set full adders final.doc

On Single-Electron Technology Full Adders

Mawahib H. Sulieman1, Member, IEEE, and Valeriu Beiu2, Senior Member, IEEE

1 College of Engineering and 2 College of Information TechnologyUnited Arab Emirates University, Al-Ain, P.O. Box 17555, UAE

[email protected] and [email protected]

Abstract — This paper reviews several full adder (FA) designs in single electron

technology (SET). In addition to the structure and size (i.e. number of devices), this paper tries

to provide a quantitative and qualitative comparison in terms of delay, sensitivity to (process)

variations, and complexity of the design. This will allow for a better understanding of the

advantages and disadvantages of each solution. An optimization of a SET FA (combining one

of the SET FAs with a static buffer), together with a new SET FA design (based on capacitive

SET threshold logic gates), will also be described and compared with the other SET FAs.

Index Terms — Full adders, sensitivity to variations, single electron technology,

threshold logic.

I. INTRODUCTION

The scaling of Complementary Metal Oxide Semiconductor (CMOS) transistors has so far

provided lower cost and higher performance circuits. However, further progress of integration

scale will be hindered by a variety of physical effects. The most important effects are the

increase in power consumption and the decrease in reliability [1]–[4]. These problems have

motivated an active search in quest for both short- and long-term solutions. The former include

non-classical CMOS (e.g., FinFET and vertical MOSFET [5], as well as SOI [6]), while the latter

M. Sulieman and V. Beiu: SET FAs 1

mailto:[email protected]

mailto:[email protected]

include several emerging technologies (see [1], [2]). Single-electron-technology (SET) is one of

these new technologies, distinguished by a very small device size and ultra-low power

dissipation. While promising for helping solve the power consumption challenge, SET is

expected to be sensitive to variations (like many other nanodevices [1], [2]), and to background

charge [7]. Such problems might be tackled by using fault-tolerant designs [8], but a closer

understanding of SET’s sensitivity to variations is clearly needed. Still, room temperature SET

devices have been demonstrated [9]–[15], while the low gain of SET devices suggest that they

should be combined with CMOS [7], [16]–[18], or even with resonant tunneling devices (RTDs),

and might lead to hybrids like SET-CMOS, SET-RTD, or even SET-CMOS-RTD. That is why, a

better understanding of the SET (e.g., by including sensitivity to variations) is considered both

timely and useful.

One of the SET logic circuits which has recently been presented in several studies is the

classical full adder (FA) [19]–[22]. By reviewing these articles it was apparent that SET FA

structures differ both in their mode of operation, and in the number of elementary components.

However, delays were reported for only two (out of the four) designs, while sensitivity to

variations was not investigated. The importance of such performance measures have strongly

motivated this work which provides both qualitative and quantitative comparison of the reported

SET FAs. Further, a modified SET FA (combining one of the SET FAs already reported [22]

with a static buffer), and a new SET FA design [23] are also presented. The first approach

modifies the SET FA introduced in [22], by using a static buffer. [Why a static buffer is needed]

The use of a static buffer was suggested in [24], where it was used only for implementing

Boolean gates. The same static buffer was recently used in [25] for implementing latches and

flip-flops, but the threshold logic gates (TLGs) used are different from the ones used in the SET


FA introduced in [22]. We have determined the optimal capacitors values for the SET FA from

[22] when using a static buffer, and the results are reported in this paper. Finally, another (novel)

SET FA design augments the one presented in [19] in a way that simplifies its structure and

reduces the total number of components. To obtain quantitative measures of performance, all the

SET FAs were simulated under identical conditions using SIMON [26], a widely used Monte

Carlo simulator for single electron devices and circuits that takes cotunneling into account.

The simulation results, together with the SET FA structures, were used to compare all these

SET FAs in terms of:

The number of junctions and capacitors, which could affect an important characteristic of

future nanoelectronics, their overall reliability [1], [2], [4].

The delay.

The “design complexity,” estimated in terms of control and voltage supply requirements.

The sensitivity to variations.

The remainder of the paper is organized as follows. Section II presents the background

material about FAs and threshold logic gates (TLGs). The different types of SET FAs are

described in Section III. The simulation procedure for characterizing the SET FAs is described in

Section IV, with Section V providing both quantitative and qualitative comparisons of the SET

FAs. Concluding remarks are provided in Section VI.

II. FULL ADDERS AND THRESHOLD LOGIC

The FA is a basic arithmetic block which has three inputs: the addend a (1-bit precision),

the augend b (1-bit precision), and the carry-in ci; and two outputs: the sum s, and the carry-out

co. An established method for computing s and co is


s = (a b ci), (1)

co = (a b) (a ci) (b ci), (2)

where we have used the following notation: for logic AND, for logic OR, and for logic XOR,

while x’ will represent the complement of a signal x.

The classical FA designs are based on transmission gates, complementary pass-transistor

logic and static CMOS (see [27] for an in-depth review of many CMOS FAs). However, most of

the SET FA solutions are based on TLGs.

A TLG is the simplest artificial neuron which computes the weighted sum of its inputs

(Σwixi), and compares this sum with a threshold value θ. If the sum is larger than the threshold θ,

the TLG outputs a one, otherwise the output will be zero. Mathematically, a TLG implements a

linearly separable function

, (3)

where wi is the weight associated with xi, θ the threshold, and n the fan-in [28]. In practice, the

(integer) threshold θ is reduced by 0.5 as the simplest method to improve on the noise margins,

but more evolved methods are possible (see [29]–[31]). In the remainder of this paper, a TLG

will be represented by its series of weights and practical threshold (w1,…,wn; θ – 0.5). A

particular TLG having all the weights equal, w1 = w2 = … = wn = 1, for n = 2k + 1 (for any

integer k ≥ 1) is known as a majority gate (MAJ).

