Let us consider a graphene sheets and a SiO_{2} substrate separated by a vacuum gap with thickness *d*≪*λ*_{T}=*cћ*/*k*_{B}*T*. Assume that the free charge carriers in the graphene sheet move with drift velocity *v*≪*c* along the *x*-axis (*c* is the light velocity) relative to the dielectric plate. In addition to the intrinsic friction due to scattering against impurities and phonons, during drift motion of the electrons in the graphene sheet, on the electrons act the extrinsic friction due to the interaction with the surface phonon–polaritons in the dielectric. Assuming that the electrons in the graphene sheet can be described by the Fermi-liquid theory, a drift motion of the free charge carriers produces a similar modification of the reflection amplitudes and consequently and the fluctuating electromagnetic field as in the case of moving graphene sheet. In this case, the theory of the Casimir friction between moving bodies [10] can be used to calculate the friction force acting on the electrons in graphene due to the drift motion. The force that acts on the electrons in the sheet can be calculated from the Maxwell stress tensor *σ*_{xz}, evaluated at the surface of the sheet at *z*=0:

$$\begin{array}{l}{\sigma}_{xz}=\frac{1}{4\pi}{\displaystyle {\int}_{0}^{\infty}\text{d}\omega}\\ \text{\hspace{1em}}{\displaystyle \int \frac{{d}^{2}q}{{\mathrm{(}2\pi \mathrm{)}}^{2}}{\mathrm{[}<{E}_{x}{E}_{z}^{\ast}>+<{E}_{x}^{\ast}{E}_{z}>+<{B}_{x}{B}_{z}^{\ast}>+<{B}_{x}^{\ast}{B}_{z}>\mathrm{]}}_{z\text{\hspace{0.17em}}=\text{\hspace{0.17em}}0}}\end{array}$$(1)

where <…> denotes statistical average over the fluctuating electromagnetic field. According to [4], [6], [9], [10], [31], the Casimir friction *F*_{x}=*σ*_{xz} between the moving media is determined by

$${F}_{x}={F}_{xT}+{F}_{x0}\mathrm{,}$$(2)

where the temperature-dependent term *F*_{xT}, and the zero-temperature contribution *F*_{x}_{0} are given by

$$\begin{array}{l}{F}_{xT}=\frac{\hslash}{{\pi}^{3}}{\displaystyle {\int}_{0}^{\infty}\text{d}{q}_{y}}{\displaystyle {\int}_{0}^{\infty}\text{d}{q}_{x}{q}_{x}{e}^{-2qd}}\\ \text{\hspace{1em}}\{{\displaystyle {\int}_{0}^{\infty}\text{d}\omega \mathrm{(}\frac{\text{Im}{R}_{1}\mathrm{(}\omega \mathrm{)}\text{Im}{R}_{2}\mathrm{(}{\omega}^{+}\mathrm{)}}{|1-{e}^{-2qd}{R}_{1}\mathrm{(}\omega \mathrm{)}{R}_{2}\mathrm{(}{\omega}^{+}\mathrm{)}{|}^{2}}\times \mathrm{[}{n}_{1}\mathrm{(}\omega \mathrm{)}-{n}_{2}\mathrm{(}{\omega}^{+}\mathrm{)}]+\mathrm{(}1\leftrightarrow 2\mathrm{)}\mathrm{}\mathrm{)}}\\ \text{\hspace{1em}}-{\displaystyle {\int}_{0}^{{q}_{x}v}\text{d}\omega \mathrm{(}\frac{\text{Im}{R}_{1}\mathrm{(}\omega \mathrm{)}\text{Im}{R}_{2}\mathrm{(}{\omega}^{-}\mathrm{)}}{|1-{e}^{-2qd}{R}_{1}\mathrm{(}\omega \mathrm{)}{R}_{2}\mathrm{(}{\omega}^{-}\mathrm{)}{|}^{2}}{n}_{1}\mathrm{(}\omega \mathrm{)}+\mathrm{(}1\leftrightarrow 2\mathrm{)}\mathrm{}\mathrm{)}}\}\mathrm{,}\end{array}$$(3)

