Magnetic-field control of near-field radiative heat transfer and the realization ofhighly tunable hyperbolic thermal emitters

E. Moncada-Villa1, V. Fernandez-Hurtado2, F. J. Garcıa-Vidal2, A. Garcıa-Martın3, and J. C. Cuevas2∗1Departamento de Fısica, Universidad del Valle, AA 25360, Cali, Colombia

2Departamento de Fısica Teorica de la Materia Condensada and Condensed Matter Physics Center (IFIMAC),Universidad Autonoma de Madrid, E-28049 Madrid, Spain and3IMM-Instituto de Microelectronica de Madrid (CNM-CSIC),Isaac Newton 8, PTM, Tres Cantos, E-28760 Madrid, Spain

(Dated: June 22, 2015)

We present a comprehensive theoretical study of the magnetic field dependence of the near-fieldradiative heat transfer (NFRHT) between two parallel plates. We show that when the plates aremade of doped semiconductors, the near-field thermal radiation can be severely affected by theapplication of a static magnetic field. We find that irrespective of its direction, the presence of amagnetic field reduces the radiative heat conductance, and dramatic reductions up to 700% can befound with fields of about 6 T at room temperature. We show that this striking behavior is dueto the fact that the magnetic field radically changes the nature of the NFRHT. The field not onlyaffects the electromagnetic surface waves (both plasmons and phonon polaritons) that normallydominate the near-field radiation in doped semiconductors, but it also induces hyperbolic modesthat progressively dominate the heat transfer as the field increases. In particular, we show thatwhen the field is perpendicular to the plates, the semiconductors become ideal hyperbolic near-fieldemitters. More importantly, by changing the magnetic field, the system can be continuously tunedfrom a situation where the surface waves dominate the heat transfer to a situation where hyperbolicmodes completely govern the near-field thermal radiation. We show that this high tunability can beachieved with accessible magnetic fields and very common materials like n-doped InSb or Si. Ourstudy paves the way for an active control of NFRHT and it opens the possibility to study uniquehyperbolic thermal emitters without the need to resort to complicated metamaterials.

I. INTRODUCTION

Radiative heat transfer is a topic with numerous funda-mental and technological implications across disciplines[1]. Presently, the investigation of radiative heat trans-fer between closely spaced objects is receiving a greatattention [2–6]. The basic challenges now are the under-standing of the mechanisms governing thermal radiationat the nanoscale and the ability to control this radiationfor its use in novel applications. During a long time, itwas believed that the maximum radiative heat transferbetween two objects was set by the Stefan-Boltzmannlaw for black bodies. However, this is only true when ob-jects are separated by distances larger than the thermalwavelength (9.6 µm at room temperature) and the ra-diative heat transfer is dominated by propagating waves.When two objects are brought in closer proximity, thethermal radiation is dominated by interference effectsand, more importantly, by the near field emerging fromthe materials surfaces in the form of evanescent waves.This was first established theoretically by Polder andVan Hove in 1971 [7] using Rytov’s framework of fluc-tuational electrodynamics [8, 9]. These researchers pre-dicted that the NFRHT could overcome the black bodylimit by several orders of magnitude, achieving what is re-ferred to as super-Planckian thermal emission. Althoughthis NFRHT enhancement was already hinted in several

∗ juancarlos.cuevas@uam.es

experiments in the late 1960’s [10, 11], its unambiguousconfirmation came only in recent years [12–27]. Theseexperiments, in turn, have triggered off the hope thatNFRHT may have an impact in different technologiessuch as heat-assisted magnetic recording [28, 29], ther-mal lithography [30], scanning thermal microscopy [31–33], coherent thermal sources [34, 35], near-field basedthermal management [22, 36–38], thermophotovoltaics[39, 40] and other energy conversion devices [41].

Presently, one of the central research lines in the fieldof radiative heat transfer is the search for materials wherethe NFRHT enhancement can be further increased. Sofar, the largest enhancements have been experimentallyreported in polar dielectrics [15, 16, 26], where the near-field thermal radiation is dominated by the excitationof surface phonon polaritons (SPhPs) [42]. Similar en-hancements have been predicted and observed in dopedsemiconductors due to surface plasmon polaritons (SPPs)[23, 27, 43, 44]. Recently, it has been predicted thathyperbolic metamaterials could behave as broadbandsuper-Planckian thermal emitters [45–47]. Hyperbolicmetamaterials are a special class of highly anisotropicmedia that have hyperbolic dispersion. In particular,they are uniaxial materials for which one of the principalcomponents of either the permittivity or the permeabilitytensors is opposite in sign to the other two principal com-ponents [48]. Hyperbolic media have been mainly real-ized by means of hybrid metal-dielectric superlattices andmetallic nanowires embedded in a dielectric host [49, 50].It has been demonstrated that these metamaterials ex-hibit exotic optical properties such as negative refrac-

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tion, subwavelength imaging and focusing, and they canbe used to do density of states engineering [49, 50]. Inthe context of radiative heat transfer, what makes thesemetamaterials so special is the fact that they can supportelectromagnetic modes that are evanescent in a vacuumgap, but they are propagating inside the material. Thisleads to a broadband enhancement of transmission effi-ciency of the evanescent modes [46]. This special prop-erty has motivated a lot of theoretical work on the useof hyperbolic metamaterials for NFRHT [51–61]. How-ever, no experimental investigation of this issue has beenreported so far, which is mainly due to the difficultiesin handling these metamaterials. In this sense, it wouldbe highly desirable to find much simpler realizations ofhyperbolic thermal emitters and, ideally, with tunableproperties.

Another key issue in the field of radiative heat transferis the active control and modulation of NFRHT. In thisrespect, several proposals have been put forward recently.One of the them is based on the use of phase-change ma-terials [62, 63], where the change of phase leads to a sig-nificant change in the material dielectric function. Thesematerials include an alloy called AIST, where the phasechange can be induced by applying an electric field [63],and VO2, which undergoes a metal-insulator transitionas a function of temperature [62]. It has also been sug-gested that the NFRHT between chiral materials withmagnetoelectric coupling can be tuned by ultrafast op-tical pulses [64]. Another proposal to tune the NFRHTis to use ferroelectric materials under an external electricfield [65], although the predicted changes are rather mod-est (< 17%). Let us also mention that very recently ithas been proposed that the heat flux between two semi-conductors can be controlled by regulating the chemicalpotential of photons by means of an external bias [66]. Soin short, although these proposals are certainly interest-ing, some of them are not easy to implement and othersare either not very efficient or they are restricted to veryspecific materials. In this sense, the challenge remainsto introduce strategies to actively control NFRHT in aneasy and relatively universal way.

In this work we tackle and resolve some of the openproblems described above by presenting an extensive the-oretical analysis of the influence of an external dc mag-netic field in the radiative heat transfer between two par-allel plates made of a variety of materials. We show thatan applied magnetic field can indeed largely affect theNFRHT in a broad class of materials, namely doped (po-lar and non-polar) semiconductors. We find that, irre-spective of its orientation, the magnetic field reduces theNFRHT with respect to the zero-field case and we showthat the reduction can be as large as 700% for fields ofabout 6 T at room temperature. This effect originatesfrom the fact that the magnetic field not only stronglymodifies the surface waves that dominate the NFRHTin doped semiconductors (both SPhPs and SPPs), butit also generates broadband hyperbolic modes that tendto govern the heat transfer as the field is increased. In

particular, when the applied field is perpendicular tothe plates surfaces, the semiconductors behave as hyper-bolic thermal emitters with highly tunable properties. Bychanging the field magnitude one can continuously tunethe system and realize situations where (i) surface wavesdominate the NFRHT, (ii) both surface waves and hyper-bolic modes contribute significantly to the near-field ther-mal radiation, and (iii) only hyperbolic modes contributeto the NFRHT and surface waves cease to exist. On theother hand, when the field is parallel to the surfaces theNFRHT is nonmonotonic as a function of the magneticfield. For moderate fields, surface waves and hyperbolicmodes coexisting, while for high fields the NFRHT islargely dominated by hyperbolic modes. We emphasizethat all these striking predictions are amenable to mea-surements and do not require the use of any complicatedmetamaterial. Thus, our work offers a simple strategy toactively control NFRHT in a broad variety of materialsand it also provides a very appealing recipe to realize hy-perbolic materials and, in particular, hyperbolic thermalemitters with highly tunable properties.

The remainder of this paper is structured as follows.Section II describes the system under study and the gen-eral formalism for the description of NFRHT in the pres-ence of a magnetic field. We then turn in Sec. III to theapplication of the general results to the case of n-dopedInSb as an example of a polar semiconductor. We discussin this section both the results for different magnetic fieldorientations and the realization of highly tunable hyper-bolic thermal emitters. Section III is devoted to the caseof Si as an example of non-polar semiconductor. SectionIV summarizes our main results and discusses future di-rections. Finally, four appendixes contain the technicaldetails of the general formalism and some additional cal-culations that support the claims in this paper.

II. RADIATIVE HEAT TRANSFER IN THEPRESENCE OF A MAGNETIC FIELD:

GENERAL FORMALISM

Our main goal is to compute the radiative heat trans-fer in the presence of an external dc magnetic field withinthe framework of fluctuational electrodynamics [8, 9].For simplicity, we shall concentrate here in the heat ex-changed between two infinite parallel plates made of ar-bitrary non-magnetic materials and that are separatedby a vacuum gap of width d, see Fig. 1(a). The magneticfield can point in any direction and following Fig. 1(a),we shall refer to the left plate as medium 1, the vacuumgap as medium 2, and the right plate as medium 3.

