Twist-induced Near-field Thermal Switch Using Nonreciprocal Surface Magnon-Polaritons
Jiebin Peng, Gaomin Tang, Luqin Wang, Rair Macêdo, Hong Chen, Jie Ren
TTwist-induced Near-field Thermal Switch UsingNonreciprocal Surface Magnon-Polaritons
Jiebin Peng, † Gaomin Tang, ∗ , ‡ Luqin Wang, † Rair Macêdo, ¶ Hong Chen, † and Jie Ren ∗ , † † Center for Phononics and Thermal Energy Science, China-EU Joint Center for Nanophononics,Shanghai Key Laboratory of Special Artificial Microstructure Materials and Technology, Schoolof Physics Science and Engineering, Tongji University, 200092 Shanghai, China ‡ Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland ¶ James Watt School of Engineering, Electronics & Nanoscale Engineering Division, Universityof Glasgow, Glasgow G128QQ, United Kingdom
E-mail: [email protected]; [email protected]
Abstract
We explore that two ferromagnetic insulator slabs host a strong twist-induced near-fieldradiative heat transfer in the presence of twisted magnetic fields. Using the formalism of fluc-tuational electrodynamics, we find the existence of large twist-induced thermal switch ratioin large damping condition and nonmonotonic twist manipulation for heat transfer in smalldamping condition, associated with the different twist-induced effects of nonreciprocal ellip-tic surface magnon-polaritons, hyperbolic surface magnon-polaritons, and twist-non-resonantsurface magnon-polaritons. Moreover, the near-field radiative heat transfer can be significantlyenhanced by the twist-non-resonant surface magnon-polaritons in the ultra-small damping con-dition. Such twist-induced effect is applicable for other kinds of anisotropic slabs with time-reversal symmetry breaking. Our findings provide a way to twisted and magnetic control innanoscale thermal management and improve it with twistronics concepts. a r X i v : . [ c ond - m a t . m e s - h a ll ] D ec key component for manipulating radiative heat flow at the nanoscale is near-field radiativeheat transfer, which can exceed Planck’s blackbody radiation limit by orders of magnitude due tothe presence of evanescent modes. Two types of surface modes have been commonly studiedin near-field heat transfer; one is surface plasmon-polaritons and the other is surface phonon-polaritons.
In addition, surface magnon-polaritons (SMPs), hybrid collective excitations dueto the coupling between magnons and electromagnetic fields, also has functional associationsto thermal management in nanotechnologies. For instance, in magnetic recording devices, a mag-netic read/write head touches above the disk surface with nanometers separation. At such a shortdistance, SMPs should play a significant role in the near-field thermal manipulation of magneticrecording devices. Moreover, due to the high gyrotropic optical effect, SMPs in uniaxial ferro-magnetic insulator (FMI) are nonreciprocal. Such nonreciprocal behavior can break Kirchhoff’slaw and paves the way for the exploitation of radiative thermal transfer at nanoscale.Recently, twistronics becomes an emerging research topic since the electronic state can be ma-nipulated through the “twist angle" between two layers, leading to flat-band superconductivity, moiré excitons, stacking-dependent interlayer magnetism and other exotic electronic proper-ties. Similar twist-induced concepts have been demonstrated in photonics, such as moiré photonicscrystal, moiré hyperbolic metasurfaces and photonic magic angles. Motivated by these ex-otic discoveries, several works have shown the development of tunable radiative heat flow betweentwo-dimensional materials and biaxial crystals through twist. With the analogous principle,we explore the effects of radiative thermal twistronics between the uniaxial FMIs with externalmagnetic fields, where the twist and nonreciprocal phenomena can both arise in the domain ofthermal management.In this Letter, we consider to manipulate near-field radiative heat transfer through the twistbetween two uniaxial FMIs. Nonreciprocal SMPs emerge at the interface between vacuum and gy-rotropic FMIs with asymmetric permeability tensor. Based on the nonreciprocity, we demonstratea large twist-induced near-field thermal switch effect with a moderate external magnetic field. Un-der ultra-small damping condition, we show an unusual twist-induced near-field thermal transfer2nhancement due to the presence of twist-non-resonant SMPs. (a)(c) 𝑇 , 𝜇 𝑇 , 𝜇 dxz y θθ x zy (b) (d) ω 𝑢 ω 𝑑 y΄x΄ θ Vacuum Bulk MP
