All media are surrounded by a fluctuating electromagnetic field due to the thermal and quantum fluctuations of the current and charge densities inside them. These electromagnetic fluctuations are related to one of the most fundamental phenomena in nature, namely Brownian motion. At present, in the connection with new experiments, the interest is reviving for the Casimir physics that includes the Casimir–van der Waals forces , , , the Casimir friction with its limiting case – quantum friction , , and the near-field radiative heat transfer , . Figure 1 illustrates the connection between the Casimir physics and Brownian motion.
At present, the Casimir friction is attracting a lot of attention due to the fact that it is one of the mechanisms of noncontact friction between bodies in the absence of direct contact . The noncontact friction determines the ultimate limit to which the friction force can be reduced and, consequently, also the force fluctuations because they are linked to friction via the fluctuation–dissipation theorem. The force fluctuations (and hence friction) are important for the ultrasensitive force detection.
In noncontact friction, the bodies are separated by a potential barrier thick enough to prevent electrons or other particles with a finite rest mass from tunnelling across it, but allowing interaction via the long-range electromagnetic field, which is always present in the gap between bodies. Noncontact friction is investigated using an atomic force microscope, a probe which is an extremely sharp tip attached to the elastic arm (cantilever). When scanning a surface, the tip of the cantilever slides above it at short distance. This device makes it possible to register the strength of the normal and lateral force components of the interaction between the surface and the probe-tip. Various mechanisms of noncontact friction are schematically illustrated in Figure 2. If the bodies are in relative motion, the fluctuating current density inside bodies will give rise to a friction which is denoted as the Casimir friction. The origin of the Casimir friction is closely connected with the van der Waals–Casimir interaction. The van der Waals interaction arises when an atom or molecule spontaneously develops an electric dipole moment due to quantum fluctuations. The short-lived atomic polarity can induce a dipole moment in a neighbouring atom or molecule some distance away. The same is true for extended media, where thermal and quantum fluctuation of the current density in one body induces a current density in other body; the interaction between these current densities is the origin of the Casimir interaction. When two bodies are in relative motion, the induced current will lag slightly behind the fluctuating current inducing it, and this is the origin of the Casimir friction (Figure 2a). The van der Waals–Casimir interaction is mostly determined by exchange of virtual photons between the bodies (connected with quantum fluctuations) and does not vanish even at zero temperature. The contribution from real photons (connected with thermal fluctuations) becomes important only at large separations between bodies. On the other hand, the Casimir friction is determined by thermal or quantum fluctuations at small or large velocities, respectively. The Casimir friction at zero temperature is denoted as quantum friction. The Casimir friction is closely related with the Doppler effect (see Figure 3). It was shown by Pendry  that the reflection amplitudes from moving metal surface are modified due to the Doppler effect. The drift motion of the charge carries in graphene will result in a modification of dielectric properties (and the Casimir force) of graphene due to the Doppler effect  (see Figure 3). If in one of two parallel graphene sheets, an electric current is induced, then the electromagnetic waves, radiated by the graphene sheet without an electric current, will experience a frequency Doppler shift in the reference frame moving with the drift velocity v of electrons in the other graphene sheet: ω′=ω−qxv, where qx is the parallel to the surface component of the momentum transfer. The same is true for the waves emitted by the other graphene sheet. Due to the frequency dependence of the reflection amplitudes, the electromagnetic waves will reflect differently in comparison to the case when there is no drift motion of electrons, and this will give rise to the change of the Casimir force. The effect of the drift motion of charge carriers in one of the graphene sheet, on the thermal Casimir force between graphene sheets, was investigated in .
The presence of an inhomogeneous tip-sample electric fields is difficult to avoid, even under the best experimental conditions . For example, even if both the tip and the sample were metallic single crystals, the tip would still have corners, and more than one crystallographic plane exposed. The presence of atomic steps, adsorbates, and other defects will also contribute to the spatial variation of the surface potential. This is referred to as “patch effect”. The surface potential can also be easily changed by applying a voltage between the tip and the sample. An inhomogeneous electric field can also be created by charged defects embedded in a dielectric sample. The relative motion of the charged bodies will produce friction which will be denoted as the electrostatic friction (Figure 2b).
A moving tip will induce the dynamical deformation of surface of substrate due to the Casimir or electrostatic interaction between the tip and surface. This dynamical deformation will excite phonons in the substrate which are responsible for phononic mechanism of noncontact friction (Figure 2c).
