L Abou-Hamdan

A Schmitt

R Bretel

S Rossetti

M Tharrault

D Mele

A Pierret

M Rosticher

T Taniguchi

K Watanabe

C Maestre

C Journet

B Toury

V Garnier

P Steyer

J H Edgar

E Janzen

J-M Berroir

G Fève

G Ménard

B Plaçais

C Voisin

J-P Hugonin

E Bailly

B Vest

J-J Greffet

P Bouchon

Y De Wilde

E Baudin

Direct observation and control of near-field radiative energy transfer in a natural hyperbolic material

Heat control is a key issue in nano-electronics, where new efficient energy transfer mechanisms are highly sought after. In this respect, there is indirect evidence that high-mobility hexagonal boron nitride (hBN)-encapsulated graphene exhibits hyperbolic out-of-plane radiative energy transfer when driven out-of-equilibrium. Here we directly observe radiative energy transfer due to the hyperbolic phonon-polaritons modes of the hBN encapsulant in intrinsic graphene devices under large bias, using mid-infrared spectroscopy and pyrometry. By using different hBN crystals of varied crystalline quality, we engineer the energy transfer efficiency, a key asset for compact thermal management of electronic circuits.

VII Acknowledgments

The research leading to these results has

I Introduction

Radiative energy transfer can be enhanced via the Purcell effect 1 , which relies on the shaping of the local electromagnetic density of states (LDOS). This can be done by engineering the geometrical and dielectric properties of the structure. For instance, materials with a negative dielectric permittivity possess a large number of polaritonic near-field modes, which substantially augments the LDOS near the surface 2 . In this case, energy transfer is in general restricted to nanometric distances due to the limited range of surface polaritons [3][4][5] .

Hyperbolic metamaterials have been proposed as a route for long-range radiative energy transfer as they host nonevanescent modes with large wavectors contributing to the large LDOS 6,7 . More recently, naturally hyperbolic materials have facilitated this approach 8,9 . In particular, the van der Waals material hexagonal Boron Nitride (hBN) possesses exceptionally well-defined hyperbolic phonon-polariton (HPhP) modes 10,11 . Consequently, the radiative thermal transfer induced by a hot source exceeds the vacuum far-field Planckian limit in graphene/hBN heterostructures [12][START_REF] Ms Ukhtary | Surface plasmons in graphene and carbon nanotubes[END_REF][START_REF] Wunsch | Dynamical polarization of graphene at finite doping[END_REF] . However, probing from the source perspective the radiative energy transfer efficiency is challenging in such structure, owing to the parasitic contribution from thermal conduction.

Graphene/hBN heterostructures are the ideal platform for studying hyperbolic radiative energy transfer [START_REF] Yang | A graphene zener-klein transistor cooled by a hyperbolic substrate[END_REF][START_REF] Schmitt | High-field 1/f noise in hbn-encapsulated graphene transistors[END_REF] : graphene is the quintessential candidate for nano-electronics 17 owing to its phonon-limited mobility at room temperature greatly surpassing that of silicon electronics. In diffusive graphene energy relaxation is mainly driven by electronphonon-impurity super-collisions 18,19 . This energy transfer pathway is silenced in high-mobility graphene devices under large electrical bias due to their characteristically weak electron-intrinsic phonon coupling 20 , so that radiative energy transfer is predominant 12,[START_REF] Wunsch | Dynamical polarization of graphene at finite doping[END_REF]21 . As hBN-encapsulated graphene has no vacuum gap, the coupling between graphene's electrons and hBN's HPhPs is maximal, owing to the atomically flat surfaces of the two materials.

Under continuous electrical driving, graphene's electrons reach a strongly out-of-equilibrium electronic distribution [START_REF] Yang | A graphene zener-klein transistor cooled by a hyperbolic substrate[END_REF]22 as interband Zener-Klein tunneling [START_REF] Yang | A graphene zener-klein transistor cooled by a hyperbolic substrate[END_REF]23 injects electron-hole pairs in the graphene channel. This effect produces electroluminescence in the mid-infrared 24 . In this case, the driving mechanism of the emission is the non-thermal electronic distribution rather than the source-reservoir temperature difference as in usual thermal transfer. Consequently, it exceeds Planckian hyperbolic radiative heat transfer [START_REF] Schmitt | High-field 1/f noise in hbn-encapsulated graphene transistors[END_REF] . The energy transfer process in the graphene/hBN heterostructure considered here, along with its electroluminescent emission of HPhP modes, is depicted in Fig. 1. The efficiency of the radiative energy transfer is then dictated by the ballisticity of the hyperbolic phonon-polaritons, as illustrated by Figure 1a andb. Utilising this ballisticity as a controlled parameter enables the engineering of radiative energy transfer.

In this study, we present a direct observation of hyperbolic radiative energy transfer by investigating hBNencapsulated graphene devices in ambient conditions under very large bias. We use mid-infrared (mid-IR) spectroscopy [25][26][27] to reveal the spectral signatures of the HPhP energy carriers, and we monitor the out-of-plane power to the device's SiO 2 substrate, unambiguously demonstrating radiative energy transfer mediated by HPhPs. In addition, we show that the out-of-plane energy transfer efficiency can be controlled by tuning the crystalline quality of the hBN encapsulant. We find that ballistic HPhP propagation leads to substantial radiative energy transfer through pristine hBN (Fig. 1a), while HPhPs in turbid hBN are localized close to the graphene layer and thus re-absorbed (Fig. 1b).

