Energy-Reversible Complementary NEM Logic Gates Kerem Akarvardar, David Elata, Roger T. Howe, H.-S. Philip Wong

Center for Integrated Systems, Stanford University, Stanford CA 94305, kerem@stanford.edu

Energy-reversible complementary nanoelectromechanical (ER CNEM) logic gates are introduced. For the

same delay, ER CNEM gates can operate at much lower supply voltages relative to conventional (CMOS-like) CNEM

gates and their reliability is significantly higher. NEM relays are attractive candidates for ultra-low power computation since their leakage is practically zero.

Indeed, there is a growing interest on realizing logic and memory functions by using NEMS-based devices [1-3]. The

characteristics of conventional (MOSFET-like) NEM relays (Fig. 1) and CMOS-based CNEM gates were analyzed in

[4]. The main drawbacks of conventional NEM relays were found to be: 1. long settling time after turn-off due to high

quality factor (Q >> 1), and 2. cantilever tip bouncing during turn-on due to high impact velocity. The ER logic gates

proposed in this work solve these problems while enabling smaller supply voltages at the same delay.

The layouts of the ER CNEM inverter and NAND gate (following the bi-stable RF switch design [5]) are

shown in Fig. 2. The structures feature, for each input, a NEM cantilever beam, which may deform laterally. In the ER

CNEM inverter, the cantilever shorts the output to VDD for Vin = 0 and to GND for Vin = VDD (Fig. 3). In the ER CNEM

NAND gate (Fig. 2b), for A or B = 0, the output is connected to VDD by one or both beams. However the connection of

the output to GND is only possible when A = B = 1. An ER CNEM NOR gate is obtained by swapping the GND and

VDD electrodes of the NAND gate.

The main advantage of the circuits in Fig. 2 is that, once the mechanism is initialized (i.e., once the beam is

pulled in towards one of the electrodes by applying a sufficiently high DC voltage or, preferably, an AC voltage at the

resonance frequency [5]) the beam can travel from one electrode to the other using the potential elastic energy that was

stored due to bending (Fig. 3). This energy (that would be dissipated by damping in a conventional relay after the beam

is released [4]) is reversible, since the beam is bent at both stable states. Electrostatic force is not required for the beam

displacement as in the conventional relay. As a result, VDD can be substantially reduced, since its role is not to actuate

(pull-in) the beam but merely to hold the beam bent (in contact) by inducing an electrostatic force that adds to attractive

surface forces (van der Waals). Also, in the ER configuration, the settling problem of the conventional relay is solved,

since at steady-state the cantilever tip is necessarily in contact with the GND or VDD electrode.

By assuming a 1D structure (Fig. 4) and realistic dimensions (achievable by the current NEMS technology

[6]), we compared the characteristics of the conventional NEM relay with those of the ER NEM inverter. As shown in

Fig. 5, in the ER NEM inverter, VDD can be as small as the pull-out (hold-down) voltage, Vpo, while the conventional

relay requires a VDD larger than the pull-in voltage, Vpi (in our example Vpo = 0.5 V, Vpi = 2.24 V).

When the damping is taken into account, VDD needs to be larger than Vpo in the ER inverter in order to

compensate the related energy loss [5]. For the dimensions we consider, the quality factor is 18.2 and VDD should be

increased by 90 mV above Vpo in order to compensate the damping loss. The ER inverter with VDD = 0.59 V and the

conventional relay with VDD = Vpi = 2.24 V achieve the same delay (10.8 ns, Fig. 6a). However, in the ER inverter the

beam velocity decreases gradually towards the end of travel leading to a much lower impact velocity (Fig. 6b) and

minimizing the tip bouncing.

The fundamental difference in the principle of operation between the two types of relays is apparent in Fig. 7

where the variation of the energy components during the switching is shown. In the conventional relay, the switching

work is done by the power supply and the energy drawn from the voltage source dominates all other components (Fig.

7a). In contrast, in the ER CNEM inverter, the largest component is the stored elastic energy (Fig. 7b). When the beam

is released from one of the electrodes, the potential elastic energy is first transformed to kinetic energy and then back to

elastic energy when the beam arrives at the opposite electrode. The substantial difference between the stored elastic

energy and the source energy (Fig. 7b) shows that the switching work is done by the elastic potential energy rather than

the voltage source. The performance of the conventional and ER relays is compared in Table 1. For the same delay, the

ER NEM inverter outperforms the conventional relay in terms of voltage, energy, and reliability.

In the ER CNEM gates, the VDD (@ Vpo in vacuum operation) can, in principle, be reduced below 100 mV as

illustrated in Fig. 8. However this would only be possible at the expense of a very high sensitivity to dimensions, which

would reduce the process margin.

