Colloquium: Sliding and pinning in structurally lubric 2D material interfaces

A plethora of two-dimensional (2D) materials have been introduced in physics and engineering in the past two decades. Their robust, membranelike sheets permit (mostly require) deposition, giving rise to solid-solid dry interfaces whose mobility, pinning, and general tribological properties under shear stress are currently being understood and controlled, both experimentally and theoretically. In this Colloquium simulated case studies of twisted graphene systems are used as a prototype workhorse tool to demonstrate and discuss the general picture of 2D material interface sliding. First highlighted is the crucial mechanical difference, often overlooked, between small and large incommensurabilities, which corresponds to, for example, small and large twist angles in graphene interfaces. In both cases, focusing on flat, structurally lubric or “superlubric” geometries, the generally separate scalings with the area of static friction in pinned states and of kinetic friction during sliding are elucidated and reviewed, tangled as they are with the effects of velocity, temperature, load, and defects. The roles of island boundaries and elasticity are also discussed, and compared when possible to results in the literature for systems other than graphene. It is proposed that the resulting picture of pinning and sliding should be applicable to interfaces in generic 2D materials that are of importance for the physics and technology of existing and future bilayer and multilayer systems.

Figure 1

(a)–(c) Some relevant 2D materials. (d) Common experimental setups involving their interfaces.

Figure 2

Sketch of 2D material interfaces. (a) Commensurate and (b) “homo” incommensurate with twist angle $\theta$. (c) “Hetero” incommensurate with lattice mismatch $a\ne b$ and zero twist. (d)–(f) Graphene-graphene moiré cell (dashed-line rhombi), shown for a small twist $\theta$, obliquely translating upon horizontal interface sliding (arrow). (g)–(i) Same approach for $h$-BN/graphene when $\theta =0$. The angle of motion of the moiré lattice relative to the graphene lattice is $\beta$ [Eq. (2)].

Figure 3

(a) Out-of-plane $z$ distance (corrugation) of a single TBG layer from the midinterface plane, fully relaxed at $T=0\mathrm{K}$ for $S$, $I$, and $L$ regimes. Red nodes mark large $AA$-like interlayer spacing, blue areas smaller $AB$-like spacing. (b) Corrugation profile along the dashed lines in (a). (c) Average (red dashed line) and minimum and maximum (pink-shaded region) relaxed interlayer spacing as a function of the twist angle $\theta$. The black dotted lines mark the analytical prediction of Eq. (3). The data were obtained as described in Sec. 3.

Figure 4

Out-of-plane moiré corrugation for graphene/$h$-BN simulated heterostructures at various twist angles: $\theta =0°$, 0.45°, and 1.5°. (a) The moiré pattern and (b) the out-of-plane corrugation magnitude along the white dashed traces in (a). (c) Maximum, minimum, and average values of the $z$ coordinate of carbon atoms in the upper graphene layer over the flat $h$-BN substrate as a function of misalignment angle. Adapted from [62].

Figure 5

Top and side views of prototypical nanofriction setups. (a) Circular island, OBC bilayer interface. (b) Boundary-free PBC bilayer interface. Note the differing moiré patterns of the 30° and 6° twist angles, respectively. (c),(d) Side view of two configurations. The slider and substrate layers are shown in blue (upper center) and red (lower center); the encapsulating stages (top and bottom layers) are black, with their vertical grip mimicked by springs $k_z$. The slider’s center of mass is pulled with velocity $v_0$, through a spring $K_{\mathrm{p}}$ representing the overall effective stiffness of the pulling system. (e)–(h) Sliding configurations for graphene/$h$-BN heterostructures in the literature ([164]; [118]; [188]).