To implement FAs using only TLGs, the sum and carry-out functions have to be

represented as linearly separable functions. The carry-out co does satisfy this condition, and can

be written as

co = sign (a + b + ci – 1.5), (4)

which is a MAJ function (w1 = w2 = w3 = 1, and θ = 1.5). On the other hand, the sum function s as


expressed in (1) is not a linearly separable function. However, it can be represented in a linearly

separable form by writing it in terms of both the carry-in (ci) and carry-out (co)

s = (a b ci) [co’ (a b ci)], (5)

which can be written as

s = sign (a + b + ci – 2co – 0.5), (6)

or equivalently (1,1,1,–2; 0.5). By simple algebra, any negative weight can be changed to a

positive weight (by inverting the associated input and modifying the threshold [28]), hence the

function can be written as

s = sign (a + b + ci + 2co’ – 2.5). (7)

This (1,1,1,–2; 0.5) solution was presented in [32], used later in [33], and rediscovered in [34].

One way to obtain the threshold equations (6) and (7) from the Boolean equation (5) is as

follows:

Equation (5) is written in a simplified sum-of-product form:

s = (a b ci) (co’ a) (co’ b) (co’ ci).

If a, b, ci are assigned a weight of 1, co’ should be assigned a weight of 2 to make all the

product terms equally weighted. This weight can be negative (using co) or positive (using

co’). Hence the threshold equation becomes either s = sign (a + b + ci – 2co – θ1), or

s = sign (a + b + ci + 2co’ – θ2).

To determine the thresholds θ1 and θ2, two weight maps, in addition to the Karnaugh map, are

created (see Fig. 1).

o In the Karnaugh map (Fig. 1(a)), don’t care (Φ) symbols are assigned to all the unfeasible

combinations of a, b, ci, and co.


o The weight map in Fig. 1(b) shows the weighted sum-of-inputs using a negative weight

for co: a + b + ci – 2co.

o The weight map in Fig. 1(c) shows the weighted sum-of-inputs using a positive weight

for co: a + b + ci + 2co’.

Comparing each of these two weight maps with the Karnaugh map, it is clear that:

o In Fig. 1(b) logic 0 corresponds to 0, and logic 1 corresponds to 1, hence the threshold is

θ1 = 0.5.

o In Fig. 1(c) logic 0 corresponds to 2, while logic 1 corresponds to 3, hence the threshold

is θ2 = 2.5.

Having all the weights and thresholds completes the specification of the threshold equations (6)

and (7). In practice, it is normally necessary to obtain either one of these equations, and the other

one can be obtained by simple algebra. Using this procedure for both of them here is only for

illustration purposes.

III. TYPES OF SINGLE-ELECTRON TECHNOLOGY FULL ADDERS

The first two types of the SET FAs proposed are based on MAJ gates [19], [20], but differ

in the way the MAJ gates are implemented. The third SET FA is based on pass-transistor logic

(PTL) [21], and the fourth one is based on TLGs [22]. The fourth one will be modified by

matching the TLG gate detailed in [22] with a static buffer, and optimizing the capacitor values.

The new SET FA we present in this article is also TLG based [23]. The different types of SET

FAs are described in the following subsections. The description will be accompanied by

simulation results to verify the functionality of each SET FA. All the simulations were done

using SIMON [26]. The input signals used in these simulations cover all the eight possible


combinations of the inputs a, b, and ci, as shown in Fig. 2, where the voltage levels (0 mV and

6.5 mV) correspond to the SET FA introduced in [19].

[We need to explain a bit how each FA functions … 1-2 paragraphs each ?!]

A. Single-Electron Transistor Majority Full Adder

In 1998, Iwamura et al. [19] introduced a SET FA based on MAJ gates (MAJ-SET). This

SET FA consists of three MAJ gates and two inverters (Fig. 3(a)). One MAJ and an inverter

produce the carry-out co, while the combination of all the gates produces the sum s. The MAJ

gate configuration (Fig. 3(b)) is based on the SET inverter proposed by Tucker [35], and

produces the complement of the MAJ function. However, it will still be referred to as MAJ for

simplicity. Starting from the inverter configuration proposed by Tucker [35], the input

capacitance is divided into three equal capacitances, which are connected to three inputs to create

a MAJ gate (see Fig. 3(b)). The MAJ consists of an input capacitor array (six input capacitors

C = 1 aF) for input summation and an inverter (four tunnel junctions Cj1 = Cj4 = 1 aF,

Cj2 = Cj3 = 2 aF; two bias capacitors Cb1 = Cb2 = 9 aF; and a load CL = 24 aF) for threshold

operation. This SET FA is quite simple, using only one voltage supply (VDD = 6.5 mV), and does

not require any external controls (i.e., other voltages or clock signals). The complete circuit

comprises 20 tunneling junctions and 37 input and bias capacitors. The sum output s of this SET

FA is shown in Fig. 4. The waveform is regular, and the output swing is comparable to the input

swing (Vi = 6.5 mV). The original paper reports a mean delay of about 0.3ns, and mentions that

for achieving small errors the FA must be operated at very low temperatures: below 0.2 K for no

errors in 100 trials, while the FA will make about 1 error at 0.5 K and about 10 errors at 1 K (in

100 trials).


B. Single-Electron Box Majority Full Adder

The MAJ implementation has been revisited in 2003 by Oya et al. [20]. In this design, the

MAJ has been implemented as an irreversible single-electron box (SEB) leading to a MAJ-SEB

FA design. The SEB has a simple structure and comprises a smaller number of devices than a

SET inverter. However, it requires additional control at the circuit level because it is

bidirectional. Figure 5(a) shows the MAJ-SEB FA. It consists of three MAJ gates, which

correspond to the three MAJs in Fig. 3(a). The other gates act as delay and/or fan-out buffers.

