Normal Metal-Superconductor Near-Field Thermal Diodes and Transistors

E. Moncada-Villa1 and J. C. Cuevas21Escuela de F́ısica, Universidad Pedagógica y Tecnológica de Colombia,

Avenida Central del Norte 39-115, Tunja, Colombia and2Departamento de F́ısica Teórica de la Materia Condensada and Condensed Matter Physics Center (IFIMAC),

Universidad Autónoma de Madrid, E-28049 Madrid, Spain(Dated: November 20, 2020)

In recent years there has been a number of proposals of thermal devices operating in the near-field regime that make use of phase-transition materials. Here, we present a theoretical study ofnear-field thermal diodes and transistors that combine superconducting materials with normal (non-superconducting) metals. To be precise, we show that a system formed by two parallel plates madeof Nb and Au can exhibit unprecedented rectification ratios very close to unity at temperaturesaround Nb superconducting critical temperature and for a wide range of gap size values withinthe near-field regime. Moreover, we also show that a superconducting Nb layer placed between Auplates can operate as a near-field thermal transistor where the amplification factor can be greatlytuned by varying different parameters such as the temperature and thickness of the Nb layer or thedistance between the Nb layer and the Au plates. Overall, our work shows the potential of the useof superconductors for the realization of near-field thermal devices.

I. INTRODUCTION

When two objects at different temperatures are sep-arated by a distance smaller than the thermal wave-length given by Wien’s displacement law (∼10 µm atroom temperature), they can exchange thermal radia-tion via evanescent waves (or photon tunneling). Sucha contribution to the near-field radiative heat transfer(NFRHT) can dominate the heat exchange for small gapsand lead to overcome the blackbody limit set by Stefan-Boltzmann’s law for the radiative heat transfer betweentwo bodies [1–3]. This NFRHT enhancement was firstpredicted by Polder and van Hove in the early 1970’s [4]making use of the so-called theory of fluctuational elec-trodynamics [5, 6]. In recent years this idea has beenthoroughly tested and confirmed in a great variety ofsystems and using different types of materials [7–31].

It is safe to say that at this stage the basic physi-cal mechanisms underlying NFRHT are relatively well-understood. For this reason, efforts in the thermal ra-diation community are now shifting towards the pro-posal and realization of novel functional devices basedon NFRHT. In this regard, a natural research line that isbeing pursued is the investigation of the near-field ther-mal analogues of the key building blocks of today’s mi-croelectronics: diodes, transistors, switches, memory el-ements, etc. Thus far, the diode or rectifier has been themost widely studied thermal device. There have beena lot theoretical proposals to achieve thermal rectifica-tion that make use of systems with dissimilar materialsthat, in turn, exhibit optical properties that depend ontemperature. The proposed material combinations in-clude SiC structures [32], doped Si films [33], dielectriccoating [34], and Si and a different material [35], justto mention a few. However, the most promising propos-als for near-field thermal diodes are based on the useof phase-transition materials [36–42]. An ideal exampleis that of vanadium dioxide (VO2), which undergoes a

phase transition from insulator below 340 K to a metalabove that temperature. This phase transition is accom-panied by a drastic change in the infrared optical prop-erties, which has a strong impact in the correspondingradiative heat transfer in systems featuring this materialas the temperature is varied across the transition tem-perature [43–45]. In fact, several experiments have al-ready demonstrated rectification between VO2 and SiO2in the far-field regime [15, 45, 46]. More importantly forour work, the first observation of thermal rectificationin the near-field regime has been recently reported be-tween a Si microdevice and a macroscopic VO2 film [47].In that work, a clear rectifying behavior that increasedat nanoscale separations was observed with a maximumrectification ratio exceeding 50% at ∼140 nm gaps anda temperature difference of 70 K. This high rectificationratio was attributed to the broadband enhancement ofheat transfer between metallic VO2 and doped Si sur-faces, as compared to the narrower-band exchange thatoccurs when VO2 is in its insulating state. From thetheoretical point of view, it has been shown that the rec-tification ratio can be boosted by nanostructuring VO2films to form, e.g., one-dimensional gratings [41, 48] andthe highest reported values for the rectification ratio inthe near-field reach about 94% [41].

