Negative-positive oscillation in interfacial friction of a ${\mathrm{In}}_2{\mathrm{Se}}_3$-graphene heterojunction

According to the classic law of Da Vinci-Amontons, a friction force $f$ was found to increase macroscopically with an external normal load $N(f=\mu N)$, with a positive definite friction coefficient μ. Here we employ first-principles calculations to predict that, when sliding the ferroelectric two-dimensional ${\mathrm{In}}_2{\mathrm{Se}}_3$ over graphene, the differential friction coefficient μ, measured by the slope of the corrugation in the sliding potential energy barrier subject to load $N$, displays an overall positive feather when the dipole is aligned toward the ${\mathrm{In}}_2{\mathrm{Se}}_3$-graphene interface; however, μ exhibits intriguing negative-positive oscillation with increasing $N$ when the dipole is aligned outward from the interface. Such striking observations can be rationalized by the van der Waals and electrostatic interaction-induced competition between the downward shift of the sliding barrier by the ${\mathrm{p}}_{\mathrm{x}}$- and $p_y$-like levels and the upward shift of the barrier by the $p_{\mathrm{z}}$ states, which is accompanied by an oscillation of the ${\mathrm{In}}_2{\mathrm{Se}}_3$-graphene interfacial charge redistribution subject to the external load. The present findings are expected to play an instrumental role in the design of high-performance solid lubricants.

Figure 1

Schematic illustrations of the geometric and electronic structures of the ${\mathrm{In}}_2{\mathrm{Se}}_3$-graphene heterojunction. (a) Top and side views of two typical contacts for the quintuple layers (1QL) ${\mathrm{In}}_2{\mathrm{Se}}_3$ on graphene (${\mathrm{In}}_2{\mathrm{Se}}_3$-graphene), with the ferroelectric polarization either pointing upward ($\mathrm{P}\uparrow$) or downward ($\mathrm{P}\downarrow$). (b) Side views of the charge transfer between ${\mathrm{In}}_2{\mathrm{Se}}_3$ and graphene. Yellow and blue denote charge accumulation and depletion, respectively. The first Brillouin zone and decomposed electronic band structures of the two types of ${\mathrm{In}}_2{\mathrm{Se}}_3$-graphene heterojunctions: (c) $\mathrm{P}\uparrow$ and (d) $\mathrm{P}\downarrow .$

Figure 2

Frictional properties of 1QL-${\mathrm{In}}_2{\mathrm{Se}}_3$ over graphene. (a) Schematic representation of the model in simulations of sliding $\mathrm{P}\uparrow$ 1QL-${\mathrm{In}}_2{\mathrm{Se}}_3$ over graphene ($\mathrm{P}\uparrow )$. Two sliding pathways along [110] and [100] are marked. (b) Sliding energy profile of the $\mathrm{P}\uparrow$ 1QL-${\mathrm{In}}_2{\mathrm{Se}}_3$ over graphene along [110] without external load. (c) and (d) Load-dependent sliding energy barriers ($E_{\mathrm{bar}}$) and differential friction coefficient $\left(\mu =df/dN\right)$ along the [110] pathway for both the $\mathrm{P}\uparrow$ and the $\mathrm{P}\downarrow$ 1QL-${\mathrm{In}}_2{\mathrm{Se}}_3$-graphene contact.

Figure 3

Decomposed energy barriers ($E_{\mathrm{bar}}$) for 1QL-${\mathrm{In}}_2{\mathrm{Se}}_3$ sliding over graphene in two typical contacts. Contributed by (a) and (d) interlayer vdW interactions, $E_{\mathrm{bar}-\mathrm{inter}-\mathrm{vdW}}$. (b) and (e) Interlayer electrostatic interactions, $E_{\mathrm{bar}-\mathrm{inter}-\mathrm{es}}$. (c) and (f) Intralayer deformation energy of the 1QL-${\mathrm{In}}_2{\mathrm{Se}}_3$ component, $E_{\mathrm{bar}-\mathrm{def}}$. (a)–(c) $\mathrm{P}\uparrow$, (d)–(f) $\mathrm{P}\downarrow$.

Figure 4

Calculated parameters of the 1QL-${\mathrm{In}}_2{\mathrm{Se}}_3$-raphene heterostructure for both $\mathrm{P}\uparrow$ and $\mathrm{P}\downarrow$ configurations under external load. (a) Energy profile of the vdW component and (b) interfacial distance $D$ of the ${\mathrm{In}}_2{\mathrm{Se}}_3$-graphene heterostructure in the optimized $E_{min}$ state. Variation of the differences of the interfacial distance ($\mathrm{\Delta }D$) and change of the thickness ($\mathrm{\Delta }T$) of the ${\mathrm{In}}_2{\mathrm{Se}}_3$ overlayer between the $E_{max}$ and $E_{min}$ states for (c) $\mathrm{P}\uparrow$ and (d) $\mathrm{P}\downarrow$.

Figure 5

Electronic structure analysis of the $E_{min}$ and $E_{max}$ states for the case of P↑ under 1.1 GPa. (a) Variation of the energy band structure of the $E_{max}$ state relative to the $E_{min}$ state, wherein the deep-purple and deep-red regimes imply that the energy band of the $E_{max}$ state will be downward- and upward-shifted 5 meV relative to the $E_{min}$ state as plotted. (b) Decomposed electronic band structure projected on substrate graphene (C) and ${\mathrm{In}}_2{\mathrm{Se}}_3$ overlayer. Four representative states (I–IV) at the high symmetry points ($K$, Г, and $M$) are marked. (c) Total $E_{\mathrm{bar}}$ and that contributed by the III and IV states at the $M$-point. (d) Electronic charges of the four states, I to IV. The isosurface of the charge density is $0.005\mathrm{e}/{\AA }^3$.

Figure 6

Electronic structure analysis. (a) Decomposed band structure contributed by graphene and ${\mathrm{In}}_2{\mathrm{Se}}_3$ for the two representative energy bands around the Fermi level, i.e., valance band (VB) and VB-1 without external loading. (b) Variation of differential electronic charge density between the $E_{max}$ and $E_{min}$ states for both the VB presented in top panels and VB-1 in bottom panels, under $N=0.55$, 1.1, 1.65, and 2.2 GPa. The isosurface of the charge density is $0.005\mathrm{e}/{\AA }^3$.