Thermal photon drag in many-body systems
TThermal photon drag in many-body systems
P. Ben-Abdallah ∗ Laboratoire Charles Fabry,UMR 8501, Institut d’Optique, CNRS,Université Paris-Sud 11, 2, Avenue Augustin Fresnel, 91127 Palaiseau Cedex, France. (Dated: April 25, 2019)We demonstrate the existence of a thermal analog of Coulomb drag in many-body systems whichis driven by thermal photons. We show that this frictional effect can either be positive or negativedepending on the separation distances within the system. Also we highlight that the persistent heatcurrents flowing in non-reciprocal systems at equilibrium are subject to this effect and the lattercan even amplify these flows.
PACS numbers: 44.40.+a, 78.20.N-, 03.50.De, 66.70.-f
Spatially separated electric conductors are coupledthrough the interactions of free charge carriers providedtheir separation distance is small enough in comparisonwith the range of Coulombic interactions. Hence, when acurrent flow in one also called drive conductor it inducesa current by Coulomb drag [1–4] in a second (passive)conductor placed close to it (Fig. 1-a). In this Letterwe demonstrate the existence of a thermal analog of thiseffect in many body systems in which heat exchangesare mediated by thermal photons. In these systems thelocal thermal fluctuations within each body give rise tooscillations of partial charges which, in turn, radiate atime-dependent electric field in their surrounding envi-ronement. This leads through the many-body interac-tions (i.e. multiscattering combined to spontaneous ab-sorption/emission of thermal photons) to an exchangeof a net heat flux between the different regions of thesystem which have different temperatures [5–10]. Afterintroducing the concept of thermal photons drag resis-tance we investigate the drag effect in a four-terminalsystem composed by two parallel pairs of nanoparticleswhen a temperature difference is held along one of thesepairs (drive) while the second pair (passive) is left free torelax. We show that the sign of the temperature differ-ence induced in the passive pair of nanoparticles can beeither positive or negative depending on the strength andthe nature of interactions between the particles. Finallywe show that the heat supercurrents in non-reciprocalmany body systems are also subject to this drag effectand the latter can amplify or inhibitate these persistentheat currents.To start, we consider a system as sketched in Fig. 1-bmade with two parallel rows of nanoparticles when theextremities of the first row are held at fixed tempera-ture with two external thermostats while others particlesrelax to their own equilibrium temperature. Practicallythe particles can be, for instance, grafted at the end ofelongated scanning probe microscope tips and they canbe heated up by conduction through the tip itself with aresistive heater integrated to the tip. The temperature ofeach particle can be controlled independently using thosethermal resistors or left free to relax to their own equi- V T h T c J d J I I d (a) (b) T T N T N+1 T d h h Figure 1. (a) Coulomb drag: a drag electric current I d in apassive conducting wire is induced by the current I flowing ina driving conductor placed close to it. (b) Radiative drag in aset of particles: a drag heat flux J d carried by thermal photonsbetween two particles is induced by a heat fux J exchangedbetween two thermostated particles in a many body system. librium temperature. In this system, all particles canexchange electromagnetic energy between them as wellas with their surrounding which can be assimilated to abosonic field at a given temperature T b . (For subwave-length separation distances the power exchanged betweenthe particles through near-field interactions is generallymuch more significant than the power exchanged withthe surrounding bath so that the latter can be neglected).In steady state regime the unthermostated particles (i.e. i (cid:54) = 1 , N ) reach their equilibrium temperature T ie . Usingthe Landauer theory in many body systems the powerexchanged between the i th and the j th particle in thisnetwork reads [11–14] ϕ ij = (cid:90) ∞ d ω π [Θ( ω, T i ) T i,j ( ω ) − Θ( ω, T j ) T j,i ( ω )] , (1)where Θ( ω, T ) = (cid:126) ω/ [ e (cid:126) ωkBT − is the mean energy of aharmonic oscillator in thermal