1 Introduction

In such an era of scaling down semiconductor devices in micro or nanometer range, RF MEMS switch with the advantages of low or near-zero power consumption, high isolation, low insertion loss, and high linearity has obtained a rapid development (Metta et al. 2018). Compared to PIN diode or FET switches, RF MEMS switch is more suitable for high frequency and miniaturized communication system. The traditional capacitive RF MEMS switch is electrostatically actuated, and it contains a coplanar waveguide to transmit signals, a big suspended membrane as actuator and an electrode to provide bias voltage. When a voltage bias is applied, the switch use mechanical movements to short or open a transmission.

Capacitance ratio and actuation voltage are two important indexes to measure the switch performance. The former reflects the RF performance of the switch, while the latter reflects the mechanical characteristic of the switch. However, the two are often mutually limited in design. A simple and effective approach to obtain high on/off capacitance ratio is to use a large displacement between the beam and signal line. However, if the height of the beam was changed too high, the reliability of MEMS switches would be degraded, then, the actuation voltage would be altered.

At present, there also have been several attempts (Angira et al. 2013; Park and Kim 2000; Park et al. 2001; Persano et al. 2010) to use high dielectric constants materials include HfO2(εr = 20), STO(εr = 30–120), Ta2O5 (εr = 32) to obtain a high on/off capacitance ratio. As a result, the capacitance ratio of switches are more than 100. Meanwhile, some approaches of designing low spring constants to get a low actuation voltage just under 10 V have been proposed, for example, Ya et al. (2013) design on COMS RF MEMS switch managed to achieve a low actuation voltage of 3 V, but the relatively wide beams cannot be easily released with membrane in one step. Yongqing et al. (2018) achieved a switch with 14 V actuation voltage, which can be mounted on frequency reconfigurable antenna. Gopalan and Kommuri (2018) achivevd a triangular series RF MEMS switch with 5.6 V actuation voltage and as low as 5 MPa stress, which can be potentially used to build the miniature phase shifter. Ma et al. (2016) design on a low actuation voltage switch of 3 V, however, the beam of 1.397 µm thickness is very thin, which will cause the instanbility. Rousstia et al. (2015) designed a RF MEMS switch, which can be combined with the array antenna and achieve a good directivity and gain of the antenna.

This letter reports a novel RF MEMS switch with a Si3N4(εr = 7.6) dielectric lyaer and a relactively lower air gap (2 µm), as shown in the Fig. 1. The MEMS switche have a Metal–insulator-metal (MIM) fixed capacitors, which can improve the capactances ratio without degrading the reliability of the switches. In order to obtain a low actuation voltage, the switch is designed in the form of bilateral drop-down electrodes and has a special beam like mechanical spring with a thickness of 1.6 µm, which can greatly reduce the spring constant. According to simulation analysis and parameter fitting, the capacitance ratio and actuationvoltage of the switch are 162 and 4.5 V respectively.

Fig. 1
figure 1

Switch diagram

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2 Circuit topology and theory analysis

2.1 Design MIM fixed capacitor to improve capacitance ratios

The circuit topology of the typical MEMS switch is as shown in Fig. 2a (Yang et al. 2014), when the MEMS switch is in up-state, the CPW-insulator-air-beam (Metal-Isulator-Air-Metal) fixed capacitors and edge capacitance Cf together form the upstate capacitance Cu; and when the switch is in down-state, the capacitance Cd is CPW-insulator-beam (Metal–insulator–Metal) fixed capacitors. The capacitive ratio Cr of the MEMS switch can be calculated from formuls (1) and (2) (Yang et al. 2014):

$$ C_{r} = \frac{{C_{u} }}{{C_{d} }} = \frac{{\frac{{\varepsilon_{0} \varepsilon_{r} A}}{{t_{d} }}}}{{\frac{{\varepsilon_{0} A}}{{g + \frac{{t_{d} }}{{\varepsilon_{r}^{{}} }}}} + C_{f} }} $$
(1)

where \(\varepsilon_{0} ,\varepsilon_{r} ,t_{d} ,g,A\) are the vacuum dielectric constant, relative dielectric constant of the dielectric layer, the thickness of the dielectric layer, the gap between the beam and the electrodes the cross-sectional area of the electrode.

