Surface roughness effects on electromechanical performance of RF-MEMS capacitive switches

Microelectronics Reliability 104 (2020) 113544

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Surface roughness effects on electromechanical performance of RF-MEMS capacitive switches

T

Hamid Nawaza, Muhammad Umar Masooda,c, Muhammad Mubasher Saleema,c, , Javaid Iqbala,c, ⁎ Muhammad Zubairb, ⁎

a

Department of Mechatronics Engineering, National University of Sciences and Technology, Islamabad, Pakistan Department of Electrical Engineering, Information Technology University of the Punjab, Lahore, Pakistan c National Centre of Robotics and Automation (NCRA), Pakistan b

ARTICLE INFO

ABSTRACT

Keywords: Reliability RF-MEMS switches Surface roughness Up-state capacitance Down-state capacitance Pull-in voltage Pull-in gap Switching time

The effect of surface roughness on the electromechanical performance of radio frequency micro-electromechanical system (RF-MEMS) based capacitive switches is an important reliability concern. This paper presents a finite element method (FEM) based simulation methodology for the estimation of surface roughness effects on the electromechanical characteristics of parallel plate capacitive RF-MEMS switch. The important electromechanical characteristics considered for the analysis of surface roughness effect include up-state capacitance, down-state capacitance, pull-in voltage, pull-in gap and switching time of RF-MEMS switch. A simple roughness model is employed consisting of semi-circles of constant radius along the surface of an RF-MEMS switch. It is shown that there are significant, but predictable shifts in electromechanical characteristics of RF-MEMS switch due to surface roughness. The results illustrate that the normalized value of up-state capacitance and pull-in gap increases, whereas the values of downstate capacitance, pull-in voltage and switching time decreases with an increase in surface roughness. The change in pull in voltage for an initial air gap of 0.4 μm and roughness of 50 nm is around 28.5% which is about quarter the value of voltage required for the actuation of smooth surface RF-MEMS switch. The results obtained through FEM based analysis in this work are in good agreement with the adopted analytical model and experimental results presented in literature.

1. Introduction RF-MEMS technology enables design and fabrication of miniature, high performance components and networks like inductors [1], varactors [2], ohmic/capacitive switches [3], resonators [4], filters [5], and antennas [6], that are being integrated with various commercial and military wireless communication, satellite and radar systems. In the last two decades, RF-MEMS switches has been studied as an alternative to traditional semiconductor switches like Positive Intrinsic Negative (PIN) diodes and Field Effect Transistor (FET) due to their small size, high quality factor, high isolation, low insertion loss and low cost [3]. These switches, with their bandwidth ranging from radio to millimeter wave frequency and operating in either shunt or series configurations, are actuated by different mechanisms including electrostatic [7], electrothermal [8], piezoelectric [9], and magneto static [10]. The RFMEMS capacitive contact switches, with electrostatic actuation, are most commonly used due to their compatibility with fabrication processes, simple adjustment with transmission lines and negligible power

⁎

consumption [3]. However, long term reliability still remains a matter of concern in the commercial utilization of RF-MEMS switches [11]. The reliability of RF-MEMS switches is hindered by several mechanical and electrical phenomena like fatigue [12], creep [13], residual stresses [14], stiction and dielectric charging [15]. Among all these reliability issues, the presence of surface roughness is a major problem which leads to dielectric charging and contact degradation of RF-MEMS switches [16]. The surface roughness in RF-MEMS switches is mainly caused by materials, chemical etching and variations in fabrication processes [17]. It can affect the performance of RF-MEMS switches leading to stiction [15], heating [18] and adhesion [19]. The rough morphology of interfaces and contacts has an adverse effect on electric field of parallel plate capacitors [17], capacitance and leakage currents of dielectric materials [20], isolation [21], frequency response [22] and contact behavior of micro switches [16], dielectric charging and breakdown of triple layer films [23], leakage currents in metal oxide semiconductor devices [24], and conductivity of metallic films [25]. An extensive research has been carried out in literature to study and

Corresponding authors. E-mail addresses: [email protected] (M.M. Saleem), [email protected] (M. Zubair).

https://doi.org/10.1016/j.microrel.2019.113544 Received 3 July 2019; Received in revised form 6 October 2019; Accepted 7 November 2019 0026-2714/ © 2019 Published by Elsevier Ltd.