SMP d = 2mm d = 100nm x' z'y' Figure 1: (a) A schematic setup for radiative heat transfer between two FMIs with vacuum sep-aration d . The bottom and top slabs have the temperature T and T , respectively. The y ( y (cid:48) )axis is along the direction of the satuation magnetisation in the bottom (top) FMI. The magneticfields in each slab is applied along the direction of the corresponding satuation magnetization. Thetwist angle θ is defined by the anticlockwise rotation of x (cid:48) y (cid:48) z (cid:48) coordinate system with respect to xyz coordinate system. (b) Dispersion relation of nonreciprocal SMP with a single vacuum-FMIinterface. (c) Energy transmission coefficient Z ( ω, q, φ = 0) with gap distance d = 2 mm. Thecyan dashed line and the red dash-dotted line are the same as in (b). The black dotted line showsthe nonreciprocal symmetric and asymmetric modes of SMPs. (d) Energy transmission coefficient Z ( ω, q, φ = 0 . π ) with gap distance d = 100 nm. The damping constant α is . in (c) and (d). Radiative heat transfer.–
We consider near-field radiative heat transfer between two FMIs withtemperatures T = T ± ∆ T / and twist angle θ [See Fig. 1(a)]. A Cartesian coordinate system xyz ( x (cid:48) y (cid:48) z (cid:48) ) is defined at the bottom (top) slab and the y ( y (cid:48) ) axis is along the direction of theapplied magnetic field and saturation magnetisation. The twist angle θ is defined as the anglebetween the y (cid:48) and y axis. We define the heat transfer coefficient κ as κ = lim ∆ T → J/ ∆ T with J the heat transfer coefficient can be expressedas κ ( T, θ ) = (cid:90) ∞ dω π ¯ hω ∂N∂T (cid:90) ∞ dq π q (cid:90) π dφ π Z ( ω, q, φ ) , (1)where q is the in-plane wave vector and φ the in-plane azimuthal angle. In the above expression, ∂N/∂T is the derivative of the Bose distribution function with respect to the temperature. Weconsider the relative heat transfer coefficient scaled by the black-body limit κ b = 4 σ b T with σ b = π k B / (60¯ h c ) . The energy transmission coefficient Z ( ω, q, φ ) with twist angle θ reads Z = Tr[( I − R † R ) D ( I − R R † ) D † ] , q < ω/c, Tr[( R † − R ) D ( R − R † ) D † ] e − | β | d , q > ω/c, (2)where β = (cid:112) ( ω/c ) − q is the out-of-plane wave vector in vacuum and I the identity matrix.The Fabry-Perot-like denominator matrix is written as D = ( I − R R e iβ d ) − . In our setup, thereflection coefficient matrix R a with a = 1 , is written as R a = r ass r asp r aps r app (3)where superscripts s and p denote the polarization states. The reflection coefficients can be cal-culated by the transfer matrix methods and the details are given in the Supplemental Material. For later convenience, we also define the integrated energy transmission coefficient, i.e. Z ( ω, φ ) ,which is the energy transmission coefficient after an integration over the wave vector q .By applying a magnetic field along the y -direction in the bottom FMI, the permeability tensorhas the form µ = µ xx µ xy µ xz µ yz µ yy µ yz µ zx µ zy µ zz = µ r − iµ i iµ i µ r , (4)4he diagonal and off-diagonal terms are, respectively, expressed as µ r = 1 + ω m ( ω + iαω )( ω + iαω ) − ω and µ i = ω m ω ( ω + iαω ) − ω with frequency ω and magnetic precession damping constant α . The magneticresonance frequencies ω = µ γh and ω m = µ γm s are due to the external magnetic field h andthe saturation magnetization m s with the gyromagnetic ratio γ . The relative permittivity of theFMI is assumed to be a constant. For the top FMI, the relative permeability tensor is expressed as µ (cid:48) = R ( θ ) µ R T ( θ ) with the rotation matrix R ( θ ) along z axis. During the numerical calculation,we adopt the parameters of yttrium iron garnet (YIG) with the relative permittivity (cid:15) = 14 . , gy-romagnetic ratio γ/ π = 28 GHz/T and saturation magnetization µ m s = 0 . T. The appliedmagnetic field µ h is taken as . T. Such set of parameters results in SMPs at microwave fre-quency range so that we consider the radiative heat transfer at the cryogenic environment (around K).
