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This paper qualitatively explores the performance limits,

Adiabatic logic circuits were introduced as a possible means to achieve ultra-low power circuits by taking advantage of the adiabatic charging principle [

On the other hand, electrostatic nanoelectromechanical (NEM) switches have already been suggested and demonstrated for use in classical logic circuits [

Nonetheless, NEM switches are ideal candidates to replace classical CMOS elements in adiabatic logic circuits [

In this paper, the basic principles of adiabatic circuits will be presented, the dissipation of adiabatic circuits will be explicitly derived based on a simplified circuit model and compared for three different device technologies, which are: the classical CMOS-based circuit, the sub-threshold CMOS-based circuit and NEMS-based circuit respectively. It will be shown that replacing a MOSFET switch by an ideal electromechanical one does contribute in reducing significantly the power dissipation. It will also be demonstrated that the contact resistance of electromechanical switches may well be a limiting factor on the performance of NEM-based adiabatic logic circuits. This paper aims to present a modeling based results that gives to a good approximation the qualitative behavior of NEMS-based adiabatic logic circuits.

This paper starts by introducing the principle of adiabatic charging and how it translates to energy saving in logic circuits in

The operation of a logic circuit typically consists of charging a capacitor through a series resistance, where the capacitor is mainly that of the fan out interconnect and the resistance is dominated by the switching element series resistance.

In conventional circuits, the capacitor charging is done under constant voltage, which is the circuit operating voltage _{dd}

In adiabatic charging, a slowly varying voltage source, usually a linearly ramped voltage, is used to charge and discharge the load capacitance [

A simplified model of a typical adiabatic logic circuit is schematically represented in _{S}

Schematic representation of an equivalent logic circuit showing the four phase power clock with equal length segments: _{S}_{L}_{S}_{L}_{1}; and that going into the variable capacitance of the NEMS switch (_{S}_{2}.

The sources of dissipation in a CMOS-based adiabatic circuit can be attributed to either adiabatic or non-adiabatic residues. While it is possible to reduce the adiabatic residues by increasing the ramp-up time

The exact performance of CMOS-based adiabatic circuits depends on the exact circuit design; a derivation is presented based on the generic logic circuit diagram shown in

The energy dissipated in a complete charge-discharge clock cycle, including the mean non-adiabatic losses, is given by the following equation [_{dd}

In Equation (2), the first term represents the adiabatic residues of a charge-discharge cycle, while the second one represents the energy dissipation due to the sudden discharge of the residual output voltage [_{leak}

Schematic representation of a CMOS implementation of an adiabatic logic function

The main leakage current component in a MOSFET is due to the sub-threshold leakage current _{leakage}_{0} is a function of the transistor size and parameters; _{t}_{DS}

Considering the four phases of the power clock shown in _{leakage}

Furthermore, the series resistance _{S}_{n} and C_{n}_{th}

By replacing the right hand sides of Equations (4) and (5) into their respective terms in Equation (2), the energy dissipation of a CMOS-based adiabatic circuit may be expressed as:

The energy dissipation as expressed in Equation (6) admits an optimum operating period _{optimum}_{th}

Combining adiabatic logic and sub-threshold mode may provide interesting energetic performance if low frequency operation is allowed. Calculation of energy is more complex in the case of sub-threshold CMOS, but by using Equation (3), it is possible to obtain the following approximate equation for the 2N-2P adiabatic gate:
_{t}

If low frequency mode is allowed (less than 1 MHz), comparison of Equations (6) and (7) defines conditions where sub-threshold mode is advantageous. Equation (7) is also plotted in

Comparison between the performance of adiabatic circuits using: conventional CMOS (solid red line) as given by Equation (2), and sub-threshold CMOS (dashed blue line) as given by Equation (7), both done for the same device parameters. The non-adiabatic residue in classical CMOS circuit is also shown for comparison (solid black line).

Nanomechanical switches are devices that rely on a beam, cantilever, or a membrane to deform under the effect of electrostatic force in order to make and break electrical contact upon the application of an external voltage. NEMS switching elements offer the property of zero leakage current [

A typical 1-dimensional reduced order model of a 3-terminal electrostatic NEMS switch, where the structure is modeled by a simple parallel plate capacitor, is schematically represented in _{pi}_{po}_{contact}_{pi}_{po}

The total energy dissipation _{Total}_{Total} = E_{Electrical} + E_{Mechanical}_{Electrical}_{Mechanical}

Based on the equivalent circuit in _{Electrical}_{1} and _{2} are the currents going through the series resistance _{S}_{L}_{S}

From Equations (9) and (10), one can remark that the NEMS parameters governing the dissipation in an adiabatic circuit are the series resistance _{S}_{dd}

Schematic illustration of a reduced order model of a nanoelectromechanical switch (

Therefore in order to obtain a comprehensive description of dissipation in NEMS-based adiabatic logic circuits, it is necessary to inject values for the switch series resistance and its transient behavior into Equations (9) and (10). These values are obtained from realistic contact mechanics models and dynamics NEMS models respectively, and are explicitly detailed below.