In 2014, Ben-Abdallah and Biehs extended the ideaof thermal diodes based on a phase-transition materialto propose the realization of a near-field thermal transis-tor [49]. Their proposed transistor featured a three-bodysystem (two diodes in series) in which a layer of a metal-to-insulator transition material (the gate) is placed atsubwavelength distances from two thermal reservoirs (thesource and the drain). In this device, the temperaturesof the reservoirs are kept fixed, while the temperature ofthe gate is modulated around its steady-state tempera-ture. Making use of an extension of the theory of fluctua-tional electrodynamics to deal with extended many-bodysystems [50], Ben-Abdallah and Biehs showed that by

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changing the gate temperature around its critical value,the heat flux exchanged between the hot body (source)and the cold body (drain) can be reversibly switched, am-plified, and modulated by a tiny action on the gate. Letus also say that these ideas have been extended to pro-pose other key elements such as a thermal memory [51],and it has also been shown that thermal logic gates canbe realized exploiting the near-field radiative interactionin N -body systems with phase-transition materials [52].These proposals are nicely reviewed in Refs. [3, 53].

Since phase-transition materials are ideally suited fornear-field thermal management devices, it is natural tothink of superconductors. Although superconductors re-quire to work at low temperatures, their use has the ad-vantage that since the thermal wavelength is inverselyproportional to temperature, the near-field regime ex-tends to gaps beyond the millimeter scale for tempera-tures around 1 K. Thus, it is easy to reach gaps or sepa-rations for which the radiative heat transfer is enhancedbeyond the blackbody limit. When a normal metal un-dergoes a superconducting phase transition (we focushere on conventional low-temperature superconductors),its optical properties change drastically in the microwaverange due to the appearance of a gap in its density ofstates (of the order of 1 meV depending on the super-conductor). This gap reduces the emissivity of the metaland in the superconducting state, one expects a substan-tial reduction of the NFRHT when a second materialis brought in close proximity. This naive idea has beenexperimentally confirmed in recent years with measure-ments of the NFRHT between parallel plates made ofsuperconductors like Nb and NbN [54, 55]. In particu-lar, it has been reported that there is a contrast in theNFRHT between the normal and the superconductingstate of a factor 5 in the case of Nb [54] and 8 in thecase of NbN plates [55]. Inspired by these experiments,Ordoñez-Miranda et al. [57] proposed the use of a low-temperature superconductor to realize a near-field ther-mal diode. In their proposal, the thermal reservoirs weremade of Nb and SiO2 and the device operated at temper-atures around the superconducting critical temperatureof Nb (9 K). In particular, they found that at temper-atures 1 and 8.7 K for the two thermal reservoirs, therectification factor reached a maximum of 71% for gapsof the order of 60 µm, which is quite high, but still belowwhat is found in proposals involving vanadium dioxide.One the the goals of this work is to show theoreticallythat the use of normal metals, instead of dielectrics likesilica, can boost the performance of near-field thermaldiodes comprising superconducting materials, even be-yond any reported value in VO2-based thermal diodes.To be precise, we shall consider an Au-Nb rectifier, seeFig. 1(b), and show that rectification ratios very close tounity are achieved in a very wide range of gap size val-ues in the near-field regime. On the other hand, we shallalso show that a Nb plate (the gate reservoir) placed inthe middle of a vacuum gap between two Au plates, seeFig. 1(c), can behave as near-field transistor with am-

Au

T3

TS(c)

(b)

δ3

d

Nb

T1(a)

d

Nb

T3

Nb Au

T1

Au

TD

d d

TG

Nb

Source Gate Drain

(1) (2) (3) (1) (2) (3)

(1) (2) (3) (4) (5)

δ1 δ5

FIG. 1. Schematic representation of the three systems con-sidered in this paper. The first one [panel (a)] consists of twoinfinite plates of a superconducting material as Nb, each onewith its respective temperature T1 and T3, and separated bya vacuum gap of size d. The second one [panel (b)] is a recti-fier composed of a Nb infinite plate exchanging radiative heatwith an infinite Au plate separated by a distance d. Finally,in panel (c), we display a near-field thermal superconductingtransistor. Source and drain, each one with fixed tempera-tures TS and TD, respectively, are assumed to be two infiniteAu slabs (δ1 = δ5 = ∞), whereas the gate is made of Nb,which undergoes a normal-superconducting phase transitionat a critical temperature, TC, as its temperature, TG, is var-ied.

plification factors that can be largely tuned by varyingdifferent parameters such as the temperature and thick-ness of the gate or the distance between the gate and thesource and drain reservoirs.

The rest of the paper is organized as follows. In Sec. IIwe introduce the different systems and devices that weanalyze in this work and explain how we model the opti-cal properties of the materials involved in these systems(Nb and Au). In Sec. III we briefly discuss the radiativeheat transfer between two parallel plates made of Nb toillustrate the impact of the superconducting phase tran-sition in the heat exchanged via radiation. Then, Sec. IVis devoted to the analysis of the radiative heat rectifi-cation in a thermal diode made of Au and Nb parallelplates. In Sec. V we study the operation of a three-bodysystem made of a Nb layer between two Au plates as anear-field thermal transistor. Finally, we summarize ourmain conclusions in Sec. VI.

