Temperature Dependence of the Response Functions of Graphene: Impact on Casimir and Casimi-Polder Forces in and out of Thermal Equilibrium

TL;DR

This work develops a first-principles, nonlocal electromagnetic description of graphene via the polarization tensor in (2+1)D quantum field theory and applies it to temperature-dependent Casimir and Casimir–Polder forces in and out of thermal equilibrium. By deriving and analyzing the longitudinal and transverse dielectric functions in both below- and above-threshold regimes, the authors reveal a distinctive double pole at zero frequency in the transverse response and quantify how $\Delta$, $\mu$, and substrates modify dispersion forces. Using the Lifshitz framework with graphene’s exact reflection coefficients, they demonstrate unusually large thermal effects at short separations for graphene–graphene and atom–graphene systems, including explicit and implicit thermal contributions and robust high-temperature asymptotics. Extending to nonequilibrium, they provide general formulas for $P_{\rm neq}$ and $F_{\rm neq}$, showing that heating or cooling graphene relative to its environment can substantially alter the forces, with practical implications for graphene-enabled nanodevices and metrology at the nanoscale.

Abstract

We review and obtain some new results on the temperature dependence of spatially nonlocal response functions of graphene and their applications to calculation of both the equilibrium and nonequilibrium Casimir and Casimir-Polder forces. After a brief summary of the properties of the polarization tensor of graphene obtained within Dirac model in the framework of quantum field theory, we derive the expressions for the longitudinal and transverse dielectric functions. The behavior of these functions at different temperatures is investigated in the regions below and above the threshold. Special attention is paid to the double pole at zero frequency which is present in the transverse response function of graphene. An application of the response functions of graphene to calculation of the equilibrium Casimir force between two graphene sheets and Casimir-Polder forces between an atom (nanoparticle) and a graphene sheet is considered with due attention to the role of a nonzero energy gap, chemical potential and a material substrate underlying the graphene sheet. The same subject is discussed for out-of-thermal-equilibrium Casimir and Casimir-Polder forces. The role of the obtained and presented results for fundamental science and nanotechnology is outlined.

Figures (7)

• Figure 1: The computational results for (a) magnitude of the real part and (b) imaginary part of the longitudinal electric susceptibility of graphene in the region below the threshold are plotted as the functions of temperature. $|{\rm Re}{\chi^{\rm L}}|$ does not depend on $\omega$ in the wide region from $\omega=10$ to $0.999999\times 10^{10}~$rad/s with except of the temperature interval $0<T<1~$K (see the inset in Figure 1(a) where the bottom and top lines are plotted for $\omega=10$ and $0.999999\times 10^{10}~$rad/s, respectively). The lines in Figure 1(b) counted from bottom to top are plotted for $\omega=10$, $10^3$, $10^5$, $10^7$, $5\times 10^8$, $9\times10^{10}$, and $0.999999\times 10^{10}~$rad/s, respectively.
• Figure 2: The computational results for (a) magnitude of the real part and (b) imaginary part of the transverse electric susceptibility of graphene in the region below the threshold are plotted as the functions of temperature. The lines representing $|{\rm Re}{\chi^{\rm T}}|$ in Figure 2(a) counted from top to bottom are plotted for $\omega=10^7$, $5\times 10^7$, $10^8$, $5\times 10^8$, and $10^9~$rad/s, respectively. The lines representing ${\rm Im}{\chi^{\rm T}}$ in Figure 2(b) counted from top to bottom are plotted for $\omega=10^7$, $5\times 10^7$, $10^8$, $5\times 10^8$, $10^9~$rad/s, $9\times10^9$, and $0.999999\times 10^{10}~$rad/s, respectively.
• Figure 3: The computational results for (a) magnitude of the real part and (b) imaginary part of the longitudinal electric susceptibility of graphene in the region above the threshold are plotted as the functions of temperature. The lines counted from top to bottom in Figure 3(a) are plotted for $\omega=1.00001\times 10^{10}$, $1.5\times 10^{10}$, $10^{11}$, $10^{12}$, and $10^{13}~$rad/s, respectively. The lines in Figure 3(b) labeled 1, 2, 3, and 4 are plotted for $\omega=1.00001\times 10^{10}$, from $1.5\times 10^{10}$ to $10^{11}$ (where the frequency-dependence is present only at low frequencies), $10^{12}$, and $10^{13}~$rad/s, respectively.
• Figure 4: The computational results for (a) magnitude of the real part and (b) imaginary part of the transverse electric susceptibility of graphene in the region above the threshold are plotted as the functions of temperature. The lines counted from top to bottom in Figure 4(a) are plotted for $\omega=1.00001\times 10^{10}$, $1.5\times 10^{10}$, $10^{11}$, $10^{12}$, and $10^{13}~$rad/s, respectively. The lines in Figure 4(b) labeled 1, 2, 3, and 4 are plotted for $\omega=1.00001\times 10^{10}$, from $1.5\times 10^{10}$ to $10^{11}$ (where the frequency dependence is lacking), $10^{12}$, and $10^{13}~$rad/s, respectively.
• Figure 5: The computational results for the normalized magnitude of the Casimir pressure between two graphene sheets are shown as the function of separation by the upper, medium and bottom lines computed at $T=300~$K exactly, at $T=300~$K with taking into account only an implicit thermal effect, and at $T=0~$K, respectively, over the separation region (a) from 2 to 250 nm and (b) from 5 to 30 nm.