Near-field radiative heat transfer in multilayered graphene system considering equilibrium temperature distribution

Ming-Jian He; Hong Qi; Yi-Fei Wang; Ya-Tao Ren; Wei-Hua Cai; Li-Ming Ruan

doi:10.1364/OE.27.00A953

1. Introduction

Since Novoselov and associates at the University of Manchester successfully prepared single-layer graphene utilizing a simple mechanical exfoliation method [1], provoked attention has been paid on graphene research. As a powerful two-dimensional (2D) material, graphene is considered as a promising candidate for plasmonic applications duo to its unique and fantastic optoelectronic characteristics [2–4]. High room-temperature electron mobility has been observed in graphene, which can be dynamically tuned through electrical gating or chemical doping. A near-field thermostat was theoretically proposed based on the tunability of graphene [5]. The 2D nature of graphene also endows it with unusual thermal properties compared with bulk materials. Concerning with thermal conductivity, there exists over 100-fold anisotropy between the in-plane and out-of-plane directions [6]. As is known, graphene surface plasmon polaritons (SPPs) can be excited in the near-field regime to transport energy in photonic channel [7]. The near-field effect may pave a new way toward improving the ability of transporting heat in the out-of-plane direction of graphene sheets.

Near-field radiative heat transfer (NFRHT), first proposed by Rytov [8] and formalized by Polder and Van Hove [9], can drastically enhance the thermal energy exchange by orders of magnitude greater than the limit predicted by Planck’s theory. In recent years, a rapid progress has been seen in the field of NFRHT, no matter in theoretical [10–17] or experimental studies [18–21]. Graphene is able to enhance and tune NFRHT when it is coated on dielectric materials [22,23], no matter in parallel-plate configurations [7,24–33] or grating geometry [34–38]. In practical applications, the performance of the near-field thermophotovoltaic system is proved to be enhanced by introducing monolayer of graphene coated on the cell or multilayered graphene embedded in the system [25,26,28,39–41]. Furthermore, experimental works in graphene matched well with the existing theoretical. It is intriguing to see that graphene can enhance NFRHT for materials with nonmatching surface excitations [42]. Recently, an experimental work successfully observes the super-Planck heat transfer between graphene sheets and validates the classic thermodynamical theory in treating graphene [43].

In pure graphene system, the radiative heat transfer between two suspended graphene monolayers has been investigated and the NFRHT is shown to be strongly mediated, enhanced, and tuned by thermally excited SPPs modes [44–46]. Moreover, a giant enhancement of the near-field radiative heat flux is observed between two graphene sheets patterned in ribbon arrays [47]. Thanks to the near-field interaction of the tunable SPPs in graphene nanodisks [48] and nanosheets, active control of propagation directions for near-field heat flux [49] and a near-field radiative thermal switch [50] are theoretically proposed. An ultrafast radiative cooling regime is predicted to take place in plasmonically active neighboring plasmon-supporting graphene nanodisks within a femtosecond timescale [51]. To sum up, the existing studies mainly focus on the cases of a couple of suspended graphene sheets or nanodisks.

In existing studies, multilayered graphene system is simplified to a couple of multilayer structures made by a periodic repetition of a graphene sheet and a dielectric slab or vacuum gap [26,27,52–55]. The temperature distribution has no choice but to be considered uniform in both periodic structures. Hence, the effect of temperature inhomogeneity on the heat transfer mechanisms is ignored when investigating multilayered graphene. As is well-known, graphene SPPs is the function of localized temperature and thereby the existing studies lack preciseness especially when the temperature of graphene changes dramatically. Nevertheless, to date no studies have reported equilibrium temperature distribution in multilayered graphene system, not to mention the corresponding heat transfer mechanisms considering temperature distribution.

In the present work, we study the NFRHT in a multilayered graphene system considering equilibrium temperature distribution. To characterize the heat transfer ability of the whole system, the whole system is evaluated with two heat transfer coefficients (HTC), transient heat transfer coefficient (THTC) and steady heat transfer coefficient (SHTC). THTC and SHTC are investigated corresponding to different geometrical parameters of the multilayered graphene system. Furthermore, a transition from diffusive to quasi-ballistic transport is investigated by changing the chemical potentials of graphene.

