Ultra-Small Coherent Thermal Conductance Using Multi-Layer Photonic Crystal

Transcription

1 Ultra-Small Coherent Thermal Conductance Uing Multi-Layer Photonic Crytal W. T. Lau*, J. -T. Shen, G. Veroni and S. Fan Edward L. Ginzton Laboratory, Stanford Univerity, Stanford, CA 94305, USA ABSTRACT A multi-layer photonic crytal can be ued to uppre coherent thermal conductance below the uum conductance value, over the entire high-temperature range. With interlacing layer of ilicon and uum, heat can only be carried by photon. The thermal conductance of the crytal would then be determined by the photonic band tructure. Partial photonic band gap that preent over mot of the thermal pectrum, a well a the uppreion of evanecent coupling of photon acro the uum layer at high frequencie, would reduce the amount heat conducting photon channel below that of the uum. Thu uch multi-layer tructure can be very efficient thermal inulator. Beide, the thermal conductance of uch tructure can exhibit ubtantial tunability, by merely changing the ize of the uum pacing. Keyword: Photonic crytal, coherent thermal tranport, thermal conductance. INTRODUCTION Thermal tranport i traditionally conidered a a diffuive proce where heat i conducted non-coherently. In thi diffuive regime, there i a finite range of the magnitude of thermal conduction, which i limited by intrinic material propertie.[] With the advancement of nano-technology, the ize of the device can be made to be comparable to the coherence length of thermal carrier, and the coherent regime of thermal tranport ha now become realizable.[-0] A it ha been peculated, when operated in the coherent regime, we can ue variou nano-tructural configuration to extend the achievable range of thermal conduction, and thu better thermal conductor or inulator can be made.[] Mot of the wor on coherent thermal tranport focu on phononic thermal conduction.[-7] Due to trong phononphonon interaction, the coherence length of phonon i very mall at room temperature. Hence thee wor need to be performed at very low temperature. On the other hand, the interaction of photon are much weaer, and thu coherent thermal tranport at high temperature hould be more readily realizable uing photon a heat carrier. In thi wor, we conider a multi-layer photonic crytal, with periodic interlacing layer of uum and a dielectric material, a hown in Fig.. The uum layer enure that thermal tranport i totally due to photon. With the dielectric material choen to be nearly lole, photon tranport would remain coherent over many crytal period. In thi coherent regime, heat conduction would be decribed by wave. Wave phenomena that could never occur in the diffuive regime, now become crucial in determining the coherent photon tranport acro uch tructure. A it turn out, there are two counteracting wave phenomena: the evanecent tunneling of photon acro the uum layer, which tend to enhance thermal conduction; and the Bragg cattering that give rie to the photonic band gap, which tend to uppre conduction. Thee two phenomena occur at different part of the photonic bandtructure, and thu dominate at different temperature range.[0] Our main reult i to obtain the thermal conducting behavior of thee multi-layer tructure a a function of the operating temperature T, and from there, exploit the poibilitie to tune the conducting characteritic by changing the device geometrie. * wlau@tanford.edu Photonic and Phononic Crytal Material and Device IX, edited by Ali Adibi, Shawn-Yu Lin, Axel Scherer, Proc. of SPIE Vol. 73, SPIE CCC code: X/09/$8 doi: 0.7/ Proc. of SPIE Vol

2 dd a=d+d y V FIG.. Schematic of the device. Layer of lole ilicon lab with refractive index n =, thicne d and infinite cro-ectional area, are periodically placed with pacing d in uum to form a multi-layer photonic crytal with periodicity a along the z-direction.. THEORY The quantity that decribed photonic thermal conduction at the coherent regime i the thermal conductance G. Unlie diffuive thermal tranport, G would not cale with the device length, ince now heat i tranported ballitically a wave, and thu the magnitude of conduction hould be independent of the propagation ditance. Mathematically, aume that heat i conducted in the z-direction, the thermal conductance per unit area GT ( ) i defined a follow:[0] d ω ( ) GT ( ) = h ( ) ( ). () j z 0 3 j ω 3 j h j( )/( BT) ( π ) z T e ω Here ω j ( ) i the photon diperion relation of the multi-layer tructure, with ω being the photon frequency, being the photon wavevector, and j being the band index which include both polarization. z i the wavevector component at the z-direction. Thi equation i a generalization to the one-dimenional expreion in Ref. []. To implify Eq. (), we note that for any function f ( ω ) that i iotropic in -pace, the following relation hold: [0] j ω ( ) 3 d j dω f( ω ) ( ) 0 3 j = f ω 0 contant z ( π ) z π ω m ( π ) d. () Here m label all poible olution of z 0 including both polarization, for a given pair of ( ω, ). i the component wavevector on the x-y plane. Note that the band indice m and j are generally not the ame. Now Eq. () can be further implified to yield: hω /( BT) dω [ hω /( T B )] e GT ( ) = B A( ω), (3) 0 3 hω /( BT) ( π ) [ e ] with A( ) ω = d contant ω m. (4) At a contant frequency, ω0 = ω( ) decribe the contant frequency urface in -pace. A( ω = ω0 ) i the magnitude of the total projected area of uch contant frequency urface onto the x - y plane, and i referred to a the projected Proc. of SPIE Vol