Figure 6

Moiré corrugation patterns, energy, and rotational torque of the OBC (island) and PBC (infinite) graphene/graphene interfaces. (a) $z$ coordinates (upper panels) and per-atom interlayer interaction $E_{\text{interlayer}}$ (lower panels) for circular islands with diameter $D=4\mathrm{nm}$ and twist angle $\theta =2.5°$, 5.0°, and 7.5°. (b) Per-atom interlayer interaction energy difference $\mathrm{\Delta }E$ relative to perfect $AB$ stacking for islands with different diameters as a function of twist angle (left panel), and for one moving island as a function of displacement along the substrate armchair (vertical) and zigzag directions (horizontal), with the periodicities $\sqrt{3}a=4.26\AA$ and $a=2.46\AA$, respectively. (c) Per-atom rotational torque of graphene islands with different radii. The three red vertical lines in (b) and (c) correspond to the twist angles shown in (a): the rotationally most unstable angle and the two locally stable angles (zero torque). (d) The same plots as in (a) for PBC infinite interfaces with twist angles $\theta =0.4°$, 1.08°, and 3.0°. (e) The same as (b) for PBC. The three gray vertical lines in (e) and (f) correspond to the twist angles shown in (d). By comparing the OBC island with PBC energy against the slider position, one can note the direction-dependent energy barriers. Despite the apparent similarity, the large island barriers are real, whereas the small PBC barriers are an artifact due to the finite cell. In addition, the OBC torque oscillations are real features of the island edge interfering with the moiré. The torque minimum near 1° signals a change of misfit regime from independent to dependent on elasticity (see the text).

Figure 7

Evolution of an interlayer energy map during sliding by $dx$ along $x$ of a circular island with a diameter of 4 nm and a twist angle $\theta =5°$. Three $AA$ nodes near the edge in (i) are highlighted and tracked as they move. When they cross the edges, the interlayer energy develops a barrier.

Figure 8

Kinetic friction traces from (a)–(c) our simulations and (d)–(f) published simulation studies. The underlying static friction $F_{\mathrm{s}}$ is zero in PBC, signaling essentially perfect structural superlubricity, but nonzero in OBC, where pinning is due to the island edges. The kinetic friction $F_{\mathrm{k}}$ is the average lateral force. The friction force of PBC systems is normalized per slider atom and can be converted to the shear stress $\tau =F/A_{\mathrm{C}}$ by dividing the per-atom area of a carbon atom: $A_{\mathrm{C}}=\sqrt{3}{a}_{\mathrm{Gr}}^2/4$. (a) Frictional transient and convergence of PBC graphene-graphene contacts evolving in time ($v_0=10\mathrm{m}/\mathrm{s}$, $T=0\mathrm{K}$). The effective lateral stiffness $K_{\mathrm{p}}$ is adjusted case by case in order to keep the underdamped inertial oscillation $\omega =\sqrt{K_{\mathrm{p}}/M}$, with $M=N{m}_{\mathrm{C}}$ the slider mass within reasonable range. The mean friction (the red dashed line; magnified in the insets), here proportional to velocity, increases as the twist angle decreases, reflecting the severe mobility drop from the $L$ to $I$ to $S$ regimes (the slider atom numbers $N$ are 11 644, 6542, and 40 838 for $\theta =30°$, 4°, and 0.4°, respectively). (b) Friction force of OBC graphene-graphene island of diameter $D=4\mathrm{nm}$ with a twist angle $\theta =30°$ as a function of the sliding distance. Upper panel: adiabatic protocol with stiff and soft pulling springs $K_{\mathrm{p}}=100$ and $1\mathrm{N}/\mathrm{m}$, respectively. Lower panels: dynamic sliding simulations ($v_0=10\mathrm{m}/\mathrm{s}$, $T=298\mathrm{K}$) and the same spring stiffness: stiff and soft, respectively. Note the evolution from smooth sliding to stick-slip upon softening. (c) OBC friction force at a small twist angle of 2°. Blue and red represent the island diameters $D=10.6$ and 14 nm, respectively. Note the evolution from stick-slip at a small size, where edge pinning dominates, to a larger size, where the edge effect disappears. (d) Time evolution of shear stress (per-area friction force) in aligned graphene/$h$-BN heterojunctions under different normal pressures. The time unit is $\sim 25\mathrm{ps}$. The incommensurability of the two lattices and the absence of edges in PBC predicts smooth sliding in this case too. Adapted from [118]. (e),(f) Friction force of graphene/$h$-BN island contacts (OBC) for 0° and 30° twist angles ([188]). The evolution from (e) stick-slip to (f) smooth sliding is associated with a dramatic shrinking of the moiré cell size with increasing twist.