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understand the influence of roughness imperfections on the electromechanical and electromagnetic characteristics of RF-MEMS switches. Greenwood (GW) et al. [26] and Archard et al. [27] presented a statistical approach for analyzing the contact behavior of rough surfaces. The GW model utilized the hertz theory [28] to model roughness by asperities of constant radius and spherical morphology. Later on the GW theory is modified to adopt for different radius and distributions [29]. Zhao et al. [20] used the Laplace equation and found that normalized correlation length of 0.2 with roughness exponent of 0.4 produces a normalized capacitance value of 1.85. Palasantzas et al. [30] reported a normalized capacitance value of 2.2 for RMS (root mean square) roughness amplitude of 1 nm and mound separation of 15 nm. Rajendra et al. [17] utilized the fractal dimensions and reported a normalized capacitance value of 1.9 for fractal dimension value of 1.6. Similarly, Zubair et al. [31,32] proposed experimentally verified fractional dimensional based modified analytical models for rough electronic surfaces. Rebeiz [3] estimated that the presence of surface roughness is highly dependent on the material used and with a contact area of 50% and roughness of only 0.04 nm capacitance degrades by 50%. Yu et al. [33] found normalized capacitance value to be 0.47 for the RMS roughness of 10 nm with pull-in voltage of 100 V. Zhihao et al. [34] found a decrease in breakdown voltage value with an increase in surface roughness. For an RF-MEMS switch, with roughness value of 16 nm and 27.4 nm, the breakdown voltage is found to be 50 and 20 V respectively. Sharma et al. [35] reported normalized down-state capacitance value of 0.75 with an applied actuation voltage of 25 V. Zhiqiang et al. [36] observed that the roughness effect on downstate capacitance is more severe as compared to the up-state capacitance and it is found that with roughness of 20 nm the down-state capacitance of RF-MEMS switch approached 26% of the theoretical value. Daniel et al. [37] examined a decrease in Eigen frequencies values and problems in closing of nano electromechanical systems (NEMS) switches due to the presence of surface roughness. Table 1 lists a brief summary of different methods presented in literature for the study of surface roughness effect on the electromechanical characteristics of RF-MEMS switches. The study of surface roughness effect on the performance of RFMEMS switches either involves multiple microfabrication trials or understanding the complex mathematical roughness models. In this paper, a FEM based approach to analyze the effect of surface roughness on both the electromechanical and electromagnetic characteristics of capacitive RF-MEMS switches is presented. Thus allowing the designer to perform the reliability based design optimization of RF-MEMS switches at the design level.

2. RF-MEMS switch design and working principle The 3D schematic and lumped mass spring model of the electrostatically actuated RF-MEMS capacitive switch is shown in Fig. 1. The structure of the RF-MEMS switch is implemented on a coplanar waveguide (CPW) and consists of a suspended metal bridge overlapping the RF transmission line. The transmission line is covered with an insulating layer known as dielectric. The top suspended bridge is supported by four beams anchored to the RF grounded electrodes. In the normal configuration, the top suspended bridge is in upstate and due to very low value of capacitance, the signal propagates through switch with negligible losses. On application of DC voltage between the suspended bridge and dielectric layer, the bridge moves downward towards the transmission line. If the value of actuation voltage is high enough and the initial distance d between the suspended bridge and signal line is reduced to ds = 2d/3, the pull-in phenomena occurs. It causes the suspended bridge to make a contact with dielectric layer reducing the gap to ideally become zero. The analytical expression for the pull-in voltage of an ideally smooth RF-MEMS switch is given as [3]:

VPI (s) =

8kd3 27 o A

(1)

where k is the spring constant of beams, A is overlapping area, d is the distance between the suspended bridge and bottom dielectric layer, εo is the relative permittivity of air with value of 8.85 × 10−12 F m−1 and ε is relative permittivity of dielectric material. The switching or pull-in time is the time required by top suspended bridge to move from equilibrium position to downstate covering the distance ‘d’ present between the bridge and dielectric layer and is given by [3]:

ts =

3.67VPI (s) Vs

(2)

where Vs is the value of static voltage applied between the bridge and dielectric layer and ω is the resonant natural frequency of RF-MEMS switch. 3. Modeling and simulation Analytical models are helpful at the design stages of MEMS devices for predicting certain design parameters and providing close approximations to the static and dynamic behavior. However, for an accurate depiction of the multiphysics behavior of MEMS devices modern numerical methods such as FEM simulations are extensively being

Table 1 Literature review summary of surface roughness effect on the electromechanical parameters of RF-MEMS switches. Author

Method

Parameters

Results

Zhao et al. [20]

Analytical

Normalized upstate capacitance value of 1.85 is obtained.

Palasantzas et al. [30]

Analytical

Rajendra et al. [17] Rebeiz et al. [3]

Numerical Statistical

Albina et al. [38]

Experimental Numerical Theoretical Experimental Experimental Experimental

Roughness exponent = 0.4 Correlation length = 0.2 RMS roughness amplitude = 1 nm Correlation length = 30 nm Mound separation = 15 nm Fractal dimension = 1.6 Roughness = 10 nm Contact area = 50% N/A RMS roughness = 2 nm Pull-in voltage = 40 V RMS roughness = 5.4 nm, 16 nm, 27.4 nm RMS roughness = 16 nm Actuation voltage = 25 V RMS roughness = 100 nm

Normalized down-state capacitance value of 0.85 of the theoretical value is obtained.

Yu et al. [33] Zhihao et al. [34] Sharma et al. [35] Zhiqiang et al. [36]

Theoretical Experimental

Normalized upstate capacitance value of 2.2 is obtained. Normalized upstate capacitance value of around 1.9 is obtained. Downstate capacitance degrades by 30–35%. Upstate capacitance increases by 15–40%.

Break down voltage is reduced to 95, 50 and 20 V respectively. Normalized down-state capacitance value of 0.75 is obtained. Down-state capacitance value of 0.32 of the theoretical value is obtained.