When a magnetic field is applied to any object, it re-sults in an optical anisotropy that can be described bythe following general permittivity tensor [67]

ε =

εxx εxy εxzεyx εyy εyzεzx εzy εzz

, (1)

3

1 0 0 1 0 1 1 0 2 1 0 3 1 0 4 1 0 51 0 - 11 0 01 0 11 0 21 0 31 0 41 0 51 0 6

W a v e l e n g t h (µm )

d

( c )

G a p s i z e (n m )

Heat t

ransfe

r coef

ficien

t (W/m

2 K)

( b )T o t a ls - p r o p a g a t i n g m o d e ss - e v a n e s c e n t m o d e sp - p r o p a g a t i n g m o d e sp - e v a n e s c e n t m o d e s

( a )d

1 0 1 1 1 0 1 2 1 0 1 3 1 0 1 41 0 - 1 61 0 - 1 51 0 - 1 41 0 - 1 31 0 - 1 21 0 - 1 11 0 - 1 01 0 - 91 0 - 8

( 2 )

Spect

ral he

at flux

(J/rad

m2 Κ)

ω (r a d / s )

d = 1 0 n md = 1 0 0 n md = 1 µm

( 1 ) ( 3 )

T 1 T 3xy z

1 0 4 1 0 3 1 0 2 1 0 1

FIG. 1. (a) Schematic representation of the system understudy: two parallel plates at temperatures T1 and T3 sepa-rated by a vacuum gap of width d. (b) Heat transfer coeffi-cient of n-doped InSb as a function of the gap at zero mag-netic field. We show the total result and the individual con-tributions of s- and p-polarized waves (both propagating andevanescent). (c) The corresponding zero-field spectral heatflux as a function of frequency (and wavelength) for threedifferent gaps.

where according to Fig. 1(a), x and y lie in the interfaceplanes and z corresponds to the surface normal. Thecomponents of the permittivity tensor depend on the ap-plied magnetic field, as we shall specify below, and on thefrequency (local approximation). Let us recall that theoff-diagonal elements in Eq. (1) are responsible for allthe well-known magneto-optical effects (Faraday effect,Kerr effects, etc.) [67]. Thus, our problem is to computethe radiative heat transfer between two anisotropic par-allel plates. This generic problem has been addressed by

Biehs et al. [68] and we just recall here the central re-sult. The net power per unit of area exchanged betweenthe parallel plates is given by the following Landauer-likeexpression [68]

Q =

∫ ∞0

dω

2π[Θ1(ω)−Θ3(ω)]

∫dk

(2π)2τ(ω,k, d), (2)

where Θi(ω) = ~ω/[exp(~ω/kBTi)−1], Ti is the absolutetemperature of the layer i, ω is the radiation frequency,k = (kx, ky) is the wave vector parallel to the surfaceplanes, and τ(ω,k, d) is the total transmission probabil-ity of the electromagnetic waves. Notice that the secondintegral in Eq. (2) is carried out over all possible direc-tions of k and it includes the contribution of both prop-agating waves with k < ω/c and evanescent waves withk > ω/c, where k is the magnitude of k and c is thevelocity of light in vacuum. The transmission coefficientτ(ω,k, d) can be expressed as [68]

τ(ω,k, d) = (3) Tr{

[1− R21R†21]D†[1− R†23R23]D}, k < ω/c

Tr{

[R21 − R†21]D†[R†23 − R23]D}e−2|q2|d, k > ω/c

,

where q2 =√ω2/c2 − k2 is the z-component of the wave

vector in the vacuum gap and the 2× 2 matrices Rij arethe reflections matrices characterizing the two interfaces.These matrices have the following generic structure

Rij =

(rs,sij rs,pijrp,sij rp,pij

), (4)

where rα,βij with α, β = s, p is the reflection amplitude forthe scattering of an incoming α-polarized plane wave intoan outgoing β-polarized wave. Finally, the 2 × 2 matrixD appearing in Eq. (3) is defined as

D = [1− R21R23e2iq2d]−1. (5)

Notice that this matrix describes the usual Fabry-Perot-like denominator resulting from the multiple scatteringbetween the two interfaces.

In Appendixes A and B we provide an alternativederivation of the central result of Eq. (3) that empha-sizes the non-reciprocal nature of our problem. More im-portantly, we show explicitly how the different reflectionmatrices appearing in Eq. (3) can be computed withina scattering-matrix approach for anisotropic multilayersystems. This approach provides, in turn, a naturalframework to analyze different issues that will be cru-cial later on such as the nature of the electromagneticmodes responsible for the heat transfer.

The result of Eqs. (2) and (3) reduces to the well-known result for isotropic media first derived by Polderand Van Hove [7]. In that case, the reflections matricesof Eq. (4) are diagonal and the non-vanishing elements

4

are given by

rs,sij =qi − qjqi + qj

(6)

rp,pij =εjqi − εiqjεjqi + εiqj

, (7)

where qi =√εiω2/c2 − k2 is the transverse or z-

component of the wave vector in layer i and εi(ω) isthe corresponding dielectric constant. Thus, the totaltransmission can be written as τ(ω,k, d) = τs(ω,k, d) +τp(ω,k, d), where the contributions of s- and p-polarizedwaves are given by

τα=s,p(ω,k, d) = (8){(1− |rα,α21 |2)(1− |rα,α23 |2)/|Dα|2, k < ω/c4Im{rα,α21 }Im{r

α,α23 }e−2|q2|d/|Dα|2, k > ω/c

,

where Dα = 1 − rα,α21 rα,α23 e2iq2d. Throughout this workwe focus on the analysis of the radiative linear heat con-ductance per unit of area, h, which is referred to as theheat transfer coefficient. This coefficient is given by

h(T, d) = lim∆T→0+

Q(T1 = T + ∆T, T3 = T, d)

∆T, (9)

where T is the absolute temperature that we assumeequal to 300 K throughout this work. Additionally, wedefine the spectral heat flux as the heat transfer coef-ficient per unit of frequency. In the following sections,we apply the general results presented here to differentmaterials and magnetic field configurations.

III. POLAR SEMICONDUCTORS: InSb

The first obvious question to be answered is: In whichmaterials can a magnetic field modify the NFRHT? Sincethe thermal radiation of an object is primarily deter-mined by its dielectric function, we need materials inwhich this function can be modified by an external mag-netic field, that is we need magneto-optical (MO) ma-terials. Focusing on room temperature experiments, theMO activity must be exhibited in the mid-infrared. Thus,doped semiconductors, where the MO activity is due toconduction electrons, are ideal candidates [69]. In thesematerials, one can play with the doping level to tunethe plasma frequency to values comparable to the cy-clotron frequency at experimentally achievable magneticfields, which is an important requirement to have sizablemagnetic-induced effects in the NFRHT (see discussionbelow). Moreover, in semiconductors the NFRHT in theabsence of field is typically dominated by surface electro-magnetic waves (both SPhPs and SPPs), which in turnare known to be strongly influenced by an external mag-netic field [69, 70]. Thus, it seems natural to expect amagnetic-field modulation of NFRHT in semiconductors.

There is a variety of semiconductors that we couldchoose to illustrate our predictions. In this section we

focus on InSb for several reasons. First, it is a po-lar semiconductor where the NFRHT in the absence offield is dominated by two different types of surface waves(SPhPs and SPPs), which allows us to study a very richphenomenology. Second, InSb has a small effective mass,which enables to tune the cyclotron frequency to valuescomparable to those of the plasma frequency with moder-ate fields. Finally, InSb has been the most widely studiedmaterial in the context of magnetoplasmons and coupledmagnetoplasmons-surface phonon polaritons. Thus, themagnetic field effect in the surface waves has been verywell characterized experimentally [71–74].

A. Perpendicular magnetic field: The realization ofhyperbolic near-field thermal emitters

Let us first discuss the radiative heat transfer betweentwo identical plates made of n-doped InSb when the mag-netic field is perpendicular to the plate surfaces, i.e.H = Hz z, see Fig. 1(a). In this case, the permittivitytensor of InSb adopts the following form [73]

ε(H) =

ε1(H) −iε2(H) 0iε2(H) ε1(H) 0

0 0 ε3

, (10)

where

ε1(H) = ε∞

(1 +

ω2L − ω2

T

ω2T − ω2 − iΓω

+ω2p(ω + iγ)

ω[ω2c − (ω + iγ)2]

),

ε2(H) =ε∞ω

2pωc

ω[(ω + iγ)2 − ω2c ], (11)

ε3 = ε∞

(1 +

ω2L − ω2

T

ω2T − ω2 − iΓω

−ω2p

ω(ω + iγ)

).

Here, ε∞ is the high-frequency dielectric constant, ωLis the longitudinal optical phonon frequency, ωT is thetransverse optical phonon frequency, ω2

p = ne2/(m∗ε0ε∞)defines the plasma frequency of free carriers of densityn and effective mass m∗, Γ is the phonon damping con-stant, and γ is the free-carrier damping constant. Finally,the magnetic field enters in these expressions via the cy-clotron frequency ωc = eH/m∗. The important featuresof the previous expressions are: (i) the magnetic field in-duces an optical anisotropy (via the modification of thediagonal elements and the introduction of off-diagonalones), (ii) there are two major contributions to the diag-onal components of the dielectric tensor: optical phononsand free carriers, and (iii) the MO activity is introducedvia the free carriers, which illustrates the need to dealwith doped semiconductors. In what follows we shall con-centrate in a particular case taken from Ref. [73], whereε∞ = 15.7, ωL = 3.62 × 1013 rad/s, ωT = 3.39 × 1013

rad/s, Γ = 5.65× 1011 rad/s, γ = 3.39× 1012 rad/s, n =1.07 × 1017 cm−3, m∗/m = 0.022, and ωp = 3.14 × 1013

rad/s. As a reference, let us say that with these parame-ters ωc = 8.02×1012 rad/s for a field of 1 T. Let us point

5

out that in this configuration, and due to the structure ofthe permittivity tensor, the transmission coefficient ap-pearing in Eq. (2) only depends on the magnitude of theparallel wave vector, which considerably simplifies thecalculation of the radiative heat transfer.

Let us now briefly review the expectations for the heattransfer in the absence of magnetic field. As we showin Fig. 1(b), the heat transfer coefficient features a largenear-field enhancement for gaps below 1 µm. For d < 100nm this enhancement is largely dominated by p-polarizedevanescent waves and the heat transfer coefficient in-creases as 1/d2 as the gap decreases, which are two clearsignatures of a situation where the heat transfer is dom-inated by surface electromagnetic waves. This can befurther confirmed with the analysis of the spectral heatflux, see Fig. 1(c), which in the near-field regime is dom-inated by two narrow peaks that can be associated toSPPs (low-frequency peak) and SPhPs (high-frequencypeak), as it will become evident below. Thus, the case ofInSb constitutes an interesting example where two typesof surface waves contribute significantly to the NFRHT.Let us now see how these results are modified in the pres-ence of a magnetic field.