Figure 5(b) shows a three-input MAJ gate based on a double-junction SEB. It consists of two

tunneling junctions connected in series (Cj = 20 aF, Rj = 200 KΩ), a bias capacitor Cb = 2 aF, an

output capacitor CL = 2 aF, three input capacitors C = 2 aF, and a bias voltage i. Three input

voltages V1, V2, and V3 are applied through the input capacitors C to the internal node int. The

input capacitors form a voltage summing network, and produce the weighted sum of their inputs

on the internal node int. The SEB produces the complementary MAJ output on the same node.

Using SEB, instead of SET inverter, reduces the number of tunneling junctions from 20 (MAJ-

SET) to 14 (MAJ-SEB), and the number of capacitors from 37 (MAJ-SET) to 29 (MAJ-SEB).

Since the proposed MAJ-SEB gate is bidirectional, the signal flow has to be controlled. A

three-phase clock has been used for this. The circuit is divided into three layers, each layer being

excited in turn by one of the three clock phases 1, 2, 3 (Fig. 5(a) and Fig. 6(a)). The clock is a

two-step pulse which is first set to an excitation voltage of 60 mV, and then to a holding voltage

of 40 mV (see Fig. 6(a) and [20]). The three clock phases should overlap, so that the output of

one stage will be evaluated while the preceding stage is still holding its outputs. The inputs are

return-to-zero signals with logic ‘1’ encoded as 4 mV and logic ‘0’ as – 4 mV (Fig. 6(b)). The

inputs are received while 3 is in the excitation period. The carry-output co is produced when 2


goes high, and is retrieved by the output buffer while 2 is in the holding period. The sum output

s is produced when 3 goes high again. The carry-out delay is two-thirds of a clock period, and

the sum delay is one clock period. The sum output s of this SET FA is shown in Fig. 7. It has a

large voltage swing, and the waveform is a return-to-zero two-step pulse, borrowing the shape of

the clock two-step pulses. Unlike the MAJ-SET, the simulation results are shown up to 90 ns

because of the one-clock-period delay explained above. The original paper reports simulations

results at 0 K using SIMON, and estimates the excitation period to be 3.7 ns (assuming zero

temperature) for an error probability of 10–10. The paper also mentions that for a holding duration

of 1 ns and 10–10 error probability the gates should be operated below 5.1 K.

C. Pass Transistor Logic Full Adder

The third SET FA design is based on pass transistor logic (PTL-SET), and was introduced

by Ono et al. [21]. In this design, multi-input SETs are used as pass transistors. Figure 8(a)

shows a two-input SET that implements the XOR function of two inputs. Given that each of the

two input capacitances is equal to C = 1 aF (Cj = 1 aF, Rj = 100 kΩ) and the voltage amplitude is

VDD = e/2C = 80 mV, the SET is ON only when one of the inputs is high, and it is OFF when

neither or both inputs are high.

The sum circuit is shown in Fig. 8(b). It consists of two multi-input transistors. The control

voltage Vcon = 0.8 V is used to change the type of SET from n-type to p-type. This is achieved by

applying a constant DC voltage of Vcon = e/2Ccon, (Ccon = 0.1 aF) which shifts the drain current

versus input voltage characteristics by 180°. The two-input transistor on the left-hand side

implements (a b) · ci’, while the right-hand-side two-input transistor produces (a b)’ · ci.

Hence, the result is a b ci, which is the sum s of the three input bits. Contrary to all the


previous solutions, which implement the carry-out co as a simple MAJ gate, this solution has a

relatively complex circuit to produce the carry-out co (see Fig. 8(c)). It consists of two subcircuits

each having two multiple input SET transistors. The lower subcircuit implements

(b · VDD + b · GND), which is obviously equal to b. The upper subcircuit implements

(a b) · ci + (a b)’ · b, which can be simplified to a · ci + b · ci + a · b, giving the carry-out co.

An inter-SET-node capacitance CM has been used as a parameter. The optimum value is

CM = 25…100 aF, while the load used was COUT = 2.5 fF. Fig. 9 shows the sum output of this

SET FA for an input swing of Vi = 80 mV. Unlike MAJ-SET and MAJ-SEB, this SET FA has a

smaller output voltage swing of about 25 mV at T = 0 K, and about 21 mV at T = 30 K (which is

the temperature reported in the paper). The original paper reports 10 mV, and a delay of 6.68 ns

and 10.8 ns for pull-down and pull-up at T = 30 K. Our simulations have shown that by reducing

both the inte-SET-node capacitance and the output node capacitance the speed can be improved.

D. Single-Electron Junction Threshold Logic Full Adder

The fourth SET FA design was proposed by Lageweg et al. [22], and is based on TLGs

implemented using single electron junctions (TLG-SEJ). This SET FA consists of two TLGs and

a buffer for each of the TLGs (Fig. 10(a)). The sum function s is implemented by a (1,1,1,–

2; 0.5) TLG, and the carry-out co is implemented by a MAJ. Figure 10(b) shows the (1,1,1,–

2; 0.5) SEJ TLG. It consists of one SEJ, and five input and bias capacitors. The input signals V1,

V2, V3 are weighted by their corresponding capacitors C = 0.5 aF, and added to the voltage across

the junction. The input signals V4 is weighted by 2C = 1 aF and subtracted from the voltage

across the junction (i.e. implementing a negative weight). The bias voltage Vb, weighted by the

capacitor Cb = 1.18 aF is used to adjust the threshold.


The advantage of this approach is the ability to directly implement negative weights. The

disadvantage is that each TLG should be augmented with a buffer in order to prevent

bidirectional effects. The SET FA described in [22] uses dynamic buffers like the one in Fig.