2. Multilayered graphene system

Here we consider a system composed of N parallel graphene sheets separated by vacuum gap, assumed to be infinite in in-plate directions, as illustrated in Fig. 1(a). The temperatures of external graphene sheets 1 and N are fixed at T₁ = 400 K and T_N = 300 K, respectively. For the sake of illustration, the distance between adjacent graphene sheets d and the chemical potential of graphene μ are both set uniform. $D = (N - 1) \cdot d$ denotes the sum of vacuum gap thickness. The heat flux transports from the hot sheet to the cold one in the out-of-plane direction. The net radiative flux received or lost by graphene sheet j can be written as a sum over the energy exchanged with every other sheet φ_l,j [56,57],

φ_{j} = \sum_{l \neq j} φ_{l, j} = \frac{1}{4 π^{2}} \sum_{l \neq j} \int_{0}^{\infty} ℏ ω n_{l, j} d ω \int_{ω / c}^{\infty} β ξ {(ω, β)}^{_{l, j}} d β

where l $\neq$ j runs from 1 to N, ћ is Planck’s constant divided by 2π and n_l,j(ω) = n_l(ω)-n_j(ω) denotes the difference between the two mean photon occupation numbers

n_{l / j} (ω) = {[\exp (ℏ ω / k_{B} T_{l / j}) - 1]}^{- 1} .

β is the component of the wave vector parallel to the interface. ξ(ω,β)^l,j describes the energy transmission coefficients between sheet l and j. The graphene sheet is modeled as a thin film with dielectric function ε_G [58].

ε_{G} (ω) = 1 + \frac{i σ_{G}}{ω d_{G} ε_{0}}

where ε₀, d_G, σ_G are permittivity of vacuum, thickness and conductivity of graphene respectively. As for the conductivity σ_G, it can be written as a sum of an intraband and an interband contribution, respectively, given by

σ_{intra} (ω) = \frac{2 i e^{2} k_{B} T}{(ω + i τ^{- 1}) π ℏ^{2}} \ln [2 \cos h (\frac{μ}{2 k_{B} T})]

σ_{inter} (ω) = \frac{e^{2}}{4 ℏ} [G (\frac{ℏ ω}{2}) + \frac{4 i ℏ ω}{π} \int_{0}^{+ \infty} \frac{G (η) - G (\frac{ℏ ω}{2})}{{(ℏ ω)}^{2} - 4 η^{2}} d η]

where G(x) = sinh(x/k_BT)/[cosh(μ/k_BT) + cosh(x/k_BT)], e and k_B are elementary charge and Boltzmann constant, respectively. In the present work, we take thickness of graphene sheet d_G = 0.34 nm and relaxation time τ = 10⁻¹³ s [59].

Fig. 1 (a) Schematic illustration of multilayered graphene system, (b) the energy transmission coefficients contour for N = 5 multilayered graphene system, (c) the flowchart of iterative procedure to calculate local equilibrium temperature profile.

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The expression of ξ(ω,β)^l,j is obtained based on many-body theory, which is based on the combination of the scattering theory and the fluctuational-electrodynamics approach in many-body systems [56]. First, the single reflection and transmission coefficients of sheet j are given as

ρ_{j} = \frac{r_{j} (1 - e^{2 i k_{z j} d_{G}})}{1 - r_{_{j}}^{2} e^{2 i k_{z j} d_{G}}}

τ_{j} = \frac{(1 - r_{_{j}}^{2}) e^{i k_{z j} d_{G}}}{1 - r_{_{j}}^{2} e^{2 i k_{z j} d_{G}}}

where k_zj is the z component of the wave vector inside graphene sheet j and r_j is the p polarized vacuum-medium Fresnel reflection coefficients of sheet j. Here we consider a number of consecutive graphene sheets having indexes from j to m (with j smaller than m). Considering reflection procedures, “+” and “-” denote the direction of propagation or decay of the outgoing field. Then the many-body scattering processes for them, representing the analogues of ρ_j and τ_j for a single sheet, are given as [56]