Fig. 2
figure 2

Circuit topology of MEMS switches: a circuit models of traditional MEMS switches; b circuit model of the proposed MEMS switch with MIM capacitors

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The formula (1) can be simplified as (2), when Cf is ignored:

$$ C_{r} = \frac{{C_{u} }}{{C_{d} }} = \frac{{\frac{{\varepsilon_{0} \varepsilon_{r} A}}{{t_{d} }}}}{{\frac{{\varepsilon_{0} A}}{{g + \frac{{t_{d} }}{{\varepsilon_{r}^{{}} }}}}}}{ = }\frac{{g\varepsilon_{r} }}{{t_{d} }} + 1 $$
(2)

In this paper, a MIM capacitor is designed on the signal line to improve capacitance ratio, as shown in Fig. 2b. When in up-state, MIM capacitor connects to a MAM capacitor in series, and the MAM is changed to resistance R in down-state. Then, the Cr can be calculated as follows, Cf is ignored:

$$ \begin{gathered} C_{r} = \frac{{C_{d} }}{{C_{r} }} = \frac{{C_{MIM} }}{{C_{MIM} \parallel C_{MAM} }} = \frac{{C_{MIM} }}{{C_{MAM} }} + 1 \hfill \\ \begin{array}{*{20}c} {} & { = \frac{{\varepsilon_{r} g}}{{t_{d} }}} \\ \end{array} \frac{{A_{MIM} }}{{A_{MAM} }} + 1 = \frac{{\varepsilon_{r} g}}{{t_{d} }}\lambda + 1 \hfill \\ \end{gathered} $$
(3)
$$ C_{MAM} = \frac{{\varepsilon_{0} }}{g}A_{MAM} ,C_{MIM} = \frac{{\varepsilon_{0} \varepsilon_{r} }}{{t_{d} }}A_{MIM} $$
(4)

where λ is equal to AMIM/AMAM, other parameters are defined as same as formulas before. As shown in Fig. 3, for traditional MEMS switches, with td is not less than 0.15um and g is more than 1um, the Cr is less than 150. Due to loading MIM fiexd capacitor, Cr is increased with the increase of the value g and λ, as shown in Fig. 3b.

Fig. 3
figure 3

The relationship among g, td, λ and Cr. a The value of Cr, when g > 1 µm, td > 0.15 µm. b The relationship among λ g and Cr

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As shown in Fig. 4, the MEMS switch loads a MIM capacitor on the signal line, in order to ensure a beam contacts the MIM capacitor when the switch is in down-state, RF and DC electrodes are separated designed.

Fig. 4
figure 4

The schematic view of the MEMS switch

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For the switch based on high-resistance silicon (εr = 11.9), the CPW line is 60/100/60(G/S/G) µm, and the size of the beam is w_beam = 220 µm, l_beam = 500 µm. Si3N4 is used as dielectric material, and the thickness of Si3N4 is 0.15 µm. The capacitance Cu of the switch without MIM capacitors can be calculated simply as 57.8fF (Yang et al. 2014), and Cr≈55. For the switch with MIM capacitors, Fig. 5 shows the relationship among the parameter of w1, w2, and Cr.

Fig. 5
figure 5

Comparison of Cr with and without MIM. a The relationship among the value of Cr and w1. b The relationship among the value of Cr and w2

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It can be found the value of Cr has been significantly improved with MIM capacitors. When w1 = 10 µm, Cr is increased with the increase of the value of w2. And when w2 = 30 µm, Cr is descreased with the increase of the value of w1. It can be found Cr = 206 with g = 2 µm, td = 0.15 µm, w1 = 10 µm, w2 = 30 µm.