10(c). This type of buffers needs a reset phase before any data can be processed. The TLG-SEJ

original solution requires four different voltages: V(TLG) = 240 mV, V(BUF) = 48 mV,

V(reset) = –53 mV, Vi = 16 mV, and the original paper reports simulations done with SIMON at

0 K.

A solution for solving some of the problems due to dynamic buffers was presented in [24],

where a static buffer was proposed (see Fig. 10(d)). The static buffer, which is Tucker’s inverter

[35], simplifies the TLG-SEJ as it does not need a reset phase anymore, and also reduces the

number of different voltages required. The design of static buffered NAND and NOR gates is

detailed in [24], where it is shown that they operate correctly for a fan-out of 4 (simulations are

done with SIMON at 0 K), but a redesigned TLG-SEJ FA with static buffers is not presented.

The SEJ TLGs using the same static buffer have very recently been used in the design of various

SET flip-flop/latches [25]. This paper presents many buffered SEJ TLGs: AND, NAND, OR,

NOR, (1,–1; 0), (–1,–1; 0.5), (1,1,–1; 0.5), (1,1,2; 0.5), and (1,–1,2; 1.5). Still, the (1,1,1,2; 2.5)

TLG required by the original TLG-SEJ FA from [22] is not presented. That is why, we have

redesigned the original TLG-SEJ from [22] using static buffers. The approach we have taken for

optimizing the buffered SEJ TLGs is original as based on our own MATLAB modules linked to

SIMON [36]. Using these modules, has allowed us to obtained a range of valid values for the

TLGs’ bias capacitors: between 11.1 aF and 11.6 aF. The values used for simulations are

Cb = 11.25 aF for both TLGs, and Cb = 4.23 aF for the buffers. The other parameter are

C = 0.5 aF, Cj1 = Cj4 = 0.1 aF, Cj2 = Cj3 = 0.5 aF, Rj = 100kΩ, CL = 9 aF, and VDD = 16 mV.


This SET FA structure (including the two static buffers) uses 10 tunneling junctions and 21

capacitors. The s output is shown in Fig. 11. The output is a full swing signal (16 mV), though

the 0 and 1 voltages are slightly shifted (by 0.8 mV) above GND and VDD.

E. Single-Electron Transistor Threshold Logic Full Adder

In this section, a new TLG-based FA (TLG-SET) is presented and analyzed (also detailed

in [23]). This SET FA is a generalization of the MAJ-SET (presented in Section III-A), where

TLGs are used instead of MAJs. By using TLGs, the circuit is reduced from three MAJs and two

inverters (Fig. 3(a)), to only two TLGs (Fig. 12(a)). The sum function s is implemented by a

(1,1,1,2; 2.5) TLG, and the carry-out co by a MAJ. Being a TLG-based solution, it uses the same

basic TLG structure as TLG-SEJ (compare Fig. 12(a) to Fig. 10(a)), but the implementation is

different as it does not require the additional buffers, and the TLGs are implemented using SETs

(as in [19] and [35]) instead of SEJs (as in [22], [24], [25]).

The implementation of the elementary TLG is based on Tucker’s inverter. Figure 12(b)

shows the (1,1,1,2; 2.5) TLG which is used to produce the sum. The carry-out co is produced by a

MAJ (see Fig. 3(b)). Both TLGs have several parameters in common, namely: Cb1 = Cb2 = 9.0 aF,

CL = 24 aF, VDD = 6.5 mV [19]. To design a TLG, the weight ratios and the threshold are used to

calculate the values of the input capacitors. The design of the (1,1,1,2; 2.5) TLG is detailed

below. The capacitors are calculated according to the weights, threshold and the sum-of-input

capacitances. This sum is determined as part of the inverter parameters which produce a step

function. Using the parameters given in [19], the sum-of-input capacitor is 3 aF. To map the

weights to the input capacitors, we first define C as the unit capacitance corresponding to a

weight of 1. Since the threshold is 2.5, three capacitor units should be larger than 1.5 aF, and two


capacitor units less than this value. Mathematically: 3C > 1.5 aF and 2C < 1.5 aF, hence

0.5 aF < C < 0.75 aF. The 3 aF sum-of-input condition, mandates that C ≤ 0.6 aF. Any value in

the range 0.5 aF < C ≤ 0.6 aF can be used in the design, and a bias capacitor should be used to

complete the sum-of-input to 3 aF. Fortunately, 0.6 aF satisfies all the conditions without a need

for a bias capacitor, hence 0.6 aF was used for this design, i.e., C1 = C2 = C3 = 0.6 aF and

C4 = 1.2 aF. This choice of capacitor values seems like a direct division of the 3 aF according to

the ratio of weights. This is true only for this particular TLG because the threshold (θ = 2.5) is

half of the sum-of-weights (1+1+1+2=5). If the threshold were 1.5, for instance, a direct division

would not produce the proper capacitor values. As indicated above, using TLGs instead of MAJs

reduces the total number of components. The number of tunneling junctions is reduced from 20

(MAJ-SET) to 8 (TLG-SET), and the number of capacitors from 37 (MAJ-SET) to 20 (TLG-

SET). Figure 13 shows the s output of this SET FA for an input swing of Vi = 6.5 mV. The

output signal is regular and has a large voltage swing, similar to that of the MAJ-SET FA [19].

IV. CHARACTERIZATION OF SINGLE-ELECTRON TECHNOLOGY FULL

ADDERS

After the initial simulations verifying the functionality of the SET FAs (presented in Figs.

3, 6, 8, 10 and 12), a second set of simulations was carried out to analyze the delay of all these

SET FAs. The simulations were run with the same input signals: b = 1, ci = 0, and a switching

between 0 and 1 every 10 ns (i.e., at a 100 MHz switching frequency). All the simulations were

done at 0 K, as in the original articles [19], [20], [22], [24], [25]. A higher operating temperature

is theoretically estimated in two of the papers, namely 1 K in [19], and 5.1 K in [20]. The only

paper reporting simulations at T = 30 K is [21].