\begin{array}{l} ρ_{_{+}}^{j \to m} = {\hat{ρ}}_{_{+}}^{j \to m} e^{- i k_{z} (d_{G} + 2 z_{m})} \\ ρ_{_{-}}^{j \to m} = {\hat{ρ}}_{_{-}}^{j \to m} e^{- i k_{z} (d_{G} - 2 z_{j})} \\ τ^{j \to m} = {\hat{τ}}^{j \to m} \exp [- (m - j + 1) i k_{z} d_{G}] \end{array}

where

\begin{array}{l} {\hat{ρ}}_{_{+}}^{j \to m} = ρ_{m} + {(τ_{m})}^{2} {\hat{ρ}}_{_{+}}^{j \to m - 1} u^{j \to m - 1, m} e^{2 i k_{z} d} \\ {\hat{ρ}}_{_{-}}^{j \to m} = ρ_{j} + {(τ_{j})}^{2} {\hat{ρ}}_{_{-}}^{j + 1 \to m} u^{j, j + 1 \to m} e^{2 i k_{z} d} \\ {\hat{τ}}^{j \to m} = {\hat{τ}}^{j \to m - 1} u^{j \to m - 1, m} τ_{m} \end{array}

and

\begin{array}{l} u^{j \to m - 1, m} = {(1 - {\hat{ρ}}_{_{+}}^{j \to m - 1} ρ_{m} e^{2 i k_{z} d})}^{- 1} \\ u^{j, j + 1 \to m} = {(1 - ρ_{j} {\hat{ρ}}_{_{-}}^{j + 1 \to m} e^{2 i k_{z} d})}^{- 1} \end{array}

To obtain the many-body scattering coefficients given above, one needs to perform iterative calculation and take ${\hat{ρ}}_{_{+}}^{j} = {\hat{ρ}}_{_{-}}^{j} = ρ_{j}, {\hat{τ}}^{j} = τ^{j}$ for a single sheet. Then the energy transmission coefficients between sheet l and j can be subsequently computed based on the obtained many-body scattering coefficients [56]

ξ^{l, j} = {\hat{ξ}}_{^{j - 1}}^{l} - {\hat{ξ}}_{^{j - 1}}^{l - 1} - {\hat{ξ}}_{^{j}}^{l} + {\hat{ξ}}_{^{j}}^{l - 1}

where

\begin{array}{l} {\hat{ξ}}_{^{γ}}^{j} = \frac{4 {| τ^{j + 1 \to γ} |}^{2} Im (ρ_{+}^{0 \to j}) Im (ρ_{-}^{γ + 1 \to N})}{{| 1 - ρ_{+}^{0 \to γ} ρ_{-}^{γ + 1 \to N} |}^{2} {| 1 - ρ_{+}^{0 \to j} ρ_{-}^{j + 1 \to γ} |}^{2}}, j < γ \\ {\hat{ξ}}_{^{γ}}^{γ} = \frac{4 Im (ρ_{+}^{0 \to γ}) Im (ρ_{-}^{γ + 1 \to N})}{{| 1 - ρ_{+}^{0 \to γ} ρ_{-}^{γ + 1 \to N} |}^{2}} \\ {\hat{ξ}}_{^{γ}}^{j} = \frac{4 {| τ^{γ + 1 \to j} |}^{2} Im (ρ_{+}^{0 \to γ}) Im (ρ_{-}^{j + 1 \to N})}{{| 1 - ρ_{+}^{0 \to j} ρ_{-}^{j + 1 \to N} |}^{2} {| 1 - ρ_{+}^{0 \to γ} ρ_{-}^{γ + 1 \to j} |}^{2}}, j > γ \end{array}

According to the many-body theory [56], the energy transmission coefficients between sheet l and j satisfy the relation $\sum_{l = 1}^{N} ξ {(ω, β)}^{_{l, j}} = 0$ and $ξ {(ω, β)}^{_{j, j}} = - \sum_{l \neq j} ξ {(ω, β)}^{_{l, j}}$ denotes the energy transmission coefficients corresponding to the energy emitted by sheet j in the presence of other sheets. The energy transmission coefficients contour for N = 5 multilayered graphene system, with d = 10 nm, μ = 0.5 eV and T = 300 K are given in Fig. 1(b).