Nonreciprocal surface magnon-polaritons.–
At a single vacuum-FMI interface, there existsSMPs of which the dispersion is nonreciprocal. The implicit dispersion relation for SMPs is β + ( µ r β − iµ i q ) / ( µ r − µ i ) = 0 . (5)where β = (cid:112) (cid:15)µ eff ( ω/c ) − q is the out-of-plane wave vector inside the FMI and µ eff = ( µ r − µ i ) /µ r . Figure. 1(b) indicates the nonreciprocal dispersion of SMPs (gray line) outside the lightcone (red dash-dotted line), together with the symmetric dispersion of bulk magnon-polations, thatis, q = √ (cid:15)µ eff ω/c (cyan dashed line). We highlight that SMPs exist at the band gap region of FMIand the high- q SMPs only exist at positive wave vector region, which is useful in manipulatingnear-field heat transfer.For the case of two FMIs with millimeter separation, SMPs from two interfaces can be coupled.Figure. 1(c) shows the energy transmission coefficient between two FMIs at zero azimuthal andtwist angle, that is, Z ( ω, q, φ = 0; θ = 0) . We can observe that there exists an asymmetric trans-mission coefficient both for bulk MPs (the region outside the light cone and inside the dispersionrelation of MPs) and SMPs (the near-unity line inside the band gap), with respect to the in-plane5ave vector. The two near-unity lines for SMPs are consistent with the implicit dispersion relationof SMPs as follows β + tanh( | β | d/ β µ r − iqµ i ) / ( µ r − µ i ) = 0 , (6) β + coth( | β | d/ β µ r − iqµ i ) / ( µ r − µ i ) = 0 . (7)In the absence of the contributions from µ i , Eqs. (6) and (7) can be reduced to dispersion relationssimilar to those of surface phonon-polaritons.In addition, the optical properties of FMI are anisotropic in the x - z plane when there is nonzeroazimuthal angle. To qualitatively analyze the anisotropic effects, we show the energy trans-mission coefficient with a nonzero azimuthal angle in Fig. 1(d), where the near-unity lines be-tween frequency ω u and ω d expand as a near-unity spot. Here, ω u and ω d are the µ -near-zerofrequencies with azimuthal angles φ = 0 and φ = 0 . π , respectively, and are determined by µ r ( ω u/d ) cos φ + sin φ = 0 . In the region between ω u and ω d , the diagonal terms of permeabilitytensor in x - z plane have the opposite sign, that is, µ xx > , µ yy > and µ zz < . It is similar totype-I hyperbolic metamaterial without considering the off-diagonal term in the permeabilitytensor. Comparing with that of φ = 0 condition, i.e., µ xx < , µ yy > and µ zz < , the twist-induced hyperbolic SMPs emerge at x - z plane when φ = 0 . π . Fig. 1(d) proves the existence ofsuch hyperbolic SMPs and also shows that it can provide more channels for radiative heat transfer.So this azimuthal-angle dependent hyperbolic mode can contribute to a enhancement of radiativeheat transfer. The coexistence of nonreciprocal and anisotropic effects in FMI is helpful for twistedand magnetic thermal management. Twist-induced Near-field Thermal Switch.–