The circuit parameters used are the ones reported in the corresponding article for each SET

FA.

For MAJ-SET: C = 1 aF; Cj1 = Cj4 = 1 aF, Cj2 = Cj3 = 2 aF, Rj = 100 KΩ; Cb1 = Cb2 = 9 aF;

CL = 24 aF; VDD = 6.5 mV.

For MAJ-SEB: C = 2 aF; Cj = 20 aF, Rj = 200 KΩ; Cb = 2 aF, CL = 2 aF; an excitation

voltage of 60 mV, and a holding voltage of 40 mV.

Considering the PTL-SET FA: C = 1 aF; Cj = 1 aF, Rj = 100 kΩ; CCON = 0.1 aF;

CM = 25…100 aF; CL = 2.5 fF; VDD = 80 mV; Vdd = 10 mV; VCON = 0.8 V. Higher speeds

can be achieved by reducing the load capacitor (the value is rather large as compared to

the other SET FAs, and did not produce the correct logic results when run at comparable

speeds). In the PTL-SET FA, all subcircuits were implemented as two-input SETs. When

a single input was needed (as in the carry-out co case), one of the inputs was grounded as

reported in [21].

With regard to the modified TLG-SEJ the circuit parameters are the ones detailed in

Sections III-D: C = 0.5 aF, Cj = 0.1 aF, Rj = 100 KΩ, Cb = 11.25 aF, and CL = 9 aF for the

carry; C = 0.5 aF, Cj1 = Cj4 = 0.1 aF, Cj2 = Cj3 = 0.5 aF, Rj = 100 KΩ, Cb1 = Cb2 = 4.23 aF,

and CL = 9 aF for the static buffer; C = 0.5 aF and C = 1.0 aF, Cj = 0.1 aF, Rj = 100 KΩ,

Cb = 11.25 aF, and CL = 9 aF for the sum; and VDD = 16 mV).

For the new TLG-SET, the circuit parameters have been determined in Section III-E as:

C = 1 aF, Cj1 = Cj4 = 1 aF, Cj2 = Cj3 = 2 aF, Rj = 100 KΩ, Cb1 = Cb2 = 9 aF, CL = 24 aF for

the carry; C = 0.6 aF and C = 1.2 aF, Cj1 = Cj4 = 1 aF, Cj2 = Cj3 = 2 aF, Rj = 100 KΩ,

Cb1 = Cb2 = 9 aF, CL = 24 aF for the sum; and VDD = 6.5 mV.

Fig. 14 shows the schematics of all the SET FA circuits in SIMON.


Concerning delay “it is usual practice for single electron circuits to represent the operation

speed by the mean of the operation delay time” [19]. That is why the results we report here are

the average delays of the sum output, i.e., from a to s. The MAJ-SEB FA is a special case in the

delay analysis. As mentioned in Section III-B, the sum delay is one clock period. Therefore, in

the case of MAJ-SEB FA the average delay was estimated as the inverse of the maximum clock

frequency that still gives the correct outputs.

Finally, another set of simulations was performed for studying the sensitivity to variations

of all these SET FAs. These simulations have been done using both SIMON and MATLAB [36].

Random errors were injected into all the capacitors and tunneling junctions that comprise these

SET FAs. A modified capacitor value was generated as C’ = C + vU(–1,1), where C is the

original value, v is the maximum allowed variation, and U(–1,1) is a random number uniformly

distributed between –1 and 1. The process of varying the selected values, and running SIMON,

was repeated 1000 times in a loop in MATLAB, while data was collected. The SET FA circuit

under test was considered to err if any of its outputs were wrong for any input combinations, in

any of the 1000 runs/iterations. Figure 15 illustrates the effects of variations on the sum outputs s

of all these SET FAs [Remark: The same outputs without variations for all SET FAs are shown

in Figs. 3, 6, 8, 10, and 12]. In Fig. 15, the 1000 results of the 1000 runs for each SET FA are

plotted (on top of each other). This was done on purpose, and explains why sometimes the lines

appear thicker, as in fact being the results of 1000 simulations plotted on top of each other. The

simulations have been run such as errors can be observed. The variations for each SET FA were

chosen such as to force the SET FA to make a few errors (out of the 1000 runs). These have also

been plotted on top of the correct ones (thicker lines). As can be seen from Fig. 17, the detection

of errors is quite straightforward for MAJ-SET, MAJ-SEB, enhanced TLG-SEJ, and TLG-SET.


For these SET FAs, the voltage values for logic 1 and 0 are almost constant, and the errors

manifest themselves as a flip of the output (i.e., the output becomes the inverse of the correct

logic value). This is in contrast to the PTL-SET FA, where there are variations in the output

voltage (see Fig. 16(c)). To detect the errors in the PTL-SET FA case, the ideal output voltage

(Vo) was divided into three regions. Logic 1 was defined as any voltage larger than 2/3 Vo, and

logic 0 as any voltage less than 1/3 Vo. The two horizontal lines bordering the light (yellow)

rectangle in Fig. 17(c), mark the lower limit of logic 1 (2/3 Vo), and the upper limit of logic 0

(1/3 Vo) for the PTL-SET FA.

V. COMPARISON

All the simulation results reported above are summarized in a compact form in Table I,

which provides a quantitative and a qualitative comparison of all these SET FAs. The second

column gives the average delay from a to s. The third column shows the amplitude of the input

voltage Vi and the load capacitance CL of each SET FA design, in addition to the output voltage

Vo obtained from simulations. The ratio between the output and the input voltage amplitudes

(Vo / Vi) is given in the fourth column. Following this is the number of tunneling junctions Cj and

input and bias capacitors C. The sixth column shows the maximum variation that can be tolerated

by each SET FA (when randomly varying all the capacitors and junctions without any

correlation). The last column shows the “design complexity” in terms of control and voltage

supply requirements of each SET FA.