II. SYSTEMS AND OPTICAL PROPERTIES

As explained in the introduction, the main goal of thiswork is to study theoretically the performance of near-field thermal diodes and transistors that make use of su-

3

perconducting materials. For this purpose, we shall firstbriefly analyze the impact of the superconducting phasetransition in the NFRHT in the case of two Nb paral-lel plates, see Fig. 1(a), in which we shall consider tem-peratures below and above the superconducting criticaltemperature of Nb, TC = 9 K. The near-field diode thatwe shall investigate is schematically depicted in Fig. 1(b)and it consists of two infinite parallel plates made of Auand Nb separated by a gap of size d. Finally, the near-field transistor that we shall analyze in detail is shownin Fig. 1(c). In this case, a Nb layer of thickness δ3, re-ferred to as gate, is placed in the middle of the vacuumgap between two infinite parallel plates made of Au, re-ferred to as source and drain. The temperatures of thesource, drain, and gate are denoted by TS, TD, and TG,respectively, and we shall assume that TS > TD. Thedistance between the gate and the source and drain isdenoted by d and we shall consider temperatures around(both above and below) TC.

The analysis of the radiative heat transfer in all thecases shown in Fig. 1 will be done within the framework ofthe theory of fluctuational electrodynamics [5, 6]. In thistheory, and within the standard local approximation, theoptical properties of the materials are fully determined bytheir frequency-dependent dielectric functions (we onlyconsider here nonmagnetic materials). In what follow,

we shall describe how we model the dielectric functionsof the two materials involved in our systems under study,Nb and Au.

The Nb dielectric function is described here followingRef. [58] which, in turn, makes use of Mattis and Bardeentheory for diffusive superconductors of arbitrary purity[59]. This dielectric function is given by

�Nb(ω) = ε∞,Nb +4πi

ωσ(ω), (1)

where ε∞,Nb is the high frequency limit for the permit-tivity, σ(ω) is the optical conductivity, τ = σdc/(�0ω

2p) is

the relaxation time, �0 is the vacuum permittivity, andωp,Nb is the plasma frequency. In the superconductingstate, the optical conductivity is given by [58]

σsc(ω) =iσdc2ωτ

(J(ω) +

∫ ∞∆

I2 dε

), (2)

with

J(ω) =

∫ ~ω+∆

∆I1 dε, ~ω ≤ 2∆∫ ~ω−∆

∆I3 dε+

∫ ~ω+∆~ω−∆ I1 dε, ~ω ≥ 2∆

, (3)

and

I1 =

[(1− ∆

2 + ε(ε− ~ω)p4p2

)1

p4 + p2 + i~/τ−(

1 +∆2 + ε(ε− ~ω)

p4p2

)1

p4 − p2 + i~/τ

]tanh

( ε2kT

),

I2 =

[(1 +

∆2 + ε(ε+ ~ω)p1p2

)1

p1 − p2 + i~/τ−(

1− ∆2 + ε(ε+ ~ω)

p1p2

)1

−p1 − p2 + i~/τ

]tanh

(ε+ ~ω

2kT

)+[(

1− ∆2 + ε(ε+ ~ω)

p1p2

)1

p1 + p2 + i~/τ−(

1 +∆2 + ε(ε+ ~ω)

p1p2

)1

p1 − p2 + i~/τ

]tanh

( ε2kT

)I3 =

[(1− ∆

2 + ε(ε− ~ω)p3p2

)1

p3 + p2 + i~/τ−(

1 +∆2 + ε(ε− ~ω)

p3p2

)1

p3 − p2 + i~/τ

]tanh

( ε2kT

), (4)

where

p1 =√

(ε+ ~ω)2 −∆2,

p2 =√ε2 −∆2,

p3 =√

(ε− ~ω)2 −∆2,p4 = i

√∆2 − (ε− ~ω)2. (5)

In these expressions, ε is the energy of the carriers, ω isthe frequency of electromagnetic waves, T is the temper-ature, and ∆ is superconducting gap, whose temperaturedependence is approximately described by [60]

∆(T ) = ∆0

(1− T

TC

)1/2(0.9663 + 0.7733

T

TC

), (6)

where TC is the critical temperature. In the normal state(T > TC), the corresponding optical conductivity is given

by [58]

σn(ω) =σdc

1− iωτ. (7)

To describe the Au layers, we use the following Drude-like relative permittivity [61]

�Au = �∞,Au −ω2p,Au

ω(ω + iγAu), (8)

where �∞,Au, ωp,Au and γAu are, respectively, the highfrequency limit of dielectric function, plasma frequency,and damping of the free carrier.

