1. Theory

Definition: RF MEMS passives are micromachined switches and variable capacitors with a movable metal beam or membrane that is pulled electrostatically toward a fixed electrode, either making an ohmic metal contact (series contact switch) or landing on a thin dielectric (capacitive shunt switch and varactor) (Soraasen; Analog Devices).
Low loss and high linearity: Because the signal path is a metal line with no semiconductor junction, insertion loss is about 0.04 to 0.2 $dB$ and isolation 20 to 60 $dB$ , depending on type and frequency (Soraasen).
Actuation: Electrostatic actuation draws almost no current but needs from a few to tens of volts, and the beam snaps down at the pull-in voltage after travelling one third of the gap (Soraasen; Analog Devices).
Speed: Mechanical motion limits switching to microseconds, such as 75 $μ s$ maximum for a commercial switch (Analog Devices).
Reliability: Lifetime is set by contact wear and hot switching in ohmic switches and by charge trapping in the dielectric of capacitive switches, with more than 10^{9} cycles demonstrated (Soraasen; Analog Devices; background knowledge for charging).

2. Working Principle

Electrostatic force and pull-in: The force $ε_{0} A V^{2} /2 g^{2}$ grows faster than the spring restoring force as the gap closes, so beyond one third of the gap the equilibrium becomes unstable and the beam closes (Soraasen; background knowledge for the derivation).
Capacitive switching: In the up state the large air gap gives a small shunt capacitance that hardly loads the line; pulled down onto the dielectric, the capacitance rises about a hundred times and shorts the signal to ground at RF (Soraasen).
Ohmic switching: Gold contacts close a metal path from DC upward, with a resistance set by the contact force and surface roughness (Soraasen; Analog Devices).
Release: When the voltage is removed, the spring force restores the beam, provided stiction and trapped dielectric charge do not hold it down (background knowledge).

3. Applications

Test and instrumentation switching: DC to 14 $GHz$ switches for automated test equipment and RF instruments, replacing electromechanical relays (Analog Devices).
Reconfigurable RF front ends: Switched capacitor banks and switches for tunable antennas, filters and matching networks (Soraasen; background knowledge).
Phased arrays and mm-wave switching: Capacitive shunt switches from 5 to 100 $GHz$ in phase shifters (Soraasen).
Current status: MEMS switches are in commercial production for test and RF applications, while solid-state switches remain preferred where switching speed or very high cycle counts under hot switching are required (Analog Devices; my reading).