The table shows that the TLG-SET adder has the smallest number of components (28),

while MAJ-SET has the largest number (57). It is clear all the SET FAs have fewer SET

transistors than classical CMOS FAs. Since each SET transistor consists of two junctions, the


number of SET transistors is #Cj/2. SET FAs comprise 4–10 SET transistors, as opposed to 18–

32 MOS transistors in CMOS FAs [27]. However, a fair comparison between SET and CMOS

FAs should include not only the active devices (i.e., transistors) but also the passive ones (i.e.,

capacitors for SET FAs). This shows that the total number of components of SET FAs (28…57)

is comparable to that of CMOS FAs (18…32).

It is also obvious that the PTL-SET FA can achieve a very small delay if load capacitances

can be reduced (to aF). It also exhibits a large tolerance to variations (±10%), and can operate at

higher temperatures. However, PTL-SET has the smallest Vo / Vi ratio (the output swing is about

25 mV for an input signal Vi of 80 mV). This is a well-known disadvantage of pass-transistor

designs in general, which also affects PTL-SET, and suggests that buffers to recover the signal

are needed (which will add to the delay).

The MAJ-SEB FA has the largest delay. In addition, this SET FA is the most complex

design when compared to all the other SET FAs. First, it requires a three-phase clock, which has

two voltage levels (60 mV, 40 mV), and secondly, the input signal should be a return-to-zero

signal (logic 1 encoded as 4 mV and 0 as – 4 mV). The output swing is large (> 97%) but

variable, taking a two-step shape like the three clock phases (see Fig. 7). This SET FA also

requires both the inputs and their complements (see Fig. 5(a)), suggesting that additional

buffering might be needed.

The enhanced TLG-SEJ FA has a full output swing. However, this SET FA requires a

buffer for each TLG to prevent bidirectional effects. This solution is quite fast and simple (no

controls and only one voltage supply), but seems to be sensitive to variations.

The MAJ-SET FA has an average delay. However, it comprises the largest number of

components (57) compared to all the other SET FA designs. In spite of these, it is quite robust to


variations. This can be explained only in part by the simple design (no controls and only one

voltage supply), but more due to the fact that the sum-of-weights of the gates used by this SET

FA is the smallest: MAJ gates with 3 inputs being simpler than a (1,1,1,2: 2.5) TLG.

Finally, the TLG-SET FA introduced in this article has the smallest number of components

(28 like the PTL-SET), and a very simple design (single supply, the same values for both the

input and supply voltage, and no control signals). The delay and output swing compare favorably

to the other SET FAs. However, it seems to be quite sensitive to variations.

A simple solution for enhancing the robustness to variations is high matching, which is

used in CMOS analog circuit designs [37], [38]. It has been used for CMOS TLGs [39], and

advocated recently for SET in [8], [18]. It consists of implementing the capacitors as several

(identical smaller) capacitors. This has similarities with designs using several islands like e.g.,

the single electron trap, the single electron turnstile, and the single electron pump [40], or the use

of an array of islands for one SET transistor [26] (which have shown that a five-junction SET

allows for a threefold increase of the operating temperature). The question we want to raise here

with high matching is if we could use such ideas to improve on the tolerance to variations. The

results are preliminary, and many more investigations are needed, but we have decided to report

them as showing one possible approach. We have used the high matching technique for the TLG-

SET FA, by dividing only the bias capacitor of each TLG into smaller identical capacitors. The

selection of the bias capacitors only is based on recent results which have shown that SET TLGs

are more sensitive to variations in the bias capacitors than in the input capacitors [36]. Our

simulation results for the SET-TLG FA show that using this low-level technique the variations

the circuit can tolerate increase from ±0.6% to ±1.1%, when nine 1 aF parallel bias capacitors are

used instead of a single 9 aF one. This matching-inspired technique can be applied to all the


MAJ-SET FAs, and also to the static buffers (inverters) of the enhanced TLG-SEJ FA. The

tolerances increase from ±1.5% to ±3.2% for the MAJ-SET FA, and from ±0.7% to ±1.0% for

the TLG-SEJ FA. Still, this technique can not be directly applied to either the MAJ-SEB or the

PTL-SET FA.


20

TABLE ICOMPARISON OF SET FULL ADDERS

Delay(ns)

Vin, Vo, CL

(mV, aF)Vo / Vi

(%)Components

(#Cj + #C)Variations (±

%)

Design Complexity

Multiple supplies

Control

MAJ-SET1998 [19]

0.2606.5 mV5.8 mV24 aF

89Cj : 20C : 37Total = 57

1.5 … 3.2(variation enhanced)

No None

MAJ-SEB 2003 [20]

8.0004.0 mV3.9 mV

1 aF> 97

Cj : 14C : 29Total = 43

1.0● Two-step clocks● Input ≠ clock

Three-phase clock

PTL-SET2002 [21](Runs at up to 30 K)

0.0480.0 mV24.6 mV

1 aF31

Cj : 12C : 16Total = 28

10.0● Two voltage supplies● Input ≠ VDD

None

TLG-SEJ [22]2002 (original)Modified in this paper

0.10016.0 mV16.0 mV

10 aF100

Cj : 10C : 21Total = 31

0.7 …1.0(variation enhanced)

No None

TLG-SET 2004This paper

0.2006.5 mV5.8 mV24 aF

89Cj : 8C : 20Total = 28

0.6 … 1.1 (variation enhanced)