2

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(a)

k

Fm

Fe

x

V

d

(b) Fig. 1. (a) 3D schematic of RF-MEMS capacitive switch. (b) Lumped elements model of RF-MEMS switch.

employed. The governing principles of RF-MEMS switches are based on electromechanics, so the analysis of electromechanical characteristics of RF-MEMS switch is carried out using FEM based coupled electrostructural simulations. Fig. 2 shows the layout for the implemented FEM methodology. The FEM model is divided into two basic parts i.e. the switch structure and the initial air gap in between top suspended bridge and dielectric layer. The beams and suspended bridge are modeled using the 8 node Solid185 element. The mechanical structural constraints are applied on the supporting beams end to restrict the displacement. The initial air gap between the top suspended plate and bottom dielectric layer is modeled using the 20 node Solid226 elements. Since the top bridge plate is symmetric with respect to center thus to minimize the meshing complexity, only a quad portion is modeled in the FEM simulations. With an applied actuation voltage, the resulting electrostatic force moves the bridge plate towards the bottom dielectric layer. The main performance parameters studied through FEM simulations for the RF-MEMS switch, with and without incorporating surface roughness include the up and downstate capacitance, pull in voltage, pull in gap and switching time. Table 2 shows the geometric parameters and material properties used for the FEM simulations of the RF-MEMS switch assuming thin film gold as a structural layer.

3.1. Case of smooth surface RF-MEMS switch To validate the accuracy of the FEM based simulations, initially electromechanical characteristics of the smooth surface RF-MEMS switch including pull-in voltage, up-state capacitance and pull-in gap are obtained using two different techniques in Multiphysics ANSYS module including transducer elements (Trans126) and coupled-field simulations. Trans126 elements provide a reduced order FEM modeling of the coupled electrostatic ̶ structural interaction utilizing the form of lumped elements. The results obtained through Trans126 and coupledfield simulations are validated through well-established analytical models for smooth surface RF-MEMS switches. Fig. 3(a) shows the effect of changing initial air gap distance between the top suspended bridge and bottom dielectric layer on the pull-in voltage value. The results perfectly follow the fact that pull-in voltage value increases nonlinearly with increasing initial air gap between the top suspended bridge and bottom dielectric layer. Thus, validating the accuracy of the FEM simulations using both Trans126 and coupled-field simulations. Fig. 3(b) shows that the up-state capacitance values vary non-linearly with increasing initial air gap value between the suspended bridge and dielectric layer. Fig. 3(b) also shows an abrupt change in capacitance value as soon as air gap starts decreasing. With an air gap change from 3

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Geometry design

Properties definition

Imported from CAD file

Element type

Modeled in software environment

Material Properties

Meshing

Start

Downstate capacitance (At pull in state)

End

Post processing

Vertical displacement

Upstate capacitance (At upstate)

Electrostatic Force

Mechanical domain

Electrical Loads

Constraints

Loadings, boundary conditions and solution Fig. 2. Layout for the coupled-field electro-structural FEM simulations of the RF-MEMS switch.

3.2.1. Effect of surface roughness on upstate capacitance The upstate position defines the point at which no contact exists between the top suspended bridge and bottom rough dielectric layer. Fig. 4 shows the schematic of the RF-MEMS switch in the upstate with top suspended metal bridge and bottom rough dielectric layer. The total number of semi-circles N covering the bottom dielectric surface is given by:

Table 2 Dimensions and mechanical properties of the RF-MEMS switch and initial air gap. Parameter

Symbol

Value

Length of the bridge Width of the bridge Height of the bridge Length of the beam Width of the beam Height of the beam Structural layer material Young's modulus [13] Poisson's ratio [13] Density [13] Young's modulus (air) Poisson's ratio (air)

L W H l w h – E ν ρ E ν

90 μm 90 μm 4.8 μm 30 μm 10 μm 1.8 μm Gold 98,500 MPa 0.42 19.32 × 10−15 kg/μm3 10−3 MPa 0

N=

l×w D2

(3)

where l is the length of dielectric layer, w is the width of dielectric layer and D is the diameter of the semi-circle. The analytical model representing the capacitance between the suspended bridge and a semicircle on a rough dielectric layer is given as [41]:

Cr = 2

1 μm to 0.4 μm the change in capacitance is approximately 2.5 times compared to the value at 1 μm. This indicate significance of initial air gap on performance of RF-MEMS switch since the capacitance in turn will determine pull-in voltage and pull-in gap of RF-MEMS switch. Fig. 3(c) shows that the variation in the pull-in bridge displacement (pull-in gap) varies linearly with the increasing initial air gap value between the top suspended bridge and bottom dielectric layer. However, for higher values of the initial air gap there is a significant mismatch between the analytical and FEM simulation results.

