Thermal radiation inside hyperbolic media
3.1 Thermal radiation inside nondispersive hyperbolic media
An ideal HM is nondissipative and nondispersive. Although such materials cannot exist because they violate Kramers–Kronig re-lations [14, 44], their analysis is rather simple and they reveal important properties. Therefore, we look at such ideal HMs in this section. Analogous to the classical BB we call the ideal HMs “hyperbolic blackbodies”. The “classical” in classical BB does of course not refer to classical physics which, different from quantum physics, ignores quantization effects but simply refers to Planck’s theory which is nowadays more than 100 years old.
The density of states and the heat flux inside uniaxial media have been derived in Section 2.5. These results are summarized and discussed. In addition, further thermal properties are studied.
3.1.1 Energy density and density of states
In dielectric anisotropic media the DOS DoD for the ordinary modes (OMs) and DeD for the extraordinary modes (EMs) are given by the expressions
DDo(ω) = ω2 π2c3
⊥√ ⊥
2 =DsBB⊥
√⊥, (3.7) DDe(ω) = ω2
π2c3 k√
⊥
2 =DBBp k√
⊥, (3.8)
which where already derived by Eckhardt [78], for instance. On the other hand, in the hyperbolic case we obtain
DIo(ω) = ω2
for type I and type II HMs, respectively.
Note that we have introduced cutoff wavenumbers kk,max and k⊥,max respectively, which are denoted by kmax whenever a dif-ferentiation is not necessary. kmax can be a function of fre-quency even for nondispersive media and is determined by the real (atomic or meta) structure of the medium. For an ideal HM kmax is infinite so that the DOS diverges as was pointed out previously [59, 60]. However, for any real structure kmax is a finite quantity (see Section 1.2). For hyperbolic metamateri-als (HMMs) it is mainly determined by the unit-cell size of the meta structure. Note further that the DOS of the EMs of type
I and type II HMs coincide for kk,max ωcp|⊥|, respectively would be proportional to ωc we would retrieve the usual quadratic behavior of the DOS with respect to frequency. We want to emphasize that the approximate expression of the DOS for a non-ideal dispersive material given in Refs. [59,60] is proportional to k3max, which is a consequence of the dispersive nature of the medium and is examined in the next section.
Let us first have a look at the expressions for the energy density inside an ordinary uniaxial material. In this case we obtain
UDo = ⊥
The relations between the internal energy, the free energy and the entropy have the familiar forms
FDo/e=−1
3UDo/e and SDo/e= 4 3
UDo/e
T . (3.16)
Note that these relations are the same as for a usual BB
be-cause the DOS of the field inside a dielectric uniaxial medium is proportional toω2.
On the contrary, in type I and type II HMs we have seen that the DOS of the EMs is linear in ω as in a 2-dimensional photon gas in vacuum. It follows that the relations between the thermody-namic properties of the photon gas are radically different in that case. Indeed, we obtain (assuming no frequency dependence of kmax)
The zeta function value isζ(3)≈1.202. Furthermore, we find FI/IIe ≈ −1
2UI/IIe and SI/IIe ≈ 3 2
UI/IIe
T . (3.20)
Hence,U and F are proportional to T3 and not anymore toT4. S is proportional toT2 and notT3. This result is a direct con-sequence of the linear dependence of the electromagnetic DOS inside HMs with respect to ω.
Naturally, for the OMs we find
UIo=UDo and UIIo = 0. (3.21) Note that for type II HMs the internal energy of the OMs is zero, since there are no OMs in such a material. The internal energy of the OMs in a type I HM is just the same as in a dielectric uniaxial medium. Hence, the relations between the thermodynamic potentials are the same as in a dielectric uniaxial medium. However, in typical HMs the maximal wavevector is much larger than the vacuum wavevector kmax ωc making the material properties dominated by EMs.
Another consequence of the linearity of the DOS with respect to ω inside a HM is a spectral shift of Wien’s frequencyωmax(resp.
wavelength λmax) at which the energy distribution function has its maximum. For both type I and type II HMs we find after a straightforward analysis of the integrand in (3.17) that this
maximum is reached when (again assuming largekmax)
~ωmax, HM
kBT = Ωmax, HM≈1.595 or 2πlc
λmax, HM ≈3.921 (3.22) whereas for a usual BB ~ωmax, BBkBT ≈2.821 and λmax, BB2πlc ≈4.965.
Hence, we see that Wien’s frequency is shifted toward smaller values and the vacuum wavelength to larger values.