The system reaches its steady state when the net flux received by each internal sheet vanishes, that is, all the internal sheets reach their own equilibrium temperature T_eq,_j (j = 2, . . ., N-1). The local equilibrium temperatures T_eq,_j of the internal sheets can be solved by means of an iterative procedure (shown in Fig. 1(c)). While, different from [56], the energy transmission coefficients ξ^l,j in the present work depend not only on the properties of sheet l and j, but also the other sheets in the system, thanks to the complexity of the multilayered graphene system. The iterative procedure considering this complexity can still converge.

The equilibrium temperature profile is shown in Fig. 2 for an N = 60 graphene system (μ = 0.5 eV) for different d. The profiles tend to be horizontal as the system are getting more compact, which means the heat transfer is enhanced and the temperature distribution inside the system is tending towards homogeneity. Moreover, we roughly measure the heat transfer ability of the whole system with HTC, H = φ/ $Δ$ T, where φ is the net radiative flux transporting in the system in the steady state, i.e., φ₁ or φ_N. $Δ$ T denotes the temperature gradient of the whole system, which is the temperature difference between graphene sheet 1 and N, i.e., 100 K. The inset in Fig. 2 demonstrates an obvious attenuation of heat transfer when the system is getting much sparser. Another phenomenon is that asymmetry of temperature distribution exists, especially for tiny d, which is mainly caused by the dependence of graphene properties on temperature. The asymmetry of temperature distribution and properties of graphene sheets have links and affect each other.

Fig. 2 Equilibrium temperature profile for a system composed of N = 60 graphene sheets.

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3. Results and analysis

An interesting phenomenon observed in Fig. 2 is that there exist temperature steps near both boundaries of the system. To find out the factor that causes the unique temperature step, the heat transport regimes inside the system are quantitatively investigated by defining an effective radiative thermal conductance (RTC)

λ_{j} = \frac{Φ_{j, j + 1} d}{Δ T_{j, j + 1}}

where

Δ

T_j_,_j₊₁ is the temperature difference between sheet j and j + 1. Φ_j_,_j₊₁ denotes the net radiative flux transporting through the vacuum gap between graphene sheet j and j + 1. It can be written as a sum of the radiative flux transferring from graphene sheets 1-j to j + 1-N, i.e.,

Φ_{j, j + 1} = \sum_{m = 1}^{j} \sum_{n = j + 1}^{N} φ_{m, n}

, schematically illustrated in Fig. 3(a). The RTC can be viewed as a measure of the thermal freedom of heat flux transferring in different regions of the system.

Fig. 3 (a) Sketch of the net radiative flux, Φ_j_,_j₊₁; (b) the ratio of RTC, λ_j/λ₁.

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For the reason that temperature term exists in Φ_j_,_j₊₁ and $Δ$ T_j_,_j₊₁ of Eq. (12), it makes no sense if we utilize the known temperature profile in Fig. 2 to calculate RTC. To avoid the effect of the known temperature profile on evaluating RTC, the temperature terms in Eq. (12) are replaced by an ultimate method. The HTC between sheet n and m held respectively at temperature T and T + $Δ$ T is defined as

h_{m, n} = \frac{φ_{m, n}}{Δ T} = \frac{1}{4 π^{2}} \int_{0}^{\infty} ℏ ω \frac{n_{m} - n_{n}}{Δ T} d ω \int_{ω / c}^{\infty} β ξ {(ω, β)}^{_{m, n}} d β

Then the RTC in Eq. (12) can be rewritten in a form of HTC

λ_{j} = d \cdot \sum_{m = 1}^{j} \sum_{n = j + 1}^{N} h_{m, n}

The ratio of RTC inside the system to that next to the boundary λ_j/λ₁ is displayed in Fig. 3(b), for different d in Fig. 2. The HTC in Eq. (14) are all obtained at T = 300 K. We note that the RTC inside the system is larger than that next to the boundary. Hence, radiative heat flux transfers more smoothly and thermal energy exchange among graphene sheets is much stronger in the internal region than the inner-boundary region of the system. Then the temperature changes slightly in the internal region and changes dramatically near the boundary of the system. In addition, for tiny d, which refers to the strong near-field regime, the above phenomenon is more obviously observed in Fig. 2. It can also be well explained by the ratio of RTC plotted in Fig. 3(b), λ_j/λ₁ increases with decreasing separation distance d. For large d, the RTC is close to unit in the internal region which means considerably uniform thermal freedom for heat transfer inside the system. Hence, it results in a nearly linear temperature profile from T₁ = 400 K to T_N = 300 K and tiny temperature step for d = 1000 nm in Fig. 2.