A moving tip will induce a drag force acting on the adsorbates on the substrate surface due to the Casimir or electrostatic interaction between the tip and adsorbates. This drag force results in a drift motion of the adsorbates relative to the substrate, and dissipation due to friction between adsorbates and substrate. This mechanism of dissipation is responsible for the adsorbate drag friction (Figure 2d).
Friction is usually a very complicated process. It has recently been shown that two noncontacting bodies moving relative to each other experience a friction due to the relative motion of the thermal and quantum fluctuations inside the bodies , , , . This friction is denoted as the Casimir friction. This friction exists even at T=0 K where it is due to the relative motion of quantum fluctuations and is denoted as quantum friction. For several decades, physicists have been intrigued by the idea of quantum friction. However, until recently, there was no experimental evidence for or against this effect, because the predicted friction forces are very small, and precise measurements of quantum friction are incredibly difficult with present technology. The Casimir friction has been studied in the configurations plate plate , , , , ,  and neutral particle plate , , , , , , , , , , , , . While the predictions of the theory for the Casimir forces were verified in many experiments , the detection of the Casimir friction is still challenging problem for experimentalists. However, the frictional drag between quantum wells ,  and graphene sheets ,  and the current–voltage dependence of nonsuspended graphene on the surface of the polar dielectric SiO2  were accurately described using the theory of the Casimir friction , , . At present, frictional drag experiments , , ,  have been performed only for weak electric fields, when the induced drift motion of the free carriers is smaller than the threshold velocity for quantum friction. Thus, in these experiments, the frictional drag is dominated by the contributions from thermal fluctuations. However, the measurements of the current–voltage dependence  were performed for high electric field, where the drift velocity is above the threshold velocity and where the frictional drag is dominated by quantum fluctuations , . Application of the theory of the Casimir friction to these experiments is based on the assumption that the Fermi-liquid theory is valid for electrons in graphene. In this case, the graphene with the drift motion of electrons with the drift velocity vDrift produces the same fluctuating electromagnetic field and has the same reflection amplitudes as graphene moving with velocity v=vDrift. The hydrodynamic model for electrons in a medium was used in  to calculate the fluctuating electromagnetic field produced by the medium in the presence of the drift motion of the electrons.
Graphene, isolated monolayer of carbon, which was obtained very recently , consists of carbon atoms densely packed into a two-dimensional honeycomb crystal lattice. The unique electronic and mechanical properties of graphene are actively studied both theoretically and experimentally partly because of their importance for fundamental physics and also because of its possible technological applications , , , . In particular, the valence band and conduction band in graphene touch each other at one point named the Dirac point. Near this point, the energy spectrum for electrons and holes has a linear dispersion. Due to this linear (or “conical”) dispersion relation electrons and holes near this point behave like relativistic particles described by the Dirac equation for massless fermions.
Graphene can also be useful for the detection of quantum friction. Consider graphene located on the surface of, for example, the polar dielectric SiO2, or nearby of a second graphene sheet. In this case, the charge carriers in graphene experience additional friction due to interaction with the optical phonons in the dielectric or the electrons in the other graphene sheet. Due to the high mobility, in a strong electric field, the electrons in graphene can move with very high drift velocities (~106 m/s). At such velocities, the main contribution to the friction will arise from quantum fluctuations. Thus, quantum friction can be detected by measuring the high electric field transport properties of graphene on a polar dielectric substrate or by measuring the voltage induced by friction in a second nearby graphene sheet. In the frictional drag experiment, the Casimir friction force between the charge free carriers in the 2D structures mediated by a current density in the one 2D structure induces the electric field in the other 2D structure, which can be measured. For the graphene sheet situated nearby the polar dielectric substrate, the Casimir friction force between the charge-free carries in graphene and the surface phonon polaritons in dielectric gives rise to the change of the resistivity of graphene that also can be measured. So far, the Casimir friction was detected only using the electrical effects, which it produces. Thus, the frictional drag effect can only be observed between the two 2D conducting structures and the electrical transport in graphene can only be measured for nonsuspended graphene when the heat conductance between graphene and underlying dielectric is high. For reviews of the Casimir friction, see .
