# Dielectric functions for near-field thermal radiation

Besides the fluctuation-dissipation theorem and dyadic Green’s function, the dielectric function of materials should also be discussed to better understand near-field thermal radiation and its interactions with materials. For thermal radiation, non-linear effects are negligible and hence the polarization P is linearly related to the electric field as follows [1]:

${\textbf P}({\textbf x},\omega ){\rm =}\varepsilon _{0} \chi _{e} ({\textbf x},\omega ){\textbf E}({\textbf x},\omega )$

(1)

where $\chi _{e} ({\textbf x},\omega )$ is the electric susceptibility of the medium and $\varepsilon _{0}$ is the permittivity of vacuum. The electric susceptibility indicates the degree of polarization of a dielectric material in response to an applied electric field, depending on the microscopic structure of the medium. The electric displacement vector D can be expressed as

${\textbf D}({\textbf x}, \omega ){\rm =}\varepsilon (\omega ){\textbf E}({\textbf x},\omega )$

(2)

where $\varepsilon (\omega )$ is the dielectric function or relative permittivity of the medium and is related with the electric susceptibility as $\varepsilon (\omega ){\textbf =}\varepsilon _{0} [1+\chi _{e} (\omega )]$. Since we have assumed a local form of the dielectric function, the spatial dependence term drops out from the expressions of susceptibility and relative permittivity. This local assumption remains valid for near-field thermal radiation unless the vacuum gap is extremely small (less than 1 nm distance). In the extreme proximity, the dielectric function is not local and its dependence of wavevector must be considered [2]. Recently, Chapuis et al. [3] used two different non-local dielectric function models to calculate the near-field heat transfer between two semi-infinite gold plates and compared their results with the heat flux calculated using the Drude model for gold. They found that the non-local dielectric function saturates the near-field thermal radiation, whilst local dielectric function erroneously diverges the thermal radiation as the vacuum gap approaches zero.

Equation (2) represents the displacement of charges inside the material upon the incidence of electric waves. Thus the dielectric function is the key property in understanding the light-matter interactions, and needs to be further discussed. Under the local assumption, the following sections will discuss two models of the dielectric function, the Drude model for metals (and semiconductors) and the Lorentz model for dielectrics, respectively.

#### Drude model

The Drude model describes the frequency-dependent conductivity of metals and can also be extended to free-carriers in semiconductors. In a metal, electrons in the outermost orbits are “free” to move in accordance with the external electric field. The dielectric function of a metal can be modeled by considering the electron movement under the electric field and is related to the conductivity by [4]

$\varepsilon (\omega )=\varepsilon '+i\varepsilon ''=\left(n+{\textbf i}\kappa \right)^{2} =\varepsilon _{\infty } -\frac{{\sigma _{0} \mathord{\left/ \right. } \tau } }{\varepsilon _{0} \left(\omega ^{2} +{{\textbf i}\omega \mathord{\left/ \right. } \tau } \right)}$

(3)

where ${\varepsilon _{\infty }}$ accounts for high-frequency contributions, τ is the relaxation time (inverse of scattering rate), σ0 is the dc conductivity, and n and κ are the refractive index and extinction coefficient respectively. Based on Eq. (3), the real and imaginary parts of the dielectric function can be expressed as ${\varepsilon '=n^{2} -\kappa ^{2}}$ and $\varepsilon ''=2n\kappa$ , respectively. The plasma frequency is defined as $\omega _{{\rm p}} =\sqrt{{\sigma _{0} \mathord{\left/ \right. } \tau \varepsilon _{0} } }$ and is in the ultraviolet region for most metals. When $\omega <\omega _{{\rm p}} \, , n<\kappa$ and $\varepsilon '$ becomes negative. At very low frequencies, the real part of the dielectric function is much smaller than the imaginary part and, therefore, $n\approx \kappa$. Generally speaking, metals become highly reflective in the visible and infrared regions.

#### Lorentz model for dielectrics

Unlike metals, the electrons in a dielectric are bound to molecules and cannot move freely. In contrast to free electrons, bound charges experience a restoring force given by the spring constant in addition to the damping force given by the scattering rate. There exist different kinds of oscillators in a real material, such as bound electrons or lattice ions. The response of a single-charge oscillator to a time-harmonic electric field can be extended to a collection of oscillators. Assuming N types of oscillators in a dielectric, the corresponding dielectric function can be given as [4]

$\varepsilon (\omega )=\varepsilon _{\infty } +\sum _{j}^{N}\frac{\omega _{{\rm p},j}^{{\rm \; }2} }{\omega _{0,j}^{2} -\omega ^{2} -{{\rm i}\omega \mathord{\left/ { { \tau _{j} }} \right. } } }$

(4)

where $\omega _{{\rm p},j}^{{\rm \; }} \, , \omega _{0,j}^{{\rm \; }}$ , and $\tau _{j}^{\rm \; }$ may be viewed as the plasma frequency, resonance frequency and the relaxation time of the jth oscillator, respectively. Since the parameters for the Lorentz model are more difficult to be modeled as compared to those for the Drude model, they are considered as adjustable parameters that are determined from fitting. It can be observed from Eq. (4) that for frequencies far greater or lower than the resonance frequency, the extinction coefficient becomes negligible and the dielectrics are completely transparent. Absorption is appreciable only with an interval (1 / τj) around the resonance frequency. Therefore, the dielectric becomes highly reflective near the resonance frequency, and the radiation inside the material is rapidly attenuated or dissipated. The spectral region with a large imaginary part of the dielectric function is also called the region of resonance absorption.

## References

[1] Griffiths, D. J., 1999, Introduction to Electrodynamics, 3rd edn., Prentice Hall, Upper Saddle River, NJ.

[2] Joulain, K., 2008, “Near-Field Heat Transfer: A Radiative Interpretation of Thermal Conduction,” Journal of Quantitative Spectroscopy and Radiative Transfer, 109, pp. 294-304.

[3] Chapuis, P. O., Volz, S., Henkel, C., Joulain, K., and Greffet, J.-J., 2008, “Effects of Spatial Dispersion in Near-Field Radiative Heat Transfer between Two Parallel Metallic Surfaces,” Physical Review B, 77, p.035431.

[4] Zhang, Z. M., 2007, Nano/Microscale Heat Transfer, McGraw-Hill, New York.