$$\begin{array}{l}{F}_{x0}=-\frac{\hslash}{2{\pi}^{3}}{\displaystyle {\int}_{0}^{\infty}\text{d}{q}_{y}}{\displaystyle {\int}_{0}^{\infty}\text{d}{q}_{x}{q}_{x}{e}^{-2qd}}\\ \text{\hspace{1em}}{\displaystyle {\int}_{0}^{{q}_{x}v}\text{d}\omega \mathrm{(}\frac{\text{Im}{R}_{1}\mathrm{(}\omega \mathrm{)}\text{Im}{R}_{2}\mathrm{(}{\omega}^{-}\mathrm{)}}{|1-{e}^{-2qd}{R}_{1}\mathrm{(}\omega \mathrm{)}{R}_{2}\mathrm{(}{\omega}^{-}\mathrm{)}{|}^{2}}+\mathrm{(}1\leftrightarrow 2\mathrm{)}\mathrm{}\mathrm{)}}\text{\hspace{0.17em}},\end{array}$$(4)

where *n*_{i}(*ω*)=[exp(*ћω*/*k*_{B}*T*_{i})−1]^{−1} (*i*=1, 2), *T*_{i} is the temperature of *i*-th body, *R*_{i} is the reflection amplitude for surface *i* for *p*-polarised electromagnetic waves, and *ω*^{±}=*ω*±*q*_{x}v. The symbol (1↔2) denotes the terms that are obtained from the preceding terms by permutation of 1 and 2. The reflection amplitude for a graphene is determined by [37]

$${R}_{g}=\frac{{\epsilon}_{g}-1}{{\epsilon}_{g}}\mathrm{,}$$(5)

where the dielectric function of the sheet is determined by *ε*_{g}(*ω*, *q*)=1+*v*_{q}Π(*ω*, *q*) where *v*_{q}=2*πe*^{2}/(*qε*) is the 2D Coulomb interaction and Π(*ω*, *q*) is the 2D polarizability.

In the finite lifetime generalisation according to the Mermin approximation [38], the dielectric function is determined by

$$\epsilon \mathrm{(}\omega \mathrm{,}\text{\hspace{0.17em}}q\mathrm{)}\approx 1+\frac{\mathrm{(}\omega +i\gamma \mathrm{)}\mathrm{(}{\epsilon}_{0}\mathrm{(}\omega +i\gamma \mathrm{,}\text{\hspace{0.17em}}q\mathrm{)}-1\mathrm{)}\mathrm{}}{\omega +i\gamma \mathrm{(}{\epsilon}_{0}\mathrm{(}\omega +i\gamma \mathrm{,}\text{\hspace{0.17em}}q\mathrm{)}-1\mathrm{)}/\mathrm{(}{\epsilon}_{0}\mathrm{(}\mathrm{0,}\text{\hspace{0.17em}}q\mathrm{)}-1\mathrm{)}\mathrm{}}\mathrm{,}$$(6)

where *ε*_{0}(*ω*, *q*) is the RPA dielectric function and *γ* the damping parameter. In the study below, we used the dielectric function of graphene, which was calculated recently within the random-phase approximation (RPA) [39], [40]. The small (and constant) value of the graphene Wigner–Seitz radius *r*_{s} indicates that it is a weakly interacting system for all carries densities, making the RPA an excellent approximation for graphene (RPA is asymptotically exact in the *r*_{s}≪1 limit). The dielectric function is an analytical function in the upper half-space of the complex *ω*-plane:

$$\begin{array}{l}{\epsilon}_{0}\mathrm{(}\omega \mathrm{,}\text{\hspace{0.17em}}q\mathrm{)}=1+\frac{4{k}_{F}{e}^{2}}{\hslash {v}_{F}q}-\frac{{e}^{2}q}{2\hslash \sqrt{{\omega}^{2}-{v}_{F}^{2}{q}^{2}}}\\ \text{\hspace{1em}}\left\{G\mathrm{(}\frac{\omega +2{v}_{F}{k}_{F}}{{v}_{F}q}\mathrm{)}-G\mathrm{(}\frac{\omega -2{v}_{F}{k}_{F}}{{v}_{F}q}\mathrm{)}-i\pi \right\}\mathrm{,}\end{array}$$(7)

where

$$G\mathrm{(}x\mathrm{)}=x\sqrt{{x}^{2}-1}-\mathrm{ln}\mathrm{(}x+\sqrt{{x}^{2}-1}\mathrm{}\mathrm{)}\mathrm{,}$$(8)

where the Fermi wave vector *k*_{F}=(*πn*)^{1/2}, *n* is the concentration of charge carriers, the Fermi energy *ϵ*_{F}=*ћv*_{F}k_{F}, *v*_{F}≈10^{6} m/s is the Fermi velocity. The damping parameter *γ* is due to electron scattering against impurities and acoustic phonons in graphene sheet and can be expressed through the low-field mobility *μ*: *γ*=*ev*_{F}/(*ћk*_{F}μ). Scattering of the graphene carries by the acoustic phonons of graphene places an intrinsic limits on the low-field room temperature (*T*_{0}=300 K) mobility, given by *μ*_{0}=20 m^{2}/Vs at the graphene carriers density 10^{16} m^{−2} (see [41]), which gives *γ*=8·10^{11}s^{−1}. At other temperatures, the mobility can be obtained using the relation *μ*=*μ*_{0}*T*_{0}/*T*.