The series resistance of an electromechanical switch is in fact dominated by the resistance at the contact between the movable and fixed electrodes [

Recent literature review of electromechanical contact in nano- and microelectromechanical systems [_{applied}

Several models exist to describe the resulting interplay between the mechanical deformation of the asperities and the electrical behavior of the contact. Here, two common contact mechanics models will be considered, the Hertz contact theory [

The Hertz contact assumes perfectly elastic deformations of two asperities, while at the same time neglecting any adhesion forces that may exist at the interface between the asperities: the radius _{H}_{1}_{2}/(_{1} + _{2}) where _{1} and _{2} are the radii of the first and second asperities respectively; and ^{*}_{1}, ν_{1}, _{2}, ν_{2}, are the Young moduli and Poisson ratios of the first and second asperities respectively.

In a reduced order model approximation, the force _{applied}_{0} is the minimum electrostatic contact force when _{dd}_{pi}_{0} and α depend on the device design and fabrication. Note that Equation (12) only applies once contact is established, _{dd}_{contact}

The Johnson-Kendall-Roberts (JKR) contact model on the other hand accounts for possible adhesion forces while also considering a purely elastic contact. A first order approximation of the contact radius in the JKR model is given by:
_{JKR}

The electrical contact resistance given by Maxwell (_{Maxwell}_{Sharvin}_{e}

The expressions of the contact deformation and the contact resistance given by Equations (11) through (14), when combined together, give four electro-mechanical contact models. These contact models can be injected into Equation (9) to determine the effects of contact resistance on the total energy dissipation. The impact of these four models on the dissipation and performance of NEMS-based adiabatic logic circuits will be explored in the results and discussion section.

Schematic illustration of two asperities in contact, deformed under the effect of an applied load.

The switching behavior of nanoelectromechanical switches will take on an important role in determining the losses in a NEMS-based circuit: this is due to the fact that the device capacitance, _{S}_{0} as the NEMS switch capacitance in the initial position,

The question of the time dependent response of a NEMS switch has been treated to some length in literature for the case of step voltage actuation, which is the type of actuation envisioned for classical circuits; see for example [

However, the voltage waveform in an adiabatic logic circuit is, by definition, required to be slowly varying. This constitutes the biggest difference between classical and adiabatic circuits with respect to NEMS dynamics. This fact results in an interplay between electrical and mechanical time constants, where the mechanical time constant is defined as the switching delay of the NEMS device. Therefore, two limiting cases will be considered separately, depending on the relations between the mechanical time constant (_{Mech}_{Elec}

_{Mech}_{Elec}_{dd}

The electrical dissipation, in this case, may be obtained by decoupling the electrical and mechanical time scales by rewriting Equation (9) as follows:
_{Mech}_{0}. While the second term represents the effect of a varying switch capacitance, which takes place upon mechanical commutation, under the effect of the now stable bias voltage _{dd}

The transient current in Equation (17b) and the resulting dissipation were derived explicitly in [

By combining Equations (16) and (17b), and including the mechanical losses, the total energy dissipated per switching cycle may be expressed as:
_{0} is the mechanical resonance frequency of the spring-mass system that models the NEMS switch. The

Schematic representation of the necessary clock signal for NEMS-adiabatic circuits working in the limit of (_{Mech}_{Elec}

_{Mech}_{Elec}

It is important to note that, for this case, if the contact is to be established beyond pull-in, then a transient dissipation term will have to be introduced again into the dissipation equations, therefore negating the advantages that are sought to be obtained under this mode of operation. Therefore, for this case, only switches operating before pull-in will be considered.

To obtain the response of a NEMS switch, and hence the dissipation, it is necessary to solve the following governing normalized nonlinear differential equation [

Equation (19) can only be solved numerically, and it is therefore difficult to provide a general solution for different values of ramp-up periods.

An approximate solution is therefore derived based on the hypothesis that the nanomechanical structure is moving slowly compared to its mechanical time constant (which is the underlying assumption for this case). More precisely, if _{0}, where

From what is already introduced in the previous section, it is clear that the two most influential features of NEMS-based adiabatic circuits are: the contact resistance and the dynamic response of the switch.