II Hyperbolic phonon-polariton fingerprint in the far-field

We begin by experimentally revealing the HPhP modes that are actively involved in the energy transfer in a graphene device under large bias. Although these modes are confined to the near-field of the hBN layers, they can be measured in the far-field if sufficient local scatterers are present, leading to a measurable scattered signal in the far-field. For example, in the TR1 device (Fig. 2a, inset), significant scattering of the HPhPs occurs at the edge contact metallizations and the edges of the heterostructure (marked by red arrowheads). The measured mid-IR emission spectrum of TR1 under large bias is given in Fig. 2a along with that of a device in which scatterers are absent (hereafter referred to as TR2). While the latter only presents spectral features of the SiO 2 substrate, the spectrum of TR1 is dominated by a peak inside hBN's second Reststrahlen band (ℏω II = 168.7 -198.4 meV, orange-shaded region in Fig. 2) at an energy of ∼ 190 meV, with other peaks of lower amplitude within the Reststrahlen band. These peaks correspond to multiple HPhP branches resulting from the multiple orders of Fabry-Pérot resonances in the hBN thin film cavity, that have been previously revealed via scattering-scanning near-field optical microscopy (s-SNOM) measurements 10,11,28 . This observation is further confirmed by the calculation of the imaginary part of the reflection coefficient of the heterostructure in Fig. 2b, whose maxima indicate the HPhP mode branches. By looking at the spectral behaviour within SiO 2 's Reststrahlen band (green-shaded region), we observe a resonance at SiO 2 's surface phonon-polariton energy, indicated by a peak in the spectral flux at around 145 meV, which also appears in the corresponding calculations (see the weak local maximum of Im(r p ) in SiO 2 's Reststrahlen band in panel b). This confirms the role of scatterers in Radiative origin of the out-of-plane energy transfer: scattering of HPhPs by defects. a, Mid-infrared (mid-IR) spectrum of an electrically-biased hBN-encapsulated graphene device (TR1, inset). The small dimensions of this device's graphene channel (9.2 µm × 15.5 µm) lead to increased scattering of the HPhP modes of the heterostructure by the edge contacts and etches in the hBN layer (marked by red arrowheads in the inset). The mid-IR spectrum of a larger graphene device (TR2, 35 µm × 35 µm, inset), in which scattering is absent, is shown for comparison. b, Imaginary part of the reflection coefficient (rp) for p-polarized light incident upon the heterostructure consisting of TR1's polariton active layers (sketched in panel c). The polariton dispersion curves of the heterostructure can be visualized from the maxima of Im(rp). See Supplementary information, section III for full details. The dashed green lines enclose the Reststrahlen band of SiO2, while the dashed orange lines enclose hBN's second Reststrahlen band, in which Re(εx) = Re(εy) < 0 and Re(εz) > 0, where εx,y and εz are the in-plane and out-of-plane dielectric functions of hBN, respectively. c, Penetration depth δz of HPhPs in the z-direction (see main text) calculated for the layers of TR1 (top sketch). d, Purcell factor above a 30 nm-thick hBN cavity, given by

FP = 3 4π 2 λ n 3 Q V ,
where λ is the free-space wavelength, n is the index of refraction of hBN, and Q and V are the quality factor and mode volume of the hBN cavity. FP is computed for an in-plane dipole p situated 3.5 Å (i.e., the inter-layer spacing in hBN) above the hBN layer and is normalized to its minimal value to elucidate its enhancement within hBN's Reststrahlen band (shaded region).

improving the sensitivity to near-field polariton modes. We verified that the mid-IR signal from the HPhP modes is unrelated to hBN's thermal emission by characterizing the device's blackbody radiation at various temperatures via an electrically controlled hot sample holder (see supplementary information for full details). hBN's polariton peak (∼ 190 meV) appears in the measured thermal emission spectrum only for temperatures above ∼ 200 • C, while the hBN temperature under large bias, measured by Stokes anti-Stokes Raman thermometry (see Ref. 24 ), is < ∼ 75 • C. Monitoring the HPhP mid-IR signal versus bias voltage shows that the onset of emission matches with the threshold of Zener-Klein tunneling extracted from DC transport measurements. Furthermore, we demonstrate in the supplementary information (section IV) the electroluminescent nature of the HPhP emission in the framework of the generalized Kirchhoff law 29 , following a similar approach reported in Ref. 24 .

We gain further insight regarding the ballistic HPhP-mediated energy transfer by examining the polariton penetration depth along the out-of-plane direction, which at the graphene/hBN interface is given by δ

z = |k z | -1 = ε x ω 2 c 2 - k 2 x εz -1
, where ε x and ε z are the in-plane and out-of-plane dielectric functions of hBN, ω is the angular frequency, and k x and k z are the in-plane and out-of-plane polariton wavevectors. As shown in Fig. 2c, the polariton penetration depth near the transverse optical phonon energy of hBN (ℏω hBN TO ∼ 169 meV) is less than 25 nm, i.e., shorter than the thickness of the bottom hBN layer (30 nm) separating the graphene layer and the SiO 2 back-gate in TR1. At higher energies, the penetration depth increases steadily, reaching a maximal value of 282 nm near the longitudinal optical phonon energy (ℏω hBN LO ∼ 198 meV). Concordantly, most of the energy is channeled through the lowest-order HPhP branch at this energy (see Fig. 2b). Note that this energy also corresponds to the largest Purcell factor, i.e. the largest LDOS enhancement (see Fig. 2d).

III Pyrographic signature of electroluminescent energy transfer

A direct indication of out-of-plane radiative energy transfer (Fig. 1a) in high-mobility graphene devices with an SiO 2 /Si back-gate can be readily obtained by monitoring the device's far-field mid-IR emission as a function of electrical bias. At large bias, beyond the threshold for Zener-Klein tunneling, a fraction P out-of-plane of the total electrical power injected into the device's channel is radiated towards the SiO 2 back-gate. This results in a slight warming of the back-gate, which is subsequently signaled by the incandescent mid-IR emission of the SiO 2 substrate as depicted schematically in Fig. 3a. We first perform spectroscopy of the measured mid-IR signal of a large graphene device with an SiO 2 /Si back-gate under large bias (TR2), which reveals signatures of the emission of the SiO 2 layer. Indeed, as shown in Fig. 2a (orange curve), the recorded spectrum possesses two peaks at the transverse and longitudinal optical phonon frequencies of SiO 2 (ω SiO2

TO and ω SiO2 LO , respectively) and a dip in the middle of the Reststrahlen band (shaded region), that results from enhanced reflectance in this spectral region. We also verify through a spatial scan of the detected mid-IR signal (P MIR ), that the signal is most intense at the channel location (see Fig. 3b, bottom-left), indicating that the detected signal originates from the heated SiO 2 substrate beneath the channel. Note that in this spectral range, the mid-IR signal emitted from the graphene channel is negligible with respect to that of the substrate's incandescence.

