Superconducting Optoelectronic Neurons IV: Transmitter Circuits

Jeffrey M. Shainline, Adam N. McCaughan, Sonia M. Buckley, Richard P. Mirin, and Sae Woo NamNational Institute of Standards and Technology, 325 Broadway, Boulder, CO, 80305

Amir Jafari-SalimHYPRES, Inc., 175 Clearbrook Rd, Elmsford, NY 10523, USA

(Dated: May 9, 2018)

A superconducting optoelectronic neuron will produce a small current pulse upon reaching thresh-old. We present an amplifier chain that converts this small current pulse to a voltage pulse sufficientto produce light from a semiconductor diode. This light is the signal used to communicate betweenneurons in the network. The amplifier chain comprises a thresholding Josephson junction, a re-laxation oscillator Josephson junction, a superconducting thin-film current-gated current amplifier,and a superconducting thin-film current-gated voltage amplifier. We analyze the performance of theelements in the amplifier chain in the time domain to calculate the energy consumption per photoncreated for several values of light-emitting diode capacitance and efficiency. The speed of the am-plification sequence allows neuronal firing up to at least 20 MHz with power density low enough tobe cooled easily with standard 4He cryogenic systems operating at 4.2 K.

I. INTRODUCTION

A physical substrate for cognition must enable inte-gration of information across multiple spatial and tem-poral scales [1, 2]. We have argued that light is ideal forcommunication in neural systems [3, 4]. In the previouspapers in this series [5, 6] we have designed supercon-ducting circuits that receive photonic signals and con-vert them into an integrated supercurrent. Such synap-tic receivers must partner with a light source to gener-ate a neuronal firing event when the integrated currentreaches a threshold. In Ref. 5 we propose a Josephsonjunction (JJ) could be used as a thresholding element. Inthis work, we show that the flux quantum generated bya thresholding JJ can trigger an amplification sequenceresulting in an optical pulse containing one to 100,000photons. These superconductor/semiconductor hybridcircuits achieve electrical-to-optical transduction and fa-cilitate communication with high fanout and light-speeddelays in complex networks of superconducting optoelec-tronic neurons. We refer to these amplifier circuits as thetransmitter of the neuron. Connectivity between theseneurons via dielectric waveguides is discussed in Ref. 7.

The superconducting optoelectronic neurons describedin this series of papers are referred to as loop neurons.The loop neuron in which this transmitter operates isshown in Fig. 1(a). Operation is as follows. Photons fromafferent neurons are received by single-photon detectors(SPDs) at a neuron’s synapses. Using Josephson circuits,these detection events are converted into an integratedsupercurrent that is stored in a superconducting loop.The amount of current added to the integration loopduring a synaptic photon detection event is determinedby the synaptic weight. The synaptic weight is dynam-ically adjusted by another circuit combining SPDs andJJs. When the integrated current from all the synapsesof a given neuron reaches a threshold, an amplificationcascade begins, resulting in the production of light froma waveguide-integrated light-emitting diode (LED). The

photons thus produced fan out through a network of pas-sive dielectric waveguides and arrive at the synaptic ter-minals of other neurons where the process repeats. Thetransmitter circuit which receives the threshold signaland produces a pulse of light is the subject of this pa-per.

II. CONCEPTUAL OVERVIEW

A large-scale neural system must achieve excellentcommunication and exceptional power efficiency. Opticalsignals are ideal for communication, and superconduct-ing detectors enable power efficiency. A central techni-cal challenge in designing superconducting optoelectronichardware is to produce optical signals at telecommuni-cation wavelengths with superconducting electronic cir-cuits. The superconducting energy gap [8] is in the mil-livolt range, making it difficult for superconducting cir-cuits to produce the one volt needed to appreciably al-ter the carrier concentrations in semiconductor electronicand optoelectronic devices. This voltage mismatch makesit difficult for superconducting electronic circuits to in-terface with CMOS logic [9] and memory [10].

A common approach to increase voltage in supercon-ducting circuits is to place JJs in series [11]. The achiev-able voltage scales as the superconducting energy gapmultiplied by the number of junctions. Order one thou-sand junctions must be utilized to drive the light sourceswe intend to employ [3, 12, 13]. While superconductingcircuits operate at low voltage, they can sustain plentyof current. Supercurrents can be converted to voltagewith a resistor. A microamp across a megaohm pro-duces a volt. A small meandering wire of a supercon-ducting thin film can easily produce a megaohm resis-tance in the normal-metal state. Thus, we can convert acurrent-biased superconducting wire into a voltage sourceby switching the wire between the superconducting andnormal states.

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ron

LED

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ature

(iii)

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time

curr

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(b)

thresholdlearning rate

gain

Se

Si

W

W

T

FIG. 1. (a) Schematic of the neuron showing excitatory (Se)and inhibitory synapses (Si) connected to an integrating com-ponent with a variable threshold. The wavy, colored arrowsare photons, and the straight, black arrows are electrical sig-nals. The output of the thresholding integrator is input tothe transmitter, T. The dashed box encloses the transmitterthat is the focus of this work. (b) Sequence of events dur-ing neuronal firing event. (i) Current threshold is reached inthe neuronal thresholding loop, causing the relaxation oscilla-tion junction to produce transient current to the nTron gate.(ii) Current from the relaxation oscillator junction drives thegate of the nTron normal, causing the nTron channel currentto be diverted to the gate of the hTron. (iii) The current fromthe nTron through the gate of the hTron drives the channelof the hTron normal, resulting in a voltage pulse across thehTron. (iv) The LED is in parallel with the hTron, so thevoltage across the hTron results in a voltage across the LED.This voltage is sufficient to forward-bias the p−n junction toproduce light.

Such a voltage source is not commonly used in su-perconducting electronics because it tends to be slowand consume more energy per operation than a JJ.The advantages of superconducting electronics for dig-ital computing are largely speed and efficiency, whilethe advantages of superconducting optoelectronic net-works are largely communication and efficiency. Whilesuperconducting electronics aspire to operation above100 GHz, superconducting optoelectronic networks com-prising neurons firing up to 20 MHz would be a radicalincrease relative to the maximum frequency of 600 Hz inbiological neural systems [2, 14, 15]. The goal of neu-ral rather than digital computing liberates us to use su-perconducting devices slower than JJs for events thatonly occur once per neuronal firing. The speed and effi-ciency of JJs are still leveraged by loop neurons to distin-guish synaptic efficacy during each synaptic firing event[5]. Thus, we can use a switching element leveragingthe superconductor-to-normal phase transition in a neu-ral context, provided it operates every neuronal firingevent and not every synaptic firing event.

An effective means of breaking the superconductingphase is to heat the wire locally. Thermal devices aregenerally slow and power hungry, but the small volume,small temperature change, and small heat capacity at lowtemperature [16, 17] enable switching times on the orderof a nanosecond with femtojoules of energy. While suchan amplifier is not suitable for the high speed and lowpower of flux-quantum logic [18–21], a single firing of theswitch can produce thousands of photons, making it veryuseful and efficient in this neural context.