To study the twist-induced thermal switch mediatedby the nonreciprocal SMPs, the thermal switch ratio R κ ( θ ) is defined as R κ ( θ ) = κ ( θ ) /κ min (8)where κ min is the minimal heat transfer coefficient by changing the twist angle θ .6 =0.01 α =0.001 α =0.1 θ =0 θ =0.5π θ =1.5π θ =π θ =π (a) (c) (b)(d) ( , ) × × (cm −1 ) 𝜙 -𝜋 𝜋 Figure 2: (a) Twist-induced near-field thermal switch ratio as a function of twist angle with differ-ent damping constants α . (b) The contour for integrated energy transfer coefficient in ω - φ spaceat single vacuum-FMI interface with different twist angles. (c)-(d) The spectral function of heattransfer coefficient with different damping constants and twist angles.7ig. 2(a) shows the switch ratio with different damping constants α . It can be seen that theswitch ratio is maximal at the parallel configuration ( θ = 0 ). At large damping conditions, thegreen-dotted line in Fig. 2(a) indicates that the switch ratio reaches about . The physical mecha-nism of such a large switch ratio can be related to the match or mismatch of the integrated energytransmission coefficient in the ω - φ space. As shown in Fig. 2(b), the overlap region of the in-tegrated energy transmission coefficient reaches maximal value in parallel configuration. Withincreasing or decreasing the twist angle θ , the central region of the integrated energy transmissioncoefficient at the twisted FMI will shift left or right in ω - φ space and the overlap between two FMIsreaches the minimum value in anti-parallel configuration. These twist-induced mismatch effectsresult in a large thermal switch ratio.Under a small damping, the switch ratio is nonmonotonic with respect to the twist angle, asindicated by the red solid line in Fig. 2(a). Such angle-dependent behavior is similar to the ther-mal magnetoresistance between two magneto-optical plasmonic particles at a large applied mag-netic field. To explore this different angle dependence at small damping condition, we show thespectral function κ ω by varying the twist angle in Figs. 2(c) and 2(d). The twist angle stronglymodulates the height and the width of the spectral function peaks at < θ < π/ . However,when π/ < θ < π , the high-frequency peak in spectral function almost disappear and the widthof the low-frequency peak becomes broader with θ increasing. Such results qualitatively indicatethat there are several nonreciprocal SMPs taking part in the heat transfer with different angle de-pendence. The isofrequency contour for energy transmission coefficient at q x - q y space in Fig. 3(a)numerically verify that statement and we show three kinds of SMPs: elliptic SMPs, hyperbolicSMPs, and twist-non-resonant SMPs. The different twist-induced tunneling and competitionbetween those modes lead to above nonmonotonic twist manipulation for heat transfer.Figure 3(a) shows the different twist-induced energy transmission coefficient of the above men-tioned SMPs in q x - q y space. Due to the nonreciprocal properties of SMPs, the tunneling of threekinds of SMPs only occur at a positive q x region, except in antiparallel configuration. On theone hand, the vertical slice contours in Fig. 3(a) indicate that there is a transition between hy-8 (GHz) θ π 𝑞 𝑥 𝑞 𝑦 𝑞 𝑥 𝑞 𝑦 ① ② ③
