# The Cost of Energy-Efficiency in Digital Hardware: The Trade-Off between Energy Dissipation, Energy–Delay Product and Reliability in Electronic, Magnetic and Optical Binary Switches

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## Abstract

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## Featured Application

**This work has applications in the benchmarking of binary switches for energy-efficient nanoelectronics.**

## Abstract

## 1. Introduction

## 2. Field-Effect-Transistor Switch

_{1}and Q

_{2}) in the channel. The switching action changes the amount of charge from Q

_{1}to Q

_{2}, or vice versa, resulting in the (time-averaged) flow of a current:

_{1}to Q

_{2}(or vice versa). This current will cause energy dissipation of the amount:

_{1}and Q

_{2}and thereby impairs our ability to distinguish between bits 0 and 1. If ΔQ is too small, then thermal generation and recombination can randomly change the amount of charge in the channel by an amount comparable to ΔQ, causing random switching. Therefore, a larger ΔQ translates to both stronger error-resilience and better reliability. This makes it obvious that there is a direct relation between reliability and energy–delay product; if we reduce the energy dissipation or energy-delay product by reducing ΔQ, then we will invariably make the switch less reliable.

_{d}, and larger energy–delay product, E

_{d}Δt, if we desire more reliability (i.e., a larger η) [4].

^{−15}[5]. In modern day FINFETs, the gate capacitance may vary between 10 and 20 aF for a 0.5 μm wide gate [6], and τ is on the order of 100 ps. Using the values in Equations (5) and (6), we get that the minimum energy dissipation and the minimum energy–delay product that we can expect in this type of FET device, while maintaining minimum acceptable reliability, are ~10 aJ and 10

^{−27}J-s, respectively.

## 3. Nanomagnetic Switches

_{c}is the clock period and τ

_{0}is the inverse of the attempt frequency (which is in the range of 1 ps–1 ns) [8].

^{−15}at a clock frequency of 1 GHz, the minimum energy dissipation would be 0.14–0.17 aJ according to Equation (7). We caution that this is an overly optimistic estimate, since it assumes that the minimum energy dissipation needed to switch is the anisotropy energy barrier within the nanomagnet and no consideration has been made of additional energy losses due to Gilbert damping, etc.

**Landau–Lifshitz–Gilbert–Langevin**simulation (also known as stochastic Landau–Lifshitz–Gilbert (s–LLG) simulation). To do this, we solve the following equation:

_{s}and the second to last term is the Slonczewski torque exerted by the same current. The coefficients a and b depend on device configurations and, following [9], we will use the values a = 1, b = 0.3. Here, $\overrightarrow{m}\left(t\right)$ is the time-varying magnetization vector in the soft layer normalized to unity, m

_{x}(t), m

_{y}(t), and m

_{z}(t) are its time-varying components along the x-, y-, and z-axis, respectively (see Figure 2 for the Cartesian axes), ${\overrightarrow{H}}_{\mathrm{demag}}$ is the demagnetizing field in the soft layer due to its elliptical shape, and ${\overrightarrow{H}}_{\mathrm{thermal}}$ is the random magnetic field due to thermal noise [10]. The different parameters in Equation (3) are: $\gamma =2{\mu}_{B}{\mu}_{0}/\hslash $ (gyromagnetic ratio), α is the Gilbert damping constant, ${\mu}_{0}$ is the magnetic permeability of free space, M

_{s}is the saturation magnetization of the cobalt soft layer, kT is the thermal energy, Ω is the volume of the soft layer which is given by $\Omega =\left(\pi /4\right){a}_{1}{a}_{2}{a}_{3}\text{}\left[{a}_{1}=\mathrm{major}\text{}\mathrm{axis},\text{}{a}_{2}=\mathrm{minor}\text{}\mathrm{axis}\text{}\mathrm{and}\text{}{a}_{3}=\mathrm{thickness}\right]$, Δt is the time step used in the simulation, and ${G}_{{}_{\left(0,1\right)}}^{x}\left(t\right)$, ${G}_{{}_{\left(0,1\right)}}^{y}\left(t\right)$ and ${G}_{{}_{\left(0,1\right)}}^{z}\left(t\right)$ are three uncorrelated Gaussians with zero mean and unit standard deviation [10]. The quantities ${N}_{d-xx},{N}_{d-yy},{N}_{d-zz}\text{}\left[{N}_{d-xx}+{N}_{d-yy}+{N}_{d-zz}=1\right]$ are calculated from the dimensions of the elliptical soft layer following the prescription of ref. [11]. The nanomagnet soft layer is assumed to be made of cobalt with saturation magnetization M

_{s}= 8 × 10

^{5}A/m and α = 0.01. Its major axis = 800 nm, minor axis = 700 nm and thickness = 2.2 nm. We assume that the spin polarization in the injected current is ξ, which we take to be 30%. The spin current is given by $\xi {\overrightarrow{I}}_{s}\left(t\right)=\xi \left|{\overrightarrow{I}}_{s}\left(t\right)\right|\widehat{y}$.