This high-impedance, phase-change switch is called anhTron [22]. The heating operation required to switch thehTron from the superconducting to metal state can beachieved through dissipation in a Joule heater. Fairlylarge current is required to provide sufficient power.While the hTron is equipped to deliver voltage to drivethe semiconductor light source, the thresholding JJ [5]will provide only fluxons when the neuron’s thresholdis reached. These fluxons are insufficient to thermallyswitch the state of the hTron. An intermediate currentamplifier is therefore required to convert fluxons from thethresholding JJ into current across the Joule heating el-ement of the hTron. For this step of the amplificationprocess, an nTron [23] will suffice. The nTron is a three-terminal thin-film superconducting current amplifier. Asmall current in the constricted gate can exceed the localcritical current and drive the path from source to drainnormal. An impedance on the order of a kilohm canbe produced quickly, and the current from the source todrain will be diverted to a load. In the present case, thatload is the ≈ 10 Ω resistive element of the hTron.

A summary of the circuit operation is shown in Fig.1. The loop neuron of Fig. 1(a) is the subject of this se-ries of papers, and the transmitter circuit enclosed in thedashed box is the subject of this paper. The sequenceof events in the operation of the transmitter is shown inFig. 1(b). The neuronal thresholding loop described in

3

Jth

Jjtl

Jro

r1

r2

L1

Ith

ILED

nTron

hTronNI

FIG. 2. Circuit diagram of the amplifier chain under consid-eration showing the thresholding JJ, voltage-state JJ, nTron,hTron, and LED.

Ref. 5 reaches threshold due to integrated current fromsynaptic firing events, and the thresholding JJ producesone or more fluxons. These fluxons enter the gate of thenTron, exceeding the gate critical current. The nTronsource/drain current is then diverted to the hTron gate.Joule heating in the hTron gate produces a temperatureshift of several kelvin, switching the hTron to the high-impedance state. The hTron source-drain current sud-denly experiences 1 MΩ, and the current rapidly switchesto first charge the LED capacitor and then, once suffi-cient voltage is achieved across the capacitor, current isdriven through the diode. This current generates photonsthrough electron-hole recombination. These photons arethe optical source for neural communication.

A circuit diagram of the amplifier chain is shown inFig. 2. This circuit is relatively complex and uses a sig-nificant amount of power. It may be possible to developa more elegant solution to the problem. Any solutionmust meet several operating criteria: 1) The transmittermust threshold on a low-current signal from the neuronalthresholding (NT) loop; 2) The amplifier chain must con-vert this signal to the voltage necessary to produce lightfrom a semiconductor diode; 3) The operation shouldhappen at least as fast as the recovery of the single-photon detectors in the loop neuron synapses (a few tensof nanoseconds); 4) A number of photons appropriateto communicate with the neuron’s synaptic connectionsmust be produced; 5) The number of photons createdmust be dynamically variable with a bias current; 6)The energy of a firing event must be low enough that anensemble of neurons in realistic network operation havepower density low enough that heat can be removed withstandard cryogenic systems or when immersed in liquidhelium; 7) Total power consumption must be low enoughto enable scaling to neural systems with billions of inter-acting nodes with firing rates up to 20 MHz. The trans-mitter circuits presented here satisfy all these criteria.

We begin with the design of the hTron driving the LEDand work backward through the circuit of Fig. 2.

I1

Ipn

p-n

rLED

rhT

CL

hT

ILED

hTron channel

LEDV

2

FIG. 3. Circuit diagram of the LED driven by the hTron.For this analysis, the hTron channel is modeled as a variableresistor (rhT) in series with an inductor (LhT). The LED ismodeled as a capacitor, C, in parallel with a variable resistor(labeled p−n). The hTron variable resistor switches under theinfluence of heat produced by Joule heating in the gate resis-tive element, discussed in Sec. IV (see also Appendix B). TheLED variable resistor models the DC current-voltage charac-teristic of the p − n junction (see Appendix A). The hTronchannel is biased with ILED. When the channel is driven nor-mal, rhT switches from 0 Ω to 800 kΩ. The current charges upthe capacitor until the voltage is ≈ 1 V, at which point currentbegins to flow through the p− n junction, producing light.

III. DRIVING THE LIGHT-EMITTING DIODE

The operation of the amplifier chain shown in Fig. 2has been described qualitatively in Sec. II, but to reach aquantitative design we must ensure the voltage requiredto drive the LED can be sustained for a duration com-mensurate with the number of photons required. Thisduration will depend on the drive current, the capaci-tance of the LED, the efficiency of the LED, and thenumber of synaptic connections served by the neuron.Once the drive requirements of the LED are understood,we can proceed to design an hTron that can meet thesedrive requirements. The first task is to explore the oper-ation of the LED.

The circuit under consideration is shown in Fig. 3. TheLED is modeled as a variable resistor in parallel with acapacitor. The variable resistor is modeled with the DCI − V characteristic of a p − n junction, as described inAppendix A. The hTron is modeled as a variable resis-tor in series with an inductor. This variable resistor istreated as a step function with zero resistance abruptlyswitching to 800 kΩ. This model is intended to capturethe behavior of the hTron driving the LED as the su-perconducting channel of the hTron is driven above itstransition temperature. A thermal model of the hTron ispresented in Sec. IV.

The equations of motion for the circuit of Fig. 3 aregiven in Appendix A. Solving these equations, we obtainthe circuit currents and voltages as a function of time.From these quantities, we can determine the energy dis-sipated during a firing event as well as the number ofphotons produced. The number of photons produced is

4

calculated as

Nph =ηqee

∫Ipndt, (1)

where ηqe is the quantum efficiency of the diode, and e isthe electron charge. The efficiency of the circuit of Fig. 3is calculated as

ηRC = hνNph/ERC , (2)

where h is Planck’s constant, ν is the frequency of a pho-ton (taken to be c/1.22 µm [12]), and ERC is the totalenergy dissipated by the RC circuit of Fig. 3 during afiring event as calculated with the model of Appendix A.

At least two other factors will contribute to the effi-ciency of the circuit in Fig. 3. First, carriers injected intothe p − n junction may recombine non-radiatively. Thisloss mechanism is captured in the internal quantum ef-ficiency, ηqe. Second, light generated by electron-holerecombination events may not couple to guided mode[12] of the axonal waveguide [4, 7], and will thereforenot couple to the synaptic terminals. This loss mech-anism is captured in the waveguide coupling efficiency,ηwg. The total LED efficiency is given by 1/ηLED =1/ηRC + 1/ηqe + 1/ηwg.

Circuit performance is shown in Fig. 4. Voltage tran-sients across the p − n junction are shown in Fig. 4(a),where Two values of capacitance and two values of quan-tum efficiency are considered. For each of the four traces,the hTron pulse duration was chosen to produce 10,000photons. In this paper and the next in this series [7], weassume that 10 photons are generated per out-directedsynaptic connection to compensate for loss, to reducenoise, and to implement learning functions. Thus, volt-age pulses such as these are appropriate to neurons without-degree of 1000. If the LED capacitance can be madeas low as 10 fF, and ηqe can be made as high as 0.1,the hTron gate must only be normal for 2.9 ns. Underthese operating conditions, 25 fJ of energy is dissipated,of which 64% is dissipated as current through the p − njunction (ηRC = 0.64). If the LED performance is worse,with C = 100 fF and ηqe = 0.01, the drive must be ap-plied for 29 ns. This operation dissipates 251 fJ of energy,also with 64% through the junction.

General trends for the number of photons producedby the circuit as a function of the time the hTron gateis above Tc are shown in Fig. 4(b) for several values ofcapacitance and quantum efficiency. In these calcula-tions, the bias current (ILED) is fixed at 10 µA. The ca-pacitance determines the minimum hTron pulse durationnecessary to achieve the voltage necessary for photon pro-duction, and the quantum efficiency determines the slope(y-intercept of this log-log plot) for longer hTron pulses.