All calculations in this work were performed with thefollowing parameters for Nb: ε∞,Nb = 4 [58], ∆0 =1.764kBTC, TC = 9 K, σdc = 1.7 × 107 S/m, andωp = 8.8× 1015 rad/s [54]. For the Au layers, �∞,Au = 4,

4

- 1 2- 9- 6- 3

Re{��

(T)}x1

04

T = 7 K T = 5 K T = 3 K

T = 8 . 9 K T = 8 K

( a )

( b )n o r m a l s t a t e g o l d

n o r m a l s t a t e g o l d

0 . 0 0 . 4 0 . 8 1 . 2 1 . 60369

Im{��

(T)}x1

05

ω ( 1 0 1 3 r a d / s )FIG. 2. Real, panel (a), and imaginary part, panel (b), of thedielectric functions of Nb [see Eq. (1)] and Au [see Eq. (8)]employed in this work. The dielectric function of Nb is shownfor different temperatures in the superconducting phase, aswell as in the normal state.

ωp,Au = 1.71× 1016 rad/s, and γAu = 1.22× 1014 rad/s.These parameters are consistent with the experimentalones reported in Ref. [62] for cryogenic temperatures. InFig. 2 we display the frequency dependence of the realand imaginary part of the dielectric functions of Nb andAu computed with those parameter values. In particular,we show the Nb dielectric function for different temper-atures inside the superconducting phase, as well as fortemperatures above TC, i.e., in the normal state.

III. NB PARALLEL PLATES

Before discussing the functional devices, diode andtransistor, it is convenient to analyze the impact of thesuperconducting phase transition in the NFRHT. For thispurpose, we revisit here the case of two Nb parallel plates,see Fig. 1(a), which has been analyzed, both theoreti-cally and experimentally, in Ref. [54]. Within the theoryof fluctuational electrodynamics, the net power per unitarea (heat flux) exchanged via radiation by two infiniteparallel plates, see Fig. 1(a), is given by [4]

Q =

∫ ∞0

dω

2π[Θ1(ω)−Θ3(ω)]

∫ ∞0

dk

2πk[τ13s + τ

13p

],

(9)where Θi(ω) = ~ω/[exp(~ω/kBTi)−1], Ti is the absolutetemperature of the layer i, ω is the radiation frequency,k is the magnitude of the wave vector parallel to thesurface planes, and τ13β (ω, k, d) is the total transmission

probability of the electromagnetic propagating (k < ω/c)and evanescent (k > ω/c) waves, given by

τ13β (ω, k, d) =

(

1−|r21β |2)(

1−|r23β |2)

|1−r21β r23β e2iq2d|2, k < ω/c

4Im(r21β )Im(r23β )e

−2Im(q2)d

|1−r21β r23β e2iq2d|2, k > ω/c

.(10)

Here, qi =√�iω2/c2 − k2 is the the wave vector compo-

nent perpendicular to the plate surfaces in the vacuumgap and c is the velocity of light in vacuum. The reflec-tion amplitudes rijβ are given by the Fresnel coefficients

rijs =qi − qjqi + qj

and rijp =�jqi − �iqj�jqi + �iqj

. (11)

The corresponding linear thermal conductance per unitof area, usually referred to as heat transfer coefficient, isgiven by

h =

∫ ∞0

dω

2π

∂

∂T

[~ω

e~ω/kBT − 1

] ∫ ∞0

dk

2πk[τ13s + τ

13p

].

(12)In the case of two black bodies, which is achieved whenτ ijβ = 1 for all frequencies for propagating waves, thisresult reduces to the Stefan-Boltzmann law

hBB = 4σT3, (13)

where σ is the Stefan-Boltzmann constant (not to be con-fused with a conductivity).

In Fig. 3(a) we show the results for the gap depen-dence of the heat transfer coefficient of two Nb paral-lel plates for different temperatures across the supercon-ducting phase transition (TC = 9 K). One can see thatthe heat transfer coefficient is greatly enhanced in thenear-field regime (d < 105 nm) and it saturates for smallgaps (d < 100 nm). Notice also that there is a pro-nounced temperature dependence, especially for temper-atures below TC. This strong dependence is due to boththe impact of the superconducting phase transition andthe fact that the temperature itself is being changed byan amount comparable to its absolute value. To disentan-gle those two dependencies, it is convenient to normalizethe heat transfer coefficient by the corresponding resultfor two black bodies, as we do in Fig. 3(b). With thisnormalization, we see that the blackbody limit is greatlyovercome in the near-field regime (by almost 5 orders ofmagnitude at T ∼ 10 K and gaps d < 100 nm). More im-portantly for this work, there is a very strong reductionof the NFRHT upon decreasing the temperature belowTC. For instance, for small gaps the NFRHT for 3 Kis about 10 smaller than for 9.1 K. In simple terms, thisdramatic effect can be explained by the presence of a gapin the spectrum of a superconductor that naturally leadsto a strong reduction of the emissivity of the material atlow frequencies, see Fig. 2(b). On the other hand, and inorder to give an insight into the NFRHT in the supercon-ducting phase, we show in Fig. 3(c) the gap dependence