16 GHz13 GHz ④ θ = 0 θ = π/4 θ = π/2 θ = 3π/4 θ = π (b) ⑤ θ = 0 θ = π/4 θ = π/2 θ = 3π/4 θ = π α =0.01 α = 0.001 θ =0 (a) α = 0.01 ,( , ) x y q q ( , ) - - - - - 𝜙 𝜙 𝜙 𝜙 𝜙 × (cm −1 ) × (cm −1 ) Figure 3: (a) Twist-induced energy transmission coefficient with different frequency in q x - q y space.Left-vertical slice figures are the energy transmission coefficient with zero twist angle with fre-quency increasing. Right-transverse slice figures are the energy transmission coefficient at fixedfrequency with the twist angle increasing. (b) Integrated energy transmission coefficient in ω - φ space with different twist angles and damping constants.9erbolic SMPs and elliptic SMPs with an increase of frequency. We highlight that the isofre-quency contours of the energy transmission coefficient can be almost flat at ω ≈ GHz andresult in a sharp peak in the spectral function (Figs. 2(c) and 2(d)). In that scenario, such flat-tening transition behavior allows the SMPs bands of each individual FMI hybridize and stronglycoupled to each other with large wavenumbers and involves a dramatic increase of the local den-sity of states for near-field radiative heat transfer. On the other hand, Fig. 3(a) also indicates thatthe elliptic SMPs and hyperbolic SMPs propagate at the open-angle ( − φ m < φ < φ m ), where φ m = arctan (cid:112) / [ µ i ( ω ) − µ r ( ω )] . But the twist-non-resonant SMPs emerge when ω < GHzand is not bounded by the open-angle φ m because it originates in the twist-induced anisotropic in x − z plane. The horizontal slice figures in Fig. 3(a) demonstrate the twist-induced effects of threekinds of SMPs: monotonically decreasing for elliptic SMPs and hyperbolic SMPs and nonmono-tonic dependence for twist-non-resonant SMPs at < θ < π . The competition mechanism amongthree kinds of modes can be understood from the integrated energy transmission coefficient in ω - φ space with different damping constants (Fig. 3(b)). When α = 0 . , elliptic SMPs, and hyper-bolic SMPs play an equal role for radiative heat transfer comparing with twist-non-resonant SMPs,which leads to an almost monotonically decreased thermal switch ratio. In the small damping con-dition, i.e., α = 0 . , the twist-non-resonant SMPs will play the dominant role for radiative heattransfer, which is induced by the optical gyrotropy and leads to a θ anisotropy in the radiative heattransfer.Besides, we find an optimal damping constant for maximizing the heat transfer coefficient inFig. 4(a): the magnitude of heat current can be enhanced almost one order in ultra-small damp-ing condition comparing with the isotropic case and the heat flux is monotonically decreased atantiparallel configuration ( θ = π ). Based on fluctuation electrodynamics, heat flux between twosemi-infinite systems is proportional to the imaginary part of the permeability and the magnitudeof heat current could be reduced to zero when the damping constant approach zero or a large value.But the heat transfer coefficient between two FMIs reaches a fixed value in zero damping constantlimit. We demonstrate that twist-non-resonant SMPs play the dominant role in ultra-small damp-10 ① ②③ ④ (a) (b) ( , , ) q Figure 4: (a) Heat transfer coefficient as a function of damping constant α with different twistangle. Gray-solid line is the heat transfer coefficient between two isotropic slab i.e. µ xx = µ yy = µ zz . (b) Energy transmission coefficient in ω - q space with different azimuthal angle φ . ( 1 (cid:13) - 5 (cid:13) )means that the azimuthal angles φ are from . π to . π with step . π , respectively. The twistangle θ is zero and the damping constant α is 0.001.ing conditions and the mechanism is slightly different from near-field radiative heat transfer inmultilayer structure due to multiple surface-states