Nanoelectromechanical Memory (NEMS Cantilever and Relay Memory)

1. Theory

Definition: A nanoelectromechanical (NEMS) memory cell is an electrostatically actuated beam (cantilever or bridge) whose two mechanical positions, open and in contact, are the two memory states.
Placement: The state is stored in the position of a moving part, not in charge, magnetization, polarization or a filament.
Non-volatility: If contact adhesion exceeds the beam’s restoring force, the beam stays closed with no applied voltage.
Beam mechanics: Cantilever stiffness $k = E w t^{3} /4 L^{3}$ and resonance $f_{0} \propto (t / L^{2}) E / ρ$ set the switching voltage and speed.
Pull-in instability: Electrostatic force grows as $1/ (g - x)^{2}$ while spring force grows linearly, so the beam snaps down at one third of the gap.
Hysteresis: The release voltage is below the pull-in voltage, and adhesion widens the gap between them.

2. Working Principle

Write closed: A gate voltage above $V_{P I} = 8 k g^{3} /27 ε_{0} A$ pulls the beam into contact.
Hold: Adhesion (van der Waals, $\propto 1/ d^{3}$ ) keeps the beam closed without power.
Write open: A reverse or release action overcomes adhesion and the beam springs back.
Read: The contact resistance (Holm or Sharvin constriction resistance) distinguishes closed from open.
Speed: Closing time is of order the mechanical period, nanoseconds to microseconds, much slower than transistors.
Energy and voltage: Switching energy $\approx \frac{1}{2} C_{a c t} V^{2}$ is small, but pull-in voltages above 1 V exceed modern logic supplies.

3. Applications

Current status: Research devices; not in commercial memory products.
Harsh-environment memory and logic (target): Near-zero leakage, high-temperature and radiation-tolerant storage where transistors fail.