Actuation

Name	Formula	Related Formulae	Variables/Constants
Electrostatic actuation and pull-in voltage	$V_{p} = \frac{8 k g _{0}^{3}}{27 ε _{0} A}, g_{p u l l - in} = \frac{2}{3} g_{0}$	$F_{e} = \frac{ε _{0} A V ^{2}}{2 g ^{2}} = k (g_{0} - g)$	$V_{p} : p u l l - in v o l t a g e$ $k : s p r in g co n s t an t o f t h e b e am or m e mb r an e$ $g_{0} : ini t ia l g a p$ $ε_{0} : p er mi tt i v i t y o f f r ee s p a ce$ $A : e l ec t r o d e a r e a$ $g : g a p$ $F_{e} : e l ec t r os t a t i c f or ce$ $V : a c t u a t i o n v o l t a g e (p a r a l l e l - pl a t e p u l l - in, ba c k g r o u n d k n o w l e d g e); S or aa se n : t h e m e mb r an e b eco m es u n s t ab l e an d c l oses o n ce i t ha s m o v e d o n e t hi r d o f t h e g a p, w i t h K t h e s p r in g co n s t an t, g_{0} t h e g a p an d A = W w t h e a r e a (t h e p u l l - in e x p r ess i o n i t se l f i s mi ss in g in t h e e x t r a c t), f i x e d - f i x e d b e am s t i f f n ess a f t er V a r a d an, a c t u a t i o n v o l t a g es t ha t s h o u l d s t a y b e l o w t e n s o f v o l t s an d f o l d e d s p r in g s t ha t a l l o w u n d er 5 V f or m o bi l e sy s t e m s; A na l o g D e v i ces : t h e A D GM 1304 g e n er a t es an 80 V a c t u a t i o n v o l t a g e f r o m a 3.0 t o 3.6 V s u ppl y w i t h an in t e g r a t e d d r i v er$ $(co m p u t e d : an a ss u m e d k = 10 N/m, g_{0} = 3 μ m an d A = 100 b y 100 μ m g i v e V_{p} = 30 V)$
Switching time	$t_{s} \approx \frac{3.67 V _{p}}{V _{s} ω _{0}}, ω_{0} = \frac{k}{m _{e f f}}$	$t_{s} defined from 50 per cent of the control voltage to 90 or 10 per cent of the RF envelope$	$t_{s} : s w i t c hin g t im e$ $V_{s} : a ppl i e d v o l t a g e (ab o v e V_{p})$ $ω_{0} : m ec hani c a l r eso nan ce an g u l a r f r e q u e n cy$ $m_{e f f} : e f f ec t i v e m o v in g ma ss (R e b e i z - t y p e es t ima t e f or l o w d am p in g, ba c k g r o u n d k n o w l e d g e); S or aa se n : s w i t c hin g s p ee d i s t h e t im e f r o m 50 p er ce n t o f t h e co n t r o l v o l t a g e t o 90 (o n) or 10 (o f f) p er ce n t o f t h e R F e n v e l o p e; A na l o g D e v i ces : o n an d o f f t im es o f 75 μ s ma x im u m f or t h e A D GM 1304$ $(co m p u t e d : an a ss u m e d g o l d b e am o f 300 b y 100 b y 1 μ m w i t h 40 p er ce n t e f f ec t i v e ma ss, 2.3 \times 1 0^{- 10} kg, an d k = 10 N/m r eso na t es a t 33 kHz, so V_{s} = 1.4 V_{p} g i v es t_{s} o f ab o u t 13 μ s)$