3 area in thi paper.[0] Thi quantity A( ω ) encapulate all information of the photonic band tructure in Eq. (3). Setting A( ω) = 4π, Eq. (3) recover the univeral quantized thermal conductance GT ( ) = π B T/(6 h ) for any type of heat carrier operated in the ingle-mode regime at any homogeneou one-dimenional tructure.[-4] For threedimenional tructure, photon can propagate in all direction and the detail of the photonic band tructure would affect the overall thermal conductance. In order to highlight the effect of the band tructure, we would compare the conductance value GT ( ) with that of the uum, where A ( ω) = π( ω/ c) and G ( T) = π B T /(5 h c ). Note that both polarization are included in the expreion of A ( ω ) and G ( T ). 3. RESULT AND DISCUSSION We would now ue Eq. (-4) to obtain the thermal conduction behavior of our multi-layer tructure in Fig.. We denote the index of the dielectric to be n, it width to be d, the uum width to be d and the lattice contant to be a = d + d. Due to ymmetry of our tructure, for each value of ( ω, ), there are two independent polarization mode, the -polarized mode, with the electric field along the direction z / ; and the p-polarized mode, with the magnetic field along the direction of z /. Here i conerved in all layer and =. The diperion relation for both polarization can be reolved by [] γ P ξγ co( λa) = co( γd)co( γd) ( )in( γd)in( γd) P γ + γ, (5) ξ where γ j=, = ( njω/ c), P = and P = n for the - and p- polarized field repectively. A real olution for λ repreent a Bloch tate of propagating photon mode and thoe value of ( ω, ) ξ =, p that give λ R contitute the photonic band. The collection of thee value of ( ω, ) ξ =, p give rie to a projected band diagram.[] In Fig. a, we how the reult for the cae when d = d, with n =, which approximate the refractive index of ilicon at infrared frequencie and room temperature. Proc. of SPIE Vol

4 (b)4 3 oic < M <nco/c ---O<<o/c 0 << nco!c - - -polarization ppolarization total w (ic/a).5 FIG.. (a) The projected band diagram for the -polarized mode for the tructure hown in Fig., with d = 0.5a. The dotted line are the uum = ω / c and the material = nω / c light-line. Thee light line divide up the photonic band into two region: 0 < < ω / c (brown), and ω / c< < nω / c (green). The inet how the projection of the contant frequency urface on the x-y plane for ω = 0.6( πc/ a). (b) The olid (blue) line how A ( ω)/ A( ω ) for -polarization. The dahed (brown) and dotted (green) line repreent the contribution in the range 0 < < ω / c, and ω / c< < nω / c repectively. (c) The olid (blac), dahed (blue), and dotted (orange) line are projected area of the total A( ω ), -polarized mode A ( ω ), and p-polarized mode A ( ω ) repectively. All projected area are normalized to A ( ω ). p In the projected band diagram, the Bloch tate exit only in the region nω / c, i.e. only in the region below the dielectric light line. The Bloch tate can be ubdivided by the uum light line. Thoe in the region 0 ω / c are extended tate in both the dielectric and uum layer. Thoe tate in the region ω / c nω / c are extended only in the dielectric layer but evanecent in uum. Proc. of SPIE Vol