2.2 Electrodes topology and spring constant analysis

As shown in Fig. 6, in order to reduce dielectric charging effects and increase switches service life, the electrodes sit on opposite sides of the center signal line. x1 and x2 represent the coordinate positions of the electrodes. In this form of electrodes, the spring constant k of MEMS switches can be calculate as follows (Deng et al. 2016)

$$ \begin{gathered} k_{1} = - \frac{P}{{\frac{2}{EI}\int\limits_{{l/2 + x_{2} }}^{{l/2 + x_{1} }} {\frac{\rho }{48}(l^{3} - 6l^{2} a + 9la^{2} - 4a^{3} )da} }} \hfill \\ \begin{array}{*{20}c} {} & = \\ \end{array} - \frac{{48EI(x_{1} - x_{2} )}}{{ - \frac{1}{4}l^{3} x_{1} + \frac{1}{4}l^{3} x_{2} + lx_{1}^{3} - lx_{2}^{3} - x_{1}^{4} + x_{2}^{4} }} \hfill \\ \end{gathered} $$
(5)
$$ k_{2} = \frac{P}{{2\int\limits_{{l/2 + x_{2} }}^{{l/2 + x_{1} }} {\frac{\rho }{2S}(l - {\text{a}})da} }}{ = }\begin{array}{*{20}c} { - \frac{4S}{{x_{1} - x_{2} + l}}} & {} \\ \end{array} $$
(6)
$$ k = k_{1} + k_{2} $$
(7)
$$ I{ = }\frac{{wt^{3} }}{12} $$
(8)
$$ P = 2\rho (x_{1} - x_{2} ) $$
(9)
$$ S{ = }\sigma (1 - v)tw $$
(10)

where k1 is the spring constant caused by stiffness of the beam, k2 is the spring constant caused by biaxial residual stress of the beam, ρ is uniformly distributed load of the beam, S is the tensile force of biaxial residual stress, σ is biaxial residual stress, v is the Possion ratio, E is the Young’s modulus, and w is the width of the beam.

Fig. 6
figure 6

Electrodes topology of the MEMS switch

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The actuation voltage of the MEMS switch can be given by (11):

$$ V_{p} = \sqrt {\frac{8k}{{27\varepsilon_{0} (2w(x_{1} - x_{2} ))}}g^{3} } $$
(11)

The relationship between the spring constant and the coordinates of the electrodes is shown in Fig. 7. It can be seen the spring constant is increased with the increase of the area of electrodes. For the rectangle beam with Au (E = 78Gpa, v = 0.44), the length is 500 µm, the weight is 220 µmm, the gap is 2 µm and σ = 10 Mpa, it can be found when x1 = 180 µm and x2 = 70 µm, k1 = 7.6 N/m, k = 31.34 N/m, the spring constant caused by biaxial residual stress is dominant.

Fig. 7
figure 7

The relationship between the x1, x2 and k1, k

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As shown in Fig. 8, the actuation voltage Vp is changed with the change of values of (x1, x2), and a minimum value of Vp can be obtained in the range of values of (x1, x2). When the values of (x1, x2) are in regions 1 and 2, Vp ≤ 10 V.

Fig. 8
figure 8

The relationship between the x1, x2 and Vp

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As shown in Fig. 9, the actuation voltage Vp and the spring constant caused by biaxial residual stress is increased as σ increases. For the rectangle beam with Au (E = 78 Gpa, v = 0.44), the length is 500 µm, the weight is 220 µmm, the gap is 2 µm, x1 = 180, x2 = 70;when Vp ≤ 20 v, σ ≤ 30 Mpa.

Fig. 9
figure 9

The relationship between the σ and the k2 and Vp

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3 Design and simulation

3.1 Low spring constant switch design and analytical pull-in model

This paper designs the switch structure as shown in Fig. 10a. The beam adopts a structure similar to mechanial spring, which is fixed on the ground by four anchor points. Cantilever beams are normally insensitive to the biaxial residual stress but are very sensitive to the stress gradient due to their free-end condition. Slotting the beam can not only reduce the spring constant but also increase the inductance of the switch.

Fig. 10
figure 10

a Detailed dimensions of membrane part in the switch, b switch with simple mechanical spring

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For simplicity, we adjusted ‘paralle-plate approximation’ to the moveble plate, as seen in Fig. 10b, and spring-mass model can be used to determine the relationships between the pull-in voltage and the dimensional parameters of the switch (Yang et al. 2014). The spring constants of the membrane can be described as follows.