As implied by the gain drive current depicted inFig. 1(a), we would like a means to dynamically controlthe number of photons produced in a firing event. Wecan achieve this by varying the bias current diverted tothe LED when the channel of the hTron becomes resis-tive, ILED. In Fig. 4(c) we show Nph as a function of ILED

ILED

[µA]

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1

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Nph x 1

000

1 fF10 fF20 fF30 fF

40 fF50 fF60 fF 70 fF 80 fF 90 fF 100 fF

5 6 7 8 9 10 11 12 13 14 15

Time with T > Tc [ns]

10-1 100 101 102 1030

1

2

3

4

5

log 10

(Nph)

C = 100 fFC = 10 fFC = 1 fF

η qe = 10

-3

η qe = 10

-2

η qe = 10

-1

η qe = 1

1050 15 20 25 30 35Time [ns]

0

1

Vol

tage

[V

]

100 fF, ηqe = 0.01

10 fF, ηqe = 0.1

10 fF, ηqe

= 0.01

100 fF, ηqe = 0.1

(a)

(b)

(c)

FIG. 4. Time-domain analysis of the circuit of Fig. 3. (a)Voltage transients for two values of LED capacitance (10 fFand 100 fF) and two values of LED quantum efficiency (0.1and 0.01). For each of the four calculations, the hTron channelresistance was modeled as a square pulse of duration necessaryto achieve 10,000 photons. Traces slightly shifted in y fordisambiguation. (b) The number of photons produced by theLED as a function of the time the hTron channel was heldabove Tc. For short pulse durations, the traces group basedon the capacitance, while for long pulse durations the tracesgroup based on LED quantum efficiency. (c) The numberof photons produced by the LED as a function of the currentbias for several values of the LED capacitance. Here the hTronchannel was held above Tc for 10 ns. The slope of the tracesranges from 617 photons per µA with C = 1 fF to 584 photonsper µA with C = 100 fF.

5

for various values of the LED capacitance in a case wherethe hTron channel is driven normal for 10 ns and the LEDquantum efficiency is assumed to be 0.01 [24]. This cur-rent bias provides a means to change the strength of neu-ronal activity. With fewer photons produced, the prob-ability of reaching distant connections diminishes. Withmore photons produced, the neuron makes a strongercontribution to network activity. This current bias maybe modified by a supervisory user or by internal activitywithin the network.

The efficiency of the light-production circuit is consid-ered with the rest of the circuit in Sec. VI. Based on thecalculations of this section, we know the hTron drive re-quirements for light production across a range of capaci-tance and quantum efficiency values. If a neuron needs toproduce 10,000 photons to communicate to 1000 synap-tic connections, an hTron biased with a channel currentof 10 µA must produce a resistance of 800 kΩ for 1 ns -100 ns, depending on the achievable LED performance.We now proceed to consider the operation of the hTronwhen driven by the current from an nTron.

IV. THE HTRON VOLTAGE AMPLIFIERDRIVEN BY THE NTRON CURRENT

AMPLIFIER

As discussed in Sec. III, in order to produce the voltageacross the LED required to generate light, a resistance of800 kΩ must be established rapidly and sustained for 1 ns- 100 ns. One way to achieve this is to switch a length ofsuperconducting wire to the normal metal state. This canbe straightforwardly accomplished by raising the temper-ature of a superconducting wire above Tc. An hTron is adevice that switches a channel resistance from zero ohmsto a large value when current is driven through the gate[22]. The current through the gate dissipates power in aresistive element through Joule heating, and this powerlocally raises the channel above Tc. We show a schematicin Fig. 5(a), wherein a resistive layer is separated from ananowire meander by a thin insulator.

While thermal devices can be slow and inefficient insome contexts, there are several reasons why the hTronis suitable for the present purpose. First, the device hasa compact footprint of 5 µm ×5 µm (see Appendix B), soa very small mass must be heated. Second, the specificheat of all materials involved falls as T 3, so at the desiredoperating temperature of 4.2 K, the specific heat is ordersof magnitude smaller than at room temperature. Third,the required temperature swing is small (≈ 2 K), so littlepower dissipation is required to raise the device above Tc.These factors taken together make the hTron a suitabledevice to achive the voltage necessary to produce lightfrom a semiconductor diode with the power and speedrequired for the neural application under consideration.

To quantify the performance of the hTron when driv-ing the LED of Sec. III, we consider the transient dy-namics of the thermal circuit shown in Fig. 5(b). A heat

heater

Superswitch

SiO2

aSi

Lower spacerSuperswitchUpper spacer

Heater

5 µm

r1

T1

Q

C1

r2

T2

C2

r3

T3

C3

r4

T4

C4

(c)

(b)

(a)

FIG. 5. The hTron voltage amplifier. (a) Schematic of thethin-film configuration with a normal-metal resistive layer(gate) over the superconducting nanowire meander (channel).(b) Thermal circuit model used to calculate the dynamics ofthe device. (c) Cross-sectional view of layers involved in cal-culation.

source Q is delivered to the stack of materials shown inFig. 5(c). The equations of motion and material param-eters used in these calculations are given in Appendix B.Figure 6(a) shows temperature transients of the super-conducting layer when power is delivered to the hTrongate long enough to drive the hTron channel normal for1 ns and for 10 ns. In this plot, the hTron gate is drivenwith square current pulses to illustrate the temporal dy-namics of the thermal components. With this model, wesee the hTron can switch to the resistive state in roughly1 ns. While this time scale is not suitable for many op-erations in superconducting digital electronics, it is morethat fast enough for a neuronal firing event in the systemunder consideration.

While a square pulse is most efficient for driving thehTron, exponentially rising and decaying pulses are morestraightforward to achieve. The nTron current amplifierwe plan to employ to produce such pulses was introducedin Ref. 23. The nTron is a three-terminal, thin-film de-vice. In the off state, a supercurrent flows from the sourceto drain, and the gate is in the superconducting state.The gate comprises a small constriction, and when thecurrent delivered to the gate exceeds the critical current,it is driven locally to the normal state, and Joule heat-ing spreads the normal domain. This quickly causes thechannel between the source and drain to be driven nor-

6

Time [ns]0 5 10 15

4

5

6

7T

emper

ature

[K

]

Tc

156 fJ27 fJ

1 ns 10 ns

0

2

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12

14

Hea

ter

Pow

er [µW

]4.7 ns

30 ns

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per

ature

[K

]

0 10 20 30 40 50 60 70 80Time [ns]

Tc

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(b)

FIG. 6. Time-domain analysis of the hTron. (a) A squarecurrent pulse is driven through the gate of the hTron. Twotraces are shown for cases when the channel was driven normallong enough to keep the the temperature above Tc for 1 ns and10 ns. (b) An exponential current pulse is driven through thegate of the hTron, as would be the case when the nTron isused to drive the hTron. In the case shown, the L/r timeconstant of the nTron is 30 ns, and the channel of the hTronis held above Tc for 4.7 ns.

mal across the entire wire width, leading to a few squaresof resistance (roughly 1 kΩ), at which point the source-drain current is diverted to a load. In the present case,the load is the gate of the hTron (10 Ω in this study, theresistance of the upper layer in Fig. 5(a) and (c)). Whenthe gate current to the nTron ceases and the channel re-turns to the superconducting state, the channel currentreturns with the L/r time constant of the system. Thus,the current pulse from the nTron to the hTron will havean exponential time dependence.

