Thermal Emission in the Near Field from Polar Semiconductors and the Prospects for Energy Conversion R.J. Trew, K.W. Kim, V. Sokolov, and B.D Kong Electrical and Computer Engineering North Carolina State University
Fundamental Questions: Can incoherent thermal energy be converted to coherent radiation? Is it possible to harvest this energy and convert it to useable dc energy?
Thermal Radiation Generally, thermal radiation has been considered ISOTROPIC BROAD BAND INCOHERENT => Similar to the WHITE NOISE due to the uncorrelated process of photon spontaneous emission Planck s Black Body Radiation: I B ( ω T ), = 3 π c 3 ω [ exp( ω k T ) 1] B
Near-Field Spectra of Thermal Emission Thermal sources can show partially coherent spectra in the near-field Evanescent surface wave plays an important role for near-field spectra Evanescent surface wave carries sub-wavelength spatial information since it decays exponentially
Near-Field vs. Far-Field Near-Field Partially coherent: quasimonochromatic Strong intensity Evanescent mode: decays exponentially away from the source Far-Field Incoherent: nearly white noise Weak Intensity Propagating mode What makes the differences in the near-field spectra? => Thermal excitation of surface polaritons leading to a resonant (monochromatic) behavior
Phonon Polariton Polaritons: Quasi-particles resulting from strong coupling of electromagnetic waves with an electric or magnetic dipole-carrying excitation Phonon-Polaritons: Polaritons resulting from the coupling of an infrared photon with a polar phonon Polar solid (dielectric/semiconductor) can cause strong polar phonon-photon coupling
Surface Electromagnetic Waves Surface EM waves: A particular type of waves that exists at the interface between two different media - propagates along the interface - decays exponentially in the perpendicular direction Surface waves due to coupling between the EM field and resonant polarization oscillation in the material => Surface polaritons Surface plasmon polariton: coupling with the surface charge density wave Surface phonon polariton: coupling with the surface infrared optical phonon
Surface EM Waves Vacuum (ε 1, z > 0) z H y E k x Dielectric [ε(ω) =ε' (ω) + i ε" (ω), z < 0] k ω c = εω ( ) : Dispersion relation of surface EM wave Approximated ε(ω) for real ω when εω ( ) + 1 ωlo ω iγ LOω ωlo ω ε ( ω) = ε ε ω iγ Surface EM wave: Only TM (Transverse Magnetic) modes exists in the non-magnetic media Available frequency range for surface phonon polariton wave excitation: ω TO ~ ω [ε(ω) S < - ε 1 for real k ] ω TO TO ω ω TO ω
Dispersion Relation for Surface Waves Range of surface wave frequency and wave numbers: where k c = ω TO /c Approximated limiting frequency: ω ω = ω s ω 0 s < ω <, < k <, TO ω s TO ε 0 ε + 1 1 + 1 when The large density of states at ω s causes quasimonochromatic radiation of near-field thermal emission k c ε ω ω ω ω LO ( ω) ε TO
Calculated Dispersion Relations ω, 10 1 s -1 ω, 10 1 s -1 ω, 10 1 s -1 ω, 10 1 s -1 64 ω 0 s =64.48 6 60 58 InP 55 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 54 ω 0 s =54.48 53 5 51 GaAs 140 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 130 ω 0 s =135.0 10 110 180 0.4 0.5 0.6 0.7 0.8 0.9 1.0 ω 0 s =178.7 170 160 SiC 150 0.5 0.6 0.7 0.8 0.9 1.0 k, 10 4 cm -1 GaN Dispersion relation for surface phonon polaritons: k ω c = εω ( ) εω ( ) + 1 Dielectric function for an isotropic material: ω ω iγ ω TO i TO εω LO LO ( ) = ε ω ω γ ω