coupling. The intrinsic relation between twist-non-resonant SMPs and the heat transfer coefficient is demonstrated at Fig. 4(b): the near-unityregion in energy transmission coefficient contour with different azimuthal angle can fill in the gi-ant area in ω − q space and the local density of states for twist-non-resonant SMPs can be boostedwithout the constraint of ultra-small damping condition. It also demonstrates that the local densityof states for elliptic SMPs and hyperbolic SMPs (the thin-solid line in Fig. 4(b)) is not be enhancedand plays little contribution for heat transfer in a ultra-small damping condition. As a whole, thedifferent α and θ dependence of elliptic SMPs, hyperbolic SMPs, and twist-non-resonant SMPsresult in above twist-induced manipulation for near-field radiative heat transfer.To conclude, we have studied twist-induced near-field radiative heat transfer between two FMIsthrough nonreciprocal SMPs. We find a large and nonmonotonic twist-induced near-field thermalswitch ratio. In addition, the near-field radiative heat transfer can be enhanced by the contribu-11ions from the twist-non-resonant SMPs under ultra-small damping condition. Our results provideinsights for active near-field heat transfer control by engineered twists. Acknowledgement
J.-P., L.-W., H.-C., and J.-R. are supported by the National Key Research Program of China (GrantNo. 2016YFA0301101), National Natural Science Foundation of China (No. 11935010, No.11775159 and No. 61621001), the Shanghai Science and Technology Committee (Grants No.18ZR1442800 and No. 18JC1410900), and the Opening Project of Shanghai Key Laboratory ofSpecial Artificial Microstructure Materials and Technology. G.-T. thanks the financial support fromthe Swiss National Science Foundation (SNSF) and the NCCR Quantum Science and Technology.Rair Macedo acknowledges support from the Leverhulme Trust and the University of Glasgowthrough LKAS funds.
Supporting Information Available
In the Supplemental Material, we derive the dispersion relation and the reflection coefficients ofthe surface magnon polariton.
Dispersion relation of surface magnon polariton
To find the dispersion relation of surface magnon polaritons (SMP) at a single vacuum-FMI inter-face, we employ Maxwell equations, ∇ × E = − ∂ t B , (9) ∇ × H = ∂ t D , (10)with B = µ µ H and D = (cid:15) (cid:15) E . The SMP is transverse electric (TE or s ) polarized and thetransverse magnetic (TM or p -polarized) mode does not exist for the case where only single FMI12lab is considered. By applying an in-plane magnetic field along the y -direction, the electric fieldsin vacuum E and in FMI E propagate along the x -direction and decay along the z -direction withthe expressions E ( x, z, t ) = ˆ yEe iqx − iβ z e − iωt , Im( β ) < , (11) E ( x, z, t ) = ˆ yEe iqx + iβ z e − iωt , Im( β ) < , (12)where q is the in-plane wave vector along the x -direction. The out-of-plane wave vector in vacuumand FMI are denoted as β and β , respectively. The corresponding magnetic fields are expressedas B / ( x, z, t ) = iω (ˆ x∂ z − ˆ z∂ x )( E / · ˆ y ) . (13)From H = ( µ µ ) − B , the magnetic field strengths are H = iωµ [ˆ x∂ z − ˆ z∂ x ]( E · ˆ y ) , (14) H = iωµ ( µ r − µ i ) [ˆ x ( − iµ i ∂ x + µ r ∂ z ) + ˆ z ( − µ r ∂ x − iµ i ∂ z )]( E · ˆ y ) . (15)Using Eq. (10) in both