Having established that the detected signal originates from the region beneath the device's graphene channel, we may now quantify the contribution of out-of-plane radiation to the device's total power budget. The total power balance is given by P elec = P in-plane + P out-of-plane , where P elec = V ds I ds (V ds , drain-source bias voltage; I ds , drain-source current) is the electrical power and P out-of-plane (P in-plane ) is the power transferred in the out-of-plane (in-plane) direction. We calculate the device's equivalent thermal resistance, R th , by solving the heat equation. This resistance is determined by dividing the average temperature increase of the SiO 2 layer beneath the channel by the out-of-plane power P out-of-plane cast into it. (see SI, section III). The temperature increase ∆T SiO2 is determined by measuring the device's mid-IR signal (see Fig. 3b, right), where the temperature dependence of the mid-IR signal is calibrated by placing the device on a hot plate and measuring its thermal emission at various temperatures (see supplementary information section II for more details). As a result, we obtain the distribution of the temperature increase as a function of electrical bias, which is on the order of ∆T SiO2 ≃ 20 -40 • C beyond the Zener-Klein threshold (see Fig. 3c).

Following this model, P out-of-plane and the ratio (P out-of-plane /P elec ) of the out-of-plane transferred power to the total electrical power injected into the device can be computed as a function of electrical bias and doping, and the result is presented in Figs. 3d ande. The maps reveal that the investigated out-of-plane energy transfer mechanism plays a significant role in the device's total power budget, with a maximal contribution of about 71 %. We underscore here that this corresponds to an out-of-plane radiative power per surface area of ∼ 19.5 kW/cm 2 . This value is three orders of magnitude larger than the highest cooling power experimentally reported for a parent mechanism, namely the electroluminescent cooling in conventional semi-conductors [30][31][32] . Interestingly, the out-of-plane radiative power is highest at large bias near the charge-neutral point (V g = 0) and falls off away from it due to Pauli blocking. Furthermore, the onset of out-of-plane energy transfer matches with the threshold of Zener-Klein tunneling extracted from DC transport measurements, as indicated by the dashed line in Fig. 3d. In fact, P out-of-plane follows the same trend as that of the radiated power (P MIR ), whose onset is also given by the Zener-Klein tunneling threshold. In contrast, the electrical power shows the expected decrease near the charge-neutral point at large bias (see Fig. SI-6a).

IV Engineering hyperbolic radiative energy transfer

Having demonstrated hyperbolic radiative energy transfer from graphene through hBN, we show how to engineer such energy transfer. To do so, we borrow a well-known concept from far-field optics in disordered media, which states that light in a highly turbid material can be back-scattered toward the source 33 . Far-field radiative energy transfer is therefore strongly reduced in such materials, in a manner that is analogous to the effect of clouds on the Earth's heat budget. This concept has been successfully used in the context of thermal conduction, where nanometer-scale acoustic defects were used as scatterers to control energy transfer 34 . Here we exploit this idea to control hyperbolic radiative energy transfer by tuning the density of scattering sites in hBN. Remarkably, hBN can be readily grown with such scattering sites while maintaining its extraordinary properties as a graphene encapsulant.

We demonstrate this property by considering two different sources of the hBN crystals used in our devices: in addition to the hBN fabricated under high pressure and high temperature 35 (denoted by HPHT-hBN in the following), we fabricated devices with hBN crystals prepared at lower and isostatic pressure through a polymer-derived ceramics (PDC-hBN in the following) route 36 . While devices using either hBN crystals present similar transport properties 37 , such as large electronic mobility µ > 8 m 2 /V.s in ambient conditions, the PDC route being a more recent synthesis method, the hBN produced contains more scattering sites. Indeed, as investigated by cathodoluminescence, crystal defects presumably involving carbon or oxygen interstitial atoms are present in such crystals 38 so that radiative energy transfer is expected to be less efficient (see Fig. 1b). This is confirmed by a near-IR determination of graphene's electron temperature (see Fig. 4a), in which the electronic temperature is noticeably higher in the PDC-hBN device (reaching temperatures of T ∼ 900 K at a maximal field of 0.7 V/µm, while we consistently measure T ∼ 600 K on all devices made with HPHT-hBN). This behaviour is also corroborated by 1-10 GHz noise thermometry measurements (see SI), in which we compare the noise temperature of a device made with PDC-hBN and another made with HPHT-hBN. Engineering energy transfer efficiency by tuning hBN crystalline quality. a, Near-IR graphene electron temperature versus bias for the two devices Lyon1 (PDC-hBN) and F161 (HPHT-hBN), measured around a wavelength λ = 1.5 µm, for a doping n = 6 × 10 11 cm -2 . b, Mid-IR characterization of a device made with PDC-hBN on a Si/SiO2 back gate. Vertical energy transfer under large bias is monitored through the incandescence of the SiO2 substrate, with a much weaker temperature increase ∆TSiO 2 < ∼ 1 • C, which corresponds to an out-of-plane radiative power per surface area of ∼ 0.47 kW/cm 2 ., i.e., only 2.4 % of that of devices made with HPHT-hBN.

Such HPhP-opaque materials open the door to new regimes with extremely out-of-equilibrium electronic distribution.

To further verify the suppression of HPhP energy transfer with PDC-hBN we conducted a similar study as the one reported in Section III, that is, we performed mid-IR spectroscopy on a PDC-hBN device on an SiO 2 /Si back-gate and monitored the out-of-plane energy transfer through the incandescence of SiO 2 . As shown in Fig. 4b, the temperature increase ∆T SiO2 is < ∼ 1 • C at large bias, which is substantially smaller than the increase ∆T SiO2 ≃ 20 -40 • C reported for the HPHT-hBN device TR1, resulting in a reduction in out-of-plane energy transfer down to only 0.47 kW/cm 2 . Interestingly, far-field measurements performed on both hBN crystals revealed similar far-field optical properties, which suggests that scattering sites in PDC-hBN are present on a subwavelength scale.