Near-Field Spectra Supported by Surface Waves Classical Spectral Energy Density (Plank s Black-body Radiation) 3 ω I B ( ω, T ) = θ ( ω, T ) N( ω) = 3 π c exp ω k T 1 Thermal emission from a semi-infinite slab (z < 0) of homogeneous and isotropic polar solid into vacuum/air (z>0) including the near-field spectra I(, z ω, T) = θω (, T) N(, z ω, T) ω dk N(, z ω, T) = 8 π ε ( ω) dz g (, ω z, z ) c k ' '' ε ( ω) = ε ( ω) + iε ( ω) ' g (, ω z, z ) αβ Mean energy of a quantum oscillator θ ( ω, T ) = ω exp( ω k T ) 1 B : dielectric function [ ( ) ] B Density of oscillation modes per unit volume ω π c ( ω ) = 3 3 0 '' ' ' 4 αβ αβ, = ( π ) xyz,, k : -D Fourier transform of the electromagnetic Green tensor (, rr, ' ω) N G αβ
Fluctuation Dissipation Theorem The total energy is the summation of electric and magnetic energy such as u ω, z = u ω, z + u ω, z, z > 0 ( ) prop ( ) evan ( ) E E E where 1 1 kdk ( ) ( ) ( ω, = ) 0 ω, 1 1 u z u T r r prop s p E 0 γ1 and 1 kdk u z u T r k r k z ( ω, ) = ( ω, ) Im( ) + ( 1) Im( ) exp[ Im( γ ) ] evan s p E 0 1 1 1 1 0 γ 1 ( ) 1 i( k ) 1 k ( ε k ) 1 γ 1 = 1 k γ = γ 1 1 = 0 Wave vector component for z- direction of each region and propagating and evanescent mode r s 1 r p 1 = = ( γ1 γ ) ( γ1+ γ ) ( εγ1 γ) ( εγ + γ ) 1 Fresnel reflection factors for s- and p-polarization
Relevant Material Parameters Material ω LO, 10 1 s -1 ω TO,10 1 s -1 γ LO,10 1 s -1 γ TO,10 1 s -1 ε ε InP 65.11 57.17 0.3581 0.60 9.5 GaAs 54.84 50.44 0.3800 0.3800 10.9 E 1 73.10 7.60 0.5843 0.60 90.80 8.80 0.3581 0.5843 E 3 118.7 107.3 1.100 E 3.077 0.8859 Al O 3 E 4 170.9 119.4.800 0.945 A 1 96.30 74.90 73 0.9990 3.07 A 166.1 109.8.900 0.5655 GaN 140.1 104. 1.508[9] 1.508[9] 5.3 SiC 18.7 149.5 0.900 0.900 6.7
Near-Field Thermal Emission SiC 1.5x10-7 1.0x10-7 z=100µm u(ω,z) (ev s/cm 3 ) 5.0x10-8 6.0x10-7 4.0x10-7.0x10-7 3.0x10-3.0x10-3 z=µm z=100nm 1.0x10-3 0 100 00 300 ω (10 1 s -1 ) Thermal emission spectra of a semi-infinite SiC sample at three different locations from the surface (z = 1000 µm, µm, and 0.1 µm) with T=300K
Near-field Thermal Emission Cubic GaN u(ω,z) (ev s/cm 3 ).0x10-7 1.5x10-7 z=100µm 1.0x10-7 5.0x10-8 8.0x10-7 6.0x10-7 4.0x10-7.0x10-7 8.0x10-3 6.0x10-3 4.0x10-3.0x10-3 0 100 00 300 ω (10 1 s -1 ) z=µm z=100nm Thermal emission spectra of a semi-infinite cubic GaN sample at three different locations from the surface (z = 1000 µm, µm, and 0.1 µm) with T=300K
Near-field Thermal Emission Al O 3.0x10-7 1.5x10-7 z=100µm u(ω,z) (ev s/cm 3 ) 1.0x10-7 5.0x10-8 1.5x10-6 1.0x10-6 5.0x10-7 8.0x10-3 6.0x10-3 4.0x10-3.0x10-3 z=µm z=100nm 0 100 00 300 ω (10 1 s -1 ) Thermal emission spectra of a semi-infinite Al O 3 sample at three different locations from the surface (z = 1000 µm, µm, and 0.1 µm) with T=300K
Near-Field Thermal Emission GaAs 1.5x10-7 z=100µ m 1.0x10-7 5.0x10-8 u(ω,z) (ev s/cm 3 ) 3.0x10-6.0x10-6 1.0x10-6.0x10-1.5x10-1.0x10-5.0x10-3 0 100 00 300 ω (10 1 s -1 ) z=µ m z=100nm Thermal emission spectra of a semi-infinite GaAs sample at three different locations from the surface (z = 1000 µm, µm, and 0.1 µm) with T=300K.
Near-Field Thermal Emission InP u(ω,z) (ev s/cm 3 ) 1.5x10-7 z=100µm 1.0x10-7 5.0x10-8 4.0x10-6.0x10-6 4.0x10-3.0x10 -.0x10-1.0x10-0 100 00 300 ω (10 1 s -1 ) z=µm z=100nm Thermal emission spectra of a semi-infinite InP sample at three different locations from the surface (z = 1000 µm, µm, and 0.1 µm) with T=300K.