No None

VI. CONCLUSIONS

Recent SET FA designs have been reviewed, while an enhanced SET FA together with a

new SET FA design have been presented. These SET FAs differ significantly in their mode of

operation: from pass transistor logic to TLG, including MAJ. The SET FAs were simulated

under similar conditions to obtain quantitative estimates of their delays, and sensitivity to

variations. This has allowed for a better understanding of the advantages and disadvantages of

each solution. The example of MAJ logic design shows an interesting tradeoff between

simplifying the gate and simplifying the circuit. The TLG-SET FA we have introduced has the

smallest number of components, and a very simple design (single supply, the same values for

both the input and supply voltage, and no control signals). The sensitivity analysis of the SET

FAs presented in this paper is still crude, but suggests that the variations-reliability relation does

not depend on the number of components, but more on the principle of operation of the gates,

pointing to the fact that, for being most effective, redundancy should be considered at the lowest

possible level, i.e. the device level. A solution inspired from the high matching techniques used

in analog CMOS circuits has been advocated, but unfortunately it can not be directly applied to

all SET FAs. Many more investigations are needed to characterize SET sensitivity to variations,

and better unified modeling of failures in devices, gates, and interconnects—based on different

probabilistic models that include correlations—have to be developed.

21

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26

Valeriu Beiu (S’92–M’95–SM’96) received the M.Sc. degree in computer engineering from the

“Politehnica” University of Bucharest, Romania, in 1980, and the Ph.D. degree (summa cum

laude) in electrical engineering from the Katholieke Universiteit Leuven, Leuven, Belgium, in

1994.

Upon graduation, for two years he was involved with high-speed CPUs and FPUs with the

Research Institute for Computer Techniques, Bucharest, Romania, prior to rejoining the

“Politehnica” University of Bucharest. He was a founder and Chief Technical Officer (CTO) of

RN2R (1998-2001), and an Associate Professor with the School of Electrical Engineering and

Computer Science, Washington State University (2001-2005). Since 2005 he is a Visiting

Professor with the School of Computing and Intelligent Systems, University of Ulster, UK, and

Chair of Computer System Engineering, College of Information Technology, United Arab

Emirates University, Al-Ain, UAE. He was the principal investigator of 34 research contracts.

He holds 11 patents. He has received 32 grants, given over 100 invited talks, and authored over

130 papers in refereed journals and international conferences. He authored a chapter on digital

integrated circuit implementations of neural networks in the Handbook of Neural Computation

(New York: Oxford Univ. Press, 1997), and two forthcoming books: one on the VLSI

complexity of discrete neural networks, and another one on emerging brain-inspired nano-

architectures. His main research interests are VLSI-efficient designs (i.e., ultrahigh speed, very

low power, and highly reliable), and emerging nanoarchitectures (massively parallel,

adaptive/reconfigurable, regular, fault-tolerant, and neural inspired) as well as their optimized

designs inspired by cellular and systolic arrays, artificial neural networks, or combinations of

these.

27

Dr. Beiu is a member of the International Neural Network Society (INNS), the European

Neural Network Society (ENNS), the Association for Computing Machinery (ACM), and the

Marie Curie Fellowship Association (MCFA). He has organized 15 conferences and 32

conference sessions. He was the Program Chairman of the IEEE Los Alamos Section (1997). He

was the recipient of five Best Paper Awards. He was also the recipient of five fellowships:

Fulbright (1991), Human Capital and Mobility (1994–1996) with King’s College London

(“Programmable Neural Arrays” project), Director’s Funded Postdoc (1996–1998) with Los

Alamos National Laboratory (“Field Programmable Neural Arrays” project, under the

Deployable Adaptive Processing Systems initiative), and Fellow of Rose Research (1999–2001).

FIGURE CAPTIONS

Fig. 1. Fig. 1. Mapping of a Boolean function to a threshold function. (a) Karnaugh map of the

Boolean function s = (a b ci) [co’ (a b ci)]. (b) Weight map showing the sum-of-

weights a + b + ci – 2co. (c) Weight map showing the sum-of-weights ai + bi + ci + 2co’.

Fig. 2. The sequence a, b, ci, used for all SET FAs (the voltage levels are those used in [19]).

Fig. 3. (a) MAJ based SET FA (MAJ-SET). (b) A MAJ gate based on SET inverter.

Fig. 4. Sum output of the MAJ-SET FA.

Fig. 5. (a) MAJ based SET FA (MAJ-SEB). (b) A MAJ gate based on SEB.

Fig. 6. Control and input signals for MAJ-SEB FA. (a) Three phase clock for controlling the

MAJ-SEB FA. (b) Input signal for the MAJ-SEB FA (return-to-zero signal).

Fig. 7. Sum output of the MAJ-SEB FA.

Fig. 8. PTL SET FA. (a) SET XOR gate. (b) PTL Sum gate. (c) PTL Carry gate.

28

Fig. 9. Sum output s of the PTL SET FA. The output swing is about 25 mV (for an input swing

of 80 mV).

Fig. 10. (a) Threshold logic FA (TLG-SEJ). (b) A TLG based on SEJ. (c) SET dynamic

(active) buffer. (d) SET static buffer (inverter).

Fig. 11. Sum output s of the enhanced TLG-SEJ FA using static buffers.

Fig. 12. (a) Threshold logic full adder (TLG-SET). (b) A TLG based on SET inverter.

Fig. 13. Sum output s of the TLG-SET FA.

Fig. 14. SIMON schematics for: (a) MAJ-SET (see also Fig. 2 in [19]); (b) MAJ-SEB (see also

Fig. 8 in [20]); (c) PTL-SET (see also Fig. 3 in [21]); (d) TLG-SEJ with static buffer; and

(e) TLG-SET (see Fig. 12).

Fig. 15. Errors caused by random variations in all the capacitors and junctions. (a) MAJ-SET.

(b) MAJ-SEB. (c) PTL-SET. (d) TLG-SEJ. (e) TLG-SET.