Conversely, using isotopically pure 39 hBN with boron-11 and nitrogen-14, in which there is no mass disorder and phonon-polariton propagation length is doubled 40 , leads to increased radiative energy transfer. This aligns with prior research on the faster cooling dynamics of graphene and larger interfacial conductance 41 . We verified by means of noise thermometry (see SI) that high-mobility graphene devices fabricated with boron-isotopically pure hBN present enhanced radiative energy transfer under large bias, validating our ability to tune HPhP energy transfer by engineering the hBN crystal.

V Conclusion

In summary, we have directly observed near-field radiative energy transfer mediated by the HPhP modes of hBN in hBN-encapsulated graphene devices under large bias. The mechanism for this energy transfer was shown to be electroluminescence from these HPhP modes, whose signature was observed in the far-field as a result of scattering by defects. The onset of such energy transfer was concomitant with the Zener threshold for interband carrier injection. We also determined the fraction of power channeled through this radiative transfer mechanism by monitoring the temperature increase of the underlying SiO 2 substrate via pyrometry. Radiative energy transfer was the dominant energy transfer mechanism accounting for ∼ 71 % of the device's total power budget. We then demonstrated the engineering of the radiative energy transfer by tuning the turbidity of the hyperbolic hBN encapsulant, and as such the ballisticity of HPhPs. This was confirmed by comparing several hBN sources using multiple probes including mid-IR pyrometry, electronic noise and near-IR thermometries.

While it is well known that far-field radiative energy transfer is affected by the turbidity of the propagating medium 33 , our experimental study demonstrates for the first time that the concepts of optics in disordered media can be applied to the transport of radiative energy by near-field modes. This is possible because the hyperbolic phonon-polaritons involved in radiative energy transfer are extremely subwavelength electromagnetic propagating modes. In addition, controlling energy transfer by engineering a material's turbidity is appealing from a technological point of view: for example, it has been shown that the acoustic turbidity of a material makes it possible to control its thermal conduction without altering its macroscopic properties 34,42 . In our experiments, we have shown that, while radiative energy transfer is effectively suppressed when introducing defects in hexagonal Boron Nitride, its performance as an electronic encapsulant remains exceptional, thus validating this concept. This observation bodes well for applications in nanothermy and microelectronics.

VI Authors contributions

I Sample fabrication and properties

The hexagonal-boron-nitride (hBN)-encapsulated graphene heterostructures are fabricated with the standard pick-up and stamping technique, using a polydimethylsiloxane(PDMS)/ polypropylene carbonate (PPC) stamp. When needed, the gate electrode is first fabricated on a high-resistivity Si substrate covered by a 285 nm-thick SiO 2 layer. It consists of a pre-patterned gold pad (80 nm thick) designed by laser lithography and Cr/Au metalization. The hBN/Graphene/hBN heterostructure is then deposited on top of the back-gate. This is followed by acetone cleaning of the stamp residues, Raman spatial mapping, and AFM characterization of the stack. Graphene edge contacts are then created by means of laser lithography and reactive ion etching, securing low contact resistance < ∼ 1 kΩ.µm. Finally, metallic contacts to the graphene channel are designed via a Cr/Au Joule evaporation. The transistors' dimensions are maximized to get the highest optical signal, while their high mobility µ > ∼ 4 m 2 .V -1 .s -1 at room temperature secures a moderate channel electric field E = V /L > ∼ 10 5 V/m for the threshold of electroluminescent near-field radiative energy transfer. The properties of the different devices are summarized in Table I

II Pyrography of an hBN-encapsulated graphene device with an SiO 2 back gate: midinfrared experimental details

The mid-infrared (mid-IR) characterization techniques employed in this study have been outlined in detail elsewhere in Ref. 1 . We only illustrate here the calibration procedure used to determine the temperature increase of the device's SiO 2 back-gate due to near-field radiative energy transfer.

The far-field mid-IR emission of the device is collected via an IR microscope and optically chopped before reaching a liquid nitrogen-cooled MCT detector (see Fig. SI-1). Spectroscopy or integrated signal mapping is then performed on the detected signal, as outlined in Ref. 1 .

The out-of-plane radiative power of the device P out-of-plane is determined from the device's detected mid-IR signal as a function of electrical bias (Fig. SI-2c). This is done by calibrating the amplitude of the detected mid-IR signal with respect to the sample temperature by placing the sample on a hot sample holder and measuring its thermal emission at various temperatures (see Fig. SI-2a andb). As a result of this calibration procedure, we find that the temperature increase of the SiO 2 substrate as a function of bias ranges between ∆T SiO2 ≃ 20 -40 • C beyond the Zener-Klein threshold (Fig. SI-2d).

Finally, we note here that the spectra reported in Figs. 2 and4 of the main text were measured via infrared spatial modulation spectroscopy (IR-SMS) 2-4 , as described in Ref. 1 .

III Modeling the out-of-plane energy transfer in a graphene/hBN heterostructure

In this section, we detail various thermal models that describe energy transfer in a graphene/hBN device with an SiO 2 /Si back-gate, starting from the simplest case (in which only the thermal resistance of the substrate is considered) then making our way through a more refined approach, in which the whole structure is modeled. The latter model is utilized to obtain the results presented in the main text.

The out-of-plane radiative power can be calculated via a simple thermal model (introduced in Ref. 5 ) which only takes into account the thermal resistances of the SiO 2 and Si layers of the device's back-gate (see Fig. SI-3). In this framework, the out-of-plane power is given by

P out-of-plane = ∆T SiO2 R th = ∆T SiO2 R SiO2 + R Si , (1)
where R th is the total effective thermal resistance of the layers and R SiO2 and R Si are the effective thermal resistances of the SiO 2 and Si layer, respectively (see Fig. SI-3), which are given by

R SiO2 = t SiO2 2κ SiO2 LW . (2)
and,

R Si = 1 2κ Si √ LW , (3)
In the above equations t SiO2 = 285 nm is the thickness of the SiO 2 layer, κ SiO2 = 1.4 W/m.K and κ Si = 150 W/m.K are the thermal conductivities of SiO 2 and Si, and L = 35 µm and W = 35 µm give the dimensions of the device's graphene channel (see Fig. SI-3). From Eqs. ( 2) and (3), we obtain a total thermal resistance R th ≃ 178 K/W. This results in an out-of-plane power P out-of-plane that reaches 68 % of the total electrical power injected into the device.