equilibrium at tempera-ture T and T i,j ( ω ) denotes the transmission coefficient,at the frequency ω , between the two particles. When the a r X i v : . [ c ond - m a t . m e s - h a ll ] A p r T T T T h d (a) S g n ( 𝐑 𝐝 ) 𝐥 𝐨 𝐠 ( 𝐑 𝐝 ) ( K . W - ) h h (m) h ∆ 𝐓 𝐞 SiC (b) T T T T h d T T T T h d S g n ( 𝐑 𝐝 ) 𝐥 𝐨 𝐠 ( 𝐑 𝐝 ) ( K . W - ) SiC Au SiC Au h h (m) h S g n ( 𝐓 𝐞 ) 𝐥 𝐨 𝐠 ( 𝐓 𝐞 ) Figure 2. Drag resistance between two dimers of (a) SiCnanoparticles and (b) mixed SiC/gold nanoparticles ( R =100 nm ) under a primary temperature difference ∆ T = T − T = 50 when T = 300 K and d = 2 R with respect tothe dimers separation distance h . Insets: temperature dif-ference ∆ T e = T − T induced in the second dimer. In-sets:temperature difference with respect to h . The dashedline in the inset of Fig. 2b shows the sign of the temperaturedifference. particles are small enough compared with their thermalwavelength λ T i = c (cid:126) / ( k B T i ) ( c is the vacuum light ve-locity, π (cid:126) is Planck’s constant, and k B is Boltzmann’sconstant) they can be modeled by simple radiating elec-trical dipoles. In this case the transmission coefficientbetween the dipole i and j is defined as [15, 16] T i,j ( ω ) = 43 ( ωc ) ImTr (cid:2) ˆ α j G ji
12 ( α i − α † i ) G † ji (cid:3) , (2)where ˆ α i denotes the polarizability tensor of the i th par-ticle and G ij is the dyadic Green tensor between the i th and the j th particle in the N-dipoles system [17].From expression (1) the net thermal power receivedby each particle is simply given by summation over allincoming power (including the power coming from the thermostats) so that φ i = (cid:88) j (cid:54) = i ϕ ji . (3)Then, the net power exchanged between the first and the N th particle also called driving power reads J = φ N − φ = (cid:88) j (cid:54) = N ϕ jN − (cid:88) j (cid:54) =1 ϕ j . (4)The calculation of these powers requires theknowledge of equilibrium temperatures ¯ T =( T e , ..., T ( N − e , T ( N +1) e , ..., T Ne ) t which can beobtained from the energy balance equations at equilib-rium φ th + (cid:80) j (cid:54) =1 ϕ j = 0 ,φ thN + (cid:80) j (cid:54) = N ϕ jN = 0 , (cid:80) j (cid:54) = ii (cid:54) =(1 ,N ) ϕ ji = 0 , (5)where φ th and φ thN stand for the power injected fromthe external thermostats into the first and the N th par-ticles. Using the thermal conductance G ij = ∂ϕ ij ∂T thissystem can be recasted into the following form φ th + (cid:80) j (cid:54) =1 G j ( T j − T h ) = 0 ,φ thN + (cid:80) j (cid:54) = N G jN ( T j − T c ) = 0 , (cid:80) j (cid:54) = ii (cid:54) =(1 ,N ) G ji ( T j − T i ) = 0 . (6)This set of equations also takes the form of followingblock system I ... A · · · · · · · · · ... C (cid:18) ¯ Φ ¯ T (cid:19) = (cid:18) ¯U¯V (cid:19) (7)where A = (cid:18) G , · · · G N, G ,N · · · G N,N (cid:19) and ¯ U =( (cid:80) j (cid:54) =1 G j T , (cid:80) j (cid:54) = N G jN T N ) t , C ij = − (cid:80) k (cid:54) = i G ki δ ij + G ji (1 − δ ij ) with i, j (cid:54) = (1 , N ) and ¯ V = − (( G , T + G N, T N ) , ( G , N T + G N, N T N )) t . In this system I isthe × unit matrix and = (cid:18) · · · · · · (cid:19) t is the zeromatrix of format (2 N − × . It turns out that theequilibrium temperatures reads ¯ T = C − ¯ V , (8)and the powers ¯ Φ = ( φ th , φ thN ) t coming from the twothermostats is simply given by the following expression ¯ Φ = ¯ U − A ¯ T . (9)Note that this reasoning can easily be extended to ar-bitrary many body systems with more than two ther-mostated particles.Given the power J transmitted from particles 1 to Nin the active layer the temperature difference ∆ T e =( T N +1 ,e − T N,e ) induced in the passive layer is propor-tional to J and we can introduce the thermal drag resis-tance as R D = ∆ T e J (10)which is a direct measure of many-body interactions be-tween the two parallel layers. In the particular case oftwo parallel dimers the equilibrium themperatures read ¯ T = 1∆ Σ j (cid:54) =4 G j ( G T + G T ) + G ( G T + G T )Σ j (cid:54) =3 G j ( G T + G T ) + G ( G T + G T ) (11)with ∆ = Σ j (cid:54) =3 G j Σ j (cid:54) =4 G j − G G . This leads after astraighforward calculation to the induced temperaturedifference between particles 3 and 4 T − T = G G − G G ∆ ( T − T ) . (12)As for the primary current it writes J = Σ j (cid:54) =2 G j ( T j − T ) − Σ j (cid:54) =1 G j ( T j − T ) , (13)with T = T h and T = T c the temperatures set bythe two thermostats. In Fig. 2-a we show the dragresistance between two dimers of silicon carbide (SiC)nanoparticles for different separation distances with re-spect to the primary temperature difference. We describethe dielectric properties of SiC by means of a Drude-Lorentz model [18] (cid:15) ( ω ) = (cid:15) ∞ ω − ω L + iγωω − ω R + iγω with (cid:15) ∞ = 6 . , ω L = 1 . × rad/s, ω T = 1 . × rad/s and γ = 0 . × rad/s. As expected this resistance is pos-itive and decays monototically with the separation dis-tance H between the two dimers. On the contrary in amixed crossed system (Fig. 2-b) composed by SiC/goldnanoparticles (the dielectric permittivity of gold parti-cles is given by the Drude model (cid:15) ( ω ) = 1 − ω p ω ( ω + iγ ) with ω p = 13 . × rad.s − and γ = 4 × s − [18])the situation radically changes and a negative drag effectappears in the intermediate regime between the near andfar-field regimes. In this region the induced temperaturedifference ∆ T e becomes negative. As shown in Fig. 3this negative drag effect is due to a better coupling alongthe diagonal between the two SiC particles compared tothe vertical coupling between the SiC and the gold par-ticles so that the th particle is heated up furthermorethan th particle. This negative drag effect results fromthe spectral mismatch of SiC and gold particles. × 𝟏𝟎 −𝟏𝟑 h h (m) Figure 3. Thermal conductance at T = 300 K between theparticles in the same system as in Fig. 2-b with respect tothe dimers separation distance h between the two dimers. It is worthwile to note that this drag effect also impactsthe supercurrents which exist at equilibrium in somemany-body systems at fixed temperature. As demon-strated by Fan et al. [11] a persistent directional heatcurrent can arise in non-reciprocal networks. In this case,when the many-body system is held at a given temper-ature T , it follows from the general expression (1) thatthe body i and j in this system still exchange an energyflux [14] ϕ eqij = (cid:90) ∞ d ω π Θ( ω, T )[ T i,j ( ω, B ) − T j,i ( ω, B )] . (14)Notice that while the non-equilibrium flux given by ex-pression (1) depends on the temperature difference be-tween the particles i and j , this flux is related to theoptical non-reciprocity within the system and it can ex-ist even at thermal equilibrum.To illustrate how this persistent heat current is af-fected by the frictional effect let us consider a systemmade with two parallel magneto-optical networks of reg-ularly distributed InSb nanoparticles along parallel ringsas sketched in (Fig. 4) and submitted to an external mag-netic field applied perpendicularly to these rings. Thedielectric tensor associated to InSb particles takes thefollowing form [19, 20] ¯¯ ε = ε − iε iε ε
00 0 ε (15)with ε ( B ) = ε ∞ (1 + ω L − ω T ω T − ω − i Γ ω + ω p ( ω + iγ ) ω [ ω c − ( ω + iγ ) ] ) , (16) ε ( B ) = ε ∞ ω p ω c ω [( ω + iγ ) − ω c ] , (17) ε = ε ∞ (1 + ω L − ω T ω T − ω − i Γ ω − ω p ω ( ω + iγ ) ) . (18)Here, ε ∞ = 15 . is the infinite-frequency dielectric con-stant, ω L = 3 . × rad.s − is the longitudinal opticalphonon frequency, ω T = 3 . × rad.s − is the trans-verse optical phonon frequency, ω p = ( ne m ∗ ε ε ∞ ) / is theplasma frequency of free carriers of density n = 1 . × cm − and effective mass m ∗ = 7 . × − kg , Γ =5 . × rad.s − is the phonon damping constant, γ =10 rad.s − is the free carrier damping constant and ω c = eB/m ∗ is the cyclotron frequency. Thus, thepolarizability tensor for a spherical particle can be de-scribed, including the radiative corrections, by the fol-lowing anisotropic polarizability [21] ¯¯ α i ( ω ) = (¯¯ − i k π ¯¯ α i ) − ¯¯ α i , (19)where ¯¯ α i denotes the quasistatic polarizability of the i th particle which reads for spheres of radius R in vacuum ε h ¯¯ α i ( ω ) = 4 πR (¯¯ ε − ¯¯1)(¯¯ ε + 2¯¯1) − . (20)In Fig. 4-a we show how the supercurrent is modifiedwith respect to the separation distance between the ringsunder the action of an external magnetic field of Tesla.At large separation distance we see that the supercur-rent spectrum displays three distinct resonant peaks inthe midinfrared around µm . Those peaks correspondto the resonances of isolated InSb particle [14] as shownin the inset of Fig.4-b. When