In Fig. 2(a) we show the heat transfer coefficient as afunction of the gap size for different values of the perpen-dicular magnetic field. There are three salient features:(i) the far-field heat transfer is fairly independent of themagnetic field, (ii) in the near-field regime (below 300nm) the magnetic field suppresses the heat transfer byup to a factor of 3 (see inset), and (iii) by increasingthe field, the heat transfer coefficient tends to saturateat around 6 T, although it is slightly reduced upon fur-ther increasing the field above 10 T (not shown here).The strong modification of heat transfer due to the mag-netic field is even more apparent in the spectral heat flux.As one can see in Fig. 2(b), the magnetic field not onlydistorts and reduces the height of the peaks related tothe surface waves, but it also generates a new peak thatshifts to higher frequencies as the field increases. Thisadditional peak appears at the cyclotron frequency andits presence illustrates the high tunability that can beachieved. Notice, for instance, that for a field of 6 T thethermal emission at the cyclotron frequency is increasedby almost 3 orders of magnitude with respect to the zero-field case.

To shed more light on these results it is convenient toexamine the transmission of the p-polarized waves, whichcan be shown to dominate the heat transfer for any field.We present in Fig. 3 this transmission as a function ofthe magnitude of the parallel wave vector, k, and the fre-quency for a gap d = 10 nm and different values of themagnetic field. As one can see, at low fields the trans-mission maxima are located around a restricted area ofk and ω, clearly indicating that surface waves dominatethe NFRHT. Notice also that their dispersion relationis modified by the field, see Fig. 3(b). By increasingthe field, those areas are progressively replaced by areaswhere the maximum transmission is reached for a broad

1 0 0 1 0 1 1 0 2 1 0 3 1 0 4 1 0 51 0 01 0 11 0 21 0 31 0 41 0 51 0 6

W a v e l e n g t h (µm )

( b )

G a p s i z e ( n m )

Heat t

ransfe

r coeff

icient

(W/m

2 K)

( a )

H z = 0 TH z = 2 TH z = 4 TH z = 6 TH z = 8 T

H z = 0 TH z = 2 TH z = 6 TH z = 1 0 T

1 0 1 3 1 0 1 41 0 - 1 4

1 0 - 1 3

1 0 - 1 2

1 0 - 1 1

1 0 - 1 0

1 0 - 9

1 0 - 8

Spect

ral he

at flux

(J/rad

m2 K)

ω ( r a d / s )

H z = 8 T

1 0 0 1 0 1 1 0 2 1 0 31 . 01 . 52 . 02 . 53 . 0

h�(0)

/ h�(H

z)

G a p s i z e (n m )

1 0 2× ×2 1 0 2

2 1 0 1×

FIG. 2. (a) Heat transfer coefficient for n-doped InSb as afunction of the gap for different values of the magnetic fieldperpendicular to the plate surfaces. The inset shows the ra-tio between the zero-field coefficient and the coefficient fordifferent values of the field in the near-field region. (b) Thecorresponding spectral heat flux as a function of the frequency(and wavelength) for a gap of d = 10 nm and different valuesof the perpendicular field. The solid lines correspond to theexact calculation and the circles to the uniaxial approxima-tion where the off-diagonal terms of the permittivity tensorare assumed to be zero.

range of k-values and finally, the surface waves are re-stricted to the reststrahlen band ωT < ω < ωL for thehighest fields, see Fig. 3(d). What is the nature of thesemagnetic-field-induced modes?

To answer this question and explain all the results justdescribed, it is important to realize that the off-diagonalelements of the permittivity tensor do not play a majorrole in this configuration. This is illustrated in Fig. 2(b)where we show that the approximation consisting in set-ting ε2 = 0 in Eq. (10) reproduces very accurately theexact results for the spectral heat flux for arbitrary mag-netic fields. This means that the polarization conversionis irrelevant and the plates effectively behave as uniaxialmedia where their permittivity tensors are diagonal: ε =diag [εxx, εxx, εzz], where εxx = ε1 and εzz = ε3. Withinthis approximation, which hereafter we refer to as uniax-

6

HMII

SPhP

SPPSPP

HMI

SPhPHMII

HMI

SPhPHMII

HMI

HMISPhP

H = 2 T z H = 4 T z H = 6 T z H = 0 T z

(a) (b) (c) (d)HMII

FIG. 3. The transmission coefficient for p-polarized waves as a function of the magnitude of the parallel wave vector andfrequency for InSb and a gap of d = 10 nm. The different panels correspond to different values of the magnetic field that isperpendicular to the surfaces. The horizontal dashed lines separate the regions where transmission is dominated by surfacewaves (SPPs and SPhPs) or hyperbolic modes of type I and II (HMI and HMII). The white solid lines correspond to theanalytical dispersion relation of the surface waves of Eq. (12).

ial approximation, it is easy to compute the dispersion re-lation of the surface electromagnetic modes in our geome-try (see Appendix D). In the electrostatic limit k � ω/c,the dispersion relation of these cavity modes is given by

kSW =1

dln

(±εxx −

√εxx/εzz

εxx +√εxx/εzz

), (12)

with the additional constraint that both εxx and εzz mustbe negative. In the zero-field limit, this expression re-duces to the known result for cavity surface modes inisotropic materials [26]. As we show in Fig. 3, see whitesolid lines, the dispersion relation of Eq. (12) nicely repro-duces the structure of the transmission maxima in thosefrequency regions in which surfaces waves are allowed(εxx, εzz < 0). It is worth stressing that this dispersionrelation describes in a unified manner both the SPPs thatappear below the reststrahlen band and the SPhPs dueto the optical phonons. More importantly, this disper-sion relation tells us that the magnetic field reduces theparallel wave vector of the surface waves and restricts thefrequency region where they exist. Indeed, at high fieldsthe SPPs disappear, while the SPhPs are restricted tothe reststrahlen band, Fig. 3(d). These two effects areactually the cause of the reduction of the NFRHT in thepresence of a magnetic field. But what about the othermodes that appear by increasing the field? Their naturecan also be understood within the uniaxial approxima-tion. As we show in Appendix C, the allowed values forthe transverse component of the wave vector inside theseuniaxial materials are given by qo =

√εxxω2/c2 − k2 for

ordinary waves and qe =√εxxω2/c2 − k2εxx/εzz for ex-

traordinary waves. The dispersion of the extraordinarywaves can be rewritten as

k2x + k2

y

εzz+

q2e

εxx=ω2

c2, (13)

a dispersion that becomes hyperbolic when εxx and εzzhave opposite signs [48]. This is exactly what happens in

our case in certain frequency regions at finite field. Thisillustrated in Fig. 3(b-d), where we have indicated thehyperbolic regions defined by the condition εxxεzz < 0.Notice that those regions correspond exactly to the areaswhere the transmission reaches its maximum for a broadrange of k-values. This fact shows unambiguously thatour InSb plates effectively behave as hyperbolic materi-als. More importantly, and as it is evident from Fig. 3,we can easily modify the hyperbolic regions by changingthe field. Thus, we can change from situations wherethe hyperbolic modes (HMs) coexist with both typesof surface waves to situations where the HMs dominatethe NFRHT, which is what occurs at high fields, seeFig. 3(d). Moreover, contrary to what happens in mosthybrid hyperbolic metamaterials, we can have in a sin-gle material HMs of type I (HMI), where εxx > 0 andεzz < 0, and HMs of type II (HMII), where εxx < 0 andεzz > 0, see Fig. 3(b-d).

Let us recall that what makes HMs so special in thecontext of NFRHT is the fact that, as it is evident fromtheir dispersion relation, they are evanescent in the vac-uum gap and propagating inside the hyperbolic materialfor k > ω/c (HMI) or k >

√|εzz|ω/c (HMII). Thus, they

are a special kind of frustrated internal reflection modesthat exhibit a very high transmission over a broad rangeof k-values that correspond to evanescent waves in thevacuum gap [46]. As shown in Ref. [46], the number ofHMs that contributes to the NFRHT is solely determinedby the intrinsic cutoff in the transmission, which has theform τ(ω, k) ∝ exp(−2kd) for k � ω/c. From this con-dition it follows that the heat flux due to HMs scales as1/d2 for small gaps, as the contribution of surface waves.This explains why the appearance of HMs as the fieldincreases does not modify the parametric dependence ofthe NFRHT with the gap size. Notice, however, that inspite of the high transmission of these HMs, their appear-ance does not enhance the NFRHT because they replacesurface waves that possess even larger k-values (notice

7

that the conditions of HMs and surface waves are mutu-ally excluding). Thus, we can conclude that the NFRHTreduction induced by the magnetic field is due to boththe modification of the surface waves and their replace-ment by HMs that, in spite of their propagating natureinside the material, turn out to be less efficient transfer-ring the radiative heat in the near-field region than thesurface waves.

Let us point out that within the uniaxial approxi-mation, the heat transfer can be obtained in a semi-analytical form. In this case, the transmission coefficientis given by the isotropic result of Eq. (8), where the re-flections coefficients adopt now the form

rs,s21 = rs,s23 =q2 − qo

q2 + qo(14)

rp,p21 = rp,p23 =εxxq2 − qe

εxxq2 + qe. (15)

The uniaxial approximation is also useful to under-stand the high field behavior of the NFRHT. The ten-dency to saturate the thermal radiation as the field in-creases is due to the to the fact that the cyclotron fre-quency becomes larger than the plasma frequency andthe last term in the expression of εxx = ε1, see Eq. (11),progressively becomes more irrelevant. Thus, the per-mittivity tensor becomes field-independent and the heattransfer is simply given by the result for uniaxial media,where εzz = ε3 has the form in Eq. (11), but εxx = ε1does not contain the last term in the first expressionof Eq. (11). We find that the strict saturation of theNFRHT occurs at around 20 T and there is an interme-diate regime, between 6 and 20 T, in which the near-fieldthermal radiation slightly increases upon increasing thefield (not shown here), leading to a nonmonotonic behav-ior. This behavior is due to an increase in the efficiencyof the HMs that dominate the NFRHT in this high-fieldregime.

To conclude this subsection, let us explain why the far-field heat transfer is rather insensitive to the magneticfield. For gaps much larger than the thermal wavelength(9.6 µm), the heat transfer is dominated by propagatingwaves and, as we show in Fig. 4, the spectral heat fluxin the absence of field exhibits a broad spectrum with apeak at around 1.5 × 1014 rad/s. Indeed, the spectrumis very similar to that of a dielectric with a frequency-independent dielectric constant ε = ε∞1, see dotted linein Fig. 4. As we illustrate in that figure, the presenceof a magnetic field only modifies this spectrum in a sig-nificant way in a small region around the cyclotron fre-quency. This fact leads to a tiny modification of the heattransfer upon the application of an external field. Asit can be seen in the inset of Fig. 4, the magnetic fieldreduces the far-field heat transfer coefficient as the mag-netic field increases, but this reduction is quite modestand, for instance, it amounts to only 2.5% at a very highfield of 20 T.