5 From the projected band diagram, we can extract the projected area A( ω ) by umming over all the available thermal channel that conduct heat in the z-direction, with photon tate at all propagating angle conidered. The reult for - polarized mode i hown in Fig. (b). Due to cylindrical ymmetry of our tructure, m () m, ( ω) A ( ω) = π d, where () m, ( ω) () m, ( ω ) and () m, ( ω ) are the minimum and maximum value of of the m th -polarized band at ω. To facilitate comparion, for each frequency, we normalize the reult with A ( ω ) in the plot. Since A ( ) ω contain contribution from both polarization, the reference uum level for -polarized field i A ( ω)/ A ( ω ) = /. Now at low frequency our tructure behave a a uniform media with an effective dielectric contant. Thi i the pectral region where a large portion of photon mode would evanecently couple acro the uum layer, in addition to the mode that propagate through the uum. Note that at the limit ω 0, A ( ω)/ A ( ω) ( n d + n d )/( a) > /. On the other hand, A ( ω)/ A ( ω ) < / above certain tranition frequency. At thi pectral range, evanecent coupling would vanih and all photon tate would conit only of extended tate in uum. In addition, now partial photonic band would eep emerging over the entire high frequency range and thi would reduce the available photon tate below that of the bare uum. The tranition from the low- to high-frequency feature typically occur at the frequencie where the evanecent decay length of the mode are comparable to the ize of the uum layer, and it would roughly occur at the firt partial band gap for mot of the device geometrie. A hown in Fig. c, the ame tranition behavior would occur for the p- polarized mode a well, and thu would appear in the total projected area A( ω) = A ( ω) + A ( ω). p Now we can obtain the thermal conducting behavior GT ( ) of our tructure, by ubtituting A( ω ) into Eq. (3). The reult, after normalized with the value of G ( T ) at each temperature, i hown in Fig. 3. The curve of GT ( )/ G ( T) a a function of T generally follow the hape of A( ω)/ A ( ω ) in ω, with fluctuation moothened by averaging over the energy ditribution of photon. The total normalized thermal conductance of the tructure GT ( )/ G ( T ) i much larger than at low temperature, and decreae to below at higher temperature. Thi tranition can be undertood from the pattern of A( ω ). At low temperature, mot photon would populate a mall bandwidth of frequency cloed to ω = 0, where A( ω)/ A ( ω ) >, thu GT ( )/ G ( T ) >. At high temperature, photon would pread over to a wide range of frequency, where A( ω)/ A ( ω ) <. A a reult, GT ( )/ G ( T ) <. Thi high-temperature reult i epecially ignificant. It how that photonic crytal can be thermally more inulating than bare uum. From Fig. 3, at ufficiently high temperature, the total thermal conductance of the device would jut be 0.54 G ( T ). Proc. of SPIE Vol

6 4 3\ polarization ppolarization - total T [hc/(ba)] FIG. 3. Normalized photonic thermal conductance GT ( )/ G( T ) for the tructure hown in Fig. with d = 0.5a. The contribution of the -polarized mode, drawn in dahed line (blue), and the p-polarized mode, drawn in dotted line (orange) are ummed to give the total conductance, drawn in olid line (blac). The inet how a wider range of temperature. Since the photonic band tructure i controlled by the lattice configuration of the crytal, by varying the uum thicne, we can dratically tune the thermal conducting behavior uing the ame et of dielectric lab. In Fig. 4, we plot GT ( )/ G ( T ) for crytal with d = 9d and d = 0.d, in addition to cae d = d that we have obtained earlier. At any fixed temperature, the normalized thermal conductance generally decreae a the uum pacing d increae, for the range of d that we conidered. In particular, we can chooe a temperature Ta = 0.0( hc/ Bd) where thi change in conductance i mot abrupt, and plot [ Gd ( )] T= T/[ G ] a T= T a we continuouly vary d a. The reult, a hown at the inet of Fig. 4, how that the conductance can be tuned over an order of magnitude by merely adjuting the uum pacing. Another obervation from Fig. 4 i that, at ufficiently high temperature, the normalized thermal conductance would all aymptotically converge to the value of 0.54 G and tay at a contant over the entire high-temperature range, regardle of the device geometry. Thi limiting value of conductance i in fact equal to the average value of A( ω)/ A ( ω ) at high frequencie, and i thu related to the relative ize of all the partial photonic band gap over the entire photon pectrum. The exact geometry of the tructure, neverthele, would determine the normalized temperature where uch convergence occur. The larger the uum pacing, the lower would be the normalized temperature for thi limiting value to be reached. A a reult, by operating at different normalized temperature, we can freely elect the operating mode of our device: either to act a a thermal channel with highly tunable conductance, or to act a a thermal inulator with a contant conductance that i reilient to geometric variation. Proc. of SPIE Vol