Note that this simple model is the same as that reported in Ref. 5 , however, in our case, there is an additional factor of 2 in the denominator of Eq. ( 2) to account for the fact that the thermal radiation emitted by the device results from the average temperature of the SiO 2 layer. A more detailed approach for calculating P out-of-plane involves solving the general heat equation for a more complicated geometry. In its three-dimensional Cartesian form, the transient heat conduction equation is expressed as

ρC p ∂T ∂t = ∇ • (κ∇T ) + Q th , (4)
where ρ represents the material density, C p the specific heat capacity, T the temperature, t the time, κ the thermal conductivity, ∇• the divergence and Q th the heat generation term.

We consider in what follows a steady-state scenario with a cylindrical geometry (Fig. SI-4), where the graphene transistor under large bias generates a disk-shaped heat flux (represented in red) on top of the SiO 2 /Si substrate. In this case the heat equation is given by

1 r ∂ ∂r rκ rr (r, z) ∂T ∂r + ∂ ∂z κ zz (r, z) ∂T ∂z = -Q th , (5)
where κ rr (r, z) and κ zz (r, z) are the spatially dependent thermal conductivities (taking into account uniaxial materials such as hBN).

Table II summarizes the thermal conductivities of the different materials considered in the heat transfer problem. We considered two different geometries: 6 400 hBN (out-of-plane) 7 8 Au 320

Material Thermal conductivity (W/m.K) SiO2 1.4 Si 150 hBN (in-plane)
Table II: Thermal conductivities of the materials.

-The simple geometry considered in the model of Ref. 5 , where a silicon dioxide layer extends from z = 0 to z = -285 nm on top of a semi-infinite (thickness 525 µm) Si substrate. A small disk-shaped heat flux is applied at the position of the graphene channel under bias (disk between r = 0 and r = R, with the radius of the disk R = 19.7 µm chosen to mimic the total surface of the graphene transistor TR2 under study). Both layers extend from r = 0 to r = 1 mm and have distinct thermal properties. Such geometry is displayed in Fig. SI-4.

-A more realistic geometry shown in Fig. SI-5, where we add (i) an additional layer of hBN situated on top of the SiO 2 layer and located radially from r = R to r = 2R (hBN hollow disk) and vertically from z = 0 to z = 80 nm. This layer models the fraction of hBN outside of the graphene region that has been etched to create edge contacts. We also add (ii) gold metallizations, with an Au disk located atop the hBN hollow disk (thickness 150 nm) and an outer Au disk located everywhere outside the hBN region (from r = 2R to r = 1 mm). These two disks represent the gold access that we deposit for the electrodes. The aforementioned steady-state heat equation is solved numerically using the Finite-Element Method (FEM) in Mathematica. The solution domain is discretized into finite elements and the equation is solved over these elements. The boundary conditions define the fixed heat flux on top and the fixed temperature at the bottom of the structure. Finally, the spatial variation in the thermal conductivity is encoded by defining a piecewise function for k r and k z to account for these variations across different regions of the domain. Table III compares the different scenarios and compares the obtained values for the thermal resistance with the value obtained with the simple model introduced in the main text, where we consider the average temperature for the SiO 2 layer. It underlines that the simple model outlined at the beginning of this section is in excellent agreement with a more refined treatment of the whole structure, and confirms that the out-of-plane radiative power reaches almost 70% of the total dissipated electrical power. Note that this result relies on three assumptions:

-SiO 2 layer is thin enough so that we consider it almost transparent, making the emission power additive with the SiO 2 layer depth -Interference effects are negligible in the contribution from each layer depth to the far field -The elevation in temperature is small compared to room temperature so that we can use the linearized version of the emitted power with respect to temperature elevation. We also assume an affine temperature profile within SiO 2 (which is confirmed by simulations). (P out-of-plane +P in-plane ) , where P in-plane is the in-plane power.

The out-of-plane power resulting from the solution of the full thermal problem is plotted in Fig. SI-6b along with the ratio between P out-of-plane and the total electrical power P elec injected into the device (Fig. SI-6c), which gives the contribution of out-of-plane radiative near-field energy transfer to the device's total power budget (Fig. SI-6a).

IV Electroluminescence origin of the measured mid-IR signal

We proceed to show in this section the electroluminescent nature of the mid-IR signal emitted by the devices under study here, following the approach adopted in Ref. 1 . Namely, we will experimentally rule out the possibility that the detected signal originates from the incandescence of hBN's optical phonons or from graphene's hot electrons. To do so, we measure the temperature of the two aforementioned quasi-particles, which is then used to estimate their blackbody emission. The temperature of hBN's optical phonons as determined via Raman Stokes-anti-Stokes thermometry is ∼ 75 • C (see Ref. 1 for full details). In Fig. SI-7, we compare the mid-IR spectrum of the TR1 device under electrical bias with that of its mid-IR thermal emission when the whole device is heated via a hot plate to the temperature of hBN's phonons (348 K). This comparison reveals that the main polariton peak at ∼ 190 meV, which is involved in the out-of-plane energy transfer in the device under bias (see main text), is notably absent from the thermal emission spectrum (see Fig. SI-7, green curve). The polariton peak of the SiO 2 substrate (∼ 145 meV) is likewise missing in the thermal emission spectrum. The main peak in the device's incandescence spectrum corresponds to a quarter-wave resonance of the heterostructure at around 170 meV (see Ref. 1 ). Thus, incandescence from hBN's optical phonons can be dismissed as the origin of the mid-IR signal.