In summary, we introduced for the first time the ER CNEM logic gates and demonstrated their superiority

over the conventional relays in terms of voltage, energy, and reliability. The ER CNEM logic a promising candidate for

low standby and low operating power applications. References: [1] Q. Li et al., Nanotechnology 18, 315202, 2007. [2] W. Y. Choi et al., IEDM, pp. 603-606, 2007. [3] J. E. Jang et al., Nature

Nanotechnology, 3, pp. 26-30, 2008. [4] K. Akarvardar et al., IEDM, pp. 299-302, 2007. [5] H. Yang, L. Pakula, P. J. French, 14th European

MicroMechanics Workshop, pp. 33-36, 2003. [6] M.-S. Kim et al., ISDRS, pp. 1-2, 2007. Acknowledgments: DARPA and FCRP C2S2.

978-1-4244-1942-5/08/$25.00 ©2008 IEEE 69

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Fig. 1. Cross-section of the

conventional (MOSFET-like)

NEM relay.

out = in

in

VDD

GND

insulator cantilever

floating

electrode

out = AB

A

VDD

VDD

GND

B

Fig. 2. Top view of the ER CNEM logic gates: (a)

inverter, (b) NAND gate. Input current is zero, due to the

insulating region near the beam tip. The floating electrode

in (b) serves to obtain a conductive path between GND

and the output only when VA = VB = VDD.

out = VDD

in = GND

GND

VDD

out = GND

in = VDD

VDD

GND

Fig. 3. Cantilever profiles at steady-

state in the ER CNEM inverter.

S DG

air gap

cantilever beam

S

insulating substrate

Fig. 4. 1D structure used for the

analysis of the ER CNEM inverter. L

= beam length, W = beam width, h =

beam thickness, g0 = half-gap, g1 =

minimum (2-D equivalent) beam to

electrode distance determined by the

tip configuration of the cantilever (in

Figs. 2 and 3), k = equivalent spring

constant, b = damping coefficient.

(a) (b)

elastic energy storage due to bending

Table 1. Comparison between the ER

inverter and conventional relay for

the parameter set in Figs. 5 and 6.

Fig. 6. Step voltage response of the

normalized beam position (a), and the

beam velocity (b) for the ER NEM inverter

and the conventional relay, both using the

parameters in Fig. 5. The step voltage

amplitude is VDD = 0.59 V in ER inverter

and VDD = 2.24 V = Vpi in the conventional

relay. In (a), the distance is normalized to

2g0-2g1 for the ER inverter, and to g0-g1 for

the conventional relay. The delay is the

same for both structures (10.8 ns).

Characteristics take into account

electrostatic, elastic, van der Waals and the

damping forces (Q = 18.24, W = 100 nm).

Fig. 7. Step voltage response of the

energy components: (a) conventional

relay, (b) ER CNEM inverter.

Parameters as in Figs. 5 and 6.

Time (ns)

Energy (fJ)

source

elastic

kinetic

vdW

damping

Time (ns)

Norm

alized Beam Position

Conventional

Energy-

Reversible

Time (ns)

Velocity (m/s)

Conventional

Energy-reversible

impact velocity

Time (ns)

Energy (aJ)

source

elastic

kinetic

vdW damping

Fig. 5. Normalized beam position as a function

of VGS in a conventional relay. The behavior is

predicted by a 1-D model (consisting of half of

the structure in Fig. 4) and taking into account

electrostatic, elastic, and van der Waals (vdW)

forces. Silicon beam is assumed. g0 = 25 nm, h

= 25 nm, L = 650 nm, g1 = 4.5 nm, (k = 0.645

N/m). The addition of a second fixed electrode

to the conventional relay as in Fig. 4 enables to

reduce the minimum operating voltage, VDD,

from Vpi (2.24 V) to Vpo (0.5 V).

Fig. 8. Variation of the minimum supply voltage

(= Vpo) as a function of the minimum electrode

to beam distance, g1, in the ER CNEM inverter

whose parameters are given in Fig. 5 (damping

force is neglected). VDD can be reduced below

100 mV (leading the holding force be dominated

by the van der Waals forces) at the expense of a

very high sensitivity to dimensions.

(a)

(b)

(a)

(b)

g1 (nm)

Minim

um V

DD(V

)

2

GND

VDD

g0

cantilever

L

g1g1

g0 h

k

W

limit stops

b

VDD

VpiVpo

Gate to Source Voltage, VGS (V)

Norm

alized Position

g1/g0

2/3

VDD

VpiVpo

Gate to Source Voltage, VGS (V)

Norm

alized Position

g1/g0

2/3

640 aJ44.5 aJSource Energy

555 nA16.9 nAMaximumDisplacement

Current

8.72 m/s0.49 m/sImpact Velocity

2.24 V0.59 VSupply Voltage

10.8 ns10.8 nsDelay

ConventionalEnergy-

Reversible

640 aJ44.5 aJSource Energy

555 nA16.9 nAMaximumDisplacement

Current

8.72 m/s0.49 m/sImpact Velocity

2.24 V0.59 VSupply Voltage

10.8 ns10.8 nsDelay

ConventionalEnergy-

Reversible

978-1-4244-1942-5/08/$25.00 ©2008 IEEE 70

Authorized licensed use limited to: National University of Singapore. Downloaded on February 10, 2010 at 22:34 from IEEE Xplore. Restrictions apply.