1 2 3 40 . 0

0 . 5

1 . 0

1 . 5

W a v e l e n g t h (µm )

Spect

ral he

at flux

(10−1

4 J/rad

m2 Κ)

ω (1014 r a d / s )

1 8 . 8 9 . 4 6 . 3 4 . 7

H z = 0 T H z = 4 T H z = 8 T H z = 1 2 T H z = 1 6 T H z = 2 0 T

1 3� � � ∞= =

4 6 8 1 0 1 2 1 4 1 6 1 81 . 0 0 01 . 0 0 51 . 0 1 01 . 0 1 51 . 0 2 01 . 0 2 5

h (0)

/ h (H

z)

H z ( T )

FIG. 4. Far-field spectral heat flux for InSb as a functionof the frequency (and wavelength) for different values of theperpendicular field. These spectra have been computed fora gap d = 1 m. The dotted line corresponds to the resultfor plates made of a dielectric with a frequency-independentdielectric constant equal to ε∞. The inset shows the corre-sponding ratio between the zero-field heat transfer coefficientand the coefficient for different values of the field.

B. Parallel magnetic field

Let us now turn to the case in which the magneticfield is parallel to the plate surfaces. For concreteness,we consider that the field is applied along the x-axis,H = Hxx, but obviously the result is independent ofthe field direction as long as it points along the surfaceplane, as we have explicitly checked. In this case, thepermittivity tensor of InSb adopts the form

ε(H) =

ε3 0 00 ε1(H) −iε20 iε2(H) ε1(H)

, (16)

where the ε’s are given by Eq. (11). Let us emphasizethat in this case the transmission coefficient appearingin Eq. (2) depends both on the magnitude of the parallelwave and on its direction, which makes the calculationsmore demanding. Let us also say that we consider herethe same parameter values for the n-doped InSb as in theexample analyzed above.

The results for the magnetic field dependence of theheat transfer coefficient for the parallel configuration aresummarized in Fig. 5(a). As in the perpendicular case,the far-field is barely affected by the magnetic field, thenear-field thermal radiation is suppressed by the field,and at high fields the NFRHT tends to saturates. Inter-estingly, it saturates to the same value as in the perpen-dicular configuration. In spite of the similarities, thereare also important differences. In this case, the NFRHTis much more sensitive to the field and a significant reduc-tion is already achieved at 1 T. Notice also that in thiscase the heat transfer coefficient is clearly nonmonotonic

8

1 0 0 1 0 1 1 0 2 1 0 3 1 0 4 1 0 51 0 0

1 0 1

1 0 2

1 0 3

1 0 4

1 0 5

1 0 6

G a p s i z e (n m )

h�(0)

/ h�(H

x)

( b )

( a ) He

at tran

sfer c

oeffic

ient (W

/m2 K)

G a p s i z e ( n m )

H x = 0 TH x = 1 TH x = 2 TH x = 6 T

H x = 8 TH x = 1 2 T

1 0 1 3 1 0 1 41 0 - 1 4

1 0 - 1 3

1 0 - 1 2

1 0 - 1 1

1 0 - 1 0

1 0 - 9

1 0 - 8

Spect

ral he

at flux

(J/rad

m2 K)

ω ( r a d / s )

H x = 0 TH x = 1 TH x = 2 TH x = 6 T

H x = 8 TH x = 1 2 T

1 0 2× ×2 1 0 2 2 1 0 1 W a v e l e n g t h ( µm )

1 0 0 1 0 1 1 0 2 1 0 31234567

FIG. 5. (a) Heat transfer coefficient for n-doped InSb as afunction of the gap for different values of the magnetic fieldapplied along the surfaces of the plates. The inset shows theratio between the zero-field coefficient and the coefficient fordifferent values of the field in the near-field region. (b) Thecorresponding spectral heat flux as a function of the frequency(and wavelength) for a gap of d = 10 nm and different valuesof the parallel field.

and the maximum reduction is reached at around 6 T. Fi-nally, notice also that the reduction is more pronouncedthan in the perpendicular case and the NFRHT can bediminished by up to a factor of 7 with respect to the zero-field case, see inset of Fig. 5(a). This more pronouncedreduction in the parallel configuration is also apparent inthe spectral heat flux, as one can see in Fig. 5(b). No-tice that also in this case there appears a high-frequencypeak that is blue-shifted as the field increases. This peakappears at the cyclotron frequency and it has the sameorigin as in the perpendicular case.

Again, to understand this complex phenomenology,it is convenient to examine the transmission of the p-polarized waves, which dominate the NFRHT for anyfield. Since in this case the transmission also dependson the direction of k, we choose to analyze the two mostrepresentative directions. In the first one, the in-planewave vector k is parallel to the field, i.e. k = (kx, 0),and in the second one, k is perpendicular to the field,

i.e. k = (0, ky). The transmission of p-polarized wavesfor these two directions is shown in Fig. 6 as a functionof the magnitude of the wave vector and as a functionof the frequency for different values of the field. As onecan see, the transmission exhibits very different behav-iors for these two directions. While for k ‖ H the sit-uation resembles that of a perpendicular field (see dis-cussion above), for k ⊥ H it seems like the transmis-sion is dominated by surface waves that are severelyaffected by the magnetic field (with the appearance ofgaps in their dispersion relations). These very differentbehaviors can be understood with an analysis of boththe surface waves and the propagating waves inside thematerial in these two situations. In the case k ‖ H,one can show that a uniaxial approximation, similar tothat discussed above, accurately reproduces the resultsfor the transmission found in the exact calculation. Inthis case, the permittivity tensor can be approximatedby ε = diag [εxx, εzz, εzz], where εxx = ε3 and εzz = ε1.Within this approximation, the dispersion relation of sur-face waves in the electrostatic limit k � ω/c is also givenby Eq. (12) (see Appendix D). As we show in Fig. 6(a-c),this dispersion relation nicely describes the structure ofthe transmission maxima in the regions where the surfacewaves can exist (εxx, εzz < 0). On the other hand, as weshow in Appendix C, the allowed values for the transversecomponent of the wave vector inside these uniaxial-likematerials are given by qo =

√εzzω2/c2 − k2 for ordinary

waves and qe =√εxxω2/c2 − k2εxx/εzz for extraordi-

nary waves. Again, the dispersion of these extraordinarywaves is of hyperbolic type when εxx and εzz have oppo-site signs. In Fig. 6(a-c) we identify the frequency regionswhere the HMs exist with the condition εxxεzz < 0, re-gions that progressively dominate the transmission as thefield increases. Thus, we see that for k ‖ H the situa-tion is very similar to that extensively discussed in thecase in which the field is perpendicular to the materials’surfaces.

On the contrary, the situation is very different fork ⊥ H. In this case, there are no HMs andno uniaxial approximation can describe the situation.As we show in Appendix C, the allowed q-valuesare given by qo,1 =

√εxxω2/c2 − k2 and qo,2 =√

(ε2yy + ε2yz)ω2/(c2εyy)− k2, which both describe waves

with no hyperbolic dispersion. On the other hand, thedispersion relation of the surface waves in the electro-static limit is given by

kSW =1

2dln

((ηyy − 1 + iηyz)(ηyy − 1− iηyz)(ηyy + 1 + iηyz)(ηyy + 1− iηyz)

), (17)

where ηyy = εyy/(ε2yy + ε2yz) and ηyz = −εyz/(ε2yy + ε2yz).

As we show in Fig. 6(d-f), this dispersion relation ex-plains the complex structure of the transmission maximain this case. We emphasize that this dispersion relationis reciprocal in our symmetric geometry and for this rea-son we only show results for ky > 0. Notice that thisdispersion is very sensitive to the magnetic field and al-

9

HMI

SPP

HMII

SPhPHMI

HMII

SPhPHMI

HMII

HMIISPhP

H = 2 T H = 4 T H = 6 T

(a) (b) (c)HMI

(d) (e) (f)

x x x

H = 1 T H = 2 T H = 3 T

k = (k ,0)x k = (k ,0)x k = (k ,0)x

k = (0,k )y k = (0,k )y k = (0,k )y

x x x

FIG. 6. The transmission coefficient for p-polarized waves as a function of the magnitude of the parallel wave vector andfrequency for InSb and a gap of d = 10 nm. In all cases the field is parallel to the plate surfaces, H = Hxx. The panels (a-c)correspond to different values of the magnetic field for wave vectors parallel to field, k = (kx, 0), while panels (d-f) correspondto wave vectors perpendicular to the field, k = (0, ky). The horizontal dashed lines separate the regions where transmission isdominated by surface waves (SPPs and SPhPs) or hyperbolic modes of type I and II (HMI and HMII). The white solid linescorrespond to the analytical dispersion relation of the surface waves of Eq. (12) in panels (a-c) and of Eq. (17) in panels (d-f).

ready fields of the order of 1 T strongly affect the sur-face waves. Notice also the appearance of gaps in thedispersion relations, a subject that has been extensivelydiscussed in the case of a single interface [69, 70]. Over-all, the field rapidly reduces the k-values of the surfacewaves and restricts the regions where they can exist. Thisstrong sensitivity of the surface waves with k ⊥ H is thereason for the more pronounced reduction of the NFRHTfor this field configuration.

In general, for an arbitrary direction k = (kx, ky) thesituation is somehow a combination of the two typesof behaviors just described. The complex interplay ofthese behaviors for different k-directions is responsiblefor the nonmonotonic dependence with magnetic field,along with the change in efficiency of the HMs uponvarying the field. On the other hand, at very high fieldsthe cyclotron frequency becomes much larger than theplasma frequency and the off-diagonal elements of thepermittivity tensor become negligible. At the same time,the field-dependent terms in the diagonal elements alsobecome very small. Thus, the systems effectively becomeuniaxial and field-independent and the heat transfer isidentical to the case in which the field is perpendicular.Finally, in the far-field regime, the heat transfer is notsensitive to the magnetic field for the same reason as inthe perpendicular configuration.