Figure 9

Dependence of per-atom kinetic friction on the twist angle, where the red dots and blue circles represent the simulation results and theoretical results estimated from Eq. (4). The parameters used here are $\zeta =2{\mathrm{ps}}^{-1}$ and $v_0=10\mathrm{m}/\mathrm{s}$.

Figure 10

Area dependence of nanoscale and mesoscale kinetic and static friction of 2D materials from simulations (upper panels) and experimental kinetic friction data (lower panels) across the scale. (a) Graphene-graphene contacts with a large twist angle $\theta =30°$ ($v_0=30\mathrm{m}/\mathrm{s}$, $T=300\mathrm{K}$). The green and red data points represent the OBC circular island static friction and kinetic friction, and the blue data points indicate the superlubric regime kinetic friction (the static friction is zero). The green, blue, and red dashed lines are power-law fits $F\sim A^{\alpha }$ with exponents 0.25, 0.5, and 1. (b) Graphene-graphene contacts with small twist angle 2° ($v_0=10\mathrm{m}/\mathrm{s}$, $T=300\mathrm{K}$). The green and red data points represent the static and kinetic friction of OBC islands, and the red dotted curve is the sinusoidal fitting whose oscillations reflect successive moiré entering and exiting the island ([188]). Two typical cases are labeled, with diameters of 10.6 and 14 nm, respectively, at local friction maximum and minimum. (c) Graphene–$h$-BN contacts with twist angle $\theta =30°$ and 0° ($v_0=10\mathrm{m}/\mathrm{s}$, $T=300\mathrm{K}$). The red ($\theta =0°$) and blue data points ($\theta =30°$) represent the kinetic friction for the OBC graphene island [Fig. 5]. Replotted with data from [188]. Comparing (b) and (c) with (a), note that the lowest friction values, where the edge does not contribute to friction, scale linearly with area, indicating structural superlubricity at all special island diameters of the sequence $D_n=\sqrt{3}\lambda (n/2+1/8)$, with the integer $n\ge 2$. The largest friction values, which are edge related, instead scale sublinearly, indicating proportionality to static friction (the red dashed upper envelopes). The scaling exponent $1/4$ is particular to the circular shape. Most other shapes, possessing straight edge portions, scale with exponent $1/2$. (d)–(k) Collection of 2D material kinetic friction data. The red circles represent data extracted and replotted from original papers ([37]; [30]; [94]; [137]; [188]; [149]; [100]). The upper green dashed lines show sublinear scaling of the highest friction points. The lower blue dashed lines underline the linear scaling of the lowest friction points, which is analogous in experiments and simulations.

Figure 11

Velocity dependence of kinetic friction of structurally lubric contacts. (a) Three friction regimes of a pinned nanoslider as a function of velocity: thermolubric, stick-slip, and ballistic. From [85]. (b) A direction comparison between the experimental results (left pink region) and $\theta =30°$ simulation results (right blue region). Circles within the low velocity range represent experimental kinetic friction data ([164]; [198]; [105]). The dashed lines are logarithmic fits. The MD simulation data and color are the same as in (c). The blue dashed (straight) line indicates linear (superlubric, thermolubric, or ballistic) friction scaling. Note the large velocity difference between simulations and experiments, whereby a high velocity ballistic regime is rarely reached in experiments and no low velocity thermolubric regime is reached in simulations. (c) Simulated kinetic friction in superlubric (blue circles, $K_{\mathrm{p}}=100\mathrm{N}/\mathrm{m}$) and stick-slip (red circles, $K_{\mathrm{p}}=1\mathrm{N}/\mathrm{m}$) sliding for large twist angle $\theta =30°$. For the larger stiffness the island friction is viscous in the entire range. The purple (black) squares display kinetic friction for small (large) island sizes at a small twist angle $\theta =2°$. All lines are power-law fits. Note that the velocity scaling is size dependent, with the larger island close to linear and the smaller one sublinear.

