnm.imech.ac.cn (Zhao YP) the switch is shown in Fig. 2. The supports from the Key Project from the Chinese Academy of The switch can be fabricated using surface micromaSciences (No. KJCX2-SW-L2), projects from the NSFC (Nos chining techniques, electroplating techniques and dry 19928205, No. 50131160739 and No. 10072068), and the National releasing technique compatible to the (millimeter wave 973 Project (Grant No. G1999033103) are gratefully acknowledged. The authors would like to thanks Prof. Yilong Hao integrated circuit) MMIC fabrication processes. A brief fabrication sequence for the RF MEMS switches described of Institute of Microelectronics, Peking University, for helpful in Fig. 1 is shown in Fig. 3 [11]. A ground plate and a discussion.

Fig. 1. Schematic of one kind of RF MEMS switches
Fig. 4. 1D lumped model for pull-in test
Fig. 2. SEM of a xed-xed RF MEMS switch
parameters can be simplied to great extent by the following analysis in this paper. The essential of the problem of the RF MEMS switches is the coupling of the mechanics and the electrostatics, which governs the behavior of the structure of RF MEMS switches [12]. The device controls mechanically electrical current or signal, by using the on/off impedance ratio. While a voltage is applied between the movable and the xed ground plates of the switch as shown in Fig. 1, the movable plate subjected to the electrostatic force moves down onto the xed ground plate. When the pull-in voltage is reached, the switch is in the down state, or the blocking state, when no voltage is applied it is in the up state, or pass-though state [13].
Fig. 3. Fabrication sequences of RF MEMS switches shown in Fig. 1
3 Existing models Electrostatic pull-in is a famous instable behavior of an elastic supported structure with regard to parallel-plate electrostatic actuation [14]. Despite the pull-in event is sudden and sharp, the actuation voltage can be accurately measured at wafer level by the standard electrical test equipment with a microscope. Existing models have no sufcient accuracy, so that there has developed an emergent requirement for more accurate model to approach the design of RF MEMS switches. 3.1 1D lumped model [1416] The 1D lumped model approximates pull-in structure by a single rigid parallel-plate capacitor suspended above a xed ground plane by an ideal linear spring as shown in Fig. 4. The model is the simplest and most intuitive analytically, but its accuracy is very poor. Its purpose is for rst-cut analysis to gain physical insight, explore design options and understand overall behavior. The pull-in voltage Vpi can be expressed by the following equation s 8Keff G; 1 Vpi 27e0 A
where e0 is the vacuum permittivity, Keff the effective spring constant, G0 and A the initial gap and the overlapping area between the movable and the xed plates, respectively.
transmission line were formed on substrate (shown in Fig. 3(a)). High dielectric layer was deposited on top of the formed transmission line (shown in Fig. 3(b)). A patterned seed layer was formed for electroplating the metal posts, and a photoresist was deposited on the ground plate and the dielectric layer (shown in Fig. 3(c)). Metal posts formed on seed layer (shown in Fig. 3(d)). A hinge and a movable plate were formed, and a mass was deposited on the top of the formed plate (shown in Fig. 3(e)). The movable plate and hinge were released by etching the sacricial layer (photoresist) (shown in Fig. 3(f)). In theoretical analysis, the parameters of RF MEMS switches involve materials constants (such as elastic modulus, Poissons ratio and so on), geometrical dimensions of the xed-xed beam moment of inertia of the cross-section, the gap between the movable and the xed ground plates, and electrostatics. The analysis of these

3.2 2D distributed model [1519] Compared with the 1D lumped model, the 2D distributed model can contain the inuences of the fringing eld, residual stress, axial stress, and so on. Thus it can attain more accurate solutions. The 2D distributed model generally includes the cantilever (shown in Fig. 5) and xedxed beam (shown in Fig. 6) structures. The governing equation of the beam as shown in Fig. 5 or Fig. 6 is
4 _ e0 V 2 B ~ ow ; 2 EI 4 ox 2G0 w2 _ ~ where E, I , w, B are the effective modulus, the effective moment of inertia of the cross-section, the deection and the width of the beam, respectively. V is the voltage applied between the movable and the ground plates on the xed substrate. Following assumptions have been supposed to simplify the analysis:
residual stress, fringing eld effect and axial stress into account simultaneously. Some research about residual stress and fringing eld effect is referred in Sect. 3.2 in modeling. In this section, in addition to further interpretation of residual stress and the fringing effect, another important factor axial stress, is analyzed. Then an improved model is established, involving these important factors. The detailed schematic of xed-xed bridge structure is shown in Fig. 7, where w is the deection of the xed-xed beam, B, t and L and are the width, the thickness, and the initial length of the xed-xed beam, respectively.
4.1 Residual stress and fringing field effect
(i) Residual stress Residual stress, due to the mismatch of both thermal expansion coefcient and crystal lattice period (i) residual stress in the xed-xed beam is ignored. (ii) error derived from nonhomogeneous distribution of between substrate and thin lm, is unavoidable in charge, after the movable xed-xed beam deects, is surface micromachining techniques, so that accurate and reliable data of residual stress is crucial to the proper ignored. design of the MEMS devices concerned with the (iii) fringing eld effect is ignored. techniques [20, 21]. Therefore the residual stress is an (iv) small deection is assumed. attractive research topic in the development of the In theoretical modeling, some models considering residual Microsystem Technology (MST). Considering the fabricastress are established [15], which are represented as the tion sequence of RF MEMS switches in Fig. 3, the residual following governing equation stress is very important and inevitable to the device. Residual force can be expressed as 2 _o w ow eV B