2 The Casimir Friction in Graphene Systems
Let us consider a graphene sheets and a SiO2 substrate separated by a vacuum gap with thickness d≪λT=cћ/kBT. 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  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 σxz, evaluated at the surface of the sheet at z=0:
where the temperature-dependent term FxT, and the zero-temperature contribution Fx0 are given by
where ni(ω)=[exp(ћω/kBTi)−1]−1 (i=1, 2), Ti is the temperature of i-th body, Ri is the reflection amplitude for surface i for p-polarised electromagnetic waves, and ω±=ω±qxv. 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 
where the dielectric function of the sheet is determined by εg(ω, q)=1+vqΠ(ω, q) where vq=2πe2/(qε) is the 2D Coulomb interaction and Π(ω, q) is the 2D polarizability.
In the finite lifetime generalisation according to the Mermin approximation , the dielectric function is determined by
where ε0(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) , . The small (and constant) value of the graphene Wigner–Seitz radius rs 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 rs≪1 limit). The dielectric function is an analytical function in the upper half-space of the complex ω-plane:
where the Fermi wave vector kF=(πn)1/2, n is the concentration of charge carriers, the Fermi energy ϵF=ћvFkF, vF≈106 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=evF/(ћkFμ). Scattering of the graphene carries by the acoustic phonons of graphene places an intrinsic limits on the low-field room temperature (T0=300 K) mobility, given by μ0=20 m2/Vs at the graphene carriers density 1016 m−2 (see ), which gives γ=8·1011s−1. At other temperatures, the mobility can be obtained using the relation μ=μ0T0/T.
The reflection amplitude for the substrate
where ϵd is the dielectric function for substrate. The dielectric function of amorphous SiO2 can be described using an oscillator model 
where parameters ω0,j , γj, and σj were obtained by fitting the actual ϵ for SiO2 to the above-mentioned equation and are given by ϵ∞=2.0014, σ1=4.4767×1027 s−2, ω0,1=8.6732×1013 s−1, γ1=3.3026×1012 s−1, σ2=2.3584× 1028 s−2, ω0,2=2.0219×1014 s−1, and γ1=8.3983×1012 s−1.
For v<dkBT/ћ (at d=1 nm and T=300 K for v<4·104 m/s), the main contribution to the friction (3) depends linearly on the sliding velocity v so that the friction force FxT=Γv, where at T1=T2=T the friction coefficient Γ is given by
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 ωeh(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 ωeh(q)−qxv. For v>ωeh(q)/qx, the excitation energy will be negative. Thus, for velocities larger than critical velocity (vcr=ωeh(q)/qx) as a result of such excitation, the excitation can be created with energy ωph(q)=qxv−ωeh(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 ωph(q)=qxv−ωeh(q)>0 will create the surface phonon–polariton with the energy ωs in the dielectric. In the case of graphene, the energy of the electron–hole pair excitation ωeh(q)≈vFq, where vF is the Fermi velocity. Thus, resonance arises when qxv≈2vFq+ωs, which requires that v>vF+ωsd≈106 m/s.
3 Using Graphene to Detect Quantum Friction
Quantum friction determines the ultimate limit to which the friction can be reduced. In order to detect quantum friction, it is necessary to reduce the contribution to friction from other mechanisms up to unprecedented levels. However, even in noncontact friction experiments , , when two bodies are not in direct contact, there are several contribution to the friction . Moreover, quantum friction dominates over thermal friction at velocities v>dkBT/ћ (at d=1 nm and room temperature: v>105 m/s). However, at present even for a hard cantilever, the velocity of the tip cannot exceed 1 m/s .
We recently proposed  that it should be possible to detect quantum friction in graphene adsorbed on an amorphous SiO2 substrate (Figure 4). The electrons, moving in graphene under the action of an electric field, will experience an intrinsic friction due to interaction with the acoustic and optical phonons in graphene and an extrinsic friction due to interaction with the optical phonons in the SiO2 substrate. In high electric fields, the electrons move with high velocities, and in this case, the main contribution to the friction arises from the interaction with the optical phonons in graphene and in SiO2. However, the frequency of the optical phonons in graphene is approximately four times larger than in SiO2. Therefore, the main contribution to the friction will result from the interaction with the optical phonons in SiO2. Thus, this frictional interaction determines the electrical conductivity of graphene at high electric field.
The dissipated energy due to the friction results in heating of the graphene and is transferred to the SiO2 substrate via the near-field radiative heat transfer process and direct phononic coupling. Using the theories of the Casimir friction and the near-field radiative heat transfer, we formulate a theory that describes these phenomena and allows us to predict experimentally measurable effects. In comparison with the existing microscopic theories of transport in graphene , , our theory is macroscopic. The electromagnetic interaction between graphene and a substrate is described by the dielectric functions of the materials that can be accurately determined from theory and experiment.