The temperature of graphene's hot electrons is measured by characterizing the device's near-infrared incandescence as outlined in Ref. 1 spectral flux at the polariton peak, marked by a cross in Fig. SI-8a, is fitted with a Planck blackbody distribution using the electron temperatures listed in Table IV. This comparison shows that the measured spectral flux with increasing bias is about two times larger than that expected from incandescence in the bias range 6 -8 V (see Fig. SI-8b), thus precluding graphene's incandescence as a possible source of the measured signal. Note that this fitting procedure represents a worst-case scenario, in which it is assumed that the measured mid-IR flux at a low value of bias (4.3 V) is equal to that of the incandescence flux and the evolution of the two fluxes is compared as a function of increasing bias. We conclude from the analysis above that the mid-IR signal emitted by the device under electrical bias is electroluminescent in nature.

V Hyperbolic phonon-polariton dispersion curves in hBN-encapsulated graphene

The polaritonic dispersion curves of our samples were obtained by evaluating the poles of the reflection coefficient r p of the sample's multi-layer heterostructure for p-polarized light. A similar approach was utilized in Refs. 8,9 .

Our samples consist of a mono-layer graphene encapsulated by two thin hBN flakes and placed on a SiO 2 /Si substrate with a 285 nm-thick SiO 2 layer and an Si layer of bulk thickness. If we consider the SiO 2 thin film on the Si substrate only, then the surface phonon-polaritons supported by this layered system should exhibit two polariton branches ω + and ω -with large wavevector asymptotic values of their dispersion curves reached when Re(ε SiO2 ) = -1 for the upper polariton ω + , and Re(ε SiO2 ) = -ε Si = -11.7 for the lower polariton branch ω - 10,11 , where ε SiO2 and ε Si are the dielectric functions of SiO 2 and Si, respectively. Nevertheless, the large damping inherent to amorphous SiO 2 precludes the excitation of the lower polariton branch, as the minimal value of the real part of ε SiO2 is min Re(ε SiO2 ) ≈ -2.43 (see Fig. SI-10 (c)). The polaritonic response, in this case, is independent of the SiO 2 layer thickness, i.e., the response is the same as that of bulk SiO 2 . Thus, to fully describe the polaritonic response of our samples, it is sufficient to consider a four-layered hBN/graphene/hBN/SiO 2 heterostructure, as sketched in Fig. SI-9.

We use a Drude-Lorentz model with a single oscillator to describe the in-plane (ε x = ε y ) and out-of-plane (ε z ) dielectric functions of hBN, as well as the dielectric function of SiO 2 (ε SiO2 ), that is

ε µ (ω) = ε µ ∞ + ε µ 0 -ε µ ∞ (ω µ TO ) 2 -ω 2 -iγ µ ω (ω µ TO ) 2 with µ = x, z, or SiO 2 . (6)
Here, ω µ TO is the transverse optical phonon frequency, γ µ is a damping constant, and ε µ ∞ and ε µ 0 are the values of ε µ (ω) for ω ≫ ω µ TO , and ω ≪ ω µ TO , respectively. The longitudinal optical frequency ω µ LO is the frequency for which ε µ (ω) = 0 and is given in terms of ω µ TO by the Lyddane-Sachs-Teller relation 12 :

ω µ LO = ω µ TO (ε µ 0 /ε µ ∞ ) 1/2
. The real and imaginary parts of the dielectric functions of hBN and SiO 2 are shown in Fig. SI-10.

The influence of the graphene mono-layer is entailed through the inclusion of its sheet conductivity σ. For low energies (ℏω ≪ E F , where E F is the Fermi energy), the sheet conductivity of graphene is given by the Drude conductivity, σ D as follows 13

In the above relation, Γ = ev 2 F /µ e E F , represents the phenomenological scattering rate or damping constant of electrons due to impurity scattering, in which µ e is the electron mobility, and v F ≈ 10 6 m/s is the Fermi velocity. In general, the sheet conductivity of graphene can be cast in the following form 9,[START_REF] Wunsch | Dynamical polarization of graphene at finite doping[END_REF]

σ(k x , ω) = - ie 2 4ℏ ξ v 2 F k 2 x -ξ 2 1 + G ℏξ + 2E F ℏv F k x -G ℏξ -2E F ℏv F k x - 2i π e 2 ωE F (ℏv F k x ) 2 , (8)
where, G(η) = -1 π η 1 -η 2 -cos -1 (η) , ξ = ω + i Γ ℏ , and k x is the in-plane polariton wavevector. Assuming that the thicknesses of the encapsulating hBN layers are d

The addition of the second film yields the total reflection coefficient of the stack:

k z = ε x ω 2 c 2 - k 2 x ε z , k z0,3 = ε 0,3 ω 2 c 2 -k 2 x , Im(k z , k z0,3 ) > 0, Q 1,2 = ε x k z , Q 0,3 = ε 0,3 k z0,3 , ε 0 = 1, ε 3 = ε SiO2 , and S = 1 ϵ • σ(k x , ω) ω . (11)
In the definitions above, ϵ • is the permittivity of free space, and k z and k z0,3 give the out-of-plane polariton wavevector for j = 1, 2 and j = 0, 3, respectively. The polariton branches, visualized from the maxima of the imaginary part of the reflection coefficient r p (Eq. ( 10)), are shown in

VI Noise thermometry comparison of the three hBN crystals

Following the approach described in Refs. [START_REF] Yang | A graphene zener-klein transistor cooled by a hyperbolic substrate[END_REF][START_REF] Schmitt | High-field 1/f noise in hbn-encapsulated graphene transistors[END_REF] we perform noise thermometry characterization of hBN-encapsulated graphene transistors under large bias, to compare the efficiency of near-field radiative energy transfer associated with

Fig 1 :

 1 Fig 1: Electroluminescent radiative energy transfer of graphene in pristine and turbid hBN encapsulate. a, Energy transfer from graphene to the substrate (SiO2 or Au) follows a 4-step process. (Step 1) Electrical Zener-Klein interband pumping under large applied bias. (Step 2) Intraband thermalization. (Step 3) Interband emission of a hyperbolic phonon-polariton (HPhP) of hBN. (Step 4) Ballistic HPhP propagation eventually reaches the substrate. (Step 5) HPhP energy transfer to the substrate results in a local temperature increase. b, In the case of turbid hBN, steps 1, 2, and, 3 are identical, but HPhPs are localized near the graphene layer due to scattering sites in the crystal (represented by pink bubbles) at step 4, leading to their re-absorption by the graphene electron gas at step 5. Consequently, in this case, the substrate remains cold.