Let us conclude this section with two brief comments.First, as it is obvious from the discussions above, an-other way to modulate the NFRHT is by rotating the

magnetic field, while keeping fixed its magnitude. Actu-ally, we find that for any field magnitude, the NFRHTis always smaller in the parallel configuration. Thus, onecan increase or decrease the near-field thermal radiationby rotating appropriately the magnetic field. Second, wehave focused here in the case of doped InSb, but similarresults can in principle be obtained for other doped polarsemiconductors such as GaAs, InAs, InP, PbTe, SiC, etc.

IV. NON-POLAR SEMICONDUCTORS: Si

In the previous section we have seen that when the fieldis parallel to the surfaces, one can have hyperbolic emit-ters, but the HMs always coexist to some degree withsurface waves (even at the highest field). We show inthis section that in the case of non-polar semiconduc-tors, where phonons do not play any role, it is possibleto tune the system with a magnetic field to a situationwhere only HMs contribute to the NFRHT. For this pur-pose, we choose Si as the material for the two plates.As mentioned in the introduction, it has been predicted[43, 44], and experimentally tested [23, 27], that in dopedSi the NFRHT in the absence of field can be dominatedby SPPs even at room temperature. Let us see now howthis is modified upon applying a magnetic field.

The dielectric properties of doped Si are similar tothose of InSb, the only difference being the absence ofa phonon contribution. Thus, the dielectric functions of

10

1 0 0 1 0 1 1 0 2 1 0 3 1 0 4 1 0 51 0 0

1 0 1

1 0 2

1 0 3

1 0 4

1 0 5

( b )

G a p s i z e ( n m )

Heat t

ransfe

r coef

ficien

t (W/m

2 K) ( a )

H z = 0 TH z = 2 TH z = 4 T

H z = 8 TH z = 1 2 T

H z = 0 TH z = 2 TH z = 4 TH z = 1 2 T

1 0 1 2 1 0 1 31 0 - 1 3

1 0 - 1 2

1 0 - 1 1

1 0 - 1 0

Spect

ral he

at flux

(J/rad

m2 K)

ω ( r a d / s )

H z = 8 T

1 0 0 1 0 1 1 0 2 1 0 31 . 0 01 . 0 51 . 1 01 . 1 51 . 2 01 . 2 5

h�(0)

/ h�(H

z)

G a p s i z e (n m )

1 0 3 1 0 2 W a v e l e n g t h ( µm )

FIG. 7. (a) Heat transfer coefficient for n-doped Si as a func-tion of the gap for different values of the magnetic field per-pendicular to the plate surfaces. The inset shows the ratiobetween the zero-field coefficient and the coefficient for differ-ent values of the field in the near-field region. (b) The cor-responding spectral heat flux as a function of the frequency(and wavelength) for a gap of d = 10 nm.

Eq. (11) now read

ε1(H) = ε∞

(1 +

ω2p(ω + iγ)

ω[ω2c − (ω + iγ)2]

),

ε3 = ε∞

(1−

ω2p

ω(ω + iγ)

), (18)

while ε2(H) remains unchanged. Using the results ofRef. [43] for the dielectric constant of doped Si, we focuson a room temperature case where the electron concen-tration is n = 9.3×1016 cm−3, ε∞ = 11.7, γ = 8.04×1012

rad/s, m∗/m = 0.27, and ωp = 9.66 × 1012 rad/s. Wehave chosen this doping level to have a situation in whichthe plasma frequency is not too high so that we can af-fect the NFRHT with a magnetic field, and not too low sothat the NFRHT in the absence of field is still dominatedby SPPs.

The results for the heat transfer coefficient and spectralheat flux for a perpendicular magnetic field are displayed

in Fig. 7. Although there are several features that aresimilar to those of the InSb case, there are also some no-table differences. To begin with, notice that now higherfields are needed to see a significant reduction of theNFRHT (the required fields are around an order of mag-nitude higher than for InSb) and the reduction factors areclearly more modest, see inset of Fig. 7(a). This is mainlya consequence of the smaller cyclotron frequency in theSi case for a given field due to its larger effective mass.Another consequence of the small cyclotron frequency isthe fact that there is no sign of saturation of the NFRHTfor reasonable magnetic fields. On the other hand, thespectral heat flux at low fields is dominated this time bya single broad peak that originates from SPPs (see dis-cussion below). As the field increases, the peak height isreduced and the peak itself is broadened and deformed.As we show in what follows, this behavior is due to theappearance of HMs that at high fields completely replacethe surface waves.

Again, we can gain a further insight into these resultsby analyzing the transmission of the p-polarized waves fordifferent fields, which is illustrated in Fig. 8. As one cansee, the transmission is dominated by evanescent waves(in the vacuum gap) in a frequency region right belowthe plasma frequency. The origin of the structure of thetransmission maxima can be understood with the uniax-ial approximation discussed above in the context of InSb.Again, this approximation reproduces very accurately allthe results for arbitrary perpendicular fields (not shownhere). Within this approximation, one can see that atlow fields the transmission is dominated by SPPs, as weillustrate in Fig. 8(a-b) in which we have introduced thedispersion relation of the SPPs given by Eq. (12). Assoon as the magnetic magnetic field becomes finite, thesystem starts to develop HMs of type I in a tiny fre-quency region right above the region of existence of theSPPs, see Fig. 8(b). The origin of these HMs is identi-cal to that of the InSb case, but the main difference inthis case is that upon increasing the field, one reaches acritical field value (of 4.36 T for this example) for whichthe surface waves cease to exist and the transmission iscompletely dominated by HMs turning the Si plates into“pure” hyperbolic thermal emitters, see Fig. 8(c-d).

For completeness, we have also studied the heat trans-fer in the parallel configuration and the results for theheat transfer coefficient and spectral heat flux are shownin Fig. 9. In this case the results are rather similar tothose of the perpendicular configuration. In particular,contrary to the InSb case we do not find a nonmono-tonic behavior. Moreover, the NFRHT reduction is notmuch more pronounced than in the perpendicular case,although one can reach reduction factors of 50% for 12T. Finally, saturation is not reached for these high fieldsfor the same reason as in the perpendicular configuration.As in the case of InSb, all these results can be understoodin terms of the modes that govern the near-field thermalradiation. In this sense, for a direction where k ‖ H,the SPPs that dominate the NFRHT at low fields are

11

(a) (b) (c) (d)

SPP SPP

H = 2 T H = 4 T H = 12 TH = 0 T

HMI

HMIHMI

SPP

z z z z

FIG. 8. The transmission coefficient for p-polarized waves as a function of the magnitude of the parallel wave vector andfrequency for Si and a gap of d = 10 nm. The different panels correspond to different values of the magnetic field that isperpendicular to the surfaces. The horizontal dashed lines separate the regions where transmission is dominated by surfaceplasmon polaritons (SPPs) or hyperbolic modes of type I (HMI). The white solid lines correspond to the analytical SPPdispersion relation of Eq. (12).

1 0 0 1 0 1 1 0 2 1 0 3 1 0 4 1 0 51 0 0

1 0 1

1 0 2

1 0 3

1 0 4

1 0 5

( b )

G a p s i z e ( n m )

Heat t

ransfe

r coef

ficien

t (W/m

2 K) ( a )

H x = 0 TH x = 2 TH x = 4 T

H x = 8 TH x = 1 2 T

H x = 0 TH x = 2 TH x = 4 TH x = 1 2 T

1 0 1 2 1 0 1 31 0 - 1 3

1 0 - 1 2

1 0 - 1 1

1 0 - 1 0

Spect

ral he

at flux

(J/rad

m2 K)

ω ( r a d / s )

H x = 8 T

1 0 0 1 0 1 1 0 2 1 0 31 . 01 . 11 . 21 . 31 . 41 . 5

h�(0)

/ h�(H

x)

G a p s i z e (n m )

1 0 3 1 0 2 W a v e l e n g t h ( µm )

FIG. 9. (a) Heat transfer coefficient for n-doped Si as a func-tion of the gap for different values of the magnetic field ap-plied along the surfaces of the plates. The inset shows theratio between the zero-field coefficient and the coefficient fordifferent values of the field in the near-field region. (b) Thecorresponding spectral heat flux as a function of the frequency(and wavelength) for a gap of d = 10 nm.

progressively replaced by HMIs upon increasing the field

and above 4.36 T they “eat out” all surface waves. Onthe contrary, for k ⊥ H there are no HMs and the onlymagnetic field effect is the modification of the SPP dis-persion relation. Again, the interplay between these twocharacteristic behaviors among the different k-directionsexplains the evolution of the NFRHT with the field.

Let us conclude this section by saying that the behaviorreported here for Si could also be observed for other non-polar semiconductors such as Ge.

V. OUTLOOK AND CONCLUSIONS

The results reported in this work raise numerous inter-esting questions. Thus for instance, in all cases analyzedso far, we have found that the magnetic field reduces theNFRHT as compared to the zero-field result. Is thereany fundamental argument that forbids a magnetic-field-induced enhancement? In principle, there is no such anargument. The reduction that we have found in dopedsemiconductors is due to the fact that the we have ex-plored cases where surface waves, which are extremelyefficient, dominate the NFRHT in the absence of field.In this sense, one may wonder if a field-induced enhance-ment could take place in a situation where the NFRHT inthe absence of field is dominated by standard frustratedinternal reflection modes, as it happens in metals [75].Obviously, metals are out of the question due to theirhuge plasma frequency, but one can investigate non-polarsemiconductors with a low doping level. Indeed, we havedone it for the case of Si and again, we find that the mag-netic field reduces the NFRHT and moreover, exceed-ingly high fields are required to see any significant effect.Of course, we have by no means exhausted all possibili-ties and, for instance, we have not explored asymmetricsituations with different materials. Thus, the questionremains of whether the application of a magnetic fieldcan under certain circumstances enhance the near-fieldthermal radiation.

12

The discovery in this work of the induction of hyper-bolic modes upon the application of a magnetic field mayalso have important consequences for layered structuresinvolving thin films. Recently, it has been demonstratedthat thin films made of polar dielectrics may supportNFRHT enhancements comparable to those of bulk sam-ples due to the excitation of SPhPs [26]. Since hyperbolicmodes have a propagating character inside the material,they may be severely affected in a thin film geometry bythe presence of a substrate. Thus, one could expect muchmore dramatic magnetic-field effects in systems coatedwith semiconductor thin films.

Obviously, the question remains of whether one canmodulate the NFRHT with a magnetic field in otherclasses of materials. For instance, since a magneto-optical activity is required, what about ferromagneticmaterials? Ideally, one could imagine to tune theNFRHT by playing around with the relative orienta-tion of the magnetization, following the spin-valve ex-periments in the context of spintronics.