o R ln

1+

R z

(4)

where Cr is the capacitance value, R is radius of semi-circle, and z is the distance between semi-circle roughness on bottom dielectric layer and flat surface of the suspended bridge. In the upstate, the overall capacitance can be modeled by considering two capacitors attached in series configuration. One capacitance is due to the presence of dielectric (Cdielectric) while the second capacitance is due to the air gap present between the top suspended bridge and bottom dielectric layer (Cairgap). The Cdielectric is negligible because the value is much greater than Cairgap [36]. So, the overall equivalent upstate capacitance is reduced to the value produced by air gap present between the suspended bridge and dielectric layer i.e. Ce = Cr where Ce is the equivalent capacitance value. The normalized model is used as a comparison technique between the variable values obtained at different scales. The normalized values for electromechanical parameters is obtained by taking the ratio of values obtained from rough to that obtained using smooth surface RF-MEMS switch. The normalized values of up-state capacitance can be written as [42];

3.2. Effect of surface roughness on static and dynamic response of RFMEMS switch The effect of surface roughness in MEMS devices is generally modeled using approximate mathematical models [39], by replacing the original rough surface by an equivalent rough and a flat surface [40] or by using the statistical approach with semi-circles of constant radius and Gaussian height distribution function [26]. To accommodate the roughness effect for RF-MEMS switches, the surface roughness can be simplified by considering the assumptions that (a) the dielectric surface or suspended bridge is covered uniformly with semi-circles representing the roughness (b) the semi circles are of fixed diameter (c) the height of semicircles represents the RMS value of roughness and (d) the semicircles can be related to the grain size of the materials..

CN =

NCr d d = ln Cs 2R d R

(5)

The equation clearly shows that the normalized upstate capacitance is dependent on ratio of radius of the semi-circle to initial air gap present between the top suspended bridge and bottom dielectric layer i.e. R/d. The value of R/d approaching 1 is not significant since it can cause shortening between the semi-circles representing roughness and 4

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0.07

180 Analycal TRANS126 Coupled Field

160 140

Analycal FEM Simulaon

0.06 0.05 Capacitance (pF)

Pull-in Voltage (V)

120 100 80 60 40

0.04 0.03 0.02 0.01

20

0

0 0

0.3

0.6

0.9

1.2 1.5 1.8 2.1 Inial air gap (µm)

2.4

2.7

0

3

0.3

0.6

0.9 1.2 1.5 1.8 Inial air gap (µm)

(a)

2.1

2.4

2.7

3

(b)

2 Analycal TRANS126 Coupled Field

1.8 1.6 Pull-in gap (µm)

1.4 1.2 1 0.8 0.6 0.4 0.2 0 0

0.3

0.6

0.9

1.2 1.5 1.8 2.1 Inial air gap (µm)

2.4

2.7

3

(c)

(d)

Fig. 3. Effect of initial air gap values on the electromechanical characteristics of RF-MEMS switch (a) pull-in voltage, (b) up-state capacitance, (c) pull-in gap and (d) simulated quad model of the RF-MEMS switch. Table 3 AFM readings for MEMS surface roughness.

k

Fm

Fe

x z=d-R

V

d

Surface

Asperity radius (nm)

RMS roughness (nm)

Asperity density (μm−2)

Ref.

Silicon nitride Polysilicon Polysilicon Polysilicon Gold Gold Gold Gold

31.11–63.22 N/A 220 116 N/A 1410 N/A 120, 270

2.19–14.81 1–13.6 N/A 15.8 3.6, 6.9 1.8 10.315 3.5, 2

41–158 N/A N/A 14.7 N/A 15.8 N/A N/A

[33] [45] [46] [47] [48] [47] [45] [49]

R

as reported in literature for various MEMS surfaces, measured by AFM (atomic force microscopy), are given in Table 3. Fig. 5(a) shows the quad RF-MEMS switch model with surface roughness implemented in the FEM simulations. Fig. 5(b) shows the normalized values of upstate capacitance values obtained by varying initial air gap values. The normalized value of upstate capacitance increases with the increase in the semi-circle radius. The normalized upstate capacitance value i.e. 1.82 obtained by considering roughness of 100 nm for an initial air gap of 0.4 μm is about twice the value of capacitance obtained for a smooth surface RF-MEMS switch. Similar type of numerical results is reported for a parallel plate capacitor using analytical methods [20] and fractal surfaces [17]. All the curves plotted in Fig. 5(b) overlap for roughness scale approaching 5 nm and below showing the independency of normalized capacitance on ratio of R and

Fig. 4. Schematic of rough RF-MEMS switch.

the top suspended bridge. However, at pull-in point the gap diminishes rapidly and value of capacitance overshoots which is critical for the contact behavior of RF-MEMS switch as it can cause surface deformations [43] and contact bouncing [44] thus seriously disrupting the performance of switch. In the present FEM analysis, the maximum value of R/d is chosen to be 0.25. For the values of R/d approaching 0, the normalized capacitance is the ratio of real and apparent contacting surface areas. This shows that both the combined effect of smaller air gap and large effective area are main reasons for the increase in capacitance value. Typically the values of d in RF-MEMS capacitive switches ranges from 0.4 μm to 3 μm [3], whereas radius of semi circles 5

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(a) 1.9 Inial air gap = 3 µm Normalized upstate capacitance