Fig 2 :

 2 Fig 2:Radiative origin of the out-of-plane energy transfer: scattering of HPhPs by defects. a, Mid-infrared (mid-IR) spectrum of an electrically-biased hBN-encapsulated graphene device (TR1, inset). The small dimensions of this device's graphene channel (9.2 µm × 15.5 µm) lead to increased scattering of the HPhP modes of the heterostructure by the edge contacts and etches in the hBN layer (marked by red arrowheads in the inset). The mid-IR spectrum of a larger graphene device (TR2, 35 µm × 35 µm, inset), in which scattering is absent, is shown for comparison. b, Imaginary part of the reflection coefficient (rp) for p-polarized light incident upon the heterostructure consisting of TR1's polariton active layers (sketched in panel c). The polariton dispersion curves of the heterostructure can be visualized from the maxima of Im(rp). See Supplementary information, section III for full details. The dashed green lines enclose the Reststrahlen band of SiO2, while the dashed orange lines enclose hBN's second Reststrahlen band, in which Re(εx) = Re(εy) < 0 and Re(εz) > 0, where εx,y and εz are the in-plane and out-of-plane dielectric functions of hBN, respectively. c, Penetration depth δz of HPhPs in the z-direction (see main text) calculated for the layers of TR1 (top sketch). d, Purcell factor above a 30 nm-thick hBN cavity, given by FP = 3

Fig 3 :

 3 Fig 3:Monitoring the out-of-plane energy transfer in a graphene device through its mid-infrared emission. a, Schematic depiction of out-of-plane energy transfer in a high-mobility graphene field-effect device with an SiO2 backgate. The back gate warms up slightly due to this out-of-plane energy transfer, emitting mid-infrared (mid-IR) thermal radiation. One can directly observe the device's out-of-plane energy transfer by monitoring the substrate's mid-IR emission. b, Left: microscope image (top) and spatial scan (bottom) of the spectrally integrated mid-IR signal (PMIR, integrated over the detector's bandwidth, λ = 6 -14 µm) emitted by the TR2 device under bias. The bright emission spot, between the source and drain electrodes, indicates that the signal is most intense in that region, revealing that the emission originates from beneath the device's channel. By contrast, the device's gold electrodes appear dark due to their low emissivity. Right: Scan of PMIR as a function of electrical bias. The white dashed curve indicates the device's Zener-Klein (ZK) tunneling threshold. c, Measured temperature increase in the SiO2 substrate as a function of bias. The temperature increase is determined by calibrating the detected mid-IR signal as a function of sample temperature (see SI for full details). d, Measured out-of-plane power P out-of-plane as a function of applied bias. e, Ratio between P out-of-plane and the total electrical power (P elec ) applied to the TR2 device.

Fig 4 :

 4 Fig 4:Engineering energy transfer efficiency by tuning hBN crystalline quality. a, Near-IR graphene electron temperature versus bias for the two devices Lyon1 (PDC-hBN) and F161 (HPHT-hBN), measured around a wavelength λ = 1.5 µm, for a doping n = 6 × 10 11 cm -2 . b, Mid-IR characterization of a device made with PDC-hBN on a Si/SiO2 back gate. Vertical energy transfer under large bias is monitored through the incandescence of the SiO2 substrate, with a much weaker temperature increase ∆TSiO 2 < ∼ 1 • C, which corresponds to an out-of-plane radiative power per surface area of ∼ 0.47 kW/cm 2 ., i.e., only 2.4 % of that of devices made with HPHT-hBN.

Fig. SI- 1 :

 1 Fig. SI-1: Schematic illustration of the detection technique used for characterizing the mid-IR signal (PMIR, integrated over the full spectral range of the detector) of an hBN-encapsulated graphene device (left, L × W = 35 × 35 µm 2 ) due to the radiative emission of the SiO2 back-gate.

Fig. SI- 2 :

 2 Fig. SI-2: a, Comparison between the mid-IR spectra of the hBN-encapsulated device of panel d with an SiO2 back gate under electrical bias and by heating the device via a hot plate at various temperatures T . The mid-IR signal is measured using an optical chopper (see Fig. SI-1). The plotted spectra are corrected by a reference measurement, which removes the contribution of the chopper to the detected signal, and then normalized by an instrumental response function (see supplementary information of Ref. 1 ). b, Linear fit of the spectrally-integrated signal Sint obtained with the hot plate as a function of temperature. Sint is found by integrating the spectra S from panel a over the detector's spectral bandwidth. From the comparison of panel a we find that the temperature of SiO2 back-gate varies between ∆TSiO 2 ≃ 0 -40 • C when the device is electrically biased. c, Scan of the mid-IR spectrally-integrated power PMIR emitted by the TR2 device as a function of bias. d, Increase in the temperature of the device's SiO2 back-gate (∆TSiO 2 ) as a function of electrical bias, obtained from the mid-IR signal to sample temperature calibration.

Fig. SI- 3 :

 3 Fig. SI-3: Sketch of a simple thermal model used to compute the total thermal resistance R th = RSiO 2 + RSi ruling the out-of-plane energy transfer in an hBN-encapsulated graphene device with a SiO2/Si back-gate.

Fig. SI- 4 :

 4 Fig. SI-4: Side view (left panel) and top view (right panel) of the considered simple geometry, where a disk-shaped heat flux is applied on top of a 285 nm-thick SiO2 layer and a semi-infinite Si substrate. Scales are in units of the disk radius R.

Fig. SI- 5 :

 5 Fig. SI-5: Side view of the considered more realistic geometry, where gold metallizations and a hollow hBN disk are present. Scales are in units of the disk radius R.