7 8 6 8 \.S A -- 'I - d=9d ---d=0.d C-) 0 4 d/d 8 o T{hc/(8d)] Fig. 4 Normalized photonic thermal conductance GT ( )/ G( T ) for three tructure with contant ilicon lab thicne, but different lab eparation. The temperature Ta = 0.0( hc/ Bd), where there i pronounced change in G/ G a d varie, i indicated by the vertical dahed line. Tunability of the G/ G generally decreae a temperature increae beyond T a. The inet how thi change of G/ G at Ta = 0.0( hc/ Bd) a we continuouly vary d. Latly, we would lie to comment on the practical realization of uch device. We note that the range of integration in Eq. hω/( BT) hω/( BT) (3) i et by the function B[ h ω /( BT)] e /[ e ], which approximately vanihe when ω > 0 T B / h. Thu at a temperature T, the material choen need to be tranparent for the range 0 ω 0 T B / h in order to demontrate 6 coherent thermal tranport. At room temperature of 300K, thi range would correpond to 0 ω.30 0 ( π c). We note that intrinic crytalline ilicon i indeed tranparent at thi range. In Fig. 5, we plot the pectral variation of the refractive index n and extinction coefficient κ of intrinic crytalline ilicon, uing data provided in Ref. []. Within the thermal pectrum of 300K, the minimum attenuation length ( λ /( πκ ) ) would till be of order of millimeter, which would be many crytal period for the length cale we conidered. For intance, the temperature Ta = 0.0( hc/ Bd) where the device conductance exhibit a wide range of tunability, would correpond to the abolute temperature of 88K if the ilicon lab are choen to be μ m. Thi i a length cale that i readily realizable uing tandard fabrication technologie. Thu thee coherent thermal-conducting phenomena would readily be oberved at room temperature with realitic device length cale. Proc. of SPIE Vol

8 / = + K /. (m') o 08 Fig. 5 Refractive index n and extinction coefficient κ of intrinic crytalline ilicon a a function of invere wavelength. REFERENCES [] W. Kim, R. Wang and A. Majumdar, "Nanotructuring expand thermal limit", Nanotoday 40 (007). [] K. Schwab, E. A. Henrien, J. M. Worloc, and M. L. Roue, "Meaurement of the quantum of thermal conductance", Nature (London) 404, 974 (000). [3] L. G. C. Rego and G. Kirczenow, "Quantized Thermal Conductance of Dielectric Quantum Wire", Phy. Rev. Lett. 8, 3 (998). [4] T. Yamamoto, S. Watanabe, and K. Watanabe, "Univeral Feature of Quantized Thermal Conductance of Carbon Nanotube", Phy. Rev. Lett. 9, (004). [5] G. Chen, "Phonon Wave Heat Conduction in Thin Film and Superlattice", J. of Heat Tranfer, 945 (999). [6] M. V. Simin and G. D. Mahan, "Minimum Thermal Conductivity of Superlattice", Phy. Rev. Lett. 84, 97 (000). [7] A. N. Cleland, D. R. Schmidt, and C. S. Yung, "Thermal conductance of nanotructured phononic crytal", Phy. Rev. B 64, 730 (00). [8] M. Meche, W. Guichard, and J. P. Peola, "Single-mode heat conduction by photon", Nature (London) 444, 87 (006). [9] D. R. Schmidt, R. J. Schoelopf, and A. N. Cleland, "Photon-Mediated Thermal Relaxation of Electron in Nanotructure", Phy. Rev. Lett. 93, (004). [0] W. T. Lau, J. - T. Shen, G. Veroni, S. Fan and P. V. Braun, "Tuning coherent radiative thermal conductance in multilayer photonic crytal", Appl. Phy. Lett. 9, 0306 (008). [] A. Yariv and P. Yeh, [Photonic: Optical Electronic in Modern Communication] Oxford Univerity Pre 6th edition, 007. [] E. D. Pali, [Handboo of Optical Contant of Solid] Academic Pre 998. Proc. of SPIE Vol