Figure 12

Temperature dependence of graphene-graphene kinetic friction. (a) Experimental results. The red data (lower left) are for nanosized ultrahigh vacuum sliding ([105]), while the blue data (upper right) pertain to microsized, ambient condition sliding ([198]). The dashed lines are exponential fits reflecting an activated barrier crossing. The high-temperature saturation of microsized data may be an artifact due to the 5 nN sensitivity limit of the force sensor. (b) Simulated kinetic friction of a graphene island [Fig. 5] with $D=2\mathrm{nm}$ and $\theta =30°$. Red lines, stick-slip regime [$K_{\mathrm{p}}=1\mathrm{N}/\mathrm{m}$; Fig. 8]; blue lines, superlubric regime [$K_{\mathrm{p}}=100\mathrm{N}/\mathrm{m}$; Fig. 8] mimicking the structurally superlubric case. The red and blue dashed lines are exponential and linear fits, respectively.

Figure 13

Load dependence of 2D material sliding friction. (a) Schematic diagram contrasting the friction coefficient $\mu$ and the differential friction coefficient ${\mu }_{\mathrm{D}}$. The slopes of the dashed and dotted lines represent $\mu$ and ${\mu }_{\mathrm{D}}$, respectively. (b),(c) Simulated load dependence of kinetic friction in 2°-twisted islands of different sizes: (b) $D=14\mathrm{nm}$, representative of the superlubric regime, and (c) $D=10.6\mathrm{nm}$, representative of stick-slip regime; see Fig. 10. The inset triangle in the upper right corner shows a mere 30% increase in stick-slip friction force when $v_0$ is increased 5000% under a 0.5 GPa load. (d) Simulated load dependence of large speed kinetic friction in infinite aligned graphene/$h$-BN heterostructures (an ideal structurally superlubric case). The blue dashed and red dotted lines represent $\mu$ and ${\mu }_{\mathrm{D}}$, respectively. Adapted from [118].

Figure 14

(a) Elastic deformation of the flexible substrate (black lattice underneath) in incommensurate contact with a rigid circular slider (upper blue layer). (i) Stiff substrate ($G/{\tau }_{\max }=256$). (ii) Soft substrate ($G/{\tau }_{\max }=1$). Adapted from [159]. (b) Moiré evolution of the simulated incommensurate $\mathrm{Kr}/\mathrm{Pb}(111)$ contacts. Colors range from blue (dark) when Kr atoms are maximally coincident with Pb atoms to white (bright) when they are minimally coincident. (i) Stationary system (dragging force $F=0$). (ii) System with $F=2.24\mathrm{nN}$ (just above the maximum static friction). The soliton entering from the left edge is highlighted by the red square. (iii) System after sliding of the island center by 1.0 surface lattice spacing. (iv) System when the sliding is 1.5 lattice spacings and a soliton exits the island on the right-hand side. Adapted from [184]. (c) Strain distribution of (ii) a uniformly driven vs (iii) an edge-pulled 2D harmonic lattice island submitted to a slightly less dense, and therefore incommensurate, 2D sinusoidal potential (“Frenkel-Kontorova”-type model) compared to its static state (i). Colors range from white (bright) for perfect local coincidence to black (dark) for maximal local mismatch. The edge-pulling procedure favors the formation of locally strained commensurate regions, thereby enhancing the static friction, and the ensuing stick-slip in comparison to the uniformly driven system ([117]; [180]). Note also the corresponding moiré switch from 2D to striped.

Figure 15

Sketch of various defects. (a) Slider with different shapes, i.e., a different number of edges and corners. (b) Grain boundaries at the graphene interface. Color represents the out-of-plane displacement. From [53]. (c) Step edge exposed to the sliding interface. Viewed from top to bottom, blue, black, and pink are the slider, the step edge, and the substrate, respectively. (d) Particle-contaminated graphene interface. Viewed from top to bottom, blue, red, and pink are the slider, the contaminated particle, and the substrate, respectively.