Figure 3.1: Comparison of the spectral energy density u = uo +ue of a classical blackbody (BB) vs. a “hyperbolic black-body”, assuming |⊥k| = 1, k⊥,max = Λπ with Λ = 50 nm and T = 300 K. The hyperbolic BB energy density is completely dominated by extraordinary modes. The dashed straight lines show the asymptotic behavior inω andω2 of the hyperbolic and classical BB spectrum. The arrows indicate Wien’s frequencies ωmax in both cases. [From Ref. [67], Copyright 2015 American Physical Society.]
In Fig. 3.1 the spectral energy densities u(ω) =D(ω)U(ω, T) of a “classical” BB, as derived by Planck, and of an ideal HM (“hy-perbolic blackbody”) are plotted. First, with reasonable assump-tions the hyperbolic BB exhibits an orders of magnitude larger thermal energy density. How many orders depends onkmax. The larger kmax the larger u(ω). Second, at small frequencies where the quantization of light is irrelevant (~ωkBT), and thus the energy per modeU(ω, T)≈kBT, we see the linear vs. quadratic frequency dependence of the DOS. Once the quantization be-comes significant (~ω kBT), U(ω, T) and the energy density decline exponentially. Third, we see that indeed the frequency at which u is maximal is shifted towards smaller values for the hyperbolic BB.
Noticing the spectral properties, the temperature dependences of the total energy densities are easy to understand. The amplitude of the spectral energy density scales ∝ U(ω, T) ∝T at smaller frequencies. Then, the Bose–Einstein termU(ω, T) introduces a (smooth) spectral cutoff, or in other words an effective integra-tion limit, with a cutoff frequency ∝T. If the integration limit shifts∝T the integral of a linear function scales withT2 (hyper-bolic BB), of a square function with T3 (classical BB). Together with the linearly scaling amplitude (at smaller frequencies) we get the T3 vs.T4 dependences.
It is now interesting to compare the internal energy of the EMs in a HM with that of a classical BB. From expressions (3.1) and
(3.19) we immediately get UI/IIe
UBBp ≈q|k⊥|30
π4ζ(3)k⊥,maxlc. (3.23) If Λ denotes the unit-cell size of our HM then kk,max = πΛ or k⊥,max= Λπ depending on the concrete structure. However, inde-pendent of the structure both maximum wavevectors are related by k⊥,max≈
r
k
⊥
kk,max so that we generally have UI/IIe
UBBp ∝ lc
Λ. (3.24)
At a temperature of 300 K the coherence length is lc ≈7.6 µm, while the period of realistic HMMs is typically larger than Λ≈ 10 nm (see Refs. [50, 51] for instance). In natural HMs [47] the main length scale is determined by the interatomic spacing, i. e.
Λ ≈ 1 ˚A. Albeit, nonlocality can increase the length scale at which HMs exhibit metallic properties (in one or two directions) to the nanometer range [86, 87]. Nonlocality is a spatial disper-sion manifesting itself in ak-dependent permittivity[87] which approaches the vacuum value as the spatial variationkincreases.
Hence, the internal energy of thermal radiation inside a HM can be 3 (Λ≈10 nm) to 5 (Λ≈1 ˚A) orders of magnitude larger than that of a perfect BB. The same is of course also true for the free energy and the entropy. This result suggests that the radiative heat flux inside a HM is dramatically enhanced compared to that of a classical BB.
3.1.2 Power flow
The expressions for the mean Poynting vector inside a dielectric uniaxial material are
The underlying setup, depicted in Fig. 2.4a, is a halfspace at temperatureT emitting into a second, identical halfspace at 0 K.
When⊥ =k = 1 we find again the usual BB result, i. e. Stefan–
Boltzmann’s law. On the other hand, inside a uniaxial material (as inside an isotropic material) with ⊥ > 1 and k > 1 the radiative heat flux is larger than the BB value, which is a well-known fact [88, 89].
In the case of HMs these results radically change. Before seeing this, let us first consider the OMs. For ΦoI we find of course the same relation as for the dielectric anisotropic material ΦoI = ΦoD, whereas as a consequence that there do not exist any propagating OMs in a type II HM we find ΦoII= 0.
On the contrary, for the EMs we find (assuming no frequency
dependence ofk⊥,max) In this case, we see that the heat flux is proportional toT2 and not anymore to T4 as in the “classical” Stefan–Boltzmann law.
The total and spectral heat flux are also proportional tok2⊥,max, which is due to the fact that the heat flux scales like the area of the projection of the isofrequency surface in k-space [80] (see Appendix A.2), or like the number of transversal modes [22,79], respectively. Note that the above limit is also the upper limit of heat radiation between two hyperbolic materials separated by a vacuum gap [18], when we assume the cutoff kmax is caused by the finite gap size l, kmax ∝ 1l, and not by the (meta)material properties.