1.85

Inial air gap = 2 µm Inial air gap = 1 µm

1.8

Inial air gap = 0.6 µm Inial air gap = 0.4 µm

1.75 1.7 1.65 1.6 1.55 1.5 0

10

20

30 40 50 60 70 Semi-circle radius (nm)

80

90

100

(b) Fig. 5. (a). Modeled RF-MEMS switch along with zoomed roughness profile of 100 nm used in FEM simulations. (b). Effect of roughness (semi-circle radius) on normalized capacitance of the RF-MEMS switch with varying initial air gap values.

d in this region. However, the surface area cannot be neglected since an increase of π/2% increase in area as given by Eq. (5) will result in a linear increase of capacitance between the top suspended bridge and dielectric layer. The increase in the normalized up-state capacitance value for a specific air gap can be explained by the fact that when the RMS roughness value increases, the tips of semi-circle become closer to the top suspended bridge and reduce the equivalent air gap which in turn increases the capacitance value. The effect of capacitance increase becomes abrupt when the air gap is of the order of the roughness. It is also shown in the literature that the presence of roughness of even 10 nm for an air gap of 1 μm can disrupt the value of upstate capacitance from the smooth surfaced switch by 9% [33]. The upstate capacitance results obtained through FEM simulations are validated by comparison with both the analytical model presented in [42] and experimental results for the fabricated RF-MEMS switch discussed in [50]. Fig. 6 shows the comparison of the results obtained. The upstate capacitance is experimentally measured as a function of applied voltage from the upstate to the pull in point. However, since the roughness is present on the fabricated switch during measurement of values so upstate capacitance seems to be effected by it. The difference

1.85 Analycal [38] FEM Simulaons Experimental [64]

Normalized upstate capacitance

1.8 1.75 1.7 1.65 1.6 1.55 1.5 1

11

21

31

41 51 61 71 Semi circle radius (nm)

81

91

101

Fig. 6. Comparison of normalized upstate capacitance for RF-MEMS switch obtained through FEM simulations with that of analytical and experimental.

6

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Table 4 Measured, calculated and simulated downstate capacitance values. Surface

Asperity radius (nm)

Aluminum Copper

59.18 63.22

RMS roughness (nm)

12.29 14.81

Asperity density (μm2)

50 41

of simulated and experimental results can be justified by the fact that various other reliability factors like presence of residual stress affecting the planarity of the top suspended bridge and non-uniform distribution of the roughness affects the upstate capacitance.

0.8968)0.5 ×

0

h

(h) dh

o r AANormalized

td + g

Cdielectric =

o r A (1

o r A (1

ANormalized ) g

ANormalized ) td

FEM simulations (this work)

0.25 0.22

0.2534 0.2685

0.27 0.255

d+R+

R2 3

4k (d

Rd + d 2

dr )(dr2 A

(10)

Rdr ) (11)

The presence of surface roughness also affects the switching time of the RF-MEMS switch and is given by:

tr =

3.67 Vs

4k (d

dr )(dr2 A

Rdr ) (12)

The FEM analysis is performed for rough surface RF-MEMS switch with thin film gold as structural layer by varying the initial air gap values from 3 μm to 0.4 μm and semi-circle radius from 100 nm to 10 nm. The graphs for comparison of pull-in voltage and pull-in gap for different values of surface roughness and initial air gap values are shown in Fig. 7. It is clearly depicted that the roughness of 10 nm with an initial air gap of 3 μm, 2 μm, 1 μm and 0.6 μm can shift the value of pull-in voltage by approximately 27 V, 15 V, 5 V and 3 V respectively. The change in pull in voltage for an initial air gap of 0.4 μm and roughness of 50 nm is around 28.5% which is about quarter the value of voltage required for the actuation of smooth RF-MEMS switch. The decrease in the breakdown voltage of capacitive RF-MEMS switch with an increase in RMS surface roughness values is also reported experimentally in [34]. The decrease of pull in voltage due to residual air gaps created by surface roughness in RF-MEMS switch is reported in [51]. So, the significance of roughness on RF-MEMS switches with small initial air gaps cannot be neglected. Fig. 8 shows that the normalized value of pull-in voltage decreases with an increase in the radius of semi-circles. The surface imperfections produce a serious reduction of pull-in voltage for higher values of R/d since pull-in gap and capacitance increases with the increase of semicircle radius. The normalized value of 0.83 clearly indicates that the roughness of even 1 nm and below shows a minute difference on the pull-in gap but decreases the pull-in voltage by about 15–20% as compared to smooth surface RF-MEMS switch. The normalized pull-in gap for the rough surface RF-MEMS is given as [42]:

(6)

(7)

where td is the thickness of dielectric and g is the average gap between the dielectric layer and semi-circle mean height plane. The capacitance due to the non-contact areas is further divided into series combination of capacitance values due to dielectric and residual air gap. The capacitance value due to dielectric and residual air gap for a rough surface is given as [36]:

Cairgap =

Analytical model [33]