$$ k_{1} = 4Ew_{1} \left( {\frac{{t_{a} }}{{l_{1} }}} \right)^{3} ,\quad k_{2} = 4Ew_{2}^{{}} \left( {\frac{t}{{l_{5} }}} \right)^{3} $$
(12)
$$ k_{4} = 4Ew_{4} \left( {\frac{t}{{l_{2} + l_{4} }}} \right)^{3} ,\quad k_{3} = 4Ew_{3} \left( {\frac{t}{{l_{3} }}} \right)^{3} $$
(13)

where w1, w2, w3, w4, l1, l2, l3, l4 are labeled in Fig. 10a. E is the Young’s modulus of the structural material, ta is the thickness of the anchor point, t is the thickness of the beam. Ignore the spring constant caused by biaxial residual stress of the beam, the total spring constant of the switch can be calculated as

$$ k_{{{\text{total}}}} = (k_{1} \parallel k_{2} \parallel k_{3} \parallel k_{4} ) $$
(14)

In addition, the overlap area between the beam and two bottom electrodes can be calculated as

$$ A_{total} = 2(S_{e} - N\pi \left( {R{\text{i}}} \right)^{2} ) $$
(15)

where Ri is also shown in Fig. 10a, Se is the area of elelctrode, N is the number of slots above the electrode. From (12) and (15), the actuation voltage of the switch can be calculated as

$$ V_{p} = \sqrt {\frac{{8k_{total} (g + t_{d} /\varepsilon_{r} )^{3} }}{{27\varepsilon_{0} A_{total} }}} $$
(16)

where εr and ε0 are the relative permititivity of the dielectric layer and the permittivity of an air, respectively.

For the switch beam with Au (E = 78Gpa, v = 0.44), the length is 500 µm, the weight is 220 µmm, the gap is 2 µm, Other physical dimensions of beams are shown in Table 2. As shown in Fig. 11, with the help of finite element analysis software, deformation of the beam under different thickness are analyzed, and all three models are under same loading form of both sides. When the thickness of beam are 1 µm, 1.6 µm and 2 µm, respectively, the spring constance are 0.7392 (N/m), 1.6215 (N/m) and 4.3571 (N/m), and the corresponding voltage can be calculated to 2 V, 4.5 V and 9 V.

Fig. 11
figure 11

Deformation diagram; a both sides loading and t_beam = 1 µm, b both sides loading and t_beam = 1.6  µm, c both sides loading and t_beam = 2 µm

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The maximum stress occurs in the position shown in the Fig. 12, which is the most likely point of failure in the working state. The max stress is 8 Mpa with the thickness of beam 1 µm, and it increases to 13 Mpa when the thickness of beam is 1.6 µm. The equivalent stress is smaller with t_beam = 1 µm, however, the thickness is too thin and flexed easily, which cannot fully contact with the dielectric layer, and this will results the signal that barrier effect is particularly poor. Therefore, beams of too small thickness cannot be selected.

Fig. 12
figure 12

Equivalent stress diagram; a both sides loading and t_beam = 1 µm, b both sides loading and t_beam = 1.6 µm

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Furthermore, Table 1 shows some other calculation parameters of the switch. It can be seen that the pull-down time and pull-up (Rebeiz 2004) time are decreasing with the increase of beam thickness, what’s more the beams with thin thickness are less stable. And compared to 2 µm, 1.6 µm thick beam is more economical in the process of material saving.

Table 1 Switch parameter
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3.2 Physical demensions of the proposed switch

Through the above analysis, physical demensions of the proprosed switch are shown in Table 2. In this paper, we use Au as the beam material, Si3N4 as the dielectric material, and high resistivity silicon substrate.

Table 2 Physical demensions of the switch
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3.3 Simulation of S-parameters

As shown in Fig. 13(a), the thickness of beam t_beam has a influence on resonant frequency. When the size of t_beam is increased, the resonant frequency increases gradually. Under the condition of different thickness, the isolation is less than  − 20 dB within the range of 10–30 GHz, and the extreme value reaches − 45 dB. As shown in Fig. 13b, the return loss is less than − 12 dB at the resonant frequency.

Fig. 13
figure 13

S12 and S11 for the switch. a Isolation for different thickness beam of the switch. b Returen loss for the switche with 1.6 µm thickness beam

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Figure 14 shows the relationship between the value of w7, w8 and isolation, when the size of the upper metal layer of the MIM capacitor is increased, the resonant frequency of MEMS switch is decreased due to the increase of the shun capacitance. When the size of MIM capacitor is w7 = 30 um, w8 = 10 um, the resonant frequency of MEMS switch is 21.5 GHz, and the isolation reaches − 40 dB.