Figure 3.2: Comparison of the spectral heat fluxφ=φo+φeof a classical blackbody (BB) vs. a “hyperbolic blackbody”, assuming k⊥,max= Λπ with Λ = 50 nm andT = 300 K. The hyperbolic BB heat flux is completely dominated by extraordinary modes. The dashed straight lines show the asymptotic behavior in 1 and ω2 of the hyperbolic and classical BB spectrum.
In Fig. 3.2 we plot the spectral heat flux of a classical and a hyperbolic BB. The BB heat flux φBB resembles uBB because φBB= 4cuBB. The hyperbolic BB heat fluxφI/II is different from its energy densityuI/II. Again, with reasonable assumptions the hyperbolic BB exhibits an orders of magnitude larger thermal power flow. The difference between hyperbolic and classic BB is even larger than for the energy density and the dependence on kmax stronger. At small frequencies where the quantization of light is irrelevant (~ω kBT) we see that φI/II is constant
over frequency whereas the classical BB shows the quadratic fre-quency dependence. Once the quantization becomes significant (~ω kBT), U(ω, T) and the heat flux decline exponentially.
The frequency at whichφis maximal is 0 for the hyperbolic BB, so shifted to an even smaller frequency than foru.
The T4 temperature dependence of the total heat flux for the classical BB can be understood identically to its energy density.
The T2 dependence for the hyperbolic BB follows the same ex-planation. The integration limit shifts∝T and the integral of a constant function thus scales withT. Together with the linearly scaling amplitude we get the T2 dependence.
Comparing the radiative heat fluxes, we find ΦeI/II
ΦpBB ≈ 5
2π2(k⊥,maxlc)2 ∝ lc
Λ 2
. (3.30)
Thus, the normalized heat flux is proportional to (k⊥,maxlc)2. We have seen before in Eq. (3.23) that the ratio of the energy density of a HM and that of a BB is only linear in k⊥,maxlc. This is quite astonishing, since for HMMs with a unit cell size Λ of 10 nm and for natural HMs with a unit-cell size Λ ≈ 1 ˚A we can now expect a hyperbolic BB heat flux 6 to 10 orders of magnitude larger than that of a usual BB at T = 300 K. At cryogenic temperatures lc becomes very large so that this ratio can become even much larger.
To summarize, Table 3.1 contrasts the important thermal
radi-ation properties of a classical BB and a hyperbolic BB.
classical blackbody
“hyperbolic blackbody”
density of
statesD(ω) ∝ ωc2 ∝ ωckmax
spectral heat
fluxφ(ω) ∝ ωc2U(ω, T) ∝k2⊥,maxU(ω, T) energy density U ∝T4 ∝kmaxT3
heat flux Φ ∝T4 ∝k2⊥,maxT2
~ωmax
kBT ≈2.821 ≈1.595
2πlc
λmax ≈4.965 ≈3.921
entropy S = 43UT ∝T3 ≈ 32UT ∝kmaxT2 Table 3.1: Comparison of the thermal radiation properties of a classical blackbody (BB) vs. a “hyperbolic blackbody”. The clas-sical BB has the properties described by Max Planck et al. The hyperbolic BB is an idealized (lossless, nondispersive) hyperbolic medium.
For dielectric anisotropic media we have seen that the thermo-dynamic properties of the photon gas inside such media are very similar to that of a classical BB. On the other hand, when these media are hyperbolic, the spectral energy distribution of radia-tion radically differs from that predicted by Planck and is shifted towards frequencies smaller than Wien’s frequency. We have also shown that in contrast to Stefan–Boltzmann’s law, the heat flux
radiated by these media depends quadratically on their temper-ature. Nevertheless, the magnitude of heat flux carried by these media can be several orders of magnitude larger than the flux radiated by a classical BB.
For completeness let us mention that dielectric and metallic ma-terials also possess a kmax beyond which they cannot be po-larized and alter their properties (due to nonlocality they be-come vacuum-like [86]). But these kmax correspond to atomic distances and have no effect on thermal radiation. Because of the Bose–Einstein cutoff the radiation is limited to a maximal frequency with a vacuum wavenumber corresponding to approx-imately Wien’s wavelengthλmax, BB. So for wavenumbers on the scale of kmax not to be spectrally cut off, the temperature has to be well above several 10 000 K. Under such extreme temper-atures our assumption of a solid material is not justified any more and a different approach is needed. HMs, on the other hand, feature high-kmodes even at low frequencies – below the Bose–Einstein cutoff – and hence, the radiation properties de-pend onkmax.