2G0 w2

^ Tr rBt ;
where Tr is the residual force of xed-xed beam. ^ where r is the residual stress, equal zero for cantilever, and Some models involving residual stress and fringing eld r r 1 m for xed-xed beam, where r is the biaxial ^ are analyzed [15, 16], which derived the governing equa- residual stress [15]. tion as follows
4 _ o2 w e0 V 2 B ~ ow 1 ff ; E I 4 Tr 2 ox ox 2G0 w2
(ii) Fringing eld effect A uniform magnetic eld cannot drop abruptly to zero at an edge as shown in Fig. 8(a). In actual situation, there is where ff is the fringing eld correction. The residual stress always a fringing eld existing, and a more realistic and fringing eld effect are interpreted further in detail in situation including fringing eld modication is illusSect. 4.1. trated in Fig. 8(b). If fringing eld effect is taken into account, the rst 4 order fringing-eld correction [15, 16] is denoted as

Model improvement G0 w Because of the miniaturization and special fabrication : ff 0:65 B technology, it is necessary to take some the effects such as
Fig. 5. Schematic of cantilever
Fig. 6. Schematic of xed-xed beam
Fig. 7. Detailed schematic of xed-xed bridge structure. a Side view; b Top view
Fig. 8. Electrostatic eld of xed-xed bridge structure applied by voltage. a Simplied model; b Schematic including fringing eld modication
4.2 Another important factor: axial stress The bending of a xed-xed beam involves generally a stretching. When the maximum deection is less than the thickness, small deection can be considered valid, and the stretching can be neglected. But for RF MEMS switches, the gap G0 is usually larger than the beam thickness t, so that the maximum deection at middle point is larger than t. Considering the invalidity of small deection, it is required to take axial stress of the xed-xed beam into account. Axial stress is analyzed in Fig. 9, where Ta is the axial force derived from the elongation of the xed-xed beam, and q is the electrostatic force per unit length. When a voltage is applied, the actual length of the beam is L2 x ZL "
" Z # ~ ~ o2 w Et 3 o4 w Et L dw 2 ^ dx rt 12 ox4 ox2 2L 0 dx e0 V 2 G0 w : 0:65 B 2G0 w

2 #1=2 dw 1 dx : dx

Considering L ) w, hence dw=dx2 ( 1, as a result, the elongation is approximately given by
5 Dimensionless numbers for the electromechanical model The behavior of a xed-xed beam involves more physical quantities in present model compared with the classical ones. In this section, it will be shown that the dimensionless numbers can play an important role in the present problem. By making Eq. (11) dimensionless, dimensionless numbers with evident physical signications can be obtained. Introducing the dimensionless transformation as follows x w X ; W ; 12 L G0 on w=oxn can be shown as on w G0 on W n 1; 2; 3 : 13 oxn Ln oX n Substituting Eqs. (12), (13) into Eq. (11), the governing equation can be expressed as the dimensionless form 1 ~ ~ 0 Z dW 2 ^ rtGo2 W Et 3 G0 o4 W 4EtG3 dX 2 12L4 oX 4 2L4 L dX oX " # e0 V G: 14 0:65 2GW2 B 1 W 0

1 DL % 2

ZL 2 dw dx : dx
Therefore, axial force is calculated as
~ Z ~BtDL=L EBt Ta rBt % E 2L