As is well-known, plasmon properties of graphene are not only influenced by local temperature, but also by chemical potentials. Thus we investigate the heat transfer mechanisms of the system corresponding to different chemical potentials of graphene spectrally. The spectral transfer function for p polarized evanescent modes is defined as

f_{p} (ω) = \int_{ω / c}^{\infty} τ (ω, β) d β = \int_{ω / c}^{\infty} β ξ (ω, β) d β

where ξ(ω,β) = ξ(ω,β)^1,1 denotes the energy transmission coefficients of sheet 1. As discussed above, ξ(ω,β), τ(ω,β) and f_p(ω) should be negative and the results of them shown hereafter are absolute values. With fixed temperature T = 300 K and d = 10 nm, the spectral transfer function for different chemical potentials of graphene and number of graphene sheets is shown in Fig. 4(a).

Fig. 4 (a) The spectral transfer function f_p(ω) (a.u.) contour. For μ = 0.5 eV, (b)-(d) the contours for integrand τ(ω,β) and (e) energy transmission coefficients ξ(ω,β) .

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f_p(ω) has different positions of resonance peak based on the characters of graphene SPPs modes. The results clearly reveal a many-body enhancement when N is larger than 2, especially for μ = 0.3 and 0.5 eV. A red-shift of resonance peak is also observed as a result of the many-body enhancement. The dashed line with arrow is added to demonstrate the region of many-body enhancement for μ = 0.5 eV. To study the deeper cause of the many-body enhancement, the integrand τ(ω,β) in Eq. (15) is plotted in Figs. 4(b)-4(d) for N = 2, 3, 100, respectively. The dashed lines in red denote the dispersion relation of the coupled graphene SPPs for two suspended graphene sheets [7,47,60]

1 + \frac{σ k_{z 0}}{ω ε_{0}} = \cot h (\frac{i k_{z 0} d}{2})

1 + \frac{σ k_{z 0}}{ω ε_{0}} = \tan h (\frac{i k_{z 0} d}{2})

and the dashed lines in green denote the dispersion relation of many-body coupling graphene SPPs, which is given as [52,61]

{(\frac{ε_{G}}{k_{z G}} - \frac{1}{k_{z 0}})}^{2} \cos (k_{z G} d_{G} - k_{z 0} d) - {(\frac{ε_{G}}{k_{z G}} + \frac{1}{k_{z 0}})}^{2} \cos (k_{z G} d_{G} + k_{z 0} d) - \frac{4 ε_{G}}{k_{z G} k_{z 0}} = 0

where k_z₀ and k_z_G are the z-component of the wave vector in vacuum and in graphene sheet. When N is reduced to 2, the τ(ω,β) contour matches well with the low-frequency symmetric and high-frequency antisymmetric branches for two suspended graphene sheets. In addition, the results obtained in Figs. 4(a) and 4(b) for N = 2 coincide well with the classical study of Ognjen [44]. An interesting phenomenon can be observed for triple sheets in Fig. 4(c) that another branch emerges inside the region surrounded by the dispersion relations for two suspended sheets. Moreover, the outer two branches extend outside the region without reaching the outer boundary for many-body coupling graphene SPPs. It is just the newly added branch of SPPs modes that causes the obvious three-body enhancement shown in Fig. 4(a). The τ(ω,β) contour for N = 100 in Fig. 4(d) exhibits a more remarkable effect of many-body enhancement that many-body SPPs modes fill up a single (ω,β) region. The outer boundary of the many-body graphene SPPs match well with the dispersion relation in Eq. (18). The contour of energy transmission coefficients ξ(ω,β) in Fig. 4(e) corresponding to τ(ω,β) in Fig. 4(d) gives a more intuitive display of the many-body graphene SPPs. The many-body enhancement can be divided into two parts, the extending part and the filling part, which are labeled with the unidirectional and bidirectional arrows, respectively. It is the spreading effect of ξ(ω,β) that results in the many-body enhancement and the red-shift of resonance peak of f_p(ω) in Fig. 4(a). Furthermore, many-body graphene SPPs cannot extend outside the boundary due to the multiple surface-states coupling proposed in [61]. The saturation point for large N observed in Fig. 4(a) can be well explained by the above phenomenon.