Another question of general interest for the field ofmetamaterials is if a doped semiconductor under a mag-netic field could exhibit the plethora of exotic opticalproperties reported in hybrid hyperbolic metamaterials[49, 50]. We have shown here that it can behave as ahyperbolic thermal emitter, but can it also exhibit neg-ative refraction or be used do to subwavelength imagingand focusing in the infrared? These are very importantquestions that we are currently pursuing.

So in summary, we have presented in this work a verydetailed theoretical analysis of the influence of a mag-netic field in the NFRHT. By considering the simple caseof two parallel plates, we have demonstrated that fordoped semiconductors the near-field thermal radiationcan be strongly modified by the application of an exter-nal magnetic field. In particular, we have shown that themagnetic field may significantly reduce the NFRHT andthe reduction in polar semiconductors can be as largeas 700% at room temperature. Moreover, we have shownthat when the field is perpendicular to the parallel plates,doped semiconductors become ideal hyperbolic thermalemitters with highly tunable properties. This providesa unique opportunity to explore the physics of thermalradiation in this class of metamaterials without the needto resort to complex hybrid structures. Finally, all thepredictions of this work are amenable to measurementswith the present experimental techniques, and we areconvinced that the multiple open questions that this workraises will motivate many new theoretical and experimen-tal studies of this subject.

ACKNOWLEDGMENTS

This work was financially supported by the Colom-bian agency COLCIENCIAS, the Spanish Ministry ofEconomy and Competitiveness (Contract No. FIS2014-53488-P), and the Comunidad de Madrid (Contract No.

S2013/MIT-2740). V.F-H. acknowledges financial sup-port from “la Caixa” Foundation and F.J.G-V. from theEuropean Research Council (ERC-2011-AdG proposalno. 290981).

Appendix A: Scattering matrix approach foranisotropic multilayer systems

Our analysis of the radiative heat transfer in the pres-ence of a static magnetic field is based on the combina-tion of Rytov’s fluctuational electrodynamics (FE) anda scattering matrix formalism that describes the prop-agation of electromagnetic waves in multilayer systemsmade of optically anisotropic materials. As we show inAppendix B, the radiative heat transfer can be expressedin terms of the scattering matrix of our system. Thus, itis convenient to first discuss in this appendix the scatter-ing matrix approach employed in this work ignoring forthe moment the fluctuating currents that generate thethermal radiation. Later in Appendix B, we show howthis approach can be combined with FE. We follow hereRef. [76], which presents a generalization of the formal-ism introduced by Whittaker and Culshaw in Ref. [77]for isotropic systems.

Let us first describe the Maxwell’s equations tobe solved. Assuming a harmonic time dependenceexp(−iωt), the Maxwell’s equations for non-magneticmaterials and in the absence of currents adopt the fol-lowing form: ∇· ε0εE = 0, ∇·H = 0, ∇×H = −iωε0εE,and ∇×E = iωµ0H, where the permittivity is in generala tensor given by Eq. (1). The first Maxwell’s equationis automatically satisfied if the third one is fulfilled, andthe second one can be satisfied by expanding the mag-netic field in terms of basis functions with zero diver-gence. Following Ref. [77], it is convenient to introducethe rescaling: ωε0E→ E and

√µ0ε0ω = ω/c→ ω. Thus,

the final two equations to be solved are

∇×H = −iεE, (A1)

∇×E = iω2H. (A2)

We consider here a planar multilayer system grownalong the z-direction in which the tensor ε is constantinside every layer, i.e. it is independent of the in-planecoordinates r ≡ (x, y). Thus, for an in-plane wave vectork ≡ (kx, ky), we can write the fields as

H(r, z) = h(z)eik·r and E(r, z) = e(z)eik·r. (A3)

With this notation, Eqs. (A1) and (A2) can be rewrittenas

ikyhz(z)− h′y(z) = −i∑j

εxjej(z) (A4)

h′x(z)− ikxhz(z) = −i∑j

εyjej(z) (A5)

ikxhy(z)− ikyhx(z) = −i∑j

εzjej(z), (A6)

13

and

ikyez(z)− e′y(z) = iω2hx(z) (A7)

e′x(z)− ikxez(z) = iω2hy(z) (A8)

ikxey(z)− ikyex(z) = iω2hz(z), (A9)

where the primes stand for ∂z.Now our task is to solve the Maxwell’s equations for

an unbounded layer. For this purpose, we write the mag-netic field h(z) as follows

h(z) = eiqz{φxx + φyy −

1

q(kxφx + kyφy)z

}, (A10)

where x, y, and z are the Cartesian unit vectors and qis the z-component of the wave vector. Here, φx andφy are the expansion coefficients to be determined bysubstituting into Maxwell’s equations. Notice that thisexpression satisfies ∇ ·H = 0. Now, it is convenient torewrite the previous expression in the vector notation:

h(z) = eiqz(φx, φy,−

1

q(kxφx + kyφy)

)T. (A11)

With this notation, Eqs. (A4-A6) can be written as

Ch(z) = εe(z), where C =

0 q −ky−q 0 kxky −kx 0

. (A12)

On the other hand, Eqs. (A7-A9) adopt now the form

CTe(z) = ω2h(z). (A13)

From Eq. (A12) we obtain the following expression forthe electric field

e(z) = ηCh(z), (A14)

where η = ε−1. Substituting this expression in Eq. (A13)we obtain the following equation for the magnetic field

CT ηCh(z) = ω2h(z), (A15)

which defines an eigenvalue problem for ω2. Indeed, onlytwo of the three identities obtained from this equation,one for each x, y, and z, are independent. From thefirst two identities, and using Eq. (A11), we obtain thefollowing equations determining the allowed values for q(

A2q2 + A1q + A0 + A−1

1

q

)φ = 0, (A16)

where φ = (φx, φy)T and the 2×2 matrices An are definedby

A2 =

(ηyy −ηyx−ηxy ηxx

), A1 = A(a)

1 + A(b)1 =

(−kyηzy kyηzxkxηzy −kxηzx

)+

(−kyηyz kxηyzkyηxz −kxηxz

),

A0 = A(a)0 + A(b)

0 − ω21 =

(k2yηzz −kxkyηzz

−kxkyηzz k2xηzz

)+

(k2xηyy − kxkyηyx kxkyηyy − k2

yηyxkxkyηxx − k2

xηxy k2yηxx − kxkyηxy

)− ω2

(1 00 1

),

A−1 =

(k2ykxηzx − k2

xkyηzy k3yηzx − k2

ykxηzyk3xηzy − k2

xkyηzx k2xkyηzy − k2

ykxηzx

). (A17)

This eigenvalue problem leads to the following quartic secular equation:∑4n=0Dnq

n = 0, where the coefficients aregiven by

D4 = ηxxηyy − ηxyηyx,D3 = kx [ηxyηyz + ηyxηzy − ηyy(ηxz + ηzx)] + ky [ηyxηxz + ηxyηzx − ηxx(ηyz + ηzy)] ,

D2 = k2x [ηyy(ηxx + ηzz)− ηxyηyx − ηyzηzy] + k2

y [ηxx(ηyy + ηzz)− ηxyηyx − ηxzηzx]

+kxky [ηxz(ηyz + ηzy) + ηyz(ηzx − ηxz)ηzz(ηxy + ηyx)]− ω2(ηxx + ηyy),

D1 = k3x [ηxyηyz + ηyxηzy − ηyy(ηxz + ηzx)] + k3

y [ηyxηxz + ηxyηzx − ηxx(ηyz + ηzy)]

+k2xky [ηxyηzx + ηxzηyx − ηxx(ηyz + ηzy)] + k2

ykx [ηyxηzy + ηyzηxy − ηyy(ηxz + ηzx)]

+ω2[k2x(ηxz + ηzx) + k2

y(ηyz + ηzy)],

D0 = k4x(ηyyηzz − ηyzηzy) + k4

y(ηxxηzz − ηxzηzx) + k3xky [ηxzηzy + ηyzηzx − ηzz(ηxy + ηyx)]

+k3ykx [ηyzηzx + ηxzηzy − ηzz(ηyx + ηxy)] + k2

xk2y [ηzz(ηxx + ηyy) + ηxyηyx − ηxzηzx − ηyzηzy]

+ω2[ω2 − k2

x(ηyy + ηzz)− k2y(ηxx + ηzz) + kxky(ηxy + ηyx)

]. (A18)

In general, this secular equation has to be solved numeri- cally, but in many situations of interest the allowed values

14

for q can be obtained analytically (see Appendix C). Thesolution of Eq. (A16) provides four complex eigenvaluesfor q, two lie in the upper half of the complex plane andthe other two in the lower half.

The next step toward the solution of the Maxwell’sequations in a multilayer structure is the determinationof the fields in the different layers. This can be doneby expressing the fields as a combination of forward andbackward propagating waves with wave numbers qn (withn = 1, 2), and complex amplitudes an and bn, respec-tively. These amplitudes will be determined later by us-ing the boundary conditions at the interfaces and sur-faces of the multilayer structure. Since the boundaryconditions are simply the continuity of the in-plane fieldcomponents, we focus here on the analysis of the fieldcomponents ex, ey, hx, and hy. From Eq. (A11), thein-plane components of h can be expanded in terms ofpropagating waves as follows(

hx(z)hy(z)

)=

2∑n=1

{(φxn

φyn

)eiqnzan

+

(ϕxn

ϕyn

)e−ipn(d−z)bn

}, (A19)

where d is the thickness of the layer. Here, an is the co-efficient of the forward going wave at the z = 0 interface,and bn is the backward going wave at z = d. On the otherhand, qn correspond to the eigenvalues of Eq. (A16) withIm{qn} > 0 and pn are the eigenvalues with Im{pn} < 0.

To simplify the notation, we now define two 2× 2 ma-trices Φ+ and Φ− whose columns are the vectors φn andϕn, respectively. Moreover, we define the diagonal 2× 2

matrices f+(z) and f−(d− z), such that [f+(z)]nn = eiqnz

and [f−(d − z)]nn = e−ipn(d−z), and the 2-dimensionalvectors h‖(z) = (hx(z), hy(z))T , a = (a1, a2)T , and

b = (b1, b2)T . In terms of these quantities, the in-planemagnetic-field components become

h‖(z) = Φ+ f+(z)a + Φ− f−(d− z)b. (A20)

Similarly, from Eq. (A14) it is straightforward to showthat the in-plane components of the electric field, e‖(z) =

(−ey(z), ex(z))T , are given by

e‖(z) =(A(b)

0 Φ+q−1 + A(b)

1 Φ+ + A2Φ+q)

f+(z)a

+(A(b)

0 Φ−p−1 + A(b)

1 Φ− + A2Φ−p)

f−(d− z)b, (A21)

where the A’s are defined in Eq. (A17) and we havedefined the 2 × 2 diagonal matrices q and p such thatqnn = qn and pnn = pn.