2 dw dx : dx

Multiplying both sides of Eq. (14) by 2G2 =e0 V 2 , it 0 4.3 becomes Governing equation 1 ~ 0 ~ 0 Z dW 2 Considering factors as mentioned in Sect. 4.1 and 4.2, the 1 Et 3 G3 o4 W ^ 0 rtGo2 W EtGdX 2 governing equation can be written as e 0 V 2 L4 e0 V 2 L2 oX e0 V 2 L4 oX 4 dX " # _ o2 w e0 V 2 B ~ ow 1 ff : 10 E I 4 Ta Tr Gox ox : 15 0:65 2G0 w2 B 1 W 1 W2 _ Substituting Ta , Tr , ff and I Bt3 =12 into Eq. (10), the Several dimensionless numbers are derived as follows governing equation can be expressed as ~ ~ Et 3 G03 EtG05 P1 ; P2 ; 2 L4 e0 V e 0 V 2 L^ rtG03 G0 Ff ; P3 ; P4 e 0 V 2 L2 B ~ where P1 is the ratio between the bending force EBt 3 G0 =L2 and the electrostatic force Be0 V =G0 per unit length, P2 the ratio between the axial force derived from the elongation of ~ 0 the beam EBtG3 =L4 and the electrostatic force Be0 V 2 =G2 per Fig. 9. Analytical schematic of the element of xed-xed beam 0

^ unit length, P3 the ratio between the residual force rBtG0 =L2 and the electrostatic force Be0 V 2 =G2 per unit length, and F f 0 the rst order fringing-eld correction number, equal to the ratio of G0 to B. Therefore, Eq. (15) can be rewritten as
6.Pull-in voltage Vpi Since the behavior of the switch is determined by at most four dimensionless numbers. As one of important indices The equation shows that the governing equation of a xed- of RF MEMS switches, the pull-in voltage V can be reppi xed bridge structure to an applied voltage is solely resented by these dimensionless numbers. dependent upon dimensionless number group P1 ; P2 ; (i) For the simplest model (1D), considering P3 ; Fr. ~ Keff 32EBt 3 =L3 and A LB, the pull-in voltage Vpi in Considering the ratio of the bending force to the axial Eq. (1) can be written as force per unit length of the beam as follows s 2 ~ EG3 t 3 P1 t : 22 ; 18 Vpi 27 e0 L4 P2 G0
we know that when t is more than G0 , the axial force derived from elongation is negligible compared with the bending force of the beam and can be neglected. Using dimensionless numbers in (16), the equation can be transformed into the following dimensionless form
ZoW 4 dW o2 W P2 dX 2P2 PoX 4 dX oX 0 " # 1 0:65 : Ff 1 W 1 W 2
If axial stress effect due to large deection of the xedxed bridge structure is also considered, the dimensionless form of the governing equation is turn into the form as shown in Eq. (17). By comparing Eqs. (17), (19), (20) and (21), we know that the inuences of axial stress, residual stress and fringing-eld effect are represented by P2 , P3 and Fr , respectively.
Vpi 16 p p P1 : 23 V 6 Discussions It is easy to see that the dimensionless pull-in voltage is In order to show the validity of these obtained dimenonly determined by P1 , the ratio of the bending force to sionless numbers in the behavior of electromechanical RF the electrostatic force per unit length. MEMES switch, some other models and some existing (ii) For a xed-xed bridge structure, when residual solutions of pull-in voltage will be related to these ones in stress and fringing-eld effect are taken into account, pullthis section. in voltage Vpi obtained in [16] is s 6.1 ~ 32Et 3 G3 8^tG3 r 0 Some models of RF MEMS switches ; 24 Vpi e0 L4 e0 L1 0:42 G0 For 2D model of a xed-xed bridge structure, the B governing equation as shown in Eq. (2) is recast into the and the equation can be changed into the dimensionless dimensionless form as follows form P1 o4 W 1 s : Vpi 8 4P1 PoX W : 25 V 0:42Fr Because of the ignorance of axial stress, residual stress and fringing eld effect, the equation is determined by the only It is evident that the dimensionless pull-in voltage depends dimensionless number P1 , the ratio between the bending on fringing-eld effect number, the ratio of the bending force and electrostatic force per unit length. force to the electrostatic force per unit length, and the ratio If residual stress is taken into account, the governing of the axial force to the electrostatic force per unit length. equation as shown in Eq. (3) can be rewritten as the (iii) For a xed-xed bridge structure, if ignoring axial dimensionless form stress and fringing-eld effect, an analytic solution for Vpi can be obtained as follows, using the Rayleigh-Ritz method P1 o4 W o2 W 1 2P3 : 20 [14, 15] s 6 oX 4 oX W2 p ~ 0 p3=5Et 3 G3 3^L2 r If fringing eld is also taken into account, the dimen26 Vpi : ~ L 25e0 p Et sionless form of governing equation as shown in Eq. (4) can be rewritten as By using the dimensionless numbers in (16), the equation " # can be rewritten as dimensionless form P1 o W oW 1 0:65 rs : 2P3 Ff 6 oX 4 oX W Vpi 2pW27 PP3 : V p 21