Name	Formula	Related Formulae	Variables/Constants
Cantilever spring constant and resonance	$k = \frac{E w t ^{3}}{4 L ^{3}}$ $f_{0} = \frac{1.87 5 ^{2}}{2 π} \frac{E I}{ρ A L ^{4}} \approx 0.1615 \frac{t}{L ^{2}} \frac{E}{ρ}$	$I = \frac{w t ^{3}}{12}$ $ω_{0} = 2 π f_{0}$	$k : s p r in g co n s t an t o f a r ec t an g u l a r c an t i l e v er l o a d e d a t i t s e n d$ $E : Y o u n g m o d u l u s o f t h e b e am ma t er ia l$ $w, t, L : w i d t h, t hi c k n ess an d l e n g t h o f t h e c an t i l e v er$ $f_{0} : f u n d am e n t a l r eso nan t f r e q u e n cy o f t h e c an t i l e v er$ $I : seco n d m o m e n t o f a r e a o f t h e cr oss - sec t i o n$ $ρ : ma ss d e n s i t y o f t h e b e am ma t er ia l$ $A : cr oss - sec t i o na l a r e a w t$ $ω_{0} : an g u l a r r eso nan t f r e q u e n cy (t h e e f f ec t i v e ma ss o f t h e b e am i s n o t t h e f u l l ma ss, so k an d f_{0} a r e n o t r e l a t e d b y t h e t o t a l ma ss)$
Electrostatic actuation and pull-in	$F_{e l} = \frac{ε _{0} A V ^{2}}{2 ( g - x ) ^{2}}$	$x_{P I} = \frac{g}{3}$ $V_{P I} = \frac{8 k g ^{3}}{27 ε _{0} A}$	$F_{e l} : e l ec t r os t a t i c f or ce b e tw ee n t h e b e am an d t h e g a t e e l ec t r o d e (p a r a l l e l - pl a t e m o d e l)$ $ε_{0} : v a c uu m p er mi tt i v i t y$ $A : o v er l a p a r e a o f t h e b e am an d t h e g a t e e l ec t r o d e$ $V : v o l t a g e b e tw ee n t h e b e am an d t h e g a t e$ $g : g a p b e tw ee n t h e b e am an d t h e g a t e a t z er o v o l t a g e$ $x : d e f l ec t i o n o f t h e b e am t o w a r d t h e g a t e$ $x_{P I} : d e f l ec t i o n a t w hi c h t h e b e am b eco m es u n s t ab l e an d s na p s o n t o t h e e l ec t r o d e (o n e t hi r d o f t h e g a p)$ $V_{P I} : p u l l - in v o l t a g e$ $k : s p r in g co n s t an t o f t h e b e am (f i r s t r o w)$
Release voltage and hysteresis	$V_{P O} = \frac{2 d _{0}^{2} [ k ( g - d _{0} ) - F _{a d h} ]}{ε _{0} A}$	$V_{P O} < V_{P I}$ $F_{a d h} > k (g - d_{0})$	$V_{P O} : p u l l - o u t (r e l e a se) v o l t a g e b e l o w w hi c h t h e b e am in co n t a c t s p r in g s ba c k$ $d_{0} : r e mainin g g a p b e tw ee n t h e e l ec t r o d es w h e n t h e b e am t o u c h es (t h e t hi c k n ess o f a d i e l ec t r i c l a y er or a co n t a c t d im pl e)$ $k, g : s p r in g co n s t an t an d ini t ia l g a p$ $F_{a d h} : a d h es i o n f or ce a t t h e co n t a c t (v an d er W aa l s an d s u r f a ce f or ces)$ $ε_{0}, A : v a c uu m p er mi tt i v i t y an d e l ec t r o d e o v er l a p a r e a$ $(i f t h e s t o p i s a d i e l ec t r i c o f r e l a t i v e p er mi tt i v i t y ε_{r}, r e pl a ce d_{0} b y d_{0} / ε_{r} in t h e e l ec t r os t a t i c t er m; t h e f or m u l a i s t h e f or ce ba l an ce ε_{0} A V^{2} /2 d_{0}^{2} + F_{a d h} = k (g - d_{0}))$ $V_{P I} : p u l l - in v o l t a g e (t h e d i f f er e n ce b e tw ee n V_{P I} an d V_{P O} i s t h e h y s t er es i s t ha t g i v es t h e tw o m e m or y s t a t es)$ $(w h e n t h e a d h es i o n f or ce e x cee d s t h e r es t or in g f or ce a t co n t a c t, t h e b e am s t a y s in co n t a c t a t z er o v o l t a g e an d t h e ce l l i s n o n - v o l a t i l e)$
Adhesion force between flat surfaces	$\frac{F _{v d W}}{A _{c}} = \frac{A _{H}}{6 π d ^{3}}$	$F_{a d h} \approx \frac{A _{H} A _{c}}{6 π d ^{3}}$	$F_{v d W} : v an d er W aa l s f or ce b e tw ee n tw o p a r a l l e l f l a t s u r f a ces$ $A_{c} : a r e a o f t h e s u r f a ces in c l ose co n t a c t$ $A_{H} : H amak er co n s t an t o f t h e tw o ma t er ia l s a cr oss t h e g a p$ $d : se p a r a t i o n o f t h e s u r f a ces (a f r a c t i o n o f a nan o m e t r e u p t o a f e w nan o m e t r es in a r e a l co n t a c t)$ $F_{a d h} : a d h es i o n f or ce u se d in t h e r e l e a se v o l t a g e r o w, o b t ain e d f r o m t h e r e a l co n t a c t a r e a A_{c} (w hi c h i s m u c h s ma l l er t han t h e g eo m e t r i c a r e a f or r o ug h s u r f a ces)$