Let us consider graphene and a substrate with flat parallel surfaces at separation d≪λT=cћ/kBT. Assume that the free charge carriers in graphene move with the velocity v≪c (c is the light velocity) relative to the substrate. According to , , , the frictional stress Fx acting on the charge carriers in graphene is given by (3) and (4), and the radiative heat flux Sz across the surface of substrate mediated by a fluctuating electromagnetic field is determined by
where ni(ω)=[exp(ћω/kBTi−1)]−1 (i=g, d), Tg(d) is the temperature of graphene (substrate), Ri is the reflection amplitude for surface i for p-polarised electromagnetic waves, and ω±=ω±qxv. The steady-state temperature can be obtained from the condition that the power generated by friction must be equal to the energy transfer across the substrate surface
where the second term in (13) takes into account the heat transfer through direct phononic coupling; αph is the thermal contact conductance due to phononic coupling. Theory of the phononic heat transfer across interfaces for a unsuspended graphene on a SiO2 substrate gives αph≈3·108 W m−2 K−1 .
Figure 5a shows the dependence of the current density on the electric field at the carrier concentration n=1012 cm−2 and for different temperatures. We have found that, in agreement with the experiment , the current density saturates at E~0.5−2.0 V/μm. According to the experiment, the saturation current density Jsat=nevsat≈1.6 mA/μm, and using the charge density concentration n=1012 cm−2: vsat≈106 m/s. The saturation current density depends weakly on the temperature. In Figure 5b, the contributions to the friction force from quantum and thermal fluctuations are shown separately. In the saturation region, the contribution to the friction force from quantum fluctuations dominates.
The friction force acting on the charge carriers in graphene for high electric field is determined by the interaction with the optical phonons of the graphene and with the optical phonons of the substrate. The frequency of optical phonons in graphene is a factor ~4 larger than for the optical phonon in SiO2. Thus, one can expect that for graphene on SiO2 the high-field I−E characteristics will be determined by excitations of optical phonons in SiO2. According to the theory of the van der Waals friction , the quantum friction, which exists even at zero temperature, is determined by the creation of excitations in each of the interacting media, the frequencies of which are connected by vqx=ω1+ω2. The relevant excitations in graphene are the electron–hole pairs whose frequencies ωel−hole≈vFq, while for SiO2, the frequency of surface phonon polaritons ωph≈60 meV (9·1013 s−1). Thus, measurements of the current density-electric field relation of graphene adsorbed on SiO2 give the possibility to detect quantum friction.
4 Near-field Radiative Heat Transfer between Closely Spaced Graphene and Amorphous SiO2
In this section, we investigate heat generation and dissipation due to friction produced by the interaction between moving (drift velocity v) charge carriers in graphene and the optical phonons in a nearby amorphous SiO2 and the acoustic phonons in graphene. Friction produces thermal heating of the graphene which results in near-field radiative energy transfer and phononic heat transfer between the graphene and SiO2. A self-consistent theory that describes these phenomena was formulated by us in  (see also Section 3), and it allows us to predict experimentally measurable effects. In comparison with the existing microscopic theories of energy transfer and transport in graphene , , our theory is macroscopic.