As can be seen from the above results, temperature distribution is quite different for various degrees of compactness due to the complexity of the system. To characterize the heat transfer ability of the whole system in a reasonable way, the heat transfer coefficients of the whole system at a specific temperature T are evaluated with another two HTC formulas

H_{t/s} : = \lim_{Δ T \to 0} \frac{φ_{1}}{Δ T} = \frac{1}{4 π^{2}} \sum_{j = 2}^{N} \int_{0}^{\infty} ℏ ω \frac{n_{1} - n_{j}}{Δ T} d ω \int_{ω / c}^{\infty} β (ω, β) ξ^{_{j, 1}} d β

where H_t and H_s are transient heat transfer coefficient and steady heat transfer coefficient, respectively. φ₁ denotes the net radiative heat flux lost by sheet 1, which is a sum of φ_1,_j where j runs from 2 to N. The sketches for the two HTC formulas are shown in Fig. 5(a). Assuming that the whole system is immersed at uniform temperature T and then sheet 1 is heat up from temperature T to T +

Δ

T. The temperature of sheet N is held constant at T. In a very little time, the losses in the sheets 2 to N-1 are too small and thereby their temperatures will not change, i.e. T₂ = T₃ = ... = T_N = T. In this situation, the heat transfer ability of the whole system is defined as H_t. Nevertheless, in a larger time scale, the system reaches its steady state in which the net radiative flux received by sheets 2 to N-1 vanishes, i.e., φ₂ = φ₃ = … = φ_N_-1 = 0 and φ₁ = φ_N. The sheets 2 to N-1 reach their equilibrium temperature T_eq,_j that varies between T +

Δ

T and T. Then the heat transfer ability of the whole system is revealed by H_s. In view of the above two HTC formulas, H_s will be undoubtly less than H_t and they are identical for N = 2. In the practical heat transfer procedures, the two HTC formulas have different applied scale corresponding to different heat transfer regime.

Fig. 5 (a) Sketches for THTC and SHTC, (b) THTC and SHTC as a function of separation distance d. (c) Contours of energy transmission coefficients and many-body dispersion relations.

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Transient and steady heat transfer coefficients as a function of separation distance d are given in Fig. 5(b) for N = 2, 5, 10 and 20, with T = 300 K and μ = 0.5 eV. In the strong near-field regime, H_t and H_s both have 1/d trend, while they convert to 1/d² in the weak near-field regime. The similar phenomenon has been observed for two suspended graphene sheets [45] and graphene nanodisk dimers [48]. N = 100 energy transmission coefficients ξ(ω,β) and the corresponding many-body dispersion relations (the boundary between the green and blue regions) for d = 1000 nm are presented in Fig. 5(c). The inset shows the corresponding results for two suspended graphene sheets. Combined with the results for d = 10 nm in Fig. 4, the ξ(ω,β) is found to shift to the low ω and β region as d increases. The inset in Fig. 5(b) shows the locally amplified figure of HTC in the ultra near-field regime. H_t for multilayers is larger than that of N = 2 due to many-body enhancement, while they almost coincidence in large d region. As discussed above, the many-body enhancement is mainly attributed to the filling part that fills the (ω,β) region surrounded by the symmetric and antisymmetric branches. When d is large, displayed in Fig. 5(c), the two branches shrink so much that the filling region is too small. Hence, there is no space for the many-body enhancement to work and then the H_t curves for N = 2 and multilayers are nearly coincident.