Pull-in switching time	$t_{P I} \approx \frac{π}{2} \frac{27}{8} \frac{V _{P I}}{V ω _{0}} \approx \frac{2.9 V _{P I}}{V ω _{0}}$	$V ≫ V_{P I}$ $t_{P I} \approx \frac{27}{2} \frac{V _{P I}}{V ω _{0}} \approx \frac{3.67 V _{P I}}{V ω _{0}}$	$t_{P I} : t im e f or t h e b e am t o c l ose f r o m r es t w h e n a v o l t a g e V i s a ppl i e d (l o w d am p in g)$ $V_{P I} : p u l l - in v o l t a g e$ $V : a ppl i e d v o l t a g e, w e l l ab o v e V_{P I} so t ha t t h e s p r in g f or ce c an b e n e g l ec t e d d u r in g t h e c l os in g$ $(d er i v e d f or t h e u n d am p e d l u m p e d m o d e l b y in t e g r a t in g m \overset{u}{¨} = - ε_{0} A V^{2} /2 u^{2} f r o m u = g t o 0; w i t h t h e s p r in g in c l u d e d t h e coe f f i c i e n t i s ab o u t 3.0 a t V = 2 V_{P I} an d 3.4 a t 1.3 V_{P I}; s q u eez e - f i l m d am p in g mak es i t l o n g er)$ $(t h e r e l a t e d f or m i s S e n t u r i a^{'} s t e x t b oo k v a l u e, w hi c h k ee p s t h e ini t ia l f or ce ε_{0} A V^{2} /2 g^{2} co n s t an t o v er t h e g a p; i t o v er es t ima t es t h e u n d am p e d t im e b y 27 p er ce n t f or V ≫ V_{P I}, see t h e q u es t i o n s)$ $ω_{0} : an g u l a r r eso nan t f r e q u e n cy o f t h e b e am (so t h e s w i t c hin g t im e i s o f t h e or d er o f t h e m ec hani c a l p er i o d)$
Contact resistance (Holm and Sharvin)	$R_{c} = \frac{ρ _{1} + ρ _{2}}{4 a}$	$R_{S} = \frac{4 ρ λ}{3 π a ^{2}}$ $a = (\frac{3 F R _{e f f}}{4 E ^{*}})^{1/3}$	$R_{c} : co n s t r i c t i o n (H o l m) r es i s t an ce o f a c i r c u l a r co n t a c t o f r a d i u s a b e tw ee n tw o ma t er ia l s o f r es i s t i v i t y ρ_{1} an d ρ_{2}$ $a : r a d i u s o f t h e co n t a c t s p o t$ $R_{S} : S ha r v in r es i s t an ce, w hi c h r e pl a ces R_{c} w h e n t h e co n t a c t r a d i u s i s s ma l l er t han t h e m e an f r ee p a t h λ o f t h e e l ec t r o n s$ $ρ, λ : r es i s t i v i t y an d e l ec t r o n m e an f r ee p a t h o f t h e co n t a c t ma t er ia l$ $F : co n t a c t f or ce$ $R_{e f f} : e f f ec t i v e r a d i u s o f c u r v a t u r e o f t h e co n t a c t in g s u r f a ces$ $E^{} : e f f ec t i v e e l a s t i c m o d u l u s o f t h e p ai r o f ma t er ia l s (t h e H er t z co n t a c t r a d i u s, v a l i d f or an e l a s t i c s p h er e o n a f l a t)$ $\frac{1}{E ^{}} = \frac{1 - ν _{1}^{2}}{E _{1}} + \frac{1 - ν _{2}^{2}}{E _{2}}, w i t h E_{i}, ν_{i} t h e Y o u n g m o d u l u s an d P o i sso n r a t i o o f ma t er ia l i$
Switching energy	$E_{s w} \approx \frac{1}{2} C_{a c t} V^{2}$	$E_{m ec h} = \frac{1}{2} k g^{2}$ $C_{a c t} = \frac{ε _{0} A}{g}$	$E_{s w} : e l ec t r i c a l e n er g y t o c ha r g e t h e a c t u a t i o n c a p a c i t an ce in o n e s w i t c hin g e v e n t (l o w er b o u n d, t h e c a p a c i t an ce c han g es a s t h e b e am c l oses)$ $C_{a c t} : c a p a c i t an ce b e tw ee n t h e b e am an d t h e g a t e a t z er o d e f l ec t i o n$ $V : a c t u a t i o n v o l t a g e$ $E_{m ec h} : e l a s t i c e n er g y s t or e d in t h e b e am w h e n i t i s c l ose d (n e g l ec t in g t h e co n t a c t g a p d_{0})$ $k, g : s p r in g co n s t an t an d ini t ia l g a p$ $ε_{0}, A : v a c uu m p er mi tt i v i t y an d e l ec t r o d e o v er l a p a r e a$

