Caroli formalism in near-field heat transfer between parallel graphene sheets

In this work we conduct a close-up investigation into the nature of near-field heat transfer (NFHT) of two graphene sheets in parallel-plate geometry. We develop a fully microscopic and quantum approach using the nonequilibrium Green's function method. A Caroli formula for heat flux is proposed and numerically verified. We show that our near-field-to-black-body heat flux ratios generally exhibit $1/d^{\alpha }$ dependence, with an effective exponent $\alpha \approx 2.2$, at long distances exceeding 100 nm and up to one micron; in the opposite $d\to 0$ limit, the values converge to a range within an order of magnitude. We justify this feature by noting it is owing to the breakdown of local conductivity theory, which predicts a $1/d$ dependence. Furthermore, from the numerical result, we find that in addition to thermal wavelength ${\lambda }_{th}$ a shorter distance scale $\sim 10-100$ nm, comparable to the graphene thermal length $\left(\hslash v_F/k_{B}T\right)$ or Fermi wavelength $\left(k_F^{-1}\right)$, marks the transition point between the short- and long-distance transfer behaviors; within that point, a relatively large variation of heat flux in response to doping level becomes a typical characteristic. The emergence of such large variation is tied to relative NFHT contributions from the intra- and interband transitions. Beyond that point, scaling of thermal flux $\propto 1/d^{\alpha }$ can be generally observed.

Figure 1

The schematic showing heat transfer across vacuum gap of distance $d$ between two graphene sheets.

Figure 2

A comparison of the results calculated with Caroli formula and Eq. (2). The close match is a proof of equivalence.

Figure 3

The red dashed line indicates the asymptotic behavior $1/d^{\alpha },\alpha \approx 2.2$ as $d$ becomes of micrometer scale. The blue dashed line indicates the saturation of the curve when $d$ approaches zero. $J_{zBB}=56244$ W ${\text{m}}^{-2}$.

Figure 4

(a) Heat flux ratio in different temperatures $(\mu =0.1$ eV). (b) The short-distance zoom-in.

Figure 5

The heat flux in different chemical potentials.

Figure 6

Heat flux ratio in different temperatures $(\mu =0.7$ eV).

Figure 7

Comparison of heat flux calculated using the full RPA and local conductivity models.

Figure 8

Match in plasmon dispersion between the full RPA and local conductivity models at $d=10$ nm $(\mu =0.7$ eV). Transmission $T\left(\omega \right)$ plots are weighted by $q\hslash \omega (N_1-N_2)$. (a) The full RPA calculation. (b) Local conductivity calculation. Cyan and green dashed lines stand for acoustic and optical modes in $q\to 0$ limit, respectively.

Figure 9

Disparity in plasmon dispersion between the full RPA and local conductivity models at $d=1\text{Å}$. Transmission $T\left(\omega \right)$ plots are weighted by $q\hslash \omega (N_1-N_2)$. (a),(c) The full RPA calculations with $\mu =0.1$ and 0.7 eV, respectively. (b),(d) Local conductivity calculations with $\mu =0.1$ and 0.7 eV, respectively. Cyan and green dashed lines stand for acoustic and optical modes in $q\to 0$ limit, respectively.