In Fig. 6(b) we show the temporal response of thehTron channel temperature when the gate is driven by anexponential current pulse of 1.2 mA amplitude from annTron. In this case, the rise time of the pulse is 300 ps,and the fall time is 30 ns. These time constants are con-trolled by the L/r time constants of the circuit. The falltime is set by LnT/rnT, where L = LnT the inductanceof the nTron channel, and rnT is the load of the nTron,which in this case is the gate resistance of the hTron. Toachieve the power necessary to switch the hTron, 1.2 mAfrom the nTron is required across the 10 Ω of the hTrongate.

Driving the hTron with the exponential pulses of thenTron is far less efficient than driving with square pulsesbecause power continues to be dissipated in the exponen-tial tail of the current pulse long after the temperature ofthe hTron channel drops back below Tc. In the case con-sidered in Fig. 6(b), the LnT/rnT time constant is 30 ns,and the hTron channel is held above Tc for 4.7 ns. Asquare pulse of 6 ns would achieve the same duration ofthe hTron in the resistive state. Figure 7(a) illustratesthis inefficiency. The required LnT/rnT time constant isplotted as a function of the time the hTron channel mustbe held above Tc, referred to as t>. The time t> dependson the LED capacitance, efficiency, and the number ofphotons produced during the pulse, which is determinedby the number of synapses formed by the neuron. TheLnT/rnT time constant must be nearly ten times t>. Im-proved drive circuit designs are likely possible, but weproceed with the single-nTron example to illustrate thateven with first-generation circuit designs, sufficient effi-ciency can be achieved.

We wish to connect the number of photons produced ina neuronal firing event to the τnT time constant that mustbe implemented in hardware. The number of photons re-quired depends on the number of synaptic connectionsas well as the LED capacitance and efficiency. In Ref. 7we study networks with neurons making 20-1000 synapticconnections. We would like neuronal firing events to pro-duce 10× as many photons as the neuron’s out-degree tocompensate for loss, reduce noise, and perform memoryupdate operations. We are therefore interested primar-ily in neuronal firing events producing 200-10,000 pho-tons, and large hub neurons may need to produce 100,000photons or more. Such photon numbers will require thehTron channel to be driven normal for some duration,which necessitates current drive from the nTron to thehTron gate for some proportional duration (Fig. 7(a)).We therefore must calculate the number of photons pro-duced by the LED as a function of the τnT = LnT/rnTtime constant, taking the LED capacitance and efficiencyinto account. The results of these calculations are shownin Fig. 7(b). To produce 200 photons from an LED with10 fF capacitance and 1% efficiency, τnT must be 9 ns, andto produce 10,000 photons τnT must be 100 ns. We showin Appendix C that an nTron channel inductor achievingthis time constant and carrying th requisite 1.2 mA nec-essary to drive the hTron gate can be achieved in an areacommensurate with the area required for the synapticwiring and routing waveguides comprising the dendriticand axonal arbors of the loop neurons.

At this point we have computationally demonstratedan nTron driving an hTron driving an LED. Assessing thecontributions of the nTron and the hTron to the total en-ergy of a neuronal firing event must occur in the contextof the firing circuit as a whole. We consider the variouscontributions to neuronal firing energy and photon pro-duction efficiency in Sec. VI. The next task is to showa Josephson junction in the neuronal thresholding loopdelivering the current to switch the gate of the nTron.

7

100 101 102 103 104

τnT

[ns]

log 10

(Nph)

0

1

2

3

4

5

Ene

rgy

[pJ]

0

1

0

4

8

12

0 50 100 150

τ nT [

µs]

Time with (T > Tc) [ns]

(a)

(b)

C = 100 fFC = 10 fFC = 1 fF

η qe = 0.0

01η qe

= 0.01

η qe = 0.1η qe

= 1

FIG. 7. Energy analysis of the hTron. (a) nTron time con-stant required and energy consumed as a function of the timethe hTron is held above Tc. Dots are data from discrete nu-merical calculations, and solid lines are linear fits. The en-ergy efficiency is nearly an order of magnitude higher whenthe hTron is driven by a square pulse. (b) The number ofphotons produced by the LED as a function of the nTron L/rtime constant, τnT, for several values of LED efficiency andcapacitance.

V. DETECTING NEURONAL THRESHOLD

The circuit considered for detecting threshold in theneuronal thresholding loop is shown in Fig. 2. The pa-rameters are given in Appendix D. When the current inthe neuronal integrating (NI) loop reaches the switchingcurrent of the thresholding junction, Jth, that junctionwill switch and transmit a fluxon to the relaxation oscil-lator junction, Jro. The relaxation oscillator junction isdesigned to enter a transient resistive period upon arrivalof a fluxon from Jth. The current from Jro switches thegate of the nTron, diverting the nTron channel currentto the gate of the hTron.

Spice simulations of the thresholding event are shownin Fig. 8. For these simulations, Cadence was used. InFig. 8(a), the thresholding junction switches, producing acurrent pulse from the relaxation oscillator junction. InFig. 8(b), the gate of the nTron switches, and the nTronchannel current is diverted to the gate of the hTron, re-turning with the τnT time constant.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 50 100 150Time [ns]

Cur

rent

[m

A]

Jro current in

nTron channel

hTron gate

Time [ns]0 1 2 3 4 5 6 7 8

-3

-2

-1

0

1

2

3

-8

-6

-4

-2

0

2

4

I ro [1

00 μ

A]

nTro

n ga

te c

urre

nt [10

μA

]

(a)

(b)

FIG. 8. Time-domain simulations of the threshold detectionevent. (a) A fluxon from the threshold junction Jth switchesJro. The current diverted from Jro to the gate of the nTron isplotted as a function of time. (b) The nTron source/drain cur-rent and the hTron gate current as a function of time followingthe threshold detection event. The simulation shown in Fig. 8was initiated with a DC pulse of 50 mV into a DC-to-SFQ con-verter [9, 25] that produced a fluxon. This fluxon propagatedthrough a Josephson transmission line (JTL) [9, 25] beforeswitching the voltage-state junction. This JTL is not essen-tial for operation, and it adds negligible power consumptionto the threshold detection event.

The series of events shown in Fig. 8 comprise the de-tection of neuronal threshold, which occurs when Jthswitches, and the subsequent current amplification, be-ginning with the switching of Jro, and leading to theswitching of the nTron. The 1.2 mA current pulse com-ing from the nTron and driving the gate of the hTronhas been shown in Sec. IV to be sufficient to switch thehTron, resulting in a voltage pulse and light generationfrom the LED. The L/r time constant in this simulationwas 50 ns, corresponding to 500 nH in series with the 10 Ωof the hTron gate. Figure 7(b) shows this nTron timeconstant is sufficient to produce more than 3000 photonsif the LED capacitance is 10 fF and the LED efficiencyis 1%. Increasing the nTron inductance can extend thepulse duration and thereby produce more photons. Withthe material and device parameters used in the modelof this nTron, 50 ns recovery was sufficient to keep thenTron from latching, and shorter recovery times may bepossible, depending on electro-thermal device engineer-ing. As discussed in Sec. IV, shaping the nTron pulse

8

to be square rather than exponential would have a sig-nificant impact on the light-production efficiency. Suchoperation may be achieved with circuits utilizing two ormore nTrons with feedback.

Having demonstrated threshold detection, current am-plification, and voltage amplification, we have completedthe description of the amplification chain of the neuronaltransmitter circuit. We next discuss the efficiency of theamplifier chain.

