Many-body effects in the van der Waals–Casimir interaction between graphene layers

Van der Waals-Casimir dispersion interactions between two apposed graphene layers, a graphene layer and a substrate, and in a multilamellar graphene system are analyzed within the framework of the Lifshitz theory. This formulation hinges on a known form of the dielectric response function of an undoped or doped graphene sheet, assumed to be of a random-phase-approximation form. In the geometry of two apposed layers, the separation dependence of the van der Waals-Casimir interaction for both types of graphene sheets is determined and critically compared with some well-known limiting cases. In a multilamellar array, the many-body effects are quantified and shown to increase the magnitude of the van der Waals-Casimir interactions.

Figure 1

Schematic presentation of two graphene sheets of finite thickness immersed in vacuo at a separation of $b$. The thickness of both (left and right) layers are equal to $a$. In view of later generalizations, we have labeled the left semi-infinite vacuum layer with $L$, the graphene layers with $A$, the intervening vacuum layer with $B$, and the right vacuum layer with $R$. The form of the dielectric functions of the graphene layers is given in Eq. (9).

Figure 2

Magnitude of the interaction free energy per unit area of the system composed of two undoped graphene layers immersed in vacuo at an interlayer spacing $b$ and at temperature $300\mathrm{K}$. The functional dependence of the interaction free energy on $b$ is compared with the following scaling forms: $b^{-3},b^{-4},b^{-5}$, and $b^{-4}$ as the separation increases.

Figure 3

Magnitude of the interaction free energy per unit area of the system composed of two doped graphene layers (solid curve, with the doping electron density $\rho ={10}^{16}{\text{m}}^{-2}$) compared with that of two ideal metallic sheets (dashed curve) immersed in vacuo as a function of the interlayer spacing $b$, at temperature $300\mathrm{K}$. The functional dependence of the free energy on the interlayer spacing is compared with scaling forms $b^{-2.5},b^{-3},b^{-4}$, and $b^{-2}$ in various regimes of separation.

Figure 4

Magnitude of the free energy per unit area of the system composed of two undoped (dashed line) and doped (solid line) graphene layers (with the electron density $\rho ={10}^{16}/{\text{m}}^2$) immersed in vacuo as a function of the interlayer spacing $b$, at temperature $300\text{K}$. As seen for all separations, the magnitude of the interaction free energy for doped graphene layers is greater than that of the undoped one.

Figure 5

The rescaled interaction free energy $F_{gg}\left(\eta \right)/F_{gg}(\eta =0)$ for the system composed of two doped graphene layers with different electron densities immersed in vacuum as a function of $\eta =\left({\rho }_1-{\rho }_2\right)/\left({\rho }_1+{\rho }_2\right)$ for different interlayer separations $b=1\text{nm},10\text{nm},10\mu \text{m},$ and $100\mu \text{m}$ (from top to bottom) at temperature $300\text{K}$. Note that $F_{gg}(\eta =0)$ has been calculated for ${\rho }_1={\rho }_2={10}^{16}{\text{m}}^{-2}$.

Figure 6

Hamaker coefficient for a system of two undoped graphene layers as a function of the distance between them $b$. The top solid line shows the general form of the Hamaker coefficient as defined via Eq. (15), the dashed line shows the limiting large-distance form, Eq. (16), and the dotted line shows the limiting small-distance form, Eq. (17).

Figure 7

Schematic presentation of a graphene layer of thickness $a$ (labeled by $A$) apposed to a substrate (labeled by $L$) at a separation $b$. We have labeled the intervening layer (assumed to be vacuum) with $B$, and the right one with $R$. The dielectric function of the graphene layer is defined via Eq. (9) and for substrate via Eq. (19).

Figure 8

Magnitude of the interaction free energy of the system composed of a SiO${}_2$ substrate and an undoped graphene layer, Eq. (18), plotted at temperature $300\text{K}$ as a function of the separation $b$. The functional dependence of the free energy on $b$ is compared with scaling forms $b^{-2},b^{-3},b^{-4}$, and $b^{-3}$ in various regimes of separation.

Figure 9

Magnitude of the interaction free energy of the system composed of a SiO${}_2$ substrate and a doped graphene layer, Eq. (18), is plotted at temperature $300\text{K}$ as a function of the separation $b$. Here, the functional dependence of the interaction free energy for the graphene doping electron density $\rho ={10}^{16}{\text{m}}^{-2}$ (solid line) is compared with scaling forms $b^{-2.5},b^{-3}$, and $b^{-2}$ in various regimes of separation. We also include the same plot for the doping electron density $\rho ={10}^{14}{\text{m}}^{-2}$ (dashed line).

Figure 10

Hamaker coefficient for a system of an undoped graphene layer apposed to a SiO${}_2$ substrate as a function of the distance between them $b$. The top solid line shows the general form of the Hamaker coefficient as defined via Eq. (20), the dashed line shows the limiting large-distance form, Eq. (21), and the dotted line shows the limiting small-distance form, Eq. (22).

Figure 11

Schematic picture of the system composed of $N+1$ layers of graphene ($A$) with equal thicknesses $a$ separated from each other by $N$ layers of vacuum ($B$) with equal thicknesses of $b$. The two semi-infinite substrates in the left and right end of the system are labeled by $L$ and $R$ and will be assumed to be vacuum.

Figure 12

Magnitude of the interaction free energy per unit area and number of layers, $|f_{gg}|/S$, plotted as a function of the separation $b$ between two successive undoped (bottom black dot-dashed line) and doped (top black solid line) graphene layers for a system composed of infinitely many layers of undoped/doped graphene as schematically depicted in Fig. 11. The temperature of the system is $300\text{K}$, the thickness of the graphene layers is fixed at 1 Å and we have chosen the dielectric function for the undoped case from Eq. (10) and for the doped case (with the doping electron density chosen as $\rho ={10}^{16}{\text{m}}^{-2}$), from Eq. (12). The functional form of the free energy for the undoped case is compared with scaling forms $b^{-3},b^{-4},b^{-5}$, and $b^{-4}$ in various regimes of separation. For the doped case, the free energy is compared with the scaling forms $b^{-2.5}$ and $b^{-3}$.

Figure 13

Magnitude of the interaction free energy per unit area and number of layers, $|f_{gg}|/S$, for the system composed of $N+1$ layers of undoped (bottom black dot-dashed line) and doped (top black solid line) graphene is compared with the magnitude of the interaction free energy per unit area, $|F_{gg}|/S$, for the system composed of only two undoped (green dots) and doped (red dots) graphene layers as a function of the separation between the layers, $b$. The inset shows the ratio of these two quantities for both undoped (black dot-dashed line) and doped (black solid line) cases.