Switch Performance

Name	Formula	Related Formulae	Variables/Constants
Capacitive shunt switch: up and down capacitance	$C_{u p} \approx \frac{ε _{0} A}{g _{0} + t _{d} / ε _{r}}, C_{d o w n} \approx \frac{ε _{0} ε _{r} A}{t _{d}}$	$\frac{C _{d o w n}}{C _{u p}} \approx \frac{ε _{r} g _{0}}{t _{d}} (g_{0} ≫ t_{d} / ε_{r})$	$C_{u p}, C_{d o w n} : c a p a c i t an ce w i t h t h e m e mb r an e u p an d p u l l e d d o w n$ $t_{d} : t hi c k n ess o f t h e d i e l ec t r i c o n t h e l o w er e l ec t r o d e$ $ε_{r} : i t s r e l a t i v e p er mi tt i v i t y$ $g_{0} : ai r g a p$ $A : o v er l a p a r e a (f r in g in g n e g l ec t e d); S or aa se n : C = ε A / d, a r a t i o t ha t s h o u l d e x cee d 100 w i t h 60 t o 120 t y p i c a l, a d i e l ec t r i c o f a t l e a s t ab o u t 1000 \overset{˚}{A} t ha t w i t h s t an d s 20 t o 50 V, an d p er mi tt i v i t i es o f ab o u t 7.6 f or s i l i co n ni t r i d e, 40 t o 200 f or s t r o n t i u m t i t ana t e an d ab o v e 1000 f or P Z T; s h u n t c a p a c i t i v e s w i t c h es co v er 5 t o 100 GHz w i t h in ser t i o n l oss o f 0.04 t o 0.1 dB u p an d i so l a t i o n o f 20 t o 30 dB f r o m 10 t o 50 GHz d o w n$ $(co m p u t e d : f or A = 100 b y 100 μ m, g_{0} = 3 μ m an d 1000 \overset{˚}{A} o f ni t r i d e, C_{u p} = 30 fF an d C_{d o w n} = 6.7 pF, an i d e a l r a t i o o f 228; 20 t o 50 V o n 1000 \overset{˚}{A} i s 2 t o 5 MV/cm)$
Insertion loss and isolation of a shunt switch	$∣ S_{21} ∣^{2} \approx \frac{4}{ω ^{2} C _{d o w n}^{2} Z _{0}^{2}} (down state), ∣ S_{11} ∣^{2} \approx \frac{ω ^{2} C _{u p}^{2} Z _{0}^{2}}{4} (up state)$	$I L = - 20 lo g_{10} ∣ S_{21} ∣, f_{L C} = \frac{1}{2 π L _{b} C _{d o w n}}$	$S_{21}, S_{11} : t r an s mi ss i o n an d r e f l ec t i o n$ $ω : an g u l a r f r e q u e n cy$ $Z_{0} : l in e im p e d an ce$ $I L : in ser t i o n l oss in d B$ $f_{L C} : ser i es r eso nan ce o f t h e d o w n - s t a t e c a p a c i t an ce w i t h t h e b r i d g e in d u c t an ce L_{b} (s h u n t - e l e m e n t a pp r o x ima t i o n s, ba c k g r o u n d k n o w l e d g e; l in e t h eor y o n t h e O n - C hi p T r an s mi ss i o n L in es p a g e); S or aa se n : in ser t i o n l oss an d i so l a t i o n a s 1/ S_{21} in d B, an d a r eso nan ce w h er e t h e in d u c t i v e an d c a p a c i t i v e r e a c t an ces a r e e q u a l in ma g ni t u d e, u se d t o im p r o v e i so l a t i o n; ser i es co n t a c t s w i t c h es co v er 0.1 t o 40 GHz w i t h 50 t o 60 dB i so l a t i o n a t 1 GHz an d 0.1 t o 0.2 dB in ser t i o n l oss$ $(co m p u t e d : w i t h t h e 6.7 pF an d 30 fF v a l u es a t 30 GHz in 50 Ω, i so l a t i o n i s 30 dB an d t h e u p - s t a t e r e f l ec t i o n - 17 dB)$
Ohmic contact switch: resistance and loss	$I L \approx 20 lo g_{10} (1 + \frac{R _{o n}}{2 Z _{0}})$	$P_{ma x} = 36 dBm \approx 4 W$	$R_{o n} : o n - r es i s t an ce o f t h e co n t a c t s w i t c h$ $Z_{0} : 50 Ω sy s t e m im p e d an ce$ $P_{ma x} : ma x im u m R F p o w er (ser i es - r es i s t an ce l oss, ba c k g r o u n d k n o w l e d g e); A na l o g D e v i ces A D GM 1304 : D C t o 14 GHz, in ser t i o n l oss 0.26 dB t y p i c a l a t 2.5 GHz an d 0.4 dB a t 6 GHz, i so l a t i o n 24 an d 19 dB, o n - r es i s t an ce a t m os t 2.9 Ω a t 50 mA, d r i f t in g t o u p t o 5 Ω a f t er a bi l l i o n a c t u a t i o n s a t 85^{\circ} C, 36 dBm R F p o w er, 100 V H B M o n t h e R F p in s an d 2.5 kV e l se w h er e; S or aa se n : g o l d co n t a c t s w h ose r es i s t an ce d e p e n d s o n r o ug hn ess an d co n t a c t f or ce$ $(co m p u t e d : 2.9 Ω g i v es 0.25 dB, ma t c hin g t h e 0.26 dB a t 2.5 GHz; 5 Ω g i v es 0.42 dB; 36 dBm i s 4.0 W)$