VI. PHOTON PRODUCTION EFFICIENCY

In Secs. III-V we have described circuits capable of de-tecting threshold in the neuronal threshold loop, ampli-fying the signal, and producing the voltage necessary togenerate light from a semiconductor diode. We have cal-culated the energy consumed by each element of the am-plifier chain when generating the number of photons nec-essary for neuronal communication in realistic networks.The networks we wish to consider are described in Ref. 7.As explained in Ref. 4, they will comprise a variety ofneurons with a range of numbers of synapses. The num-ber of out-directed synapses made by a neuron is referredto as the out-degree, kout. The number of photons thatmust be produced in a neuronal firing event depends onkout, and therefore so does the energy of a neuronal firingevent, which we denote by Eout(kout).

In an ideal case, a neuron could produce one pho-ton per synaptic connection with unity production ef-ficiency, the photons would reach their destination with-out loss, and they would be detected at the synapse withunity detection efficiency. In this case, we would haveEout = hνkout, where h is Planck’s constant and ν is thefrequency of the photons being generated. In an actualnetwork implemented in hardware, waveguides will havepropagation loss, and detectors will have efficiency lessthan unity. To account for these loss mechanisms, a neu-ron will produce a number of photons greater than oneper synaptic connection. We refer to this number of pho-tons as ζ. If ζ is too large, neurons are wasting power.If ζ is too small, communication will be unreliable. Thesynapses of Ref. 5 have the same response if they receiveone or more photons. Therefore, the noise is not shotnoise, as it would be if the synapse were attempting todetect the precise number of incident photons. The com-munication error can be calculated from the Poisson dis-tribution. Given an average number of incident photonson a synapse due to an upstream neuronal firing event, weuse the Poisson distribution to calculate the probabilitythat a synapse will receive zero photons due to a neuronalfiring event. If the average number of incident photonsis five, the probability that the synapse will receive zerophotons is less than 1%. A neural system is likely to beable to tolerate this level of error [26]. In the case of lossywaveguides with low-efficiency detectors, 3 dB loss maybe incurred between a neuron and a synaptic target. Wethus consider ζ = 10 to be a conservative number to use

Nph

101 102 103 104 105

-6

-7

-5

-4

-3

-2

-1

log 10

(ηhT)

log 10

(ηam

p)

log 10

(ηLE

D)

Nph

101 102 103 104 105

-6

-7

-5

-4

-3

-2

-1

Nph

101 102 103 104 105-6

-5

-4

-3

-2

-1

0(a)

(c)

(b)

C = 100 fF

C = 10 fF

C = 1 fF

ηqe = 1

ηqe = 10-1

ηqe = 10-2

ηqe = 10-3

C = 100 fF

C = 10 fF

C = 1 fF

ηqe = 1

ηqe = 10-1

ηqe = 10-2

ηqe = 10-3

C = 100 fF

C = 10 fF

C = 1 fF

ηqe = 1

ηqe = 10-1

ηqe = 10-2

ηqe = 10-3

FIG. 9. Efficiency of the photon production process as a func-tion of the number of photons produced. (a) Efficiency of theLED. (b) Efficiency of the hTron. (c) Combined efficiency ofthe LED and hTron.

in calculations of network power consumption.In addition to propagation loss and detector ineffi-

ciency, the circuits described in this work are not entirelyefficient at producing photons. If an LED produces Nph

photons during a firing event, and the LED consumesELED total energy during that firing event, the LED effi-ciency is defined by ELED(Nph) = hνNph/ηLED(Nph). Asimilar expression holds for the nTron driving the hTron.The total energy consumed by the amplifier chain duringa neuronal firing event is given by

Eamp(Nph) = ζhνNph/ηamp(Nph), (3)

where 1/ηamp = 1/ηLED + 1/ηhT. The Josephson cir-

9

cuits driving the nTron contribute negligible energy con-sumption. The calculations of Secs. III and IV provideus with numbers for ηLED and ηhT for several values ofLED capacitance and internal quantum efficiency. Thesefunctions are plotted in Fig. 9(a) and (b). Figure 9(c)shows the total amplifier efficiency, ηamp. We see thatdriving the hTron with the nTron (Fig. 9(b)) dominatespower consumption, yet the amount of time the hTronmust be on is determined by the emitter capacitance andefficiency, and therefore emitter improvement is neces-sary for system improvement. The circuits can also bemore efficient with improved thermal design of the hTron(through phonon localization) and current pulse shapingfrom the nTron.

In practice, achieving small, waveguide-integratedLEDs with low capacitance should be possible. A simpleparallel plate model indicates 1 fF should be achievable,and 10 fF should not be particularly challenging, evenwith wiring parasitics. In Fig. 9, we consider values aspoor as 100 fF. The quantum efficiency of the device isharder to predict. Waveguide-integrated light-emittingdiodes with efficiency near 0.01 have been demonstrated[24], and low-temperature operation helps significantly inthis regard. We expect quantum efficiency of 0.01 to beachievable at large scale, but in Fig. 9 we consider val-ues as poor as 10−3. Considering an LED with 10 fFcapacitance or less, if the LED efficiency is as poor 10−3,the total photon production efficiency of the amplifierchain is ηamp ≈ 10−4 when more than 200 photons areproduced. For calculating the power consumption of thenetworks described in Ref. 7, we use Eq. 3 with ζ = 10and ηamp = 10−4. We next discuss reset of the deviceand the inclusion of a refractory period.

VII. RESET AND REFRACTION

We have described the production of light by the trans-mitter circuit, and we must now consider how the circuitresets. Reset is an important part of a neuron’s function,and in the integrate-and-fire model, it is often assumedthat upon reaching threshold, the variable representingthe integrated signal is instantaneously set to some ini-tial value [27, 28]. In the circuit of Fig. 10, the switchingof Jth results in the addition of flux to the NI loop, coun-tering the applied current bias. Such operation may beuseful for reset.

Yet it behooves us to consider reset operation in thebroader context of continuous afferent synaptic activityto the loop neuron. Assume the synapses described inRef. 5 and connected to the NI loop receive inputs ata constant rate, and the synaptic integration (SI) loopshave a constant leak rate, set by an L/r time constant.At any moment in time, we would like the neuronalthresholding loop to contain a current that is a linearcombination of the currents in the SI loops (or the den-dritic integrating loops if they are employed [5]). Nowsuppose all afferent stimuli cease, and the currents in the

Jth

Jro

Lref

rref

Ith

ILED

to LED

FIG. 10. Variant on the circuit in Fig. 2. Here a refractoryperiod is implemented by cutting off current to Jth with anadditional nTron. Parameters used in the model are given inAppendix D.

SI loops decay to zero. If flux has been trapped in the NIloop due to a neuronal thresholding event, Jth will be bi-ased below its original bias point by the current of a fluxquantum, Φ0/Lni. It may be advantageous to employ abuffer junction inside the NI loop to switch in responseto the fluxon from Jth, thereby leaving the NI loop in thesame state as immediately before the thresholding event.

In this manner of operation, the SI and NT loopsalways maintain a representation of the state of thesynapses. In order to ensure spiking operation and keepthe circuit from generating a constant stream of photonswhen threshold is reached, refraction can be introducedand controlled with the circuit shown in Fig. 10. Whenthreshold is reached and the nTron switches, some of thecurrent is diverted to a choke nTron above the thresh-olding junction. The time constant, Lref/rref , sets therefractory period.