5

1 0 - 91 0 - 71 0 - 51 0 - 31 0 - 11 0 1

1 0 - 31 0 - 11 0 11 0 31 0 5

1 0 0 1 0 1 1 0 2 1 0 3 1 0 4 1 0 51 0 - 91 0 - 71 0 - 51 0 - 31 0 - 11 0 1 ( c )

( b )

( a )h (

W/m2

K)h (

W/m2

K)h /

h BB

d ( n m )

7 K 5 K 3 K

8 K 9 . 1 K 8 . 9 K 1 3 K

T = 7 K

e v a n . T Mp r o p . T Me v a n . T Ep r o p . T ET o t a l

FIG. 3. (a) Heat transfer coefficient, h, for two Nb parallelplates, see Fig. 1(a), as a function of the vacuum gap sized. The different curves correspond to different values of theabsolute temperature T . (b) The same as in panel (a), butnow the heat transfer coefficient is normalized to the corre-sponding blackbody limit, hBB [see Eq. (13)]. (c) Differentcontributions to the heat transfer coefficient of two Nb platesfrom both propagating and evanescent waves with TE andTM polarization and for a temperature of 7 K.

of the different contributions to the heat transfer coeffi-cient for the Nb parallel plates for a temperature T = 7K, including both propagating and evanescent waves forboth TE (or s) and TM (or p) waves. As expected formetals, the NFRHT is largely dominated by evanescentTE modes, which can be attributed to total internal re-flection modes as explained in detail in Ref. [61].

IV. NEAR-FIELD THERMAL RECTIFIER

Now we turn to the analysis of the thermal rectifiershown in Fig. 1(b) and composed by two parallel platesmade of Nb and Au. As explained in the introduction,there have been many proposals for near-field thermalrectifiers employing a variety of materials that undergo aphase transition as a function of temperature. In partic-

ular, Ordoñez-Miranda et al. [57] proposed a diode withterminals made of parallel plates of Nb and SiO2 and op-erating at temperatures between 1 and 8.7 K, for whichNb is superconducting. These authors reported that therectification factor (see definition below) could reach 71%for gaps on the order of 60 µm. In what follows, weshow that the performance of a superconducting near-field thermal rectifier can be boosted by using a metalas a second thermal reservoir instead of a polar dielectriclike silica.

The heat flux in an asymmetric system like that ofFig. 1(b) can be calculated with the formulas describedin the previous section. For the rectifying behavior, wecalculated the net heat flux for both forward (FB) and re-verse (RB) bias configuration. The forward (reverse) biasheat flux, QFB (QRB), was calculated by setting T1 = 1K and T3 = T1 + ∆T with ∆T > 0 (T1 = T3 + ∆T andT3 = 1 K ), i.e., in forward (reverse) bias the tempera-ture gradient is from media (1) to (3) [(3) to (1)]. Therectification factor is defined as

η =

∣∣∣∣ |QFB| − |QRB|max(|QFB| , |QRB|)∣∣∣∣ . (14)

Notice that with this definition, η is bounded between 0and 1.

In Fig. 4 we summarize our main results for the ther-mal diode of Fig. 1(b). In the upper panel we show theforward and reverse bias heat fluxes for d = 10 nm asa function of the temperature difference |∆T |, while thetemperature of the coldest plate is fixed at 1 K. The cor-responding rectification factor is also shown. Notice thatvery high values above 0.9 can be achieved for small val-ues of the temperature difference. In the lower panel weshow the results for the gap dependence of the rectifica-tion factor for various temperature differences. As onecan see, very high values are reached in the near-fieldregime in a huge range of gap size values (of about fourdecades). In particular, values as high as 98.7% are ob-tained for gaps on the order of 10 µm, see Fig. 4(b), whichto our knowledge are the highest ever reported and, inparticular, are much higher than those predicted in previ-ous proposals of superconducting thermal rectifiers [57].