Monochromatic Emission vs. Distance I(ω, z = 0.1µm ) 10-4 ev s / cm 3 10 8 6 4 InP GaAs Al O 3 GaN SiC Z = 0.1μm I(ω, z =.0µm ) 10-8 ev s / cm 3 1 10 8 6 4 InP GaAs Al O 3 GaN SiC Z =.0μm I(ω, z = 4.0µm ) 10-9 ev s / cm 3 14 1 10 8 6 4 InP GaAs Al O 3 GaN SiC Z = 4.0μm 0 0 100 00 300 400 ω, 10 1 s -1 0 0 100 00 300 400 ω, 10 1 s -1 0 0 100 00 300 400 ω, 10 1 s -1 I(ω, z = 7.0µm ) 10-9 ev s / cm 3 4 3 1 InP GaAs Al O 3 GaN SiC Z = 7.0μm I(ω, z = 8.0µm ) 10-9 ev s / cm 3 4 3 1 InP GaAs Al O 3 GaN SiC Z = 8.0μm I(ω, z = 1µm ) 10-9 ev s / cm 3 4 3 1 InP GaAs Al O 3 GaN SiC Z = 10μm 0 0 100 00 300 400 ω, 10 1 s -1 0 0 100 00 300 400 ω, 10 1 s -1 0 0 100 00 300 400 ω, 10 1 s -1 GaAs maintains monochromatic properties for the longest distance from the surface ( i.e. z ~ 7.95 μm)
Comparison of Near-Field Peak Intensity Materials ω peak,10 1 s -1 I(ω,z), 10-5 ev s / cm (ω = ω z = 0.1μm ) peak, 3 λ, μm ω s0, 10 1 s -1 InP 64.48 76.100 9.3 64.40 GaAs 54.48 59.837 34.60 54.48 Al O 3 E 90.40 4.80 0.85 88.89 GaN 135.0 19.97 13.96 135.04 SiC 178.7 8.635 10.55 178.74 Semiconductors with smaller surface phonon energies (e.g., InP, GaAs) exhibit stronger near-field emission peaks.
Peak Spectral Energy Density vs. Distance u(ω,z) ev s/cm 3 10 - InP 10-3 10-4 10-5 10-6 GaAs Al O 3 SiC GaN 0. 0.4 0.6 0.8 1.0 z (µm) Spectral energy density of thermal emission as a function of the distance z from the surface calculated at ω = ω peak and T = 300 K Emission with a shorter wavelength tends to decay faster.
Temperature Dependence u(ω p,z) (ev s/cm 3 ) 10-1 10-10 -3 10-4 10-5 10-6 InP SiC GaAs GaN (a) Al O 3 0. 0.4 0.6 0.8 1.0 z (µm) (b) 300 o K 600 o K InP GaAs SiC Al O 3 GaN 0. 0.4 0.6 0.8 1.0 The difference in the spectral energy density between the materials with large vs. small surface phonon energies decreases as the temperature goes up.
Comparison Between Candidate Materials All materials show near-field quasi-monochromatic thermal emission spectra. The height of the emission peaks varies significantly from material to material. ω θ ( ω, T ) = exp( is the main determining factor for the ω kbt ) 1 peak intensity height. => larger intensity for a smaller ω peak (i.e., surface phonons with a longer wavelength). The peak intensity exponentially decrease and disappear within the distance of one wavelength evanescent mode. The materials with a smaller ω peak tend to maintain monochromatic emission for a longer distance from the surface.
Electric Energy Density The total radiation energy can be found by integrating with respect to ω: d U z = ω u ω, z 0 π ( ) ( )
Electric Energy Density 10-7 U(z) mj/cm 3 10-8 10-9 10-10 10-11 10-1 10-13 1 0.1 1 10 100 z, µm 3.0x10-3.0x10-3 1.0x10-3 150 160 170 180 190 00 ω, 10 1 s -1 Electromagnetic energy density U(z) near a semi-infinite GaAs interface as a function of z (T = 300 K): curve 1 - Δ is centered at ω p ; curve - Δ is centered at ω max (z), corresponding to the maximum value of u(ω,z), that shifts as a function of z; dashed curve - blackbody radiation. (Δ is integration period; full width half maximum) Surface wave energy is several orders of magnitude larger than blackbody radiation
Proposed Energy Conversion Device Near-field thermal emission supported by the surface phonon polariton shows coherence in a corresponding frequency domain => Quasi-monochromatic; Spectral Coherence Spatial coherence can be achieved by using microstructures such as gratings, micro cavity or photonic crystal, etc..
Summary The polar semiconductor/insulator materials, InP, GaAs, GaN, SiC, and Al O 3, demonstrate a quasi-monochromatic thermal emission in the near-field zone. The electric energy intensity for the quasi-monochromatic thermal emission is several orders of magnitude greater than black body radiation. (GaAs at z= 100nm and T=300K: 1.46 x 10-7 mj/cm 3, Black body radiation: 1.6x10-1 mj/cm 3 ) The thermal coefficient for nanoscale radiation heat transfer is calculated and the results shows that heat transfer is drastically enhanced by surface phonon coupling. (GaAs T=300K, h c = 487.5 mj/cm at z=10nm, h c = 0.3439 mj/cm at z=100μm) The angular emissivity pattern shows that the thermal emission can be controlled to have directionality