Another means to detect threshold while avoidingadding flux to the neuronal integrating loop is to usea yTron [29] as the thresholding element instead of thethresholding JJ. The yTron has the advantage of pro-viding the non-destructive readout we seek, and a yTroncould likely drive the gate of the nTron without the needfor a latching junction. The major challenge of the yTronis that if it is going to be induced to threshold with asmall current, as appears to be beneficial from the con-sideration of multisynaptic neurons in Ref. 5, the yTrongate will require small features and a sharp corner [29]as well as low inductance. A DC SQUID [8, 9, 25] is alsoa candidate for non-destructive readout of the NT loop.In this variant, the NT loop would be a DC SQUID,and the voltage across it would represent the integratedsignal from the SI loops. When sufficient voltage wasreached, a junction in parallel would switch, leading tothe amplification sequence described in this paper.

At present it appears that several approaches to

10

threshold, reset, and refraction are viable based on cir-cuits with Josephson junctions and thin-film amplifiers.This discussion has considered readout and reset of theNT loop, but we must also consider the behavior of the SIloops after neuronal firing. Many models exist [28, 30] fortreating synaptic dynamics, and it may be advantageousto utilize different synapses within a system or within aneuron with different reset dynamics. In the present con-text of loop neurons wherein afferent synaptic activity isintegrated in an ensemble of SI loops, we assume eachsynapse has an independent leak rate set by the L/r timeconstant of that loop. We may think of an SI loop as a de-vice providing information regarding its recent history ofactivity within a sliding time window set by the time con-stant. This information represents not just the integratedsignal, but also may include higher-order temporal termsobtained through coincidence detection [5], dendritic pro-cessing [5], and short-term plasticity [6]. The dendriticarbor, comprising all the neuron’s synapses, contains in-formation about not just recent average activity, but alsohigher-order temporal statistics. Neuronal refraction isnecessary to ensure spiking activity and not latching be-havior, but if the circuits do not reset the synapses aftereach neuronal firing event, the dendritic arbor maintainsa sliding history of synaptic activity unperturbed by neu-ronal firing events. The dendritic arbor is a spatial dis-tribution of synapses that contain a broad representationof information from the frequency domain. The NI loopand NT loop preserve mirrors of the net dendritic flux. Inthe NT loop, this requires a buffer to release flux from aneuronal firing event. The threshold, determined by thedifference between the critical current of the thresholdjunction and bias to the threshold junction, experiencesfirst an absolute then a relative refractory period as cur-rent returns to the threshold junction with an L/r timeconstant.

The leak rates of the synaptic integrating loops andthe refractory period of the neuron set the time constantsthat determine the nonlinearities in the rate in/rate outtransfer function of the system and contribute to the os-cillation frequencies of the relaxation oscillators. Theleak rates of the integrating loops can be tuned from DCto MHz, enabling the turn-on nonlinearity of the ratein/rate out neuronal response to be placed across a widerange of frequencies. The refractory period can be length-ened with the Lref/rref time constant mentioned above,and it can be as short as the recovery time of the nTron.The hTron is only driven normal during a brief durationof the nTron exponential pulse, and there is a refractoryperiod associated with the current returning to the nTronchannel, as the nTron gate cannot be switched if the biasto the nTron is not above a certain value. It may be pos-sible to reduce this recovery time to 10 ns with propermaterials and design, and if the LEDs can be made effi-cient, such short pulses may produce sufficient photons todrive the neuron’s synaptic connections. In this case themaximum neuronal firing rate would be 100 MHz. Yet itremains to be demonstrated conclusively that such rapid

operation is possible. The nTron investigated in Fig. 8operates with 50 ns recovery time. We therefore take 20MHz to be the maximum oscillation frequency we con-sider for loop neurons in the power calculations presentedin Ref. 7.

VIII. DISCUSSION

We have presented a circuit that amplifies the smallcurrent pulse produced when threshold is reached in thesuperconducting neuronal threshold loop [5]. This cir-cuit produces a voltage sufficient to generate light froma semiconductor diode. We have analyzed the tempo-ral dynamics of the amplifier chain as well as the energyconsumption to arrive at a system photon production ef-ficiency. This efficiency will be used in the next paper inthis series [7] in the calculation of power consumption ofa network.

The analysis presented here points to two conclusions.The first conclusion is that to produce photons with shortpulses from the hTron, it is necessary to fabricate LEDswith low capacitance. Other integrated photonic appli-cations find that capacitance dictates the scale at whichlight becomes advantageous [31]. The second conclusionis that the overall photon production efficiency is lim-ited by the internal quantum efficiency of the LED whenproducing large numbers of photons. To be useful, su-perconducting optoelectronic neurons must function in avariety of contexts. As argued in Ref. 4, cognitive neu-ral systems are likely to utilize some neurons with rel-atively small out-degree making only local connectionsas well as other neurons with high degree making localand long-range connections. The value of capacitanceachievable by the LED will determine the lowest degreepractical to implement with photonic connectivity. Asimple parallel-plate model of the LED (Appendix A)shows that achieving 1 fF should be possible, in whichcase neurons with as few as 10 out-directed connectionsmay be practicable. Whatever this limit may be, we an-ticipate combining superconducting optoelectronic neu-rons in networks also employing purely superconductingelectronic neurons [32–36] that are well-suited to form-ing dense local clusters with low-degree connectivity, highspeed, and excellent energy efficiency.

While capacitance limits utility for low-degree op-eration, internal quantum efficiency determines systempower consumption dominated by neurons with high de-gree. The maximum achievable quantum efficiency willdepend on emitter design, materials employed, and fab-rication optimization. Design of emitters specifically forthis application has been described in Ref. 13, where itis concluded that non-resonant devices with high fabri-cation yield may achieve above 40% optical collectionefficiency. The electrical injection efficiency will dependon many factors, and a summary of candidate emittersis presented in Ref. 3. As in many systems, trade-offsmust be weighed. Compound semiconductor light emit-

11

ters are far superior to silicon light emitter in terms ofefficiency, speed, and spectral coverage, yet fabricationwith compound semiconductors is more expensive andmore difficult to scale than silicon. While silicon hasa long history as a disappointing light emitter [37, 38],it may be good enough [12] for near-term developmentof superconducting optoelectronic networks. The easeof integration with superconducting electronics and pas-sive photonic waveguides makes cryogenic silicon lightsources legitimate as a starting point for developing thisplatform. To assess the long-term potential of supercon-ducting optoelectronic networks, much more experimen-tal work on low-capacitance, high-efficiency, waveguide-integrated few-photon light-emitting diodes is needed.Further investigation of thin-film superconducting am-plifiers is also required.

The reader may wonder why we insist on incorporat-ing a light source at each neuron. Perhaps we can utilizean architecture with circulating light in waveguides thatare tapped by a modulator during each neuronal firingevent. Such a system would have the benefit of keepinglight sources out of the cryostat and separate from super-conducting circuits. In the case of shared off-chip lightsources, the new challenges become to develop very lowloss waveguides; compact, efficient modulators with lowinsertion loss that do not need to be tuned; and extensivefiber-to-chip coupling. Initial calculations indicate thatsuch an approach is less efficient unless waveguides canbe made with very low propagation loss, and modulatorscan be made with very low insertion loss. Yet device andsystem limits are far from understood. Amplifier circuitssimilar to those presented here may be useful when driv-ing the modulators of such a system.