vacuum and FMI, one has β + q = k , (16) β + q = (cid:15)µ eff k , (17)with k = ω/c and µ eff = ( µ r − µ i ) /µ r . Using the interface conditions for the magnetic fieldstrengths, H · ˆ x = H · ˆ x , the implicit dispersion relation for the SMPs can be obtained with β + ( µ r β − iµ i q ) / ( µ r − µ i ) = 0 . (18)From Eqs. (16), (17) and (18), the dispersion relation of SMP can be numerically obtained.13 eflection coefficients In this section, we obtain the reflection coefficients by taking the anisotropic effect into account.For the case where incidence plane is at an angle φ with respect to the x -axis, the effective perme-ability tensor is µ (cid:48) = R µ R T = µ xx µ xy µ xz µ yx µ yy µ yz µ zx µ zy µ zz , (19)where R is the rotation matrix with R = cos φ sin φ − sin φ cos φ
00 0 1 . (20)Although the SMP is s -polarized by considering a single FMI slab, the p -polarized mode existsbetween two FMI slabs as well due to the anisotropic permeability tensor. We focus on the interfacebetween the lower FMI slab and vacuum. The genernal form of the electric and magnetic fieldsinside the FMI slab can be written as E = ( E x , E y , E z ) e − iωt + iqx , (21) H = ( H x , H y , H z ) e − iωt + iqx , (22)where the superscript (cid:48) in the space variables x (cid:48) , y (cid:48) and z (cid:48) is dropped for simplicity. From theMaxwell equations, Eqs. (9) and (10), we can get the differential equation ddz E x E y α H x α H y = iK E x E y α H x α H y (23)14ith α = (cid:112) µ /(cid:15) and K = qµ yz /µ zz k ( µ yx − µ yz µ zx /µ zz ) k ( µ yy − µ yz µ zy /µ zz ) − q / ( k (cid:15) )0 − qµ xz /µ zz k ( − µ xx + µ xz µ zx /µ zz ) k ( − µ xy + µ xz µ zy /µ zz )0 − k (cid:15) + q / ( k µ zz ) − qµ zx /µ zz − qµ zy /µ zz k (cid:15) . (24)By solving this differential equation, we get [ E x ( z ) , E y ( z ) , α H y ( z ) , α H y ( z )] = (cid:88) m =1 c m [ u ,m , u ,m , u ,m , u ,m ] e ik m z , (25)where k m and u i,m are, respectively, the eigenvalue and eigenvector of matrix K . Since K is afour-by-four matrix, we have four eigenvalues: two of them satisfy Im( k m ) < and the other two Im( k m ) > . We take k m with Im( k m ) < , of which the subscripts are denoted as m = 1 , , toensure that the electromagnetic fields vanish at z →−∞ .In the vacuum, the incoming electric and magnetic fields can be, respectively, writen as E in = [ e s in ˆ y + e p in ( β ˆ x − q ˆ z ) /k ] e iωt − iqx − iβ z , (26) α H in = [ e p in ˆ y − e s in ( β ˆ x − q ˆ z ) /k ] e iωt − iqx − iβ z , (27)where the superscripts s and p are used to denote the polarizations. The reflected fields are ex-pressed as E re = [ e s re ˆ y − e p re ( β ˆ x + q ˆ z ) /k ] e iωt − iqx + iβ z , (28) α H re = [ e p re ˆ y + e s re ( β ˆ x + q ˆ z ) /k ] e iωt − iqx + iβ z . (29)At the interface of the vacuum side with z = 0 + , the in-plane electric and magnetic fields can be15ritten as E (cid:107) =[( e s in + e s re )ˆ y + ( e p in − e p re ) β /k ˆ x ] e iωt − iqx , (30) α H (cid:107) =[( e p in + e p re )ˆ y − ( e s in − e s re ) β /k ˆ x ] e iωt − iqx . (31)For the case of the s -polarized incoming field, that is, e p in = 0 , the interface conditions give − e p re β /k = E x ( z = 0) , (32) e s in + e s re = E y ( z = 0) , (33) − ( e s in − e s re ) β /k = α H x ( z = 0) , (34) e p re = α H y ( z = 0) . (35)From