Important Questions

Ques: How does a nanoelectromechanical (NEMS) memory store a bit?

A NEMS memory cell is a beam (a cantilever or a doubly clamped bridge) of a conductor or a semiconductor suspended over a gate and a contact electrode. The two states are the beam in the up position and in the contact position, which gives a high and a low resistance between the beam and the contact (the electrical read). The beam is pulled down by an electrostatic force and released by a lower voltage because of the pull-in hysteresis ( $V_{P O} < V_{P I}$ ). If the adhesion force at the contact exceeds the restoring force of the beam, the beam stays down with no applied voltage, and the cell is non-volatile. The state is read as the contact resistance and the write is an electrostatic pull-in or release. Reference: V. Pott et al., “Mechanical computing redux: relays for integrated circuit applications,” Proc. IEEE 98, 2076 (2010); O. Loh and H.D. Espinosa, “Nanoelectromechanical contact switches,” Nat. Nanotechnol. 7, 283 (2012).

Ques: Why is there a pull-in at one third of the gap, and how does it scale?

The electrostatic force rises as $1/ (g - x)^{2}$ while the spring force rises linearly with $x$ . The balance is stable until the deflection reaches $g /3$ , where the slopes of the two forces become equal, so beyond this point the beam snaps down at the voltage $V_{P I} = 8 k g^{3} /27 ε_{0} A$ . The pull-in voltage therefore falls as the gap is reduced ( $g^{3/2}$ ) and as the beam is made softer ( $k^{1/2}$ ), and for a cantilever $k \propto t^{3} / L^{3}$ , so a thin, long beam switches at a low voltage. The cost is a lower resonant frequency, a lower restoring force against adhesion and a higher risk of permanent stiction. Reference: S.D. Senturia, Microsystem Design, Kluwer, 2001; H.C. Nathanson, W.E. Newell, R.A. Wickstrom and J.R. Davis, “The resonant gate transistor,” IEEE Trans. Electron Devices 14, 117 (1967).

Ques: What limits the speed and the energy of a NEMS memory?

The mechanical closing time is of the order of the resonant period, $t_{P I} \approx 2.9 V_{P I} / (V ω_{0})$ for a drive voltage well above pull-in (undamped; 3.67 in Senturia, see the next question), so a stiff and light beam gives a few nanoseconds to a microsecond, much slower than a transistor. The electrical energy per operation is of the order of $\frac{1}{2} C_{a c t} V^{2}$ , which is small for a small gap and area, but the voltage is above one volt (pull-in), which is high compared with the supply voltage of a modern logic process. The device is therefore attractive for low-leakage and high-temperature applications, not for speed. Reference: S.D. Senturia, Microsystem Design, Kluwer, 2001; V. Pott et al., Proc. IEEE 98, 2076 (2010).

Ques: The pull-in time coefficient is given as 3.67 and as 2.9. Which is right?

Both come from the same undamped lumped model with the spring force neglected, which holds for $V ≫ V_{P I}$ . Senturia keeps the electrostatic force at its initial value $ε_{0} A V^{2} /2 g^{2}$ during the whole travel, which gives $t_{P I} = 27/2 V_{P I} / (V ω_{0}) = 3.67 V_{P I} / (V ω_{0})$ . The force actually grows as $1/ u^{2}$ as the gap closes, and integrating the energy balance exactly gives $(π /2) 27/8 = 2.89$ (computed). For $V ≫ V_{P I}$ the exact value 2.9 is more likely to be right, and 3.67 is an upper estimate. Close to pull-in the spring slows the start, and a numerical integration with the spring gives 3.41 at $1.3 V_{P I}$ , 3.06 at $2 V_{P I}$ and 2.91 at $5 V_{P I}$ (computed), so 3.67 is closer near pull-in. Squeeze-film damping lengthens both. Reference: S.D. Senturia, Microsystem Design, Kluwer, 2001 (constant-force estimate); the exact integral is my derivation.

Ques: Why does the contact resistance vary, and what sets it?

Two solid surfaces touch at a few asperities, so the electrical contact area is much smaller than the geometric area. For contact spots larger than the mean free path the resistance is the Holm constriction resistance $(ρ_{1} + ρ_{2}) /4 a$ , and for spots smaller than the mean free path it is the Sharvin resistance $4 ρ λ /3 π a^{2}$ . The contact radius follows from the contact force and the elasticity (the Hertz expression for a smooth sphere on a plane), so the force of the beam on the contact sets the on-state resistance, and any oxide or contamination film adds a resistance in series. Reference: R. Holm, Electric Contacts: Theory and Application, 4th ed., Springer, 1967; O. Loh and H.D. Espinosa, Nat. Nanotechnol. 7, 283 (2012).

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