With the design of the transmitter portion of the cir-cuit, the neuron model presented in this series of papersis complete. The full circuit is shown in Fig. 11. Thesynaptic receiver portion is discussed in Ref. 5, and thecircuit that updates the weights based on spike timing isdescribed in Ref. 6. The transmitter portion is the sub-ject of the present paper. Consideration of the circuitwithin an integrate and fire model is presented in Ap-pendix E.

The synaptic receiver and synaptic weight update cir-cuit in the form presented in Refs. 5 and 6 utilize sepa-rate photons for synaptic firing and synaptic update. Inthis work, circuit operation culminates by producing lightfrom a single LED, yet it may be desirable to implementtransmitters that fire different LEDs for the different op-erations. This is one way that the power used for syanpticfiring events can be decoupled from the power used forsynaptic update events. If different colors are used foreach of these operations, the same waveguide routingnetwork can be employed for the three signals (synap-tic firing, synaptic strengthening, and synaptic weaken-ing), and the demultiplexers located at each downstreamneuron can separate the signals and route them locallyto the three synaptic ports. Similarly, the spike-timing-dependent plasticity circuit of Ref. 6 requires not only

photons from the pre-synaptic neurons, but also photonsfrom the local, post-synaptic neuron. Two additionallight sources may be useful at each neuron to be utilizedlocally for synaptic update. The schematic diagram inFig. 1 shows these five light sources. While producingfive light sources instead of just one adds hardware over-head, the light sources are extremely compact comparedto the routing waveguides and inductors associated withthe synapses. The synaptic update light sources neednot fire nearly as frequently as the synaptic firing lightsources. By utilizing independent light sources, it maybe possible to reduce network area as well as power con-sumption. Such operations may be only the beginning ofutilizing color in these networks.

In the schematic of Fig. 11, we emphasize three maincurrent biases that affect the operation of each neuron. Acurrent bias into the synaptic update circuit of Ref. 6 af-fects the learning rate. A current bias into the neuronalthreshold loop [5] affects the neuron threshold. And acurrent bias into the light emitter affects the number ofphotons produced in a neuronal firing event. With eachof these currents fixed, the neurons and the network willhave rich spatio-temporal dynamics. Yet a network withthe ability to dynamically vary these currents will be ca-pable of achieving further complexity over longer time pe-riods. For example, with the learning rate current fixed,spike-timing-dependent plasticity occurs at a fixed rate.By changing this current, the network can make certainregions more adaptive at certain times, and it can makethose regions maintain synaptic weights at other times.Similarly, with a fixed gain current into the transmit-ter, the neuron will address each of its downstream con-nections with a given probability. By changing the gaincurrent, the number of photons produced in a neuronalfiring event can be adjusted, and therefore the proba-bility of reaching downstream connections can be tuned.Changing these bias currents is analogous to changingvarious neuromodulators in biological systems. The val-ues of each of these neuromodulatory control currents canbe modified with photonic or electronic signals. The con-trol currents can also be set externally for experimentsin critical dynamics or applications in neural computing,or adjusted based on internal network activity. Adapt-ing these neuromodulatory currents is likely to be usefulto achieve maximal information integration through self-organized criticality [39–43].

The description of superconducting optoelectronic loopneurons presented in Ref. 5, Ref. 6, and in the presentwork summarizes the operation of these devices in a man-ifestation that we anticipate being conducive to massivescaling and complex neural operation. The next task isto consider networks comprised of these neurons. Con-sideration of these networks is presented in Ref. 7.

This is a contribution of NIST, an agency of the USgovernment, not subject to copyright.

12

SS

NI

Jth

Jsf

Jro

Ith

Isu

Isu

ILED

SI

SB

Synapse

Weight

Threshold Transmitter

Se

Si

w

w

T

thresholdlearning rate

gain

FIG. 11. Circuit diagram of superconducting optoelectronic neuron as described in this series of papers. Large-scale neuralsystems will employ a wide variety of neurons with myriad structural and dynamical properties. The full potential of neuralcircuits based on superconducting optoelectronic hardware will only be realized with a long-term experimental and theoreticaleffort of a broad research community.

Appendix A: LED circuit model

The equations of motion for this circuit are given by

dI1dt

=1

L[V2 + r1 + ILED − (r1 + rhc)I1] ; (A1)

dV2dt

=1

C[ILED − I1 − Ipn(V2)]. (A2)

In these equations, Ipn is an analytical function whichdescribes the LED’s DC I − V characteristic. It is givenby [44]

Ipn =eA [(Dp/Lp)pn + (Dn/Ln)np]

×[eeV/kBT − 1

].

(A3)

The parameters that enter this equation are as follows.The number of acceptors, Na = 5×1019 cm−3. The num-ber of donors, Nd = 5 × 1019 cm−3. The intrinsic carrier

density, ni = 1.5 × 1010 cm−3. The number of holes onthe n side of junction, pn = n2i /Nd. The number of elec-trons on p side of junction, np = n2i /Na. The electronminority carrier lifetime, τnp = 40 ns. The hole minor-ity carrier lifetime, τpn = 40 ns. The mobility of holeson p side, µpp = 10−2 m2/V·s. The mobility of holes onn side, µpn = 10−2 m2/V·s. The mobility of electronson n side, µnn = 2.5 × 10−2 m2/V·s. The mobility ofelectrons on p side, µnp = 2.5 × 10−2 m2/V·s. The holediffusion coefficient, Dp = (kBT/e)µpn. The electron dif-fusion coefficient, Dn = (kBT/e)µnp. The hole diffusion

length, Lp =√Dpτpn. The electron diffusion length,

Ln =√Dnτnp. T = 300 K is the temperature of opera-

tion. While we plan to operate these circuits at 4.2 K, ourmeasurements indicate the model captures the operationwell at low temperature, and the model is more stablewith this room-temperature value. To approximate thecapacitance we have assumed the length of the junction

13

is 5 µm, the height of the junction is 200 nm, the intrin-sic region is 100 nm, and ε = 12 for the semiconductor.This gives C = 1 fF, which is the smallest value we haveconsidered. Fringe fields and wiring parasitics will likelyincrease this value. We have considered capacitance aslarge as 100 fF to ensure our estimates are conservative.

Appendix B: hTron circuit model

The equations of motion for the circuit in Fig. 5(b)are given by

dT1dt

=Q

C1+T2 − T1R1C1

; (B1)

dT2dt

=T1 − T2R1C2

+T3 − T2R2C2

; (B2)

dT3dt

=T2 − T3R2C3

+T4 − T3R3C3

; (B3)

dT4dt

=T3 − T4R3C4

+Tg − T4R4C4

. (B4)

In the calculations presented in Fig. 6 and 7, Q = I2rwhere I = 1.2 mA is the current through the gate of thehTron, and r = 10 Ω is the resistance of the hTron gate.Thus, Q = 14.4 µW.

To produce the plots shown in Fig. 6 and 7, we modelthe heater layer as a 10 nm-thick film of Al. The upperspacer is modeled as a 10 nm-thick film of a-Si, and thelower spacer is modeled as a 50 nm-thick film of SiO2.For the material parameters, the following values areused. The density of Al was taken to be 2700 kg/m3,and the thermal conductivity was taken to be 30 W/m·K.The density of a-Si was taken to be 2285 kg/m3, and thethermal conductivity was taken to be 0.01 W/m·K [45].The density of SiO2 was taken to be 2650 kg/m3 and thethermal conductivity was taken to be 0.01 W/m·K. Thetemperature-dependent specific heat of Al and SiO2 weretaken from Ref. 17.