Eqs. (32) and (35), we have c /c = − ( u , k + u , β ) / ( u , k + u , β ) . (36)The reflection coefficient r ss = e s re /e s in can be obtained from Eqs. (33) and (34) as r ss = ( u , β + u , k ) + ( u , β + u , k ) c /c ( u , β − u , k ) + ( u , β − u , k ) c /c . (37)From Eqs. (32) and (34), we can obtain r ps = e p re /e s in as r ps = (1 − r ss ) u , + u , c /c u , + u , c /c . (38)16or the case of the p -polarized incoming field, that is, e s in = 0 , the interface conditions give ( e p in − e p re ) β /k = E x ( z = 0) , (39) e s re = E y ( z = 0) , (40) e s re β /k = α H x ( z = 0) , (41) e p in + e p re = α H y ( z = 0) . (42)From Eqs. (40) and (41), we have d /d ≡ c /c = − ( u , β − u , k ) / ( u , β − u , k ) . (43)Notice c /c here is different from that in Eq. (36) and we denote it as d /d instead. The reflectioncoefficient r pp = e p re /e p in obtained from Eqs. (40) and (41) is expressed as r pp = ( u , β − u , k ) + ( u , β − u , k ) d /d ( u , β + u , k ) + ( u , β + u , k ) d /d . (44)From Eqs. (39) and (41), we obtain r sp = e s re /e p in as r sp = (1 − r pp ) u , + u , d /d u , + u , d /d . (45) Near-field radiative heat transfer
From the fluctuating electrodynamics, the radiative heat current with vacuum gap d is given by J = (cid:90) ∞ dω π ¯ hω ( N − N ) (cid:90) ∞ dq π q (cid:90) π dφ π Z ( ω, q, φ ) , (46)17here N i = 1 / [ e ¯ hω/ ( k B T i ) − with i = 1 , is the Bose-Einstein distribution function. The photonictransmission coefficient Z ( ω, q, φ ) reads Z = Tr[( I − R ∗ R ) D ( I − R ∗ R ) D ∗ ] , q < ω/c Tr[( R ∗ − R ) D ( R ∗ − R ) D ∗ ] e − | β | d , q > ω/c (47)where q and β = (cid:112) ( ω/c ) − q are the in-plane and out-of-plane wave vectors, respectively. Theidentity matrix is denoted as I . The reflection coefficient matrix for the interface between vacuumand the FMI i is R i = r iss r isp r ips r ipp . (48)The Fabry-Perot-like matrix reads D = ( I − R R e iβ d ) − .The heat transfer coefficient, which is defined as κ ( T ) ≡ lim ∆ T → J/ ∆ T , is expressed as κ ( T ) = (cid:90) ∞ dω π ¯ hωN (cid:48) (cid:90) ∞ dq π q (cid:90) π dφ π Z ( ω, q, φ ) , (49)where the derivative of the Bose-Einstein distribution with respect to the temperature is expressedas N (cid:48) ≡ ∂N/∂T = ¯ hω e ¯ hω/ ( k B T ) k B T [ e ¯ hω/ ( k B T ) − . (50)For the case of φ = 0 , the reflection coefficient for the s -polarized mode can be expressed as r ss = β − ( β µ r − iqµ i ) / ( µ r − µ i ) β + ( β µ r − iqµ i ) / ( µ r − µ i ) , (51)and the other reflection coefficients vanish. The photonic transmission coefficient can be expressedas, Z ( ω, q, φ = 0) = 4[Im( r ss )] e − | β | d | − r ss e − | β | d | , (52)18rom which one can obtain the nonreciprocal dispersion of the symmetric and asymmetric SMPmodes between two FMI slabs. Permeability tensor components of uniaxial FMI (a) (b) 𝜔 𝑑 𝜔 𝑢 Figure 5: (a-b) The real part of diagonal term of permeability tensor with different azimuthal angle.The damping factor is α =0.01.In our calculation, we use the parameters of YIG and rewrite the permeability tensor as below: µ = µ r − iµ i iµ i µ r (53)After a φ rotation in x − y plane, the rotated permeability tensor regards as: µ ( φ ) = µ r cos( φ ) + sin( φ ) cos( φ ) sin( φ )( µ r − − iµ i cos( φ )cos( φ ) sin( φ )( µ r −
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