The dynamic response of the switch and the way it affects power consumption are related to the mode of actuation of the switch, where a quasi-static actuation,

The difference between the two operation modes in terms of power dissipation as a function of frequency is plotted in _{0}_{dd}_{L}_{0}, a value chosen by industry standards [

Comparison of the performance of NEMS-based adiabatic circuits shown for: the dynamic mode of operation (black line) as given by Equation (18a–c), and the quasi-static mode of operation (red line) as given by Equation (20), both done for _{L}_{0}.

The first difference between the two operating modes shown in

The contact resistance is a critical feature of a NEMS switch as it dictates energy dissipation and plays a direct role as it sets the

The energy dissipation of a NEMS-based adiabatic circuit operating in the dynamic mode is calculated and plotted in ^{2} [^{−9} Ω·m, and _{e}

Even though these classical contact models are used to simulate micrometer scale contacts, recent literature considers them to be inadequate to address nanometer scale contact, and expects the actual contact resistance to be more sophisticated to simulate and to present larger values than those actually given by the above models [

Therefore, contact resistance in nanomechanical relays remains a key parameter that dictates the adiabatic energy saving term in adiabatic logic circuits, and a parameter that needs further theoretical and experimental investigation.

Plots of the contact resistance, described by Equations (11–14b), given by the different models on the performance of NEMS-based adiabatic circuits operating in the dynamic mode. The quasi-static mode is similarly affected (not shown) by the value of contact resistance values.

At this point it is worth mentioning that having a supply voltage that is higher than the pull-in voltage, _{dd}_{pull-in}

In fact a certain amount of voltage overdrive should be expected in any nanoelectromechanical integrated circuit, since fabrication variability will undoubtedly results in dispersion of the value of pull-in voltage. Therefore, in any NEMS-based logic circuit certain devices will be slightly overdriven, as it is the case with CMOS logic circuits. While this voltage overdrive does not interfere with the circuit function, it does result in an increased energy dissipation as given by equations (18a–c) and (20) depending on the mode of operation of the circuit.

Finally, we would like to provide a numerical example that compares directly the expected performance of CMOS-based and NEMS-based adiabatic (operating in the “dynamic mode”) logic circuits. In this example we compare the performance of 45 nm CMOS adiabatic circuit with a 1 fF load capacitance to that of a NEMS switch with the following properties: device capacitance = 0.1 fF, pull-in voltage = 1 V, switching time > 1 ns, feature size = 50 nm, contact resistance = 10 KΩ, and a load capacitance of 1 fF.

The NEMS device parameters listed above correspond to the best projected parameters as expected by the ITRS roadmap [

Plots comparing the performance of the classical (black) and sub-threshold (blue) CMOS adiabatic logic circuits (for 45 nm technology node and a 1fF load capacitance), and the performance of a NEMS-based adiabatic logic circuit (also with a 1fF load capacitance).

In summary, in this work, analytical expressions based on simplified circuit models were derived to obtain the performance limits of adiabatic circuits based on conventional CMOS, sub-threshold CMOS and NEMS based devices. Comparisons show that CMOS implementations are plagued by their leakage currents: when operating under a given frequency, the energy dissipation starts increasing again, and this remains the case for CMOS circuits designed and operated in the sub-threshold regime. It should be noted that the energy-optimum of sub-threshold CMOS is lower than that of a classical CMOS, but at high frequencies energy dissipation strongly increases.

It is also shown that for NEMS-based adiabatic circuits, an entire spectrum of possible operating regimes is available, from which two limiting cases were considered. In the first mode, the mechanical structure is assumed to be actuated similarly to a classical,

Furthermore, this paper identified several critical nanomechanical parameters that affect circuit behavior. In particular, the mechanical energy injected into the system limits the performance of NEMS switches operating in the dynamic mode, while adhesion energy will play a limiting role on the performance of quasi-statically operated switches. In both cases, it was also observed that the contact resistance of the NEMS device sets the magnitude of the dissipated energy; the full impact of contact resistance should be subject to further investigation, especially as it is correlated with adhesion forces.

Finally, it is of interest to note that the energy dissipated by NEMS-based adiabatic circuits is expected to be significantly lower than is the case of CMOS-based circuit, especially if low to medium circuit operating frequency is desired, a frequency range where their CMOS counterpart performs poorly due to static power loss.

The authors would like to thank Mr. Alexis Peschot for his help with the experimental characterization of the electromechanical switches.

This work was funded by the action Carnot AdiaMéca.

The authors declare no conflict of interest.