According to (3, 4, and 12) in the case when free carriers are moving relative to the substrate, both thermal and quantum fluctuations give contributions to the frictional stress and the radiative energy transfer. This situation is different from that considered in ,  where it was assumed that the free carries in graphene had vanishing drift velocity. The contribution of the quantum fluctuations to the frictional stress was investigated by us in  (see also Section 3). According to (12), the contribution to the near-field energy transfer from quantum fluctuations is determined by
As discussed in Section 3, for graphene on SiO2, the excess heat generated by the current is transferred to the substrate through the near-field radiative heat transfer, and via the direct phononic coupling (for which the heat transfer coefficient α≈108 Wm−2K−1). At small temperature difference (∆T=Tg−Td≪Td), from (13), we get
where Fx0=Fx(Td, Tg=Td), Sz0=Sz(Td, Tg=Td),
We note that, in contrast to the heat transfer between bodies at rest, for moving bodies the energy flux Sz(Td, Tg) is not equal to zero even for the case when there is no temperature difference between the bodies. The energy transfer coefficient is given by
For small velocities, Fx0~v and Sz0~v2. Thus, from (16), it follows that in the limit v→0 the energy transfer coefficient between moving bodies is not reduced to the heat transfer coefficient between bodies at rest, which is determined by This effect is due to the term Sz0 in the total energy flux which exists only between moving bodies. The energy transfer coefficient can be strongly enhanced in comparison to the heat transfer coefficient when Fx0v≈Sz0. Figure 6a shows the ratio of the energy transfer coefficient to the phononic heat transfer coefficient for d=0.35 nm and n=1016 m−2. For low and intermediate field, this ratio is larger than unity what means that in this region the near-fields radiative energy transfer gives an additional significant contribution to the heat transfer. For nonsuspended graphene on SiO2, the energy and heat transfer are very effective and the temperature difference does not rise high, even for such high electric field that saturation in I−E characteristic starts  (see Figure 6b). The radiative heat transfer between bodies at rest is determined only by thermal fluctuations, in contrast to the radiative energy transfer between moving bodies which is determined by both thermal and quantum fluctuations. Figure 6c shows that quantum fluctuations can give significant contribution to the total energy transfer for low temperatures and large electric field (high drift velocity). Similarly, in the (elecric current) saturation region quantum fluctuations give significant contribution to the total friction force which is determined, as discussed above, by the sum of the extrinsic and intrinsic friction forces (see Figure 6d). The extrinsic friction force has contributions from both thermal and quantum fluctuations. The friction force due to quantum fluctuations is denoted as quantum friction which was discussed in Section 3 (see also ).
Figure 7a shows the dependence of the energy transfer coefficient on the separation d for low electric field (v→0). At d~5 nm and T=300 K, the energy transfer coefficient, due to the near-field radiative energy transfer, is ~104 Wm−2K−1, which is ~three orders of magnitude larger than the radiative heat transfer coefficient due to the black-body radiation. In comparison, the near-field radiative heat transfer coefficient in SiO2–SiO2 system for the plate–plate configuration, when extracted from experimental data  for the plate-sphere configuration, is ~2230 Wm−2K−1 at a ~30 nm gap. For this system, the radiative heat transfer coefficient depends on the separation as 1/d2. Thus, α~105 Wm−2K−1 at d~5 nm, which is one order of magnitude larger than for the graphene-SiO2 system in the same configuration. However, the sphere has a characteristic roughness of ~40 nm, and the experiments ,  were restricted to separations larger than 30 nm (at smaller separations, the surface roughness affects the measured heat transfer). Thus, the extreme near-field-separation, with d less than approximately 10 nm, may not be accessible using a plate–sphere geometry. A suspended graphene sheet has a roughness ~1 nm , and measurements of the thermal contact conductance can be performed for separation larger than ~1 nm. For small separation, one would expect the emergence of nonlocal and nonlinear effects. This range is of great interest for the design of nanoscale devices, as modern nanostructures are considerably smaller than 10 nm and are separated in some cases by only a few Angstroms.
Figure 7b shows that at small separation there is significant difference between the radiative energy transfer coefficient and the the radiative heat transfer coefficient determined (in the absence of direct phononic coupling) by This difference vanishes for large separation because Sz0 and Fx0 rapidly decrease when the separation increases. At large separation, the friction force is dominated by the intrinsic friction and in this case α≈α0. Figure 7c shows the dependence of the radiative energy flux on electric field for d=1 nm. For this separation, the energy transfer is considerably less effective than for d=0.35 nm, which leads to a rapid increase in the temperature difference (see Figure 7d). High temperatures are achieved at low electric field (small drift velocities), where contribution to the radiative energy transfer from quantum fluctuations is very small, and the energy transfer is mainly determined by thermal fluctuations.
In this article, we have revisited the Casimir friction and near-field radiative heat transfer between a graphene sheet and a SiO2 substrate in the presence of the drift motion of electrons in graphene. Both these phenomena are strongly enhanced above threshold velocity when they are related with the resonant excitation of the electron–hole pairs in graphene and the surface phonon–polaritons in the SiO2 substrate. This enhancement exists even at zero temperature when it is determined by quantum friction. Thus, observation of these phenomena can be used to study quantum friction.
The study was supported by the Russian Foundation for Basic Research (Grant No. 16-02-00059-a).
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About the article
Published Online: 2017-01-11
Published in Print: 2017-02-01
Funding Source: Russian Foundation for Basic Research
Award identifier / Grant number: 16-02-00059-a
The study was supported by the Russian Foundation for Basic Research (Grant No. 16-02-00059-a).