We can now combine Eq. (A20) and (A21) into a singleexpression as follows(

e‖(z)h‖(z)

)= M

(f+(z)a

f−(d− z)b

)(A22)

=

(M11 M12

M21 M22

)(f+(z)a

f−(d− z)b

),

where the 2× 2 matrices Mij are defined as

M11 = A(b)0 Φ+q

−1 + A(b)1 Φ+ + A2Φ+q,

M12 = A(b)0 Φ−p

−1 + A(b)1 Φ− + A2Φ−p,

M21 = Φ+, M22 = Φ−. (A23)

The final step in our calculation is to use the scatter-ing matrix (S-matrix) to compute the field amplitudesneeded to describe the different relevant physical quan-tities. By definition, the S-matrix relates the vectors ofthe amplitudes of forward and backward going waves, aland bl, where l now denotes the layer, in the differentlayers of the structure as follows(

albl′

)= S(l′, l)

(al′bl

)=

(S11 S12

S21 S22

)(al′bl

).

(A24)The field amplitudes in two consecutive layers are re-

lated via the continuity of the in-plane components ofthe fields in every interface and surface. If we considerthe interface between the layer l and the layer l+ 1, thiscontinuity leads to(

e‖(dl)h‖(dl)

)l

=

(e‖(0)h‖(0)

)l+1

, (A25)

where dl is the thickness of layer l. From this condition,together with Eq. (A22), it is easy to show that the am-plitudes in layers l and l + 1 are related by the interfacematrix I(l, l + 1) = M−1

l Ml+1 in the following way(f+l albl

)= I(l, l + 1)

(al+1

f−l+1bl+1

)=

(I11 I12

I21 I22

)(al+1

f−l+1bl+1

), (A26)

where f+l = fl,+(dl) and f−l+1 = fl+1,−(dl+1).

Now, with the help of the interface matrices, the S-matrix can be calculated in an iterative way as follows.The matrix S(l′, l + 1) can be calculated from S(l′, l)

using the definition of S(l′, l) in Eq. (A24) and the inter-

face matrix I(l, l + 1). Eliminating al and bl we obtainthe relation between al′ , bl′ and al+1, bl+1, from which

S(l′, l + 1) can be constructed. This reasoning leads tothe following iterative relations

S11(l′, l + 1) =[I11 − f+

l S12(l′, l)I21

]−1

f+l S11(l′, l)

S12(l′, l + 1) =[I11 − f+

l S12(l′, l)I21

]−1

×(f+l S12(l′, l)I22 − I12

)f−l+1

S21(l′, l + 1) = S22(l′, l)I21S11(l′, l + 1) + S21(l′, l)

S22(l′, l + 1) = S22(l′, l)I21S12(l′, l + 1) +

S22(l′, l)I22f−l+1. (A27)

Starting from S(l′, l′) = 1, one can apply the previous

recursive relations to a layer at a time to build up S(l′, l).

15

Let us conclude this appendix by saying that from theknowledge of the S-matrix one can easily compute thefield amplitudes in every layer and, in turn, the fieldseverywhere in the system [77].

Appendix B: Thermal radiation in anisotropicmultilayer systems

In this appendix we show how the scattering matrixapproach of Appendix A can be used to describe the ther-mal radiation between planar multilayer systems made ofanisotropic materials. For this purpose, we first discusshow a generic emission problem can be formulated in theframework of the S-matrix formalism and then, we showhow such a formulation can be used to describe the ther-mal emission of a multilayer system.

1. Emission in the scattering matrix approach

For concreteness, let us assume that there is a set of os-cillating point sources, with harmonic time dependence,occupying the whole plane defined by z = z′. The corre-sponding electric current density J is given by

J(r, z) = J0δ(z − z′) = j0eik·rδ(z − z′), (B1)

where j0(k) = J0e−ik·r. This current density enters as

a source term in Ampere’s law, Eq. (A1), which nowbecomes ∇ × H = J − iεE, while Eq. (A2) (Faraday’slaw) remains unchanged. Thus, Eqs. (A4-A6) adopt nowthe following form

ikyhz(z)− h′y(z) = j0xδ(z − z′)− i∑j

εxjej(z) (B2)

h′x(z)− ikxhz(z) = j0yδ(z − z′)− i∑j

εyjej(z) (B3)

ikxhy(z)− ikyhx(z) = j0zδ(z − z′)− i∑j

εzjej(z). (B4)

The presence of the source term induces discontinu-ities in the fields across the plane z = z′, as we proceedto show. First, let us consider the effect of the in-planecomponents of the current density by putting jz = 0. Tocancel the singular term due to the source in Eqs. (B2)and (B3), there must be discontinuities in hx and hyat z = z′ equal to j0y and −j0x, respectively. All theother field components are continuous, except for ez thatexhibits a discontinuity equal to (kxj0x + kyj0y)/εzz in

virtue of Eq. (B4). Let us analyze now the role of theperpendicular component of J by putting j0x = j0y = 0.From Eq. (B4), it is clear that in this case ez must containa singularity to cancel the singular term associated tothe current source, that is ez(z) = −i(j0z/εzz)δ(z− z′)+non-singular parts. This introduces singular terms in theleft-hand side of the Maxwell Eqs. (A7) and (A8), whichare cancelled by discontinuities in ex and ey equal tokxj0z/εzz and kyj0z/εzz, respectively. Additionally, it isobvious from Eqs. (B2) and (B3) that hx and hy acquireddiscontinuities equal to −εyzj0z/εzz and εxzj0z/εzz, re-spectively. Defining the following vectors

p‖ = (j0y − εyzj0z/εzz,−j0x + εxzj0z/εzz)T (B5)

pz = (−kyj0z/εzz, kxj0z/εzz)T , (B6)

the boundary conditions on the in-plane components ofthe fields are thus

e‖(z′+)− e‖(z

′−) = pz

h‖(z′+)− h‖(z

′−) = p‖. (B7)

These discontinuity conditions can now be combinedwith the S-matrix formalism of the previous appendixto calculate the emission throughout the system. Letus consider that the emission plane defines the interfacebetween layers l and l + 1 in our multilayer structure.Thus, the boundary conditions in this interface become(

e‖(0)h‖(0)

)l+1

−(

e‖(dl)h‖(dl)

)l

=

(pzp‖

). (B8)

Using now the expression of the fields in terms of thelayer matrices (M ’s), see Eq. (A22), we can write

Ml+1

(al+1

f−l+1bl+1

)− Ml

(f+l albl

)=

(pzp‖

). (B9)

The external boundary conditions for an emissionproblem is that there should be only outgoing waves, thatis a0 = bN = 0, where 0 denotes here the first layer ofthe structure and N the last one. Using the definitionsof the S-matrices S(0, l) and S(l+ 1, N) from Eq. (A24),it follows that

al = S12(0, l)bl (B10)

bl+1 = S21(l + 1, N)al+1. (B11)

Substituting for al and bl+1 from Eqs. (B10) and (B11)in Eq. (B9) and rearranging things, we arrive at the fol-lowing central result

(M11,l+1 + M12,l+1f

−l+1S21(l + 1, N) −[M12,l + M11,lf

+l S12(0, l)]

M21,l+1 + M22,l+1f−l+1S21(l + 1, N) −[M22,l + M21,lf

+l S12(0, l)]

)(al+1

bl

)=

(pzp‖

), (B12)

which allows us to compute the field amplitudes on the left and on the right-hand side of the emitting plane.

16

From the solution of this matrix equation we can computethe field amplitude everywhere inside and outside themultilayer structure from the knowledge of the scatteringmatrix.

2. Radiative heat transfer

Let us now show that the previous results can be usedto describe the radiative heat transfer. First of all, weneed to specify the properties of the electric currentsthat generate the thermal radiation. In the frameworkof fluctuational electrodynamics [9], the thermal emis-sion is generated by random currents J inside the mate-rial. While the statistical average of these currents van-ishes, i.e. 〈J〉 = 0, their correlations are given by thefluctuation-dissipation theorem [78, 79]

〈Jk(R, ω)J∗l (R′, ω′)〉 =4ε0ωc

πΘ(ω, T )δ(R−R′)δ(ω − ω′)×

[εkl(R, ω)− ε∗lk(R, ω)] /(2i), (B13)

where R = (r, z) and Θ(ω, T ) = ~ωc/[exp(~ωc/kBT )−1],T being the absolute temperature. Let us remind thereader that with the rescaling introduced at the begin-ning of Appendix A, ω has dimensions of wave vector inour notation. Notice that in the expression of Θ(ω, T ) aterm equal to ~ωc/2 that accounts for vacuum fluctua-tions has been omitted since it does not affect the neatradiation heat flux. Notice also that we are using herethe most general form of this theorem that is suitablefor non-reciprocal systems. The fact that the currentcorrelations are local in space and diagonal in frequencyspace reduces the problem of the thermal radiation to thedescription of the emission by point sources for a givenfrequency, parallel wave vector, and position inside thestructure. Thus, we can directly apply the results derivedin the previous subsection.

Let us now consider our system of study, namely twoparallel plates at temperatures T1 and T3 separated bya vacuum gap of width d, see Fig. 10. Our strategy tocompute the net radiative heat transfer between the twoplates follows closely that of the seminal work by Polderand Van Hove [7]. First, we compute the radiation powerper unit of area transferred from the left plate to theright one, Q1→3. For this purpose, we first compute thestatistical average of the z-component of the Poyntingvector describing the power emitted from a plane locatedat z = z′ inside the left plate for a given frequency andparallel wave vector and then, we integrate integrate theresult over all possible values of z′, ω, and k, i.e.

Q1→3(d, T1) =

∫ ∞0

dω

∫dk

∫ ∞0

dz′〈Sz(ω,k, z′)〉.

(B14)A similar calculation for the power Q3→1 transferredfrom the right plate to the left one completes the com-putation of the net transferred power per unit of area.