^{2}, $2.27\times {10}^{9}$ A/m

^{2}, $1.14\times {10}^{10}$ A/m

^{2}, $2.27\times {10}^{10}$ A/m

^{2}, $3.41\times {10}^{10}$ A/m

^{2}and $4.55\times {10}^{10}$ A/m

^{2}, respectively. We generated 1000 switching trajectories for each current by solving Equation (10). We start with the initial condition ${m}_{x}\left(t=0\right)=0.1,{m}_{y}\left(t=0\right)=-0.99,{m}_{z}\left(t=0\right)=0.1$ and ran each trajectory for 20 ns with a time step of 0.1 ps. After 20 ns, each trajectory ended with a value of ${m}_{y}$ either close to +1 (switching success) or −1 (switching failure). The error probability is the fraction of trajectories that resulted in failure.

_{y}ended up close to +1 (success) or −1 (failure). Again, the simulation duration of 20 ns was sufficient to ensure that for each simulated trajectory, m

_{y}ended up close to either +1 (success) or −1 (failure) at the end of the simulation. One thousand switching trajectories were generated for each pulse width, and the error probability is the fraction of trajectories that result in failure. We observe that the error probability decreased with increasing current pulse width (longer passage of current, or slower switching), as expected.

## 4. All-Optical Switches

_{l}. If the angle of incidence exceeds the critical angle for this pair of media, the ray suffers total internal reflection. Otherwise, it is refracted into the non-linear medium. The critical angle is given by the expression:

_{i}, the minimum intensity needed to switch is given by the relation:

_{i}exceeding θ

_{c}, (i.e., with the difference θ

_{i}− θ

_{c}). Therefore, the probability will go up rapidly with increasing value of the quantity $\underset{\mathrm{fixed}}{\underbrace{{\mathrm{sin}}^{-1}\left(\frac{{n}_{0}^{(2)}-{n}_{2}^{(2)}{I}_{\mathrm{min}}}{{n}^{(1)}}\right)}}-{\mathrm{sin}}^{-1}\left(\frac{{n}_{0}^{(2)}-{n}_{2}^{(2)}I}{{n}^{(1)}}\right)$, which shows that the probability of successful switching increases with increasing intensity I, or increasing energy dissipation. Thus, there is once again a trade-off between energy dissipation and reliability.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Conduction band profile along the channel of a field effect transistor in the OFF state (solid line) and ON state (broken line).

**Figure 2.**A nanomagnet shaped like an elliptical disk has two stable magnetization orientations which can encode the binary bits 0 and 1. (

**a**) In-plane magnetic anisotropy and (

**b**) perpendicular magnetic anisotropy. (

**c**) A magnetic tunnel junction (MTJ) showing the high (OFF) and low (ON) resistance states.

**Figure 3.**Switching error probability as a function of injected current magnitude. The energy dissipated is proportional to the square of the current. The current was kept on for the entire duration of the simulation, which was 20 ns.

**Figure 4.**Switching error probability as a function of current pulse width (i.e., the duration of spin–transfer–torque). The current strength was kept fixed at 10 mA.

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**MDPI and ACS Style**

Rahman, R.; Bandyopadhyay, S. The Cost of Energy-Efficiency in Digital Hardware: The Trade-Off between Energy Dissipation, Energy–Delay Product and Reliability in Electronic, Magnetic and Optical Binary Switches. *Appl. Sci.* **2021**, *11*, 5590.
https://doi.org/10.3390/app11125590

**AMA Style**

Rahman R, Bandyopadhyay S. The Cost of Energy-Efficiency in Digital Hardware: The Trade-Off between Energy Dissipation, Energy–Delay Product and Reliability in Electronic, Magnetic and Optical Binary Switches. *Applied Sciences*. 2021; 11(12):5590.
https://doi.org/10.3390/app11125590

**Chicago/Turabian Style**

Rahman, Rahnuma, and Supriyo Bandyopadhyay. 2021. "The Cost of Energy-Efficiency in Digital Hardware: The Trade-Off between Energy Dissipation, Energy–Delay Product and Reliability in Electronic, Magnetic and Optical Binary Switches" *Applied Sciences* 11, no. 12: 5590.
https://doi.org/10.3390/app11125590