29

Fig. 2. The sequence a, b, ci, used for all SET FAs (the voltage levels are those used in [19]).

ci co

a b

00 01 11 10 ci co

a b

00 01 11 10 ci co

a b

00 01 11 10

00 0 1 00 0 –2 –1 1 00 2 0 1 3

01 1 0 01 1 –1 0 2 01 3 1 2 4

11 0 1 11 2 0 1 3 11 4 2 3 5

10 1 0 10 1 –1 0 2 10 3 1 2 4

(a) (b) (c)

Fig. 1. Mapping of a Boolean function to a threshold function. (a) Karnaugh map of the Boolean function s = (a b ci) [co’ (a b ci)]. (b) Weight map showing the sum-of-weights a + b + ci – 2co. (c) Weight map showing the sum-of-weights ai + bi + ci + 2co’.

30

a

b

ci

s

co

a

b

ci

s

co

VDD

Cb1

CL

Cb2

VDD

C

V1

C

V2

C

V3

C

V1

C

V2

C

V3

VDD

Cb1

CL

Cb2

VDD

C

V1

C

V2

C

V3

C

V1

C

V2

C

V3

(a) (b)

Fig. 3. (a) MAJ based SET FA (MAJ-SET). (b) A MAJ gate based on SET inverter.

31

Fig. 4. Sum output of the MAJ-SET FA.

32

c’

a

s

co

a’

b

b’

c

1 2 3

c’

a

s

co

a’

b

b’

c

1 2 3

Cb

CL

C

V1

C

V2

C

V3

int

Cb

CL

C

V1

C

V2

C

V3

int

(a) (b)

Fig. 5. (a) MAJ based SET FA (MAJ-SEB). (b) A MAJ gate based on SEB.

33

(a) (b)

Fig. 6. Control and input signals for MAJ-SEB FA. (a) Three phase clock for controlling the MAJ-SEB FA. (b) Input signal for the MAJ-SEB FA (return-to-zero signal).

34

Fig. 7. Sum output of the MAJ-SEB FA.

36

v1

drain

v2

sourceCC

v1

drain

v2

sourceCC

Vcon

a

b

ci

co

VDD

C

C

C

C

CC

Ccon

Ccon

Vcon

a

b

ci

co

VDD

C

C

C

C

CC

Ccon

Ccon

CM

CL

Vcon

a

b

ci

co

VDD

C

C

C

C

CC

Ccon

Ccon

Vcon

a

b

ci

co

VDD

C

C

C

C

CC

Ccon

Ccon

CM

CL

(a)

Vcon

a

b

ci’ ci

s

C

C

C

C

Ccon

Vcon

a

b

ci’ ci

s

C

C

C

C

Ccon

(b) (c)

Fig. 8. PTL SET FA. (a) SET XOR gate. (b) PTL Sum gate. (c) PTL Carry gate.

37

Fig. 9. Sum output s of the PTL SET FA. The output swing is about 25 mV (for an input swing of 80 mV).

38

abci

s

co

abci

s

coC

V1

C

V2

C

V3

V4

2C

Vb

Cb

CL

Vo

C

V1

C

V2

C

V3

V4

2C

Vb

Cb

CL

Vo

(a) (b)

Vb

Cin

Vin

Vo

CL

Vb

Cin

Vin

Vo

CL

Cb1

Cb2 CL

VDD

VDD

Vi Vo

C

C

Cb1

Cb2 CL

VDD

VDD

Vi Vo

C

C

(c) (d)

Fig. 10. (a) Threshold logic FA (TLG-SEJ). (b) A TLG based on SEJ. (c) SET dynamic (active) buffer. (d) SET static buffer (inverter).

39

Fig. 11. Sum output s of the enhanced TLG-SEJ FA using static buffers.

40

co’abci

s’

co’abci

s’

VDD

Cb1

CLCb2

VDD

2C

V1

C

V3

C

V4

V2

C

2C

V1

C

V3

C

V4

V2

C

VDD

Cb1

CLCb2

VDD

2C

V1

C

V3

C

V4

V2

C

2C

V1

C

V3

C

V4

V2

C

(a) (b)

Fig. 12. (a) Threshold logic full adder (TLG-SET). (b) A TLG based on SET inverter.

41

Fig. 13. Sum output s of the TLG-SET FA.

42

(a) (b)

(c) (d)

(e)

Figure 14. SIMON schematics for: (a) MAJ-SET (see also Fig. 2 in [19]); (b) MAJ-SEB (see also Fig. 8 in [20]); (c) PTL-SET (see also Fig. 3 in [21]); (d) TLG-SEJ with static buffer; and (e) TLG-SET (see Fig. 12).

43

Title:C:\Program Files\SIMON2.0\maj_var.epsCreator:MATLAB, The Mathworks, Inc.Preview:This EPS picture was not savedwith a preview included in it.Comment:This EPS picture will print to aPostScript printer, but not toother types of printers.

Title:C:\Program Files\SIMON2.0\Box_var.epsCreator:MATLAB, The Mathworks, Inc.Preview:This EPS picture was not savedwith a preview included in it.Comment:This EPS picture will print to aPostScript printer, but not toother types of printers.

(a) (b)Title:C:\Program Files\SIMON2.0\TLG1_16mV_var.epsCreator:MATLAB, The Mathworks, Inc.Preview:This EPS picture was not savedwith a preview included in it.Comment:This EPS picture will print to aPostScript printer, but not toother types of printers.

(c) (d)Title:C:\Program Files\SIMON2.0\TLG2_var.epsCreator:MATLAB, The Mathworks, Inc.Preview:This EPS picture was not savedwith a preview included in it.Comment:This EPS picture will print to aPostScript printer, but not toother types of printers.

(e)

Fig. 15. Errors caused by random variations in all the capacitors and junctions. (a) MAJ-SET. (b) MAJ-SEB. (c) PTL-SET. (d) TLG-SEJ. (e) TLG-SET.

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Valeriu Beiu

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set full adders final.doc

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