  Fig. SI-6: a, Total electrical power (P elec ) injected into the TR2 device as a function of bias. b, Out-of-plane dissipated power (P out-of-plane ) in the TR2 device as a function of electrical bias, computed from Pout = ∆TSiO 2 /R th , where ∆TSiO 2 ≃ 0 -40 • C and R th ≃ 171.4 K.W -1 (see Table III and Fig. SI-2d). c, Percentage giving the contribution of the out-of-plane radiative energy transfer mechanism to the total power budget obtained from the ratio P out-of-plane P elec =

  . The obtained electron temperatures are shown in Table IV as a function of bias. The mid-IR spectrum of the TR1 device is measured via IR-SMS as a function of increasing bias (Fig. SI-8a) and the resulting

Fig. SI- 7 :

 7 Fig. SI-7: Electroluminescence vs hBN incandescence. Comparison between the IR-SMS spectral flux of TR1 under electrical bias (orange curve) and its incandescence when heated via a hot plate to the temperature of hBN's optical phonons (green curve).

Fig. SI- 8 :

 8 Fig. SI-8: Electroluminescence vs graphene's hot electron incandescence. a, Sweep of the mid-infrared spectral emission of a graphene device TR1 (inset) under electrical bias, measured via IR-SMS. Legend keys to the right of panel b. b, Spectral flux at the polariton peak, marked by a cross in panel a, as a function of applied bias. The grey triangles are the Planck law fit given by AE 3 /[exp (E/kBTe) -1], in which A is a fitting constant, E = 190 meV is the polariton energy, and Te is the electron temperature given in TableIV. A is determined by equating the spectral flux at V ds = 4.3 V to the value of the measured spectral flux at the same point. The inset shows the ratio of the measured spectral flux to that obtained from the Planck law fit.

Fig. SI- 9 :

 9 Fig. SI-9: Sketch of a graphene layer encapsulated between two hBN thin films of thicknesses d1 and d2.

Fig. SI- 10 : 2 ∞= 1 .

 1021 Fig. SI-10: a & b, Real and imaginary parts of the dielectric function of hBN, obtained by applying Eq. (6) for the in-plane (x and y) and out-of-plane (z) directions. The orange-shaded regions indicate the two Reststrahlen bands of hBN. The optical constants appropriate for hBN are ε z ∞ = 2.95, ε z 0 = 3.57, ω z TO = 740 cm -1 , ω z LO = 814 cm -1 , and γ z = 4 cm -1 , for the out-of-plane direction (type I Reststrahlen band), and ε x ∞ = 4.87, ε x 0 = 6.71, ω x TO = 1360 cm -1 , ω x LO = 1596.4 cm -1 , and γ x = 5 cm -1 for the in-plane direction (type II Reststrahlen band). c & d, Real and imaginary parts of the dielectric function of SiO2. The green-shaded region indicates the Reststrahlen band of SiO2. Optical constants: ε SiO 2 ∞ = 1.97, and ε SiO 2 0 = 2.54, ω SiO 2 TO = 1096 cm -1 , ω SiO 2 LO = 1245 cm -1 , and γ SiO 2 .

1

 1 and d 2 (see Fig. SI-9), we may divide the hBN/graphene/hBN/SiO 2 heterostructure of Fig. SI-9 into the following four regions: z > 0 (j = 0, vacuum), -d 1 < z < 0 (j = 1, hBN), -d 1 -d 2 < z < -d 1 (j = 2, hBN), and z < -d 1 -d 2 (j = 3, SiO 2 ). The total reflection

Fig. SI- 11 :

 11 Fig. SI-11: HPhP-plasmon dispersion curves visualized through the maxima of the imaginary part of the reflection coefficient rp, for two different values of the Fermi energy (EF = 30 meV, left; EF = 1 eV, right) and electron mobility µe = 66, 000 cm 2 /V.s.The thicknesses of the hBN layers are given by: d1 = 52 nm and d2 = 30 nm. The dashed orange and green lines enclose the Reststrahlen bands of hBN and SiO2, respectively.

1 -r 10 r 1 p e 2ikzd1 . ( 10 )

 12ikzd110 r p = r 01 + r ′ p (1 -r 01 -r 10 )e 2ikzd1In the above two equations,r 12 = (Q 2 -Q 1 + S)/(Q 1 + Q 2 + S) and r 21 = (Q 1 -Q 2 + S)/(Q 1 + Q 2 + S)are the reflection coefficients at the graphene/hBN boundaries,r mn = -r nm = (Q n -Q m )/(Q n + Q m ) (mn ̸ = 12, 21) the reflection coefficients at the boundary between regions m and n, in which the following definitions were used

  Fig. SI-11 for two different values of the Fermi energy (0.03 eV and 1 eV). As presented in Fig. SI-11, the influence of graphene's plasmons on the HPhP branches of hBN becomes important only for high Fermi energies (E F = 1 eV) and is almost completely negligible when the graphene is close to neutrality (see Fig. SI-11, E F = 0.03 eV).

Fig. SI- 12 :

 12 Fig. SI-12: Electronic noise temperature as a function of bias for devices made with different hexagonal boron nitrides synthesized with various techniques, for the same doping range n = [0 -1] 10 12 cm -2 . (a) Electronic noise temperature of a device made with mono-isotopic boron nitride from KSU (>99% 11 B, fabricated at high temperature and atmospheric pressure). (b) Electronic noise temperature of a device made with HPHT-hBN from NIMS, Japan. (c) Electronic noise temperature of a device made with PDC-hBN from Lyon, France. Strikingly, the measured noise temperature in the KSU device is much lower than previously observed. Compared with devices made with HPHT-hBN from NIMS and PDC-hBN from Lyon, the noise temperature is lower by a factor of ∼ 10.

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AJ

Table I :

 I . Geometrical and electronic properties of the different graphene transistors.

	Device	Length Width	µ	Contact resistance
		(µm)	(µm) (m 2 /V.s)	(Ω)
	TR1	9.2	15.5	6	250
	TR2	35	35	10	150
	PDC-hBN	9	9	4	230
	HPHT-hBN	6	10	16	45
	KSU-hBN	19	6	8	400

received partial funding from the European Union Horizon 2020 research and innovation program under grant agreement No.881603 "Graphene Core 3", and from the French ANR-21-CE24-0025-01 "ELuSeM". This work has received support under the program "Investissements d'Avenir" launched by the French Government. JHE and EJ appreciate support for monoisotopic hBN crystal growth from the U. S. Office of Naval Research under award number N00014-22-1-2582.

will be publicly available on Zenodo.