larger with increasing P1 (depicted in Fig. 10). Obviously, both P2 and P3 increase with increasing P1 due to the changes of parameters t and L. In fact, the increment of the difference between the theoretical models and the experimental results is a result of ignorance of axial stress and residual stress. Therefore, to obtain more accurate models, it is necessary to take some these factors into account.
7 Concluding remarks Axial stress, residual stress, and fringing-eld effect are important in the behaviors of RF switches and can cause Fig. 10. Use of P1 to compare the experimental results with degradation or even failure of the devices [22]. An imsimple theoretical models of xed-xed bridge structures m proved xed-xed bridge structure model, involving these experimental results due to variant length and thickness of the factors, is established and discussed in this paper. In order beam; , - - - -,. theoretical analyses due to different to facilitate the analysis of the factors, four dimensionless models numbers P1 , P2 , P3 and Ff are presented. P1 is the ratio between the bending force and the electrostatic force per It is evident that the dimensionless pull-in voltage depends unit length, P2 the ratio between the axial force and the upon the ratios of the bending force and residual force to electrostatic force per unit length, P3 the ratio between the electrostatic force per unit length. residual force and the electrostatic force per unit length, (iv) Solution of pull-in voltage involving the fringing and Ff the fringing-eld correction number. Such a simeld effect is obtained [15] as follows plication not only decreases the enormous and unnecs essary repeated work, but also gives a convenient ~ 0 11:86Et 3 G3 approach of design. ; 28 Vpi GThe inuences of axial stress, residual stress and 1 0:42 B e0 L fringing-eld effect on some obtained models and soluwhich can be rewritten into the following dimensionless tions of pull-in voltage are reformulated and analyzed form using dimensionless numbers. r It should be pointed out that dimensional analysis can Vpi 11:86 P1 : 29 give only general relationships among these dimensionless V 1 0:42Fr numbers. The relative importance of each dimensionless Ignoring the factors of residual stress, fringing eld effect numbers has to be studied further by analytical, experimental and numerical studies. and axial stress, Eq. (27) can be rewritten as r Vpi 2p3p P1 ; 30 Appendix V Table 1. Parameters in the present paper and Eq. (25) can be recast as the following form Vpi 16 p Physical meaning Dimension 31 Symbol p P1 ; V B Width of beam L ~ E Effective modulus of beam ML1 T 2 which is same to Eq. (23). ff The rst order fringing-eld Dimensionless In order to show its validity to predict the pull-in correction voltage of switches, Fig. 10 illustrates the use of P1 for Gap between movable and L G0 comparison of the experimental results [15] with simple ground plates _ theoretical models of xed-xed bridge structures igMoment of the inertia of L4 I noring the residual stress, the axial stress and the cross-section fringing effect. The horizontal axis represents P1 , the Keff Effective spring constant of beam MT 2 Length of beam L ratio of the bending force to the electrostatic force per L Electrostatic force per unit length MT 2 unit length, and the vertical axis represents the dimen- q on beam 0 sionless voltage, i.e., Vpi is Vpi normalized by unit t Thickness of beam L voltage. The solid curve is the theoretical analysis of Ta Axial force of beam MLT 2 model based on Eq. (30), similarly the dashed curves are Tr Residual force of beam MLT 2 the theoretical analysis of models based on Eq. (29) and w Deection of beam L Eq. (31), respectively. V Voltage applied ML2 T 2 Q1 Vpi Pull-in voltage ML2 T 2 Q1 When P1 is less than about 30, these theoretical x Longitudinal distance along beam L models agree relatively well with the experimental results. e0 Vacuum permittivity M 1 L3 T 2 Q2 However, the difference between prediction of these ^ r Residual stress of beam ML1 T 2 theoretical models and the experimental results will be

Table 2. Dimensionless numbers obtained in the present paper

Number 3 4

Dimensionless number P1
~ Et 3 Ge0 V 2 L4 ~ EtG5 ^ rtG3
Comments Ratio between the bending force and the electrostatic force per unit length. Ratio between the axial force and the electrostatic force per unit length. Ratio between the residual force and the electrostatic force per unit length. Fringing eld correction number.
0 P2 e0 V 2 LP3 e0 V 2 L2

F f G0 B

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