VPI (r ) =

where α is the bandwidth parameter that describes the extent and shape of the roughness profile, h is the height of asperity, σ is the RMS value of roughness and ∅(h) is the Gaussian distribution function. It is shown experimentally that the contact area increases with increase in working time [16]. The real down-state capacitance is determined by evaluating the capacitance values due to contact and non-contact areas. The contact area is the point at which the semi-circles touch the top suspended bridge and the capacitance value is given as [36]:

Ccontact =

Experimental [33]

dr =

3.2.2. Effect of surface roughness on downstate capacitance In the actuated state of RF-MEMS switch, a contact is established between the top suspended bridge and bottom dielectric layer. However, due to the presence of roughness only highest asperities touch the suspended bridge. As a result, the real contact area decreases. The normalized contact area based on GW model is given as [33]:

ANormalized = 0.064(

Normalized downstate capacitance

(8) (9)

The normalized down-state capacitance value can be obtained by taking the ratios of real and apparent capacitance values. The normalized capacitance value decreases during the down-state of the RFMEMS switch. Table 4 shows the comparison of the normalized downstate capacitance values obtained through FEM simulations in this study, for surface roughness of 12.29 nm and 14.81 nm on Aluminum and copper surfaces, to that obtained through analytical model and experimental results presented in [33]. The downstate capacitance obtained through FEM simulations are in good agreement with both the analytical and experimental results, thus validating the accuracy of the FEM methodology.

dn =

1 R 1+ + 2 d

R d

2

R +1 d

(13)

Fig. 9 shows the effect of R/d ratio on the pull-in gap for the RFMEMS switch obtained through coupled-field FEM simulations. The results show that the normalized pull-in gap increases with increase in semi-circle radius and shows an increased normalized value of 1.057 for semi-circle radius of 100 nm with 0.4 μm air gap. These results show that the effect of surface roughness on the pull-in gap is very less. However, for the small initial air gap values for the RF-MEMS switch like 0.4 μm the effect of surface roughness on the pull-in gap is more pronounced. These results are consistent with the analytical model for the pull-in gap presented in [42] where for an initial air gap of 0.2 μm and roughness of 100 nm the change in pull in gap is reported to be 19%. It should be noted that the normalized pull-in gap shows approximate value of 1 as opposed to 0.83 showed by normalized pull-in voltage which shows that the pull in voltage is seriously disrupted by presence of surface roughness.

3.2.3. Effect of surface roughness on pull in gap, pull-in voltage and switching time The pull-in instability for the RF-MEMS switches occurs at the point where spring constant of electrical force becomes equal to stiffness of mechanical spring i.e. ke = km. Computing the derivative of electrostatic and mechanical forces at this point as a stroke of displacement x and solving for pull-in gap and voltage yields [42]: 7

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0

20

40

Voltage (V) 60 80 100

120

140

160

0

0

20

30

Voltage (V) 40 50

60

70

80

90

0 -0.25 Displacement (µm)

-0.5 Displacement (µm)

10

-1 -1.5 R = 10 nm R = 25 nm R = 50 nm R = 75 nm R = 100 nm Smooth

-2 -2.5

-0.5 -0.75 -1

R = 10 nm R = 25 nm R = 50 nm R = 75 nm R = 100 nm Smooth

-1.25 -1.5 -1.75 -2

-3

(a)

(b) Voltage (V) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5

Voltage (V) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0

0

-0.05 Displacement (µm)

Displacement (µm)

-0.1 -0.2 -0.3

R = 10 nm R = 25 nm R = 50 nm R = 75 nm R = 100 nm Smooth

-0.4 -0.5

-0.1 -0.15 -0.2

R = 10 nm R = 25 nm R = 50 nm R = 75 nm R = 100 nm Smooth

-0.25 -0.3 -0.35 -0.4

-0.6

(c)

(d)

Fig. 7. Comparison of pull-in voltage values for RF-MEMS switch with different roughness values for an initial air gap of (a) 3 μm, (b) 2 μm, (c) 0.6 μm and (d) 0.4 μm. 1.06

0.85

Inal air gap = 3 µm Inial air gap = 2 µm Inial air gap = 1 µm Inial air gap = 0.6 µm Inial air gap = 0.4 µm Smooth

1.055 1.05 1.045

0.75

Normalized pull in gap

Normalized pull in voltage

0.8

0.7 0.65 0.6

Inial air gap = 3 µm

0.55

Inial air gap = 2 µm

0.5

Inial air gap = 1 µm

1.03 1.025 1.02 1.015 1.01

Inial air gap = 0.6 µm 0.45

1.04 1.035

1.005

Inial air gap = 0.4 µm

1

0.4

0.995 1

11

21

31

41 51 61 71 Semi circle radius (nm)

81

91

101

1

11

21

31

41 51 61 71 Semi circle radius (nm)

81

91

101

Fig. 8. Effect of roughness (semi-circle radius) on normalized pull-in voltage values.

Fig. 9. Effect of roughness (semi-circle radius) on the normalized pull-in gap with varying initial air gap values.

