## 2 Nonequilibrium Dispersion Force on a Nanoparticle on the Source Side of Gapped Graphene  
We consider the dispersion (Casimir-Polder) force acting on a spherical nanoparticle of radius \(R\) spaced above a graphene sheet at a separation \(a\gg R\). The consideration of nanoparticles of other types (for instance, having a nonspherical shape) would need a more complicated theory using the scattering-matrix approach [58]. The area of graphene sheet is taken to be much larger than the separation to a nanoparticle squared. It is assumed that, at all temperatures \(T\) under consideration, it holds that \(R\ll\hbar c/(k_{B}T)\), where \(k_{B}\) is the Boltzmann constant (for instance, at the environmental temperature \(T_{E}=300\) K, one has \(\hbar c/(k_{B}T)\approx 7.6\)\(\upmu\)m). Under this condition, within the range of separations \(a\) considered below, the nanoparticle can be described by the static polarizability \(\alpha(0)\), which takes the form [61]  
\[\alpha(0)=R^{3}\frac{\varepsilon(0)-1}{\varepsilon(0)+2},\qquad\alpha(0)=R^{3} \tag{1}\]  
for dielectric and metallic nanoparticles, respectively, where \(\varepsilon(0)\) is the static dielectric permittivity of a nanoparticle material. Below, we assume that nanoparticles have the same temperature \(T_{E}\) as the environment, whereas the graphene sheet has the temperature \(T_{g}\), which is either lower or higher than \(T_{E}\). As distinct from [67], where the case of a pristine graphene was considered, here the graphene sheet is characterized by a nonzero mass-gap parameter \(\Delta=2 mv_{F}^{2}\), where \(m\) is the mass of quasiparticles and \(v_{F}\approx c/300\) is the Fermi velocity [21; 68; 69]. The nonequilibrium dispersion force acting on a nanoparticle is represented in the form [56; 58]  
\[F_{\text{neq}}(a,\Delta,T_{E},T_{g})=F_{M}(a,\Delta,T_{E},T_{g})+F_{r}(a, \Delta,T_{E},T_{g}), \tag{2}\]  
where \(F_{M}(a,\Delta,T_{E},T_{g})\) can be expressed as a sum over the discrete Matsubara frequencies, much as the equilibrium Casimir-Polder force [70; 71], whereas \(F_{r}(a,\Delta,T_{E},T_{g})\) is the contribution, which is given by an integral over the real frequency axis. In fact, the effects of nonequilibrium contribute to both terms on the right-hand side of (2). Because of this, it is not reasonable to call the first of them \"equilibrium\" and the second--\"nonequilibrium\", as occurs in the literature. Moreover, the division of \(F_{\text{neq}}\) into \(F_{M}\) and \(F_{r}\) is not unique and can be made in a number of ways. Below, we use the same division as in [67]. In this case, the first term in (2) is given by [67]  
\[F_{M}(a,\Delta,T_{E},T_{g})=-\frac{2k_{B}T_{E}\alpha(0)}{c^{2}}\sum_{l=0}^{ \infty}{}^{\prime}\int\limits_{0}^{\infty}k\,dke^{-2\alpha q_{l}(k)}\Big{\{} \Big{[}2q_{l}^{2}(k)c^{2}-\xi_{E,l}^{2}\Big{]}\,R_{\text{TM}}(i\xi_{E,l},k; \Delta,T_{g})-\xi_{E,l}^{2}R_{\text{TE}}(i\xi_{E,l},k;\Delta,T_{g})\Big{\}}. \tag{3}\]  
Here, \(k\) is the magnitude of the wave vector component along the graphene sheet, \(q_{l}^{2}(k)=k^{2}+\xi_{E,l}^{2}/c^{2}\), \(\xi_{E,l}=2\pi k_{B}T_{E}l/\hbar\) with \(l=0,\,1,\,2,\,\ldots\) are the Matsubara frequencies at the environmental temperature \(T_{E}\), and the prime on the summation sign multiples the term with \(l=0\) by the factor \(1/2\). The quantities \(R_{\text{TM}}\) and \(R_{\text{TE}}\) are the reflection coefficients of the electromagnetic fluctuations on a graphene sheet for the transverse magnetic (TM) and transverse electric (TE) polarizations calculated at the pure imaginary Matsubara frequencies \(\omega=i\xi_{E,l}\), but at the temperature of graphene \(T_{g}\).