To gain some physical insight into the origin of thehuge rectification ratios in our system, we show inFig. 5(a) the spectral heat flux (or power per unit of areaand frequency) as a function of frequency for the forwardand reverse bias for a case in which the temperatures ofthe cold and hot reservoirs are 1 and 8.9 K, respectively.In this case, the gap size is 10 nm. The spectral heat fluxhas the characteristic form in metallic systems in whichthe evanescent TE electromagnetic modes completelydominate the NFRHT [22, 61], which is also the casein our asymmetric configuration. Notice, in particular,that for the reverse bias there is an abrupt frequency cut-off below which the spectral function drastically drops.For the reverse bias, the Nb plate is at 1 K, i.e., deepinto the superconducting phase, and that cut-off simplycorresponds to the frequency ω0 = 2∆0/~ ≈ 4.2 × 1012

6

4 8 1 2 1 60123

1 0 0 1 0 2 1 0 4 1 0 60 . 40 . 60 . 81 . 0

|Q| (1

02 Wm

-2 )

| ∆T | ( K )

( a )

( b )

R B F B

d ( n m )

7 K3 K

9 K8 . 1 K7 . 9 K

0 . 60 . 70 . 80 . 91 . 0

�

�

FIG. 4. (a) Forward (FB) and reverse (RB) bias radiativeheat flux in the rectifier of Fig. 1(b) as a function of thetemperature difference |∆T | = |T1 − T3| and for a gap sizeof 10 nm. The corresponding rectification factor η, definedin Eq. (14), is also included. (b) Rectification factor as afunction of the vacuum gap size, d and for different values ofthe temperature difference |∆T | = |T1 − T3|. All results wereobtained for min(T1, T3) = 1 K.

rad/s that is required to break a Cooper pair. Below thisfrequency, the emissivity of the Nb plate is drasticallyreduced due to the presence of a gap in the electronicspectrum, which explains the strong reduction of the ra-diative heat transfer, as compared to the forward biasconfiguration. This interpretation is further illustratedin Fig. 5(b,c) where we show the corresponding trans-mission probability for the evanescent TE modes as afunction of the frequency and the parallel component ofthe wave vector. Notice that for the reverse bias, panel(c), the transmission is very small below ω0. This fact,together with the frequency dependence of the thermalfactor |Θ1(ω)−Θ3(ω)|, see dotted lines in panels (b) and(c), determining the electromagnetic modes available forheat transfer, see Eq. (9), explains the huge differencebetween in the net power between the reverse and theforward bias configurations.

V. NEAR-FIELD THERMAL TRANSISTOR

Let us now analyze the superconducting thermal tran-sistor depicted in Fig. 1(c) that features a Nb layer ofthickness δ3 as a gate electrode that is placed at a dis-

0 1 2 30 . 00 . 51 . 01 . 52 . 02 . 5

k ( 1 0 7 m - 1 )k ( 1 0 7 m - 1 )0 1 2 3

ω00 . 00 . 10 . 30 . 4

0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 51 0 - 2 21 0 - 1 91 0 - 1 61 0 - 1 3

R BF B

R BF B

( c )( b )

ω (1

013rad

/s)|Q ω

| (J/ra

d m2 )

ω ( 1 0 1 3 r a d / s )

( a )

FIG. 5. (a) Spectral heat flux for the forward (solid line)and reverse (dashed line) bias configurations as a function ofthe frequency and parallel wave vector for a gap size of 10nm in the system of Fig. 1(b). The temperature differenceis set to |∆T | = |T1 − T3| = 7.9 K, with a temperature of 1K for the cold reservoir. (b,c) The transmission probabilityof the evanescent TE modes corresponding to each configu-ration. The dashed and solid lines in both panels represent,

respectively, the Nb and Au light lines ω = ckRe(�1/2Nb ) and

ω = ckRe(�1/2Au ), whereas the dotted line corresponds to the

thermal factor |Θ1 − Θ3| shown here in arbitrary units. Thedash-dotted line in panel (c) indicates the photon frequencyrequired to break a Cooper pair in the superconducting state,ω0 = 2∆0/~ ≈ 4.2× 1012 rad/s.

tance d of two Au plates that act as source and drain.To compute the different heat exchanges in this three-body system, we have employed the many-body theoryput forward in Ref. [50]. According to this theory, theheat flux received by the drain, QD, and the heat flux lostby the source, QS, in the near-field thermal transistor ofFig. 1(c) are given by

QD =

∫ ∞0

dω

2π

∑β=s,p

∫ ∞0

dk

2πk[Θ13τ

13β + Θ35τ

35β

],(15)

QS =

∫ ∞0

dω

2π

∑β=s,p

∫ ∞0

dk

2πk[Θ53τ

53β + Θ31τ

31β

],(16)

where Θij = Θi(ω)−Θj(ω), and τ ijβ is the transmissionprobability of electromagnetic waves from region i to jin Fig. 1(c). For a symmetric configuration (source anddrain equidistant to gate), as considered in this work,