The switching time is also influenced by presence of surface roughness. The switching time of the switch decreases with increase of roughness since the time is mainly dependent on pull-in voltage as given by the Eq. (2). Fig. 10 shows the effect of different roughness values on the switching time of RF-MEMS switch with varying initial air gap values. The switching time of an RF-MEMS switch with an air gap

of 0.4 μm and semi-circle radius of 100 nm is reduced to half of the value required for actuation of smooth switch. 4. Discussion The normalized upstate and down-state capacitance affects both the 8

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The normalized insertion loss value increases with an increase in the upstate capacitance. Yu et al. [33] reported an increase in the normalized insertion loss due to the increase of surface roughness. The isolation of the RF-MEMS switch is obtained by replacing the upstate capacitance with down-state capacitance in the Eq. (15). The equation clearly indicates that the decrease in downstate capacitance increases the isolation. However, the normalized isolation value shows a decrease when the suspended bridge is pulled down. Yu et al. [33] experimentally reported a decrease of about 38% in the isolation value due to roughness of 10 nm only. Dielectric charging is the trapping of charges in dielectric layer of the capacitive RF-MEMS switch. The presence of surface roughness may result in two charging mechanisms namely injection/contact charging and induced/contactless charging. The injection charging mainly occurs through the contacting areas of the suspended bridge and dielectric layer during the pull in state of the RF-MEMS capacitive switch. The induced charging is mainly provoked by the translation of free intrinsic charges during the up state of the switch. During the pull in state of the RF-MEMS switches, due to the presence of roughness, small residual air gaps are created that results in both induced charging through field emission and injection charges simultaneously. The induced charging through field emission is the major cause of charging during the upstate of the switch when the switch is stressed and is placed in vacuum. The presence of surface roughness on the suspended bridge or dielectric layer enhances the process of field emission. The electric field intensity due to the presence of surface roughness can significantly increase up to an order of V/nm and is capable of initiating the field emission process. In the present case since the roughness is presented by semi-circles, the electric field between a semi-circle and flat surface is given by [52]:

11 R = 10 nm R = 25 nm R = 50 nm R = 75 nm R = 100 nm Smooth

10 9

Switching me (µs)

8 7 6 5 4 3 2 1 0 0

0.25 0.5 0.75

1

1.25 1.5 1.75 2 Inial air gap (µm)

2.25 2.5 2.75

3

Fig 10. Initial air gap and switching time for different RMS values of roughness (semi-circle radius).

electromechanical and electromagnetic characteristics of the RF-MEMS capacitive switch. The presence of surface roughness in RF-MEMS switches increases the normalized upstate capacitance while reduces the downstate capacitance. The increase in upstate capacitance in the FEM simulations can be justified by the fact that the semi-circles, representing roughness in simulation model, reduce the initial air gap and increase effective surface area between the top suspended bridge and bottom dielectric layer. The change in upstate capacitance is abrupt when the initial air gap between suspended bridge and dielectric layer is of the magnitude comparable to the surface roughness. The downstate capacitance reduces due to the presence of roughness which results in a rough contact between the suspended bridge and the dielectric layer. The residual air gap and reduced contact area results in the decrease of the downstate capacitance. The FEM simulation results, in this work, show that the pull-in voltage of the RF-MEMS switch decreases in the presence of surface roughness. This decrease in the pull in voltage in the presence of surface roughness can be attributed to the fact that 1) the increase in effective area, in the presence of surface roughness, increases the capacitance which eventually leads to the decrease in the pull-in voltage value and 2) the increase in the capacitance, in the presence of surface roughness, allows higher amount of charges to be collected at the surface of dielectric layer resulting in increase in both the electric potential and electric field between the suspended bridge and dielectric layer. As a result, the electrostatic force increases which leads to pull-in phenomenon to occur at lower actuation voltage in comparison to the smooth surface RF-MEMS switch. Moreover, the FEM simulation results show that effect of surface roughness on the pull-in gap is only significant for the small values of initial air gap distances between the top suspended bridge and bottom dielectric layer of the RFMEMS switch. As explained earlier, the normalized upstate capacitance increases with the increase in the RMS value of surface roughness. The insertion loss of the RF-MEMS switch is related to the upstate capacitance by the relation given as [3]:

S21 = 20 log

1 1 + j Cu Z /2

ERough =

ENormalized =

0.9 R d

()

R 2 d

(18)

The equation shows that the electric field due to semi-circles is clearly dependent upon the ratio of radius of semi-circle to the distance between the suspended bridge and dielectric. The normalized value of electric field obtained for a roughness of even 10 nm and an air gap of 0.4 μm is several times higher than the value obtained from its smooth counterpart. The change in electric field is observed experimentally and found to be much higher in magnitude as compared to the smooth surfaced electric field value [53]. According to electrostatic theory [54] electrical charge density on the tip of the semi-circle is maximum since the electrical charges are mostly concentrated on the center of the semicircles. Due to increase in electric field, the field emission process becomes severe causing transfer of charges in the dielectric layer. It is shown that the due to trapping of charges, an increase in electric field is observed due to which the lifetime of the RF-MEMS capacitive switch deteriorates exponentially. The trapping of charges create the phenomena of dielectric charging and results in change of capacitance and pull in characteristics. Reid [55] studied analytically and experimentally the effect of charge trapping on the capacitance voltage curve. The development of shifts of pull in and pull out voltages is accessed in [56]. It is also found that shift in curve is observed for non-uniform trapping of charges. The charges are induced through the contacting area, so there is high probability of the fact that the charges are mostly concentrated on multiple different areas as in the case of semi-circles. Moreover, a small amount of charges is also injected through micro gaps by field emission process generated due to the presence of surface roughness.