Reliability and Varactors

Name	Formula	Related Formulae	Variables/Constants
Lifetime, hot switching and dielectric charging	$N_{l i f e} \propto P^{- n} (hot switching)$	$n = \frac{l o g ( N _{1} / N _{2} )}{l o g ( P _{2} / P _{1} )}$	$N_{l i f e} : n u mb er o f cy c l es t o f ai l u r e$ $P : R F p o w er d u r in g s w i t c hin g$ $n : p o w er e x p o n e n t$ $N_{1}, N_{2} : l i v es a t p o w er s P_{1}, P_{2} (f i t f or m, m y r e a d in g); S or aa se n : t a r g e t s ab o v e 1 0^{9} cy c l es, d e m o n s t r a t e d; A na l o g D e v i ces A D GM 1304 : a t l e a s t 1 0^{9} co l d - s w i t c h e d a c t u a t i o n s (220 mA, 85^{\circ} C), 50 p er ce n t h o t - s w i t c hin g f ai l u r e a t ab o u t 5.16 \times 1 0^{9} cy c l es a t 10 dBm, 3.21 \times 1 0^{6} a t 15 dBm an d 3.9 \times 1 0^{5} a t 20 dBm, an d a m e d ian o f 7.2 y e a r s co n t in u o u s l y o n a t 50^{\circ} C; in c a p a c i t i v e s w i t c h es t h e hi g h f i e l d a cr oss t h e t hin d i e l ec t r i c inj ec t s c ha r g e t ha t s hi f t s t h e a c t u a t i o n v o l t a g e an d c an h o l d t h e m e mb r an e d o w n (ba c k g r o u n d k n o w l e d g e)$ $(co m p u t e d : n i s ab o u t 6.4 f r o m 10 t o 15 dBm an d 1.8 f r o m 15 t o 20 dBm, so t h e l i f e d oes n o t f o l l o w a s in g l e p o w er l a w)$
Analog MEMS varactor tuning range	$C (g) = \frac{ε _{0} A}{g}, \frac{C _{ma x}}{C _{min}} = \frac{g _{0}}{2 g _{0} /3} = 1.5 (stable range before pull-in)$	$C_{ma x} / C_{min} = C_{d o w n} / C_{u p} (switched or digital varactor)$	$C (g) : c a p a c i t an ce a t g a p g, C_{ma x} / C_{min} : t u nin g r a t i o, o t h er sy mb o l s a s ab o v e (ba c k g r o u n d k n o w l e d g e); S or aa se n : p u l l - in a f t er o n e t hi r d o f t h e g a p, w hi c h l imi t s co n t in u o u s p a r a l l e l - pl a t e t u nin g; t h e d o w n - t o - u p r a t i o o f c a p a c i t i v e s w i t c h es o f 60 t o 120 se t s t h e r an g e o f s w i t c h e d - c a p a c i t or bank s; M O S v a r a c t or s o n t h e M O S V a r a c t or s p a g e$ $(co m p u t e d : 1.5 in c a p a c i t an ce i s 1.22 in f r e q u e n cy f or an L C t ank w i t h o n l y t h e v a r a c t or, a g ain s t 228 i d e a l an d 60 t o 120 t y p i c a l f or a s w i t c h e d c a p a c i t or)$

Important Questions

Ques: What sets the pull-in voltage of an RF MEMS switch?

The electrostatic force on the beam, $ε_{0} A V^{2} /2 g^{2}$ , rises as the gap shrinks, while the spring force rises only linearly, so beyond one third of the gap no stable equilibrium exists and the beam snaps down. The pull-in voltage is then $8 k g_{0}^{3} / (27 ε_{0} A)$ (background knowledge). For an assumed spring constant of 10 $N/m$ , a 3 $μ m$ gap and a 100 by 100 $μ m$ electrode it is 30 $V$ (computed). Lower voltage needs a softer spring, a smaller gap or a larger electrode. Soraasen notes that actuation should stay below tens of volts and that folded springs can reach below 5 $V$ for mobile systems, but a soft spring has less force to break stiction, so low voltage trades against reliability. Reference: O. Soraasen, INF5490 RF MEMS, Lecture LN05 ‘RF MEMS switches, I’, Department of Informatics, University of Oslo (Spring 2011) (several equations missing in the extract) (fetch-tool summary).

Ques: Should actuation voltages be below tens of volts, or is 80 V acceptable?

Both positions appear. Soraasen’s course notes recommend actuation below tens of volts, and below 5 $V$ with folded springs for mobile systems, to suit integrated drivers. The commercial ADGM1304 instead actuates at 80 $V$ , generated on chip from a 3.0 to 3.6 $V$ supply by an integrated driver, so the user never sees the high voltage. A higher actuation voltage allows a stiffer beam, whose larger restoring force releases the contact reliably and resists stiction, which is probably why a product rated for at least 10^{9} cycles chose it (my reading). For integrated systems without a charge pump the low-voltage target is the more practical one, while for packaged switches with a driver 80 $V$ is acceptable. Pull-in scales as $k$ , so 80 $V$ allows about seven times the stiffness of a 30 $V$ design with the same gap and area (computed). Reference: O. Soraasen, INF5490 RF MEMS, Lecture LN05 ‘RF MEMS switches, I’, Department of Informatics, University of Oslo (Spring 2011) (several equations missing in the extract) (fetch-tool summary); Analog Devices, ADGM1304 0 Hz/DC to 14 GHz SP4T MEMS switch with integrated driver, data sheet (fetch-tool summary).

Ques: Why is the measured down-to-up capacitance ratio of a capacitive switch lower than the ideal value?

