Near-field thermal radiation transfer controlled by plasmons in graphene
It is shown that thermally excited plasmon-polariton modes can strongly mediate, enhance and tune the near-field radiation transfer between two closely separated graphene sheets. The dependence of near-field heat exchange on doping and electron relaxation time is analyzed in the near infra-red within the framework of fluctuational electrodynamics. The dominant contribution to heat transfer can be controlled to arise from either interband or intraband processes. We predict maximum transfer at low doping and for plasmons in two graphene sheets in resonance, with orders-of-magnitude enhancement (e.g. to for separations between to ) over the Stefan-Boltzmann law, known as the far field limit. Strong, tunable, near-field transfer offers the promise of an externally controllable thermal switch as well as a novel hybrid graphene-graphene thermoelectric/thermophotovoltaic energy conversion platform.
pacs:44.40.+a, 78,67.Wj, 73.20.Mf
Heat transfer between two bodies can be greatly enhanced in the near field, i.e. by bringing their surfaces close together to allow tunnelling of evanescent photon modes. For two parallel, semi-infinite, dielectric surfaces of index of refraction , maximum flux enhancement is known to be times the Planck’s black body limit Landau and Lifshitz (1980). However, particularly interesting near-field radiation transfer phenomena involve thermal excitation of various surface modes. Due to their localization and evanescent nature, it is only at sub-micron separations that these modes become relevant. Measuring near-field transfer has been experimentally difficult Hargreaves (1969); Narayanaswamy et al. (2008); Shen et al. (2009); Rousseau et al. (2009); nevertheless, the promise of order-of-magnitude enhancement over the far field Planck’s black body limit has made near-field transfer the topic of much research Joulain et al. (2005). A promising class of materials for enhancing the near-field transfer are plasmonic materials, due to high density of modes around the frequency of plasmons. The potential of graphene Novoselov et al. (2004) as a versatile and tunable plasmonic material has already been recognized in applications such as teraherz optoelectronics and transformation optics Jablan et al. (2009); Wunsch et al. (2006); Hwang and Das Sarma (2007); Ju et al. (2011); Vakil and Engheta (2011). Unlike in metals, where high plasma frequencies make thermal excitation of surface modes difficult, plasmon frequencies in graphene can be anywhere from the teraherz to the near infra-red Rana (2011). In addition, the dependence of graphene conductivity on chemical potential, which in turn can be controlled by doping or by gating, allows for a tunable plasmonic dispersion relation. Transfer between graphene and amorphous Persson and Ueba (2010); Volokitin and Persson (2011) as well as application of graphene as a thermal emitter in a near-field thermophotovoltaic (TPV) system has been reported Ilic et al. (2012). Here we analyze the contribution of plasmon-polaritons to graphene-graphene near-field heat transfer. The choice of identical coupled systems is predicated on the idea that resonant enhancement could lead to even greater heat transfer capacity. Indeed, we find maximal transfer for resonantly coupled plasmon modes (corresponding to similar doping in the two graphene sheets), which can be orders of magnitude larger than the heat transfer between two black bodies in the far field.
In general, the radiative heat transfer between two bodies at temperatures and is given by
where is the average energy of a photon at frequency (the Boltzmann factor), and is the spectral transfer function, characterizing frequency dependence of the heat exchange (i.e. how much heat is exchanged at a given frequency). In the context of fluctuational electrodynamics Rytov et al. (1987), the spectral transfer function is calculated in the following way: thermal fluctuations in the first (emitter) medium induce correlations between electric currents, which are proportional to the real part of the medium conductivity Haus (1961); next, using Green functions, we can find the electromagnetic fields in the second (absorber) medium induced by the fluctuating currents in the first Sipe (1987); finally, the radiation transfer is obtained by calculating the Poynting flux around (or the ohmic losses within) the second medium. This approach has been used to numerically calculate the near-field transfer between two half-spaces Rytov et al. (1987); Pendry (1999), as well as generalizations such as two slabs Ben-Abdallah et al. (2009), sphere and a plane Mulet et al. (2002); Narayanaswamy et al. (2008) two spheres Narayanaswamy and Chen (2008), as well as 1D periodic structures Rodriguez et al. (2011).
The system we analyze, shown in Fig. 1, consists of a
suspended graphene sheet at temperature emitting to another suspended graphene sheet held at room temperature , and a distance away. In general, the -polarization spectral transfer function for evanescent modes between two bodies is
where is the perpendicular wave-vector and is the reflection coefficient for the bottom(top) body; note that depend on , and hence the -dependence of . Integration is over the parallel wave-vector , limited only to the evanescent () modes. The spectral transfer function (2) was derived for the case of two semi-infinite slabs Joulain et al. (2005); however, it can be shown that the same expression is valid when any of the two bodies is a 2D system, such as graphene Ilic et al. (2012). Since graphene absorbs poorly () in the far field (hence is also a poor emitter), not including the propagating modes is a good approximation. The contribution of evanescent -polarized modes can also be calculated using Eq. (2), but it turns out to be negligible compared to -polarized modes, as we discuss later. We assume graphene is completely characterized by its complex optical conductivity , which depends on angular frequency , electron scattering lifetime , chemical potential , and temperature . Furthermore, the graphene conductivity is taken to be independent of the parallel wave-vector (see discussion below), and consists of the Drude (intraband) and interband conductivity, expressed respectively as Falkovsky (2008)
where , and is the chemical potential. Various electron scattering processes are taken into account through the relaxation time . From DC mobility measurements in graphene, one obtains Jablan et al. (2009) an order-of-magnitude value of .