(15)

S21(Rough) S21 (Smooth)

(17)

The normalized model of electric field i.e. the ratio of maximum electric field value obtained by rough surface to the electric field value obtained with smooth surface is given by [42]:

where S21 is the insertion loss, Cu is the upstate capacitance, Z is the impedance of the transmission line and ω is the angular frequency. As the equation implies, the increase in upstate capacitance decrease the insertion loss. The normalized value of insertion loss can then be calculated as:

S21 =

0.9V (z + R) zR

(16) 9

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5. Conclusion

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The impact of surface roughness on the up and down-state capacitance, pull-in gap, pull-in voltage and switching time of a capacitive RFMEMS switch is studied. A FEM based analysis is presented for a simple roughness structure consisting of evenly distributed semi-circles of constant radius along the surface of dielectric layer of an RF-MEMS capacitive switch. It is shown that the roughness effect on the electromechanical characteristics increases with decrease in the initial air gap present between the top suspended bridge and bottom dielectric layer of the RF-MEMS switch. The effect of surface roughness becomes more pronounced when the initial air gap is of order of the roughness scale (i.e. radius of semi-circle). The upstate capacitance increases with increase in roughness — a normalized value of 1.82 is obtained for an initial air gap of 0.4 μm and 100 nm radius of semi-circle. The pull-in voltage decreases with an increase in the roughness parameter. It has been shown that the pull-in voltage value of RF-MEMS switch with an initial air gap of 0.4 μm and 50 nm roughness decreases by 28.4% as compared to its smooth counterpart. The pull-in gap increases with the increase in radius of the semi-circle. The pull-in gap shows a normalized value of 1.057 for an air gap of 0.4 μm with 100 nm roughness. Similarly, the switching time of an RF-MEMS switch decreases with increase in roughness. The switching time of an RF-MEMS switch with an initial air gap of 3 μm and roughness of 100 nm decreases by about 24.2%. The results presented in this work show that the FEM based methodology for the estimation of surface roughness effects on the electromechanical characteristics of RF-MEMS switches, at the design level, is a good alternative to the complex analytical models and costly and time consuming experimental investigations. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. References [1] Stella Chang, Siva Sivoththaman, A tunable RF MEMS inductor on silicon incorporating an amorphous silicon bimorph in a low-temperature process, IEEE Electron Device Letters 27 (11) (2006) 905–907. [2] Melendez, Jose L., Tsen-Hwang Lin, and Byron Williams. "High Q-large tuning range micro-electro mechanical system (MEMS) varactor for broadband applications." U.S. Patent No. 6,635,919. 21 Oct. 2003. [3] Gabriel M. Rebeiz, RF MEMS: Theory, Design, and Technology, John Wiley & Sons, 2004. [4] Xiaoguang Liu, et al., High-$ Q $ tunable microwave cavity resonators and filters using SOI-based RF MEMS tuners, Journal of Microelectromechanical Systems 19.4 (2010) 774–784. [5] Kamran Entesari, Gabriel M. Rebeiz, A 12–18-GHz three-pole RF MEMS tunable filter, IEEE Transactions on Microwave Theory and Techniques 53 (8) (2005) 2566–2571. [6] Dimitrios E. Anagnostou, et al., Design, fabrication, and measurements of an RFMEMS-based self-similar reconfigurable antenna, IEEE Trans. Antennas Propag. 54 (2) (2006) 422–432. [7] Z.J. Guo, N.E. McGruer, G.G. Adams, Modeling, simulation and measurement of the dynamic performance of an ohmic contact, electrostatically actuated RF MEMS switch, J. Micromech. Microeng. 17 (9) (2007) 1899. [8] David Girbau, et al., Electrothermally actuated RF MEMS switches suspended on a low-resistivity substrate, Journal of Microelectromechanical Systems 16.5 (2007) 1061–1070. [9] Hee-Chul Lee, et al., Design, fabrication and RF performances of two different types of piezoelectrically actuated ohmic MEMS switches, J. Micromech. Microeng. 15 (11) (2005) 2098. [10] Ta-Hsuan Lin, et al., A study on the performance and reliability of magnetostatic actuated RF MEMS switches, Microelectron. Reliab. 49 (1) (2009) 59–65. [11] Muhammad Mubasher Saleem, Hamid Nawaz, A systematic review of reliability issues in RF-MEMS switches, Micro and Nanosystems 11 (2019) 1. [12] Aurelio Soma, Giorgio De Pasquale, MEMS mechanical fatigue: experimental results on gold microbeams, Journal of Microelectromechanical Systems 18.4 (2009) 828–835. [13] Aurelio Soma, Muhammad Mubasher Saleem, Giorgio De Pasquale, Effect of creep in RF MEMS static and dynamic behavior, Microsystem Technologies 22.5 (2016) 1067–1078.

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