The ideal ratio $ε_{r} g_{0} / t_{d}$ is 228 for a 3 $μ m$ gap over 1000 $\overset{˚}{A}$ of silicon nitride with a permittivity of 7.6 (computed), giving 30 $fF$ up and 6.7 $pF$ down for a 100 by 100 $μ m$ electrode (computed). Soraasen quotes 60 to 120 as typical, against a target above 100. In the down state, surface roughness and residual bow leave a thin air layer between the membrane and the dielectric, which acts in series and lowers the down capacitance (background knowledge), while fringing fields raise the up capacitance. Higher-permittivity dielectrics such as strontium titanate (40 to 200) or PZT (above 1000) raise the ideal ratio, but they suffer more from the same air gap. The ratio sets the isolation: 6.7 $pF$ at 30 $GHz$ in 50 $Ω$ gives about 30 $dB$ (computed). Reference: O. Soraasen, INF5490 RF MEMS, Lecture LN05 ‘RF MEMS switches, I’, Department of Informatics, University of Oslo (Spring 2011) (several equations missing in the extract) (fetch-tool summary).

Ques: How does hot switching shorten the life of an ohmic MEMS switch?

Switching while RF power flows lets current pass through the first microscopic contact points as they open or close, which heats and erodes them. The ADGM1304 reaches at least 10^{9} cold-switched actuations at 85 $^{\circ} C$ , but its median hot-switching life falls from about $5.16 \times 1 0^{9}$ cycles at 10 $dBm$ to $3.21 \times 1 0^{6}$ at 15 $dBm$ and $3.9 \times 1 0^{5}$ at 20 $dBm$ . Fitting $N \propto P^{- n}$ gives an exponent of about 6.4 between 10 and 15 $dBm$ but only 1.8 between 15 and 20 $dBm$ (computed), so the damage does not follow a single power law and probably changes mechanism above a threshold power (my reading). Contact resistance also drifts, from at most 2.9 $Ω$ to up to 5 $Ω$ after a billion actuations, raising the insertion loss from 0.25 to 0.42 $dB$ (computed). Reference: Analog Devices, ADGM1304 0 Hz/DC to 14 GHz SP4T MEMS switch with integrated driver, data sheet (fetch-tool summary).

Ques: How much tuning range can a MEMS varactor give?

An analog parallel-plate MEMS varactor can only move the gap by one third before pull-in, so its capacitance changes by at most 1.5 times, from $ε_{0} A / g_{0}$ to $ε_{0} A / (2 g_{0} /3)$ (background knowledge), which changes the frequency of an LC tank by at most 1.22 times (computed). That is smaller than the tuning ratio of 2.5 to 3.4 measured for accumulation-mode MOS varactors. Switched or digital MEMS varactors avoid the limit by using the full down-to-up ratio of a capacitive switch, 60 to 120 in practice according to Soraasen, in binary-weighted banks. The MEMS advantage is not range but linearity and Q, since the capacitance does not depend on the RF voltage in the way that a junction or MOS capacitance does, as long as the RF swing stays well below pull-in (background knowledge). Reference: O. Soraasen, INF5490 RF MEMS, Lecture LN05 ‘RF MEMS switches, I’, Department of Informatics, University of Oslo (Spring 2011) (several equations missing in the extract) (fetch-tool summary); MOS Varactors page of this vault.

Ques: How do MEMS switches compare in loss and isolation with each other?

Soraasen summarizes the two main types. Series ohmic contact switches work from 0.1 to 40 $GHz$ with 0.1 to 0.2 $dB$ insertion loss and 50 to 60 $dB$ isolation at 1 $GHz$ , while capacitive shunt switches cover 5 to 100 $GHz$ with 0.04 to 0.1 $dB$ loss in the up state and 20 to 30 $dB$ isolation from 10 to 50 $GHz$ . The commercial ADGM1304, an ohmic SP4T with integrated driver and package, has 0.26 $dB$ loss and 24 $dB$ isolation at 2.5 $GHz$ and 0.4 and 19 $dB$ at 6 $GHz$ , worse than the bare-switch values because of package and routing parasitics (my reading). Its loss matches a 2.9 $Ω$ series resistance, which gives 0.25 $dB$ in 50 $Ω$ (computed). Ohmic switches therefore suit wideband use down to DC, capacitive switches high frequencies. Reference: O. Soraasen, INF5490 RF MEMS, Lecture LN05 ‘RF MEMS switches, I’, Department of Informatics, University of Oslo (Spring 2011) (several equations missing in the extract) (fetch-tool summary); Analog Devices, ADGM1304 0 Hz/DC to 14 GHz SP4T MEMS switch with integrated driver, data sheet (fetch-tool summary).

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RF MEMS Passives