First we discuss the electrodynamic properties of a single suspended sheet of graphene, inherent in the -polarization reflection coefficient, which is illustrated in Figure 1b. The reflection coefficient is , where is the dielectric function of graphene Falkovsky (2008). Its pole corresponds to the dispersion relation of -polarized plasmon modes Jablan et al. (2009)
which is shown as the dashed line in Fig. 1b. Figure 1 shows plasmons exist in a strongly non-retarded regime (), indicating a tightly confined plasmon polariton mode. Graphene also supports -polarized surface modes with a dispersion relation very close to the light line Mikhailov and Ziegler (2007). However, due to the large density of states and the tightly confined nature of -polarized surface modes, it is the -polarization that dominates (as our calculations confirm) the near-field transfer.
When two parallel graphene sheets are sufficiently close (see Fig. 1a), their plasmonic modes can become coupled. The dispersion of these coupled modes is , when , so , which is exactly the pole of the integrand of the spectral transfer function (2). The integrand is illustrated in Figure 2 for different values of chemical potential. The coupling of modes is strongest when both graphene sheets have identical parameters (middle panel in Fig. 2). In that case, their individual dispersions are identical. Nevertheless, the dispersion of the combined system shows two branches that dominate the near-field spectral transfer, i.e. the implicit equation for has two explicit solutions: and for the even, and the odd mode, respectively.
The splitting of two superimposed resonances is particularly noticeable at smaller wave vectors . For larger , the splitting disappears, and the resonant matching of peaks of significantly enhances the near-field transfer. As the chemical potential of one of the sheets changes (top and bottom panel in Fig. 2), the plasmons in the two sheets move out of resonance, coupling decreases, the peaks in the integrand approach the individual (vacuum) plasmons dispersion curves, and the heat transfer becomes lower than in resonance.
Figure 3a shows a highly tunable spectral transfer function for different values of chemical potential and relaxation time.
Given the chemical potential, the relaxation time determines which processes (interband or intraband) are responsible for the peaks in spectral transfer. Since interband processes are dominant at high frequencies, all curves converge in the high frequency limit, where Drude losses are negligible. However, interband processes can play a leading role even below the absorption threshold , particularly for small chemical potential where thermal broadening of the interband threshold (on the order of few ) becomes more significant. For example, for (first peak in Fig. 3a) the similarity between and spectral transfer functions indicates that the majority of loss in graphene comes from interband processes. On the other hand, the Drude (intraband) loss term, usually important for , can become dominant at higher frequencies, for large enough (third peak). Finally, a combination of two loss processes, , can lead to a hybrid spectral transfer. While the use of -independent expression for graphene conductivity Eq. (3) for intraband processes is a good approximation Jablan et al. (2009), one must take care when applying Eq. (3) to interband transitions. As indicated in Fig. 3a, interband transitions can play a significant role in near-field transfer at low doping levels. Here, the contribution from the non-zero wave-vector becomes important since it broadens the interband threshold from to . On the other hand, this is similar to non-zero temperature effects which also broaden the interband threshold, so we do not expect a qualitatively different result with -dependent conductivity.
We quantify the heat exchange in the near-field by plotting (Fig. 3b) the integrated transfer from Eq. (1) normalized to the transfer between two black bodies in the far field. Factoring in the temperature dependence shifts the majority of the near-field transfer to lower frequencies, due to the exponentially decaying Boltzmann factor. This implies that while doping or gating might be advantageous in some applications (for example, emitter-PV cell bandgap frequency matching in near-field TPV systems Ilic et al. (2012)), near-field transfer between two graphene sheets is maximized for small values of doping, despite the stronger peak in spectral transfer for vs. (Fig. 3a). For plasmons in resonance with (left panel, Fig. 3b), we observe orders-of-magnitude increase in heat exchange, particularly at small separations ( for ), but also at separations as large as . At larger separations, we observe (not shown) the shift of the peak of the spectral transfer function to case (red line in Fig. 3a), indicating that the coupling between highly localized, large , modes becomes weaker, and the transfer is dominated by lower-frequency, less evanescent modes. The heat transfer depends in a complex fashion on the parameters of the system, and does not seem to yield a simple functional dependence on the emitter and absorber temperatures (as is the case for two black bodies). Nevertheless, there is a relative advantage (Fig. 3b) to operating at lower temperatures, as the temperature dependence of the near-field transfer appears to grow slower than the black body dependence. Finally, we note that the temperature dependence of conductivity reduces the resonant effect when two graphene sheets are at different temperatures. This reduction is more pronounced for large temperature difference, shifting the peak of the spectral transfer on the order of ; however, the relative reduction of the integrated spectral transfer function is small, with the main temperature dependence coming from the Boltzmann factor. This efficient heat exchange between two graphene sheets in the near-field, together with recently reported advances in hot carrier extraction from graphene Gabor et al. (2011), may offer a potential for a novel, hybrid thermophotovoltaic/thermoelectric solid-state heat-to-electricity conversion platform. In addition, this material system could pave the way toward an externally controllable thermal switch behavior, where one can, by means of doping or gating, tune the resonant coupling between the hot and the cold side.
The authors would like to acknowledge helpful discussions with Pablo Jarillo-Herrero, Nathan Gabor, Gang Chen, Alejandro Rodriguez and Steven Johnson. OI and MS were partially supported by the MIT S3TEC Energy Research Frontier Center of the Department of Energy under Grant No. DE-SC0001299. MJ was supported in part by the Croatian Ministry of Science under Grant No. 119-0000000-1015. This work was also partially supported by the Army Research Office through the Institute for Soldier Nanotechnologies under Contract No. W911NF-07-D0004, and the Unity through Knowledge Fund Grant Agreement No. 93/11.
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