Let us focus now on the analysis of the power emittedby a plane inside the left plate, see Fig. 10. This emitting

0 21 3

(left plate) (right plate) (vacuum gap)

a1 b0

Emitting plane

′z d

FIG. 10. Two parallel plates separated by a vacuum gap ofwidth d. The vertical dashed line inside the left plate indicatesthe position of an emitting plane that contains the radiationsources that generate the field amplitudes b0 and a1.

plane defines a fictitious interface between layers 0 and1, which are both inside the left plate. To determine thepower emitted to the right plate we first compute thefield amplitudes a1 on the right hand side of the plane.For this purpose we make of use of Eq. (B12), where inthis case l = 0 and N = 3. Taking into account thatS12(0, 0) = 0, it is straightforward to show that

a1 = [M11,1 − M12,1M−122,1M21,1]−1pz +

[M21,1 − M22,1M−112,1M11,1]−1p‖

= [M−11 ]11pz + [M−1

1 ]12p‖. (B15)

To compute Q1→3, it is convenient to calculate thePoynting vector in the vacuum gap. For this purpose, weneed the field amplitudes in that layer. From Eq. (A24),it is easy to deduce that these amplitudes are given interms of a1 as follows

a2 = DS11(1, 2)a1, (B16)

where D ≡ [1− S12(1, 2)S21(2, 3)]−1 and

b2 =[1− S21(2, 3)S12(1, 2)

]−1

S21(2, 3)S11(1, 2)a1

= S21(2, 3)a2. (B17)

It is worth stressing that the different elements of thescattering matrix that appear in the previous expressionscan be factorized into scattering matrices S containingonly information about the interfaces of the layered sys-tem, which are basically the Fresnel coefficients of thestructure, and phase factors describing the propagationbetween these interfaces. In particular, from Eq. (A27)it is easy to show that

S11(1, 2) = S11(1, 2)f+1 (z′) (B18)

S12(1, 2) = S12(1, 2)eiq2d (B19)

S21(2, 3) = S21(2, 3)eiq2d, (B20)

where q2 =√ω2 − k2 is the z-component of the wave

vector in the vacuum gap and

f+1 (z′) =

(eiq1,1z

′0

0 eiq2,1z′

). (B21)

17

Here, qi,1 (with i = 1, 2) are the z-components of thetwo allowed wave vectors in the medium 1. On the otherhand, the S-matrices can be computed directly from theinterface matrices as follows [see Eq. (A27)]

S11(1, 2) = I−111 (1, 2) (B22)

S12(1, 2) = −I−111 (1, 2)I12(1, 2) (B23)

S21(2, 3) = I21(2, 3)I−111 (2, 3). (B24)

In terms of the amplitudes a2 and b2, the fields in thevacuum gap at z = 0 are given by [see Eq. (A22)]

(e‖(0)h‖(0)

)2

=

(M11,2

[a2 − eiq2db2

]a2 + eiq2db2

), (B25)

where we have used that M12,2 = −M11,2, valid for anyisotropic system. Thus, the z-component of the Poyntingvector evaluated at z = 0 in the vacuum gap reads

Sz(ω,k, z′) =

1

4ω

õ0

ε0

{h†‖(0)e‖(0) + e†‖(0)h‖(0)

}2

=1

4ω

õ0

ε0

{a†2

(M11,2 + M†11,2

)a2−

ei(q2−q∗2 )db†2

(M11,2 + M†11,2

)b2 +

e−iq∗2db†2

(M11,2 − M†11,2

)a2 −

eiq2da†2

(M11,2 − M†11,2

)b2

}. (B26)

Moreover, since

M11,2 =1

q2

(ω2 − k2

y kxkykxky ω2 − k2

x

)≡ 1

q2A (B27)

and q2 is either real (for k < ω) or purely imaginary (for

k > ω), Eq. (B26) reduces to

Sz(ω,k, z′) =

1

2q2ω

õ0

ε0× (B28){

a†2Aa2 − b†2Ab2, k < ω

e−iq∗2db†2Aa2 − eiq2da†2Ab2, k > ω,

where the first term provides the contribution of propa-gating waves and the second one corresponds to the con-tribution of evanescent waves.

From this point on, the rest of the calculation is purealgebra and we will not describe it here in detail. Letus simply say that the basic idea is to use Eqs. (B16)and (B17) to express the Poynting vector in Eq. (B28) interms of the field amplitude a1. Then, using Eq. (B15)and the fluctuation-dissipation theorem of Eq. (B13) onecan calculate the statistical average of the Poynting vec-tor. Let us mention that the calculation can be greatlysimplified by rotating every 2×2 matrix appearing in theproblem from the Cartesian basis (x-y) to the basis of s-and p-polarized waves. This can be done via the unitarymatrix

R ≡ 1

k

(kx kyky −kx

), (B29)

which is the matrix that defines the transformation thatdiagonalizes the matrix A, i.e.

Ad ≡ RAR =

(ω2 00 q2

2

). (B30)

Finally, after integrating over all possible values of ω, k,and z′, see Eq. (B14), one arrives at the following resultfor the power per unit of area transferred from the leftplate to the right one

Q1→3(d, T1) =

∫ ∞0

dω

2πΘ(ω, T1)

∫dk

(2π)2τ(ω,k, d),

(B31)where τ(ω,k, d) is the total transmission coefficient of theelectromagnetic modes and it is given by

τ(ω,k, d) =

Tr{

[1− S12(1, 2)S†12(1, 2)]D†[1− S†21(2, 3)S21(2, 3)]D}, k < ω (propagating waves)

Tr{

[S12(1, 2)− S†12(1, 2)]D†[S†21(2, 3)− S21(2, 3)]D}e−2|q2|d, k > ω (evanescent waves)

. (B32)

Here, the 2 × 2 matrices indicated by a bar are definedas follows

D ≡ A1/2d RDRA

−1/2d (B33)

D† ≡ A−1/2d RD†RA

1/2d . (B34)

Following a similar reasoning, one can compute thepower per unit of area transfer from the right plate to

the left one and the final result reads

Q3→1(d, T3) =

∫ ∞0

dω

2πΘ(ω, T3)

∫dk

(2π)2τ(ω,k, d),

(B35)where τ(ω,k, d) is also given by Eq. (B32). Thus, thenet power per unit of area exchanged by the plates isgiven by Eqs. (2) and (3) in section II. To conclude, letus stress that in the manuscript ω is meant to be anangular frequency.

18

Appendix C: Dispersion relations

In this appendix we provide the solution of the eigen-value problem of Eqs. (A16) and (A17) that give the dis-persion relations of the electromagnetic modes that canexist inside the materials considered in this work. In par-ticular, we focus here on three cases of special interest forour discussions in the main body of the manuscript.

Case 1: ε = diag[εxx, εxx, εzz]. This situation is ofrelevance for the case in which the magnetic field is per-pendicular to the plate surfaces, see section IIIA. In thiscase, the allowed q-values are given by

q2o = εxxω

2 − k2, q2e = εxxω

2 − k2εxx/εzz. (C1)

Case 2: ε = diag[εxx, εzz, εzz]. This situation is rele-vant for the case in which the magnetic field is parallelto the plate surfaces, see section IIIB, and the allowedq-values are given by

q2o = εzzω

2 − k2, q2e = εxxω

2 − k2εxx/εzz. (C2)

Case 3: the diagonal elements of ε are εxx and εyy =εzz, while the only non-vanishing off-diagonal elementsare εyz = −εzy. This situation is relevant for the case inwhich the magnetic field is parallel to the plate surfaces,see section IIIB. In this case, the allowed q-values adoptthe following form

q2o,1 = εxxω

2 − k2, q2o,2 = (ε2yy + ε2yz)ω

2/εyy − k2. (C3)

Appendix D: Surface electromagnetic modes

We briefly describe here how we determine the disper-sion relation of the surface electromagnetic modes in oursystem and we also provide the results for some configu-rations of special interest.

Let us consider a structure containing N planar lay-ers. From Eq. (A26), it is easy to show that the fieldamplitudes in layers l and l + 1 are related as follows(

albl

)=

(f+l 0

0 1

)−1

I(l, l + 1)

(1 0

0 f−l+1

)(al+1

bl+1

)≡ I ′(l, l + 1)

(al+1

bl+1

). (D1)

Now, using this relation recursively we can relate the field

amplitudes in the first and last layers as follows(a1

b1

)=

[N−1∏l=1

I ′(l, l + 1)

](aNbN

)≡ IS

(aNbN

).

(D2)The condition for an eigenmode of the system is thata1 = bN = 0, which from the previous equation impliesthat IS11aN = 0. The condition for having a non-trivial

solution of this equation is that det IS11 = 0, which isthe condition that surface electromagnetic modes mustsatisfy. In our plate-plate geometry, the 4× 4 matrix IS

is simply given by

IS = I(1, 2)

(e−iq2d1 0

0 eiq2d1

)I(2, 3), (D3)

where let us recall that q2 =√ω2 − k2. Thus, the condi-

tion for an eigenmode of the system reads

det[I11(1, 2)I11(2, 3)e−iq2d + I12(1, 2)I21(2, 3)eiq2d] = 0.(D4)

In what follows, we provide the explicit equations sat-isfied by the dispersion relation of the surface waves inthe three cases considered in Appendix C.Case 1: In this case, Eq. (D4) leads to

e−iq2d = ±(qe − εxxq2

qe + εxxq2

), (D5)

where qe is given in Eq. (C1). This equations reduces toEq. (12) in the electrostatic limit k � ω/c.Case 2: Here, assuming that the surface wave propa-

gates along the x-direction, its dispersion relation satis-fies the following relation

e−iq2d = ±(qe − εxxq2

qe + εxxq2

), (D6)

where qe is given in Eq. (C2). In the electrostatic limit,this equation reduces to Eq. (12).Case 3: In this case, and assuming that the surface

waves propagate along the y-direction, its dispersion re-lation is given by the solution of the following equation

e−2iq2d =(ηyyqo,2 − q2 + ηyzk)(ηyyqo,2 − q2 − ηyzk)

(ηyyqo,2 + q2 + ηyzk)(ηyyqo,2 + q2 − ηyzk),

(D7)where ηyy = εyy/(ε

2yy + ε2yz), ηyz = −εyz/(ε2yy + ε2yz), and

qo,2 is given in Eq. (C3). In the electrostatic limit thisequation reduces to Eq. (17).

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