7

these transmissions probabilities are given by

τ13β =4∣∣∣τ3β ∣∣∣2 Im(ρ1β) Im(ρ5β) e−4Im(q2)d∣∣∣1− ρ13β ρ5βe−2Im(q2)d∣∣∣2 ∣∣∣1− ρ1βρ3βe−2Im(q2)d∣∣∣2 ,

τ35β =4Im

(ρ13β

)Im(ρ5β

)e−2Im(q2)d∣∣∣1− ρ13β ρ5βe−2Im(q2)d∣∣∣2 , (17)

where

ρ13β = ρ3β +

(τ3β

)2ρ1βe−2Im(q2)d∣∣∣1− ρ1βρ3βe−2Im(q2)d∣∣∣2 ,

ρiβ = −riβ1− e2iqiδi

1− (riβ)2e2iqiδi,

τ iβ =tiβt

iβeiqiδi

1− (riβ)2e2iqiδi,

ris =q2 − qiq2 + qi

, rip =�iq2 − qi�iq2 + qi

,

tis =2q2

q2 + qi, tip =

2√�iq2

�iq2 + qi,

tis =

2qiq2 + qi

, tip =

2√�iqi

�iq2 + qi.

Additionally, the transmission probabilities τ53β (τ31β ) can

be obtained after substitution 1 → 5 and 5 → 1 in τ13β(τ35β ). The corresponding net heat flux received or emit-

ted by the gate is defined as [49]

QG = |QS| − |QD| , (18)

and the amplification factor as

α =

∣∣∣∣∂|QD|∂QG∣∣∣∣ = 1|1−Q′S/Q′D| , (19)

where Q′S/D = ∂|QS/D|/∂TG. Amplification requires anegative differential conductance and it is characterizedby α > 1 [49]. In what follows, we shall choose TS = 20K, TD = 1 K, and TS > TG > TD.

In Fig. 6(a-c) we present the different powers gainedor lost by the three thermal reservoirs (drain, source andgate) as a function of the gate temperature TG for dif-ferent values of the gate thickness δ3 and a Nb-Au gapd = 500 nm. As expected, the heat received by the drain,|QD|, increases monotonically with TG, irrespective ofwhether the gate is in the normal or in the superconduct-ing state, see Fig. 6(a). On the other hand, and contraryto what one could naively expect, the heat lost by thesource, see panel (b), and the net heat in the gate, seepanel (c), do not decrease monotonically upon increas-ing TG. This peculiar behavior only takes place in thesuperconducting state (TG < TC) and it is a necessarycondition for the amplification to occur, see Eq. (19). In

3 9 1 50246

3 9 1 50246

3 9 1 5- 2024

1 3 5 7 91 0 - 21 0 01 0 2

|Q S| (W

m-2)

Q G (W

m-2)

|Q D| (W

m-2)

( d )

( b )( a )

( c )

T G ( K )

�

T G ( K )

5 n m1 0 n m

7 4 0 n m8 0 n m2 0 n m

FIG. 6. (a) Absolute value of the of the radiative heat fluxreceived by the drain in the thermal transistor of Fig. 1(c)as a function of the temperature of the gate. The differentcurves correspond to different values of the gate thickness δ3.(b) The corresponding absolute value of the heat flux lost bythe source. (c) The corresponding net flux received or lost bythe gate. (d) The corresponding amplification factor. All theresults were obtained for a vacuum gap size of d = 500 nm,and the source and drain temperatures are TS = 20 K andTD = 1 K, respectively.

Fig. 6(d) we show the corresponding value of the ampli-fication factor, defined in Eq. (19), as a function of TG.Notice that for thin Nb layers, values of α larger than1 are possible in a certain range of temperatures withinthe superconducting phase, and for very specific values ofTG the amplification factor can reach values close to 100.In those cases, our three-body system truly behaves asa near-field transistor. This is, however, never the casewhen the Nb gate is in its normal state where α tends to1/2.

To get some insight into the origin of the amplifica-tion, we show in Fig. 7(b) the derivatives of the heatexchanges in the source and drain with respect to thegate temperature, Q′S/D, for the case of δ3 = 5 nm shown

in Fig. 6(d), see also Fig. 7(a). From Eq. (19) we see thatamplification (α > 1) requires that 1 < Q′S/Q

′D < 2. The

dependence with the gate temperature shown in Fig. 7(b)explains why there is amplification in the range betweenapproximately 2.5 and 4.2 K and why there is a huge peakat around 4.2 K, which corresponds to the situation inwhich Q′S ≈ Q′D. Moreover, we show in Fig. 7(b) the dif-ferent individual contributions to Q′S following Eq. (16).There we see that the key temperature dependence ofQ′S is mainly determined by the term involving Θ31τ

31s

for TE (or s) polarization, which is related to the heatlost by the source to the combined gate-drain system.