Regarding the superconductor, we anticipate usingMoSi for this purpose due to its resistivity in the normalstate, its critical current density, and its critical temper-ature. In a film of 8 nm thickness, the sheet resistance inthe normal state is 300 Ω/. A wire of 100 nm width cansustain 16 µA critical current. A length of 2000 squaresachieves 800 kΩ. A meander of 2000 squares of a wire of100 nm width and 50 nm gap fits in a square of 5.4 µm ×5.4 µm. At this thickness, Tc = 6.2 K. A device held at4.2 K must only be raised by 2 K to switch to the normalstate. The temperature-dependent specific heat of MoSiwas taken from Ref. 16, and the thermal conductivity wastaken to be 1 W/m·K.

Of the material parameters, the values of thermal con-ductivity are the most uncertain. From Ref. 46 we know

the thermal conductivity of quartz and silica differ fromone another by orders of magnitude at low temperature,and we expect the value of thermal conductivity to de-pend significantly on the specific material and film depo-sition method used in fabrication. For functional hTronsfabricated with SiO2 spacer layers and operating between1 and 5 K, it has been observed [47] that 18 nW/µm2 isrequired to switch the hTron. When using the value of0.01 W/m·K for the thermal conductivity of the spacerlayers, we arrive at a power density of 494 nW/µm2, afactor of 27 higher than the measured value. We there-fore suspect hTrons can be more efficient than the presentmodel would predict, but further empirical investigationis required to determine the limits of hTron efficiency.The model of Eqs. B1-B4 neglects phonon reflection atmaterial interfaces. This effect may be responsible forthe discrepancy between the model and measurements.Utilizing phonon reflection or phononic crystals in designmay further increase the efficiency of the device.

Appendix C: Inductor for the nTron driving thehTron

Time constants as large as 100 ns may be necessary forthe nTron driving the hTron to produce sufficient pho-tons to serve thousands of synaptic connections. Such alarge time constant requires a large inductor, necessitat-ing many squares of meander. The large inductor mustalso carry a large current (1.2 mA) to drive the hTrongate, and therefore the wire must be relatively wide. Thearea of the meander is a concern. With rnT = 10 Ω,these nTron drive durations require LnT = 90 nH andLnT = 1 µH, respectively. The meander achieving thisinductance must carry current of 1.2 mA. If an 8 nm-thick film of MoSi is utilized for the wire, it will needto be 10 um wide to carry the current, and it will have180 pH/. In the case of the high-degree neuron pro-ducing 10,000 photons per neuronal pulse, 1 µH inductormeander will occupy 0.6 mm2. As is shown in Ref. 7, thearea occupied by a neuron of this degree is roughly thesame when calculated based on the area of optoelectronicsynaptic connections. Thus, while the inductive meanderrequired to achieve the drive duration and to keep thenTron from latching is a large component, because thereis only one per neuron (as opposed to one per synapse), itcan be straightforwardly accommodated on a dedicatedinductive wiring layer.

Appendix D: Parameters used in threshold circuit

The Ic of the JJs in the JTL is 250 µm, and the induc-tors are 2.1 pH. The Ic of the JJ just before the relaxationoscillator junction is 280 µm, and the Ic of the relaxationoscillator junction is the same. The relaxation oscillatoris biased with 140 µA. The value of L1 = 200 pH . Thevalue of r1 = 2.76 Ω. The value of r2 = 3 Ω.

14

Appendix E: Integrate and fire model of loop neuron

A standard integrate and fire model [27, 28] assumesa neuron integrates the signals from all of its synapticconnections, the integrated signal is subject to a leak,the neuron produces a spike when it reaches a thresh-old, the integrated signal is set to zero upon reachingthreshold, and the neuron enters a refractory period af-ter producing a spike during which it cannot produceanother spike. Loop neurons operating in this mannercan be constructed, yet they may not have the best per-formance. For loop neurons with synaptic integrationloops [5], it may be more advantageous for each synap-tic integration loop to have its own leak rate. This givesthe neuron a much broader range of time constants, con-tributing to richer dynamics [2, 39, 48]. In this case, abetter model treats each synapse with its own integrateand fire equation:

dI(si)i

dt= wiji −

I(si)i

τ(si)i

. (E1)

In Eq. E1, I(si)i is the current in the synaptic integra-

tion loop of the neuron’s ith synapse, wi is the synapticweight (expressed as an integer number of fluxons gen-

erated furing a synaptic firing event), τ(si)i is the Lsi/rsi

time constant of the synaptic integration loop, and thedriving signal into the synapse is given by

j(sp)i = j0

∑q

δ(t− tq), (E2)

where j0 = Φ0/L(si)i is the current added to the synaptic

integration loop by a flux quantum, and tq is the arrivaltime of the qth input spike to the synapse. The synapsesof loop neurons comprise an SPD in parallel with a JJ.The L/r recovery time of the SPD must be sufficientlylong to avoid latching. The required time constant is ma-terial dependent and usually in the range of 10 - 50 ns.During this time, the SPD is biased too low to detect aphoton, and the synaptic receiver experiences a refrac-tory period. We therefore take the maximum rate atwhich synapses can receive signals to be 20 MHz.

In this model, each synapse is capable of integrationand leak, but threshold still occurs for the neuron as awhole. The current in the neuronal integration loop [5]is a linear combination of the currents in the synaptic

integration loops:

I(ni)(t) =

nsy∑i=1

ciI(si)i (t). (E3)

Here, the coefficients ci represent the mutual inductorscouping the SI loops to the NI loop, and they representa constant, non-plastic scaling of the synaptic weight.

Firing operation is envisioned as follows. WhenI(ni)(t) reaches threshold (determined by the threshold-ing Josephson junction and its dynamic bias), the thresh-olding junction produces a fluxon that triggers the ampli-fier chain sequence described in this paper. The neuronalthresholding junction is then induced into a refractoryperiod by the self-feedback nTron circuit of Fig. 10(b)(Sec. VII) that diverts the current Ith, with exponentialrecovery. The integrated signals in the synaptic integra-

tion loops continue to decay with L(si)i /r

(si)i , but they are

not explicitly set to zero. In this mode of operation, thesynapses continue to be sensitive to stimulus during theneuron’s refractory period. Firing of the neuron does notreset the integrated signal to zero as in the standard inte-grate and fire formalism. Reset of the integrated currentscan be straightforwardly achieved with additional hard-ware and power, but for many networks it is likely notnecessary.

This synaptic leak model can be extended to variouslevels of dendritic processing with different nonlinearitiesand time constants present at different levels of dendriticloop hierarchy. Synaptic and dendritic leaks, as opposedto a single neuronal leak, have the advantage of intro-ducing many more decay rates to the repertoire of theneuron. Such time constants may be useful for givingrise to self-organized criticality, as the power law dynam-ics of systems at the critical point can be constructed asa superposition of exponential functions [39, 49]. Onedisadvantage of synaptic/dendritic leak is the computa-tional overhead in simulation. Each synapse must berepresented by an ordinary differential equation in thecoupled system. Such a description is more complete[48], but more expensive to simulate. In the limit that allsynaptic time constants are equivalent, one can model theloop neuron with the standard integrate and fire formal-ism that does not require a separate differential equationfor each synapse.

Eqs. E1 and E3 do not model coincidence detection, se-quence detection, or inhibitory dendritic processing, butthese effects can be included in more realistic models [28].

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