The detailed results for H_s and H_t as a function of the number of graphene sheets N are plotted in Fig. 6. There exist two different trends for THTC and SHTC with N. THTC owns a rapid growth when N is small and then is substantially unchanged when N is approximately greater than 20. This can be well explained by the saturation effect of many-body enhancement observed in Fig. 4. Different from THTC, SHTC steadily decreases with increasing N. According to the definition of SHTC (shown in Fig. 5(a)), the intermediate sheets act like a passive relay and the whole system can be viewed as two-parallel plates configuration. With the increase of N, the transmission distance $D = (N - 1) \cdot d$ increases and therefore H_s undoubtedly decreases according to the existing studies about two-body configuration. It is worth noting there exists a three-body enhancement when d is between 3 and 22 nm, as shown by the inset in Fig. 6 with a three-body enhancement ratio R = (H_s(3)- H_s(2))/ H_s(2). H_s(3) and H_s(2) are SHTC for three and two graphene sheets, respectively. The enhancement for three-body configuration has been observed for three plates in [62], while the enhancement is only valid when thickness of plate is approximately equal to the separation distance. Compared with that in [62], the three-body enhancement of multilayered graphene system is able to work in a larger scale. The low losses in the intermediate sheets and strong SPPs of graphene contribute to achieve the enhancement.

Fig. 6 THTC and SHTC as a function of sum number of graphene sheets N.

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To investigate how the chemical potentials of graphene impact the temperature distribution, the equilibrium temperature profiles for a system composed of N = 60 graphene sheets for different chemical potentials of graphene are given in Fig. 7. The results demonstrate that the regime of heat transport can be tuned by changing the chemical potentials of graphene. In Fig. 7(a), the equilibrium temperature profile for d = 10 nm indicates a diffusive regime when the chemical potentials of graphene is 0.1 eV. Furthermore, the heat regime undergoes a monotonic transition from diffusive to quasi-ballistic transport as the chemical potentials of graphene increases. The transition is mainly attributed to the different characteristics of graphene SPPs in the strong near-field regime, which has already been observed in Fig. 4(a). Different from that, the equilibrium temperature profile for d = 1000 nm in Fig. 7(b) lacks the transition. The equilibrium temperature profile for different chemical potentials of graphene all demonstrate the diffusive transport, which is mainly attributed to the weak heat transfer ability with the large separation distance. The increase of chemical potentials can hardly make up the decay of heat transfer ability which is caused by the large separation distance.

Fig. 7 Equilibrium temperature profile for a system composed of N = 60 graphene sheets for different chemical potentials of graphene: (a) d = 10 nm and (b) d = 1000 nm.

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4. Conclusions

In the present work, we study the NFRHT in a system composed of multiple graphene sheets based on the many-body theory. The equilibrium temperature distribution of the system is given for different separation distance. The internal temperature profile tends to be smooth, while temperature steps are observed near both boundaries of the system, especially for dense system. The unique temperature step is attributed to the weaker energy exchange in the boundary region due to the complexity of the system, which is quantitatively explained by RTC. Moreover, many-body enhancement is observed at low-frequency band which causes a red-shift of resonance peak of graphene SPPs modes. Many-body enhancement saturates as a result of the shrinkage effect of graphene SPPs modes. Compared with two-body configuration, the attenuation of heat transfer ability is weaker for many-body graphene system in steady state due to the relay effect caused by the intermediate sheets. We highlight that there exists a unique three-body enhancement as a result of the relay effect and strong SPPs of graphene. In the strong near-field regime, the heat regime undergoes a monotonic transition from diffusive to quasi-ballistic transport as the chemical potentials of graphene increases.

Funding

National Natural Science Foundation of China (51576053, 51806047).

Acknowledgments

The authors thank Philippe Ben-Abdallah for fruitful discussions and useful suggestions. A very special acknowledgment is also made to the editors and referees who make important comments to improve this paper.

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Near-field radiative heat transfer in multilayered graphene system considering equilibrium temperature distribution

Abstract

1. Introduction

2. Multilayered graphene system

3. Results and analysis

4. Conclusions

Funding

Acknowledgments

References

Cited By

Figures (7)

Equations (19)

Optics Express