8

1 0 - 31 0 - 21 0 - 11 0 01 0 11 0 2

1 2 3 4 5- 30369

123

0 . 0 0 . 2 0 . 4 0 . 6

123

�

( d )( b )

( c )( a )

Q' (1

0-2 W/m

2 K)

T G ( K )

- Q ' ( p )S , 3 1- Q ' ( s )S , 3 1

- Q ' ( p )S , 5 3

- Q ' SQ ' D

3 . 6 K3 . 8 K4 . 0 K4 . 2 K

4 . 4 K4 . 6 K

- Q ' ( s )S , 5 3 |Q (s) ω,S

,31| (1

0-13 J/r

ad m2

)

ω ( 1 0 1 3 r a d / s )

1 . 8 K2 . 0 K2 . 2 K2 . 4 K

2 . 6 K2 . 8 K

FIG. 7. (a) Amplification factor for a Nb gate with a thick-ness of 5 nm, separated from both the source and the drain bya 500 nm vacuum gap, see Fig. 6(d). (b) The correspondingderivates of the heat received by the drain, Q′D, and of theheat lost by the source, Q′S. Also included are the derivatesof each one of the terms contributing to the heat lost by thesource, see Eq. (16). (c) Absolute value of the spectral heat

flux Q(s)ω,S,31 = Θ31τ

31s for several temperatures around the

maximum amplification region in panel (a). (d) The same asin panel (c) for a gate of thickness 10 nm, in the tempera-ture region where the corresponding amplification reaches itsmaximum in the Fig. 6(d).

To understand the TG-dependence of this term, we illus-trate in Fig. 7(c) the evolution with the gate temperatureof the corresponding spectral contribution. As one cansee, the temperature dependence is only significant in thefrequency region close to ω0 = 2∆0/~ for the tempera-tures in which the amplification occurs. As we show inFig. 7(d), that temperature dependence is much weakerwhen the gate thickness is increased, which explains whythe amplification gets lost as the gate thickness increases.

For completeness, we have systematically explored therole of the gap size d between the Nb gate and the Aureservoirs. We show a summary of these results in Fig. 8where we show the amplification factor as a function ofthe gate temperature for different values of d (and alsodifferent values of the gate thickness). As one can see, theoptimal amplification occurs for gap sizes d on the orderof a few hundred nm, while it only appears at some spe-cial points of the parameter space for smaller and largergap values. In all cases, the amplification only appearsfor very thin Nb films.

VI. CONCLUSIONS

Motivated by recent theoretical proposals and experi-ments on the use of phase-transition materials to realizenear-field thermal devices that mimic electronic compo-

nents, we have presented in this work a theoretical study

1 0 - 21 0 01 0 2

1 3 5 7 91 0 - 21 0 01 0 2

3 5 7 9

��

( d )

( b )( a )

( c )

d = 1 0 0 0 n md = 2 0 0 n m

T G ( K )T G ( K )

d = 1 0 0 n md = 1 0 n m

FIG. 8. Amplification factor in the thermal transistor ofFig. 1(c) as a function of gate temperature. The differentpanels correspond to different vacuum gap sizes, while thedifferent curves in each panel correspond to different valuesof the gate thickness, δ3, whose values are the same as inFig. 6. In all cases the source and drain temperatures areTS = 20 K and TD = 1 K, respectively.

of the performance of near-field thermal diodes and tran-sistors based on the combination of superconductors andnormal metals. We have shown that the drastic reduc-tion in the thermal emission of a metal when it undergoesa superconducting transition can be utilized to realizeheat rectification in two-body systems and heat ampli-fication in three-body devices. In particular, we haveshown that a system formed by two parallel plates madeof Au and (superconducting) Nb can act as a near-fieldthermal diode that exhibits striking rectification ratiosvery close to unity in a wide range of gap size values,outperforming all the existent proposals to date. More-over, we have shown that a Nb layer placed in the mid-dle of two Au plates can amplify the heat transferred tothe drain at temperatures below Nb critical temperature.The ideas put forward in this work can be extended topropose other functional devices such as thermal logicgates and, overall, it illustrates the potential of the useof superconducting materials for near-field thermal man-agement at low temperatures.

ACKNOWLEDGMENTS

J.C.C. acknowledges funding from the Spanish Min-istry of Economy and Competitiveness (MINECO) (con-tract No. FIS2017-84057-P).

9

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Normal Metal-Superconductor Near-Field Thermal Diodes and TransistorsAbstractI IntroductionII Systems and optical propertiesIII Nb parallel platesIV Near-field thermal rectifierV Near-field thermal transistorVI Conclusions Acknowledgments References