## 3 Polarization Tensor in the Area of Evanescent Waves  
The nonequilibrium dispersion force (2) acting on nanoparticles on the source of a gapped graphene sheet can be computed by Equations (3)-(7). For this purpose, one should know the component of the polarization tensor \(\Pi_{00}\) and the combination of its components \(\Pi\) defined in (5) for a graphene sheet with the nonzero mass-gap parameter \(\Delta\). As mentioned in Section 1, the polarization tensor of graphene was found in [50; 51; 52; 53] in the framework of the Dirac model. In doing so, Ref. [50] was devoted to the case of zero-temperature, \(T=0\). In [51], the polarization tensor of graphene was obtained at nonzero temperatures at all discrete Matsubara frequencies. These results, however, did not admit a continuation to the entire plane of complex frequencies and, specifically, were inapplicable along the real frequency axis. Thus, they can be used for the calculation of the equilibrium Casimir and Casimir-Polder forces and the contribution \(F_{M}\) to the nonequilibrium force, but not the contribution \(F_{r}\). The polarization tensor of graphene with nonzero \(\Delta\) valid over the entire plane of complex frequencies was derived in [52], where the most attention was paid to the region of propagating waves \(k<\omega/c\) in connection with the topical applications to the reflectivity [73; 74; 75; 76] and conductivity [77; 78; 79; 80] properties of graphene. Below, we present a more detailed exposition of the results of [52], which are relevant to the area of evanescent waves (\(k>\omega/c\)) that determines the contribution (6) to the nonequilibrium dispersion force. Before dealing with the polarization tensor, a few remarks concerning the area of application of this quantity for the calculation of dispersion forces are in order. In [50; 51; 52; 53], the polarization tensor of graphene was derived in the framework of the Dirac model. This model provides a physically adequate description of graphene at energies below approximately 3 eV [81]. Thus, the energies \(\hbar\omega\) providing the major contribution to the dispersion force should be below this limit. The characteristic frequency determining the dispersion force is \(\omega_{c}=c/(2a)\)[70; 71]. It can easily be seen that the respective characteristic energy \(\hbar\omega_{c}\) is below 1 eV at all separations \(a>100\) nm. Therefore, at separations, say, \(a>200\) nm, one can safely use the Dirac model and its consequences in calculations of dispersion forces. This was confirmed by the fact that measurements of the dispersion interaction with graphene were found to be in very good agreement with theoretical predictions computed using the polarization tensor [82; 83]. Now, we present explicit expressions for the quantities \(\Pi_{00}\) and \(\Pi\) in the frequency region of evanescent waves \(\omega/c<k\). Similar to [30], we present these quantities as the sums of two contributions  
\[\Pi_{00}(\omega,k;\Delta,T_{g})=\Pi_{00}^{(0)}(\omega,k;\Delta)+\Pi_{00}^{(1)}( \omega,k;\Delta,T_{g}),\]  
\[\Pi(\omega,k;\Delta,T_{g})=\Pi^{(0)}(\omega,k;\Delta)+\Pi^{(1)}(\omega,k;\Delta,T_{g}). \tag{8}\]  
Here, the contributions with an upper index (0) are defined at zero-temperature, \(T=0\), whereas the quantities with an upper index (1) have the meaning of the thermal corrections. In doing so, both contributions depend on the mass-gap parameter of graphene \(\Delta\). With vanishing temperature, both \(\Pi_{00}^{(1)}\) and \(\Pi^{(1)}\) go to zero. The analytic continuation of the polarization tensor of graphene to the frequency region of evanescent waves takes different forms in the interval  
\[\frac{\omega}{c}<k\leqslant\frac{\omega}{v_{\rm F}}\approx 300\frac{\omega}{c} \tag{9}\]  
and in the interval  
\[300\frac{\omega}{c}\approx\frac{\omega}{v_{\rm F}}<k<\infty. \tag{10}\]  
First, we consider the interval (9), which is often called the plasmonic region [84].