Fig. 14
figure 14

S12 for the switch with different MIM capacitors; a different weight in the MIM capacitor w7, b different weight in the MIM capacitor w8

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We can see from Fig. 15a, when the switch is in up-state, the current can be transmitted better on the signal line, and its main energy is concentrated on both sides of the transmission line. A small amount of energy is coupled with the beam, and consumption is very small. When the switch is in down-state, Fig. 10b, the signal is blocked by the beam and cannot be transmitted to the other side, most energy is reflected back. Therefore, the current value on the side of the transmission line is higher than it in up-state. Some energy is transmitted through the beam to the ground.

Fig. 15
figure 15

Current diagram; a switch in up-state, b switch in down-state

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3.4 Analysis

In this paper, the beam structure like mechanical spring can be drastically reduced the value of spring constance due to cantilever beams are normally insensitive to the biaxial residual stress but are very sensitive to the stress gradient. Meanwhile, when the thickness of the beam is decreased, the actuation voltage decrease, however, the thickness is too thin and flexed easily, which cannot fully contact with the dielectric layer, the signal that barrier effect is particularly poor. In this paper, the thickness of the beam is 1.6 µm.

For the MEMS switches, Cu, Cd, L and R can be extracted from the return loss S11 and isolation S12. The RF MEMS switch and CPW transmission line consist of three parts and can be expressed by ABCD matrix (Pozar 2009):

$$ \left( {\begin{array}{*{20}c} A & B \\ C & D \\ \end{array} } \right) = {\text{M}}_{{1}} {\text{M}}_{{2}} {\text{M}}_{{1}} $$
(17)

where M1 represents the CPW transmission line part (the ABVD matrix consist of two M1 for the symmetry). M2 represents the lumped parameter model of the RF MEMS. They are:

$$ {\text{M}}_{1} = \left( {\begin{array}{*{20}c} {\cos \theta } & {jZ_{0} \sin \theta } \\ {j\frac{1}{{Z_{0} }}\sin \theta } & {\cos \theta } \\ \end{array} } \right) $$
(18)
$$ {\text{M}}_{2} = \left( {\begin{array}{*{20}c} 1 & 0 \\ {{\text{Y}}_{2} } & 1 \\ \end{array} } \right) $$
(19)

where θ is CPW transmission line electric length, Z0 is the characteristic impedance of transmission line, Y2 is:

$$ {\text{Y}}_{2} = \frac{1}{{(jwC_{s} )^{ - 1} + jwL + R_{s} }} $$
(20)

where Cs is Cup or Cdw when the switch stays the corresponding state. The expression L = (wCs) − 1 is satisfied when the switch operates at the resonant frequency. Rs is the loss resistance. S21 is:

$$ S_{21} = \frac{2}{{A + B/Z_{0} + CZ_{0} + D}} $$
(21)

From Eqs. (17)–(22), for the proposed switch, Cu = 26.5 fF, Cd = 4.3 pF, Cr = 162, the MIM capacitors can improve the on/off capacitance ratio without adding more less to the switches.

Table 3 shows the comparison of this paper with other developed capacitive RF-MEMS switch. It presents that the proposed switch have a lower actuation voltage with excellent electromagnetic characteristics than others.

Table 3 Comparison of developed capacitive RF-MEMS switches
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4 Conclusions

In this paper, a novel type high on/off capacitance ratio and low actuation voltage RF MEMS switch has been designed. The switch is in the form of bilateral drop-down electrodes and has a special beam like mechanical spring with a thickness of 1.6 µm, which can greatly reduce the spring constance. The actuation voltage is as low as 4.5 V by finite element analysis software simulation. MIM capacitors can improve the on/off capacitance radio and can also ensure good electrical isolation performance. From the fitted results, the on/off capacitance ratio is 162 for the MEMS switch. Through the use of HFSS software analysis, within the range of 10–30 GHz, the up-state insertion loss is better than − 0.2 dB, the down-state isolation is better than − 20 dB, and the minimum isolation is − 41 dB at resonant frequency.