the separation distancebetween the two rings is reduced to ( h = 2 R ) we ob-serve that the spectrum is broadened toward the higherwavelengths meaning so that the supercurrent can be am-plified by the interplay of two networks when the Wien’swavelength of system is in this spectral range. On thecontrary if the equilibrium temperature is sufficiently in-creased the Planck window where energy exchanges occuris shifted toward the lower wavelengths and the supercur-rent is inhibated. This broadening simply results fromthe increase in the number of coupling channels betweenthe two networks. In Fig.4-b we show that the drag effecton the persistent current also is sensitive to the magni-tude of magnetic field or in other words to the importanceof the optical non-reciprocity inside the system. When B = 1 T the supercurrent spectrum is relatively narrowand centered around the resonant wavelengths of eachInSb particle (see Inset of Fig. 4-b). However, compar-erd to what happens at B = 5 T , in this case the splittingof resonances is less pronounced. As the magntitude ofmagnetic field increases the non-reciprocity in the systembecomes more important and the splitting between theresonant modes of InSb particles become more signitfi-cant. It turns out that the supercurrent specturm widensand this widening mainly occurs at low wavelengths. Fi-nally I emphasize in this section the differences which h B h=2R h 𝛕 𝐢 𝐣 − 𝛕 𝐣 𝐢 (a) B=5 T B=1 T 𝛕 𝐢 𝐣 − 𝛕 𝐣 𝐢 (b) Figure 4. Heat supercurrent spectrum (normalized by the en-ergy of Planck oscillator) in a many-body system of 8 InSbparticles ( R = 100 nm ) regularly and identically distributedon two parallel rings under an external magnetic field B or-thogonal to the rings for (a) two separation distances with B = 5 T and for (b) two magnetic field intensities when h = 2 R . Inset of (b): resonances of an isolated InSb par-ticle shown in the ( λ, B ) plane. exist between the supercurrents interaction and the cou-pling between magnetic dipoles produced for instance byloops of electric current or by free charges oscillation insplit-ring resonators. Let us consider for clarity reasonsthe case of two circular loops of same area S which carrya primary current I ( t ) . Then this current generates amagnetic moment m = I. S ( S being the oriented sur-face of the loop) which gives rise to an electromagneticfield in its surrounding environment. Then, according toFaraday’s law this field leads to an electromotive force inthe neighboring loop which in turn induced an electriccurrent I d . In the particular case of two parallel loopsthis induced current is related to I by the simple relation I d = MR dIdt where M denotes the mutual inductance and R the electric resistance inside each loop ( M = µ πd S R , d being their separation distance and µ the vacuumpermeability). On the contrary, in the thermal problemthe persistent current is a flow of energy which is car-ried by the electromagnetic field itself and not by electriccharges. This field weakly impacts the cyclotronic mo-tion within the magneto-optical material so that it doesnot induced an electromotive force. The interplay be-tween the two systems is only due to the electric dipolescoupling. Nevertheless the circular heat flux is connectedto an orbital angular momentum of the electromagneticfield and also to a spin angular momentum [22]. Theconsequences of these quantities and of their mutual in-teraction in the system environment could be used todetect the existence of persistent heat flux in a similarway as the persistent electric currents predicted by thequantum mechanics [23, 24].In summary, we have introduced a drag effect inducedby the electromagnetic interactions in many-body sys-tems. This frictional effect quantifies the strength of elec-tromagnetic interactions in these systems and it gives in-sights on the spatial distribution of temperature in non-equilibrium situations. By introducing the concept ofdrag resistance for the thermal photons we have demon-strated the existence of regions in these systems wherethe heat flux can locally flows in an opposite direc-tion to the local temperature variation. Beside we haveshown that this drag effect also exists at thermal equi-librium between persistent heat flux. 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