The reflection amplitude for the substrate

$${R}_{d}=\frac{{\u03f5}_{d}-1}{{\u03f5}_{d}+1}\mathrm{,}$$(9)

where *ϵ*_{d} is the dielectric function for substrate. The dielectric function of amorphous SiO_{2} can be described using an oscillator model [42]

$$\u03f5\mathrm{(}\omega \mathrm{)}={\u03f5}_{\infty}+{\displaystyle \sum _{j\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1}^{2}\frac{{\sigma}_{j}}{{\omega}_{\mathrm{0,}j}^{2}-{\omega}^{2}-i\omega {\gamma}_{j}}}\mathrm{,}$$(10)

where parameters *ω*_{0,j} , *γ*_{j}, and *σ*_{j} were obtained by fitting the actual *ϵ* for SiO_{2} to the above-mentioned equation and are given by *ϵ*_{∞}=2.0014, *σ*_{1}=4.4767×10^{27} s^{−2}, *ω*_{0,1}=8.6732×10^{13} s^{−1}, *γ*_{1}=3.3026×10^{12} s^{−1}, *σ*_{2}=2.3584× 10^{28} s^{−2}, *ω*_{0,2}=2.0219×10^{14} s^{−1}, and *γ*_{1}=8.3983×10^{12} s^{−1}.

For *v*<*dk*_{B}T/*ћ* (at *d*=1 nm and *T*=300 K for *v*<4·10^{4} m/s), the main contribution to the friction (3) depends linearly on the sliding velocity *v* so that the friction force *F*_{xT}=Γ*v*, where at *T*_{1}=*T*_{2}=*T* the friction coefficient Γ is given by

$$\begin{array}{l}\Gamma =\frac{{\hslash}^{2}}{8{\pi}^{2}{k}_{B}T}{\displaystyle {\int}_{0}^{\infty}\frac{\text{d}\omega}{{\text{sinh}}^{\text{2}}\mathrm{(}\frac{\hslash \omega}{2{k}_{B}T}\mathrm{)}}}\\ \text{\hspace{1em}}{\displaystyle {\int}_{0}^{\infty}\text{d}q{q}^{3}{e}^{-2qd}\frac{\text{Im}{R}_{1}\mathrm{(}\omega \mathrm{)}\text{Im}{R}_{2}\mathrm{(}\omega \mathrm{)}}{|1-{e}^{-2qd}{R}_{1}\mathrm{(}\omega \mathrm{)}{R}_{2}\mathrm{(}\omega \mathrm{)}{|}^{2}}}\mathrm{.}\end{array}$$(11)

Let us assume that in the rest reference frame in which there is no drift motion of electrons, an electron–hole pair excitation has the energy *ω*_{eh}(*q*) and momentum **q**, then in the laboratory reference frame, in which the electron system is moving with drift velocity *v*, due to the Doppler effect the energy of this excitation will be equal to *ω*_{eh}(*q*)−*q*_{x}v. For *v*>*ω*_{eh}(*q*)/*q*_{x}, the excitation energy will be negative. Thus, for velocities larger than critical velocity (*v*_{cr}=*ω*_{eh}(*q*)/*q*_{x}) as a result of such excitation, the excitation can be created with energy *ω*_{ph}(*q*)=*q*_{x}v−*ω*_{eh}(*q*)>0. This phenomenon is reminiscent of the Cherenkov radiation that arises when an electron moves in medium with a velocity exceeding the light velocity in the medium. The difference between the two phenomena is that the Cherenkov radiation is connected with the radiation of electromagnetic waves, but the excitation that arises from drift motion of the electron in the graphene sheet result to the excitations of the surface phonon–polariton. Resonance arises when the energy gain resulting from the excitation in the moving electron system in graphene sheet *ω*_{ph}(*q*)=*q*_{x}v−*ω*_{eh}(*q*)>0 will create the surface phonon–polariton with the energy *ω*_{s} in the dielectric. In the case of graphene, the energy of the electron–hole pair excitation *ω*_{eh}(*q*)≈*v*_{F}q, where *v*_{F} is the Fermi velocity. Thus, resonance arises when *q*_{x}v≈2*v*_{F}q+*ω*_{s}, which requires that *v*>*v*_{F}+*ω*_{s}d≈10^{6} m/s.

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