1 Introduction

Hexagonal boron nitride (h-BN), which has a structural analogue of graphene, has attracted extensive attention due to its outstanding properties [1]. These superior chemical and physical properties impel us continuously to exploit the cause that h-BN exhibits superior performance to the bulk counterpart applied in transistor, solid lubricant, high-temperature environment, and energy storage [2,3,4]. Moreover, researchers have particularly payed more attentions to monolayer h-BN (the so-called white graphene [5]) due to its high in-plane mechanical strength, thermal conductivity and chemical stability, and distinct oxidation resistance performance compared to graphene [5,6,7,8]. These exceptional properties of monolayer h-BN indicate its potential as an anti-wear material.

Watanabe et al. report that h-BN film deposited on the silicon substrate shows a short lubricating endurance life due to its lower adhesion to the silicon substrate [9]. If h-BN film well adheres to substrate, they also believe that it is a good solid lubricant. Indeed, the monolayer h-BN as a anti-wear coating for Cu surface could well reduce the exceptional low friction as reported by Li et al. [8]. Besides, h-BN as a lubricant additive also plays an important role in enhancing wear resistance and reducing friction coefficient [10,11,12,13]. The same tribological theory could be concluded from these references [8,9,10,11,12,13,14] that h-BN well adhesion to substrate or worn surfaces is a key factor to obtain high wear resistance and low-friction coefficient during sliding. Thus, understanding the active mechanism of h-BN for the tribological improvements is conducive to probing the low-friction behavior. Given that the complex interactions during sliding process are not directly monitored in experiments at the nanoscale, first-principles method offers a powerful tool to capture atomic details and gain a deeper insight into the complex interactions at the nanoscale. Accordingly, Koskilinna et al. systematically investigate the tribological properties of h-BN based on first-principles calculations [15]. They point out that the friction coefficient of h-BN is enhanced with increase of the normal load and low-friction coefficient can only be obtained at low normal load. Thereafter, a question is: what are the key factors for the different friction coefficients with increase of the normal load?

Herein, in order to well answer this question, the interfaces between two h-BN monolayers with different normal loads and the static three-dimensional potential energy surfaces (3D-PES) assigned to their motion to the interlayer forces are calculated using first-principles calculations. The calculated 3D-PES show that the low friction not only could be obtained for the commensurate layers under low normal load, but also high normal load. We discuss the unusually tribological behavior in detail to well probe the active mechanism.

2 Computational Details

All calculations are carried out using CASTEP code [16]. The DFT-D method is proved to be an effective tool due to the good agreement among the geometrical parameters, the experimental and high-accuracy theoretical data [17, 18]. Thus, the generalized gradient approximation (GGA) with PBE-D scheme reported by Grimme [19], which can well account for van der Waals interactions in all calculations. A kinetic cutoff energy of 400 eV is used for the plane-wave basis set. A 10 × 10 × 1 Monkhorst–Pack mesh is used for the k-point sampling. The electron density difference is used to investigate the interaction of two h-BN layers.

The bulk lattice parameters a and c are calculated as 2.494 and 6.488 Å, respectively, which is well in consistent with the experimental data (2.503 and 6.661 Å) [20]. The calculated model as shown in Fig. 1 is constructed with the same monolayer h-BN. The periodic boundary conditions are set in the x–y plane, and interactions between periodic replicas are avoided by adding 15 Å of vacuum along the z direction. With the upper h-BN monolayer translation above the lower one, the three-dimensional potential energy surfaces (3D-PES) is constructed by collecting the interlayer interaction energy for different relative lateral positions [21]. In particular, for Fig. 1b, along x, y, or xy direction, the h-BN bilayer coincides after the upper h-BN monolayer moving 0.5 Å above the lower one.

Fig. 1
figure 1

a Top view of a h-BN monolayer, b side view of h-BN bilayer with hollow–hollow (H–H) or top–top (T–T) configuration, c top view of h-BN bilayer with hollow–top (H–T) configuration

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Next, the compressed bilayer configurations, in which the interlayer distance is fixed at different values, are considered to investigate the effect of increasing the normal load. The force along z direction as a functional of z is defined as \({F_z}= - \frac{{\partial V(z)}}{{\partial z}},\) where V(z) is the potential energy with different interlayer distances along z direction.

3 Results

As shown in Fig. 2, the potential energy and its first derivative with respect to the interlayer distance along z direction are monotonic curves. The potential energy initially kept a constant of 0 eV ranges from 4.0 to 3.5 Å, the position of the minimum energy locates at ~ 3.5 Å, which is well in consistent with previous computational results [15]. Less than 3.5 Å, the potential energy increases with decrease of interlayer distance. The variable of Fz exhibits a similar change tendency with that of potential energy.

Fig. 2
figure 2

Variable of potential energy and its first derivative with respect to the interlayer distance along z direction

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Next, to well investigate the effect of the normal load on sliding properties of h-BN bilayer, the three-dimensional potential energy surfaces (3D-PES) for H–H (T–T) configuration are calculated with different normal loads as shown in Fig. 3. In Fig. 3a, under the normal load of 0 nN, there exists low-friction path along X or Y direction, however, increasing the normal load to 1.4 nN, only one low-friction path is observed along Y direction, the low-friction path along Y direction is blocked under high normal load, as shown in Fig. 3b. Interesting, further rising the normal load to 6.0 nN, the obvious low-friction path is found along X direction (Fig. 3c). It indicates that the low friction of h-BN bilayer can be achieved at high normal load for H–H (T–T) configuration.

Fig. 3
figure 3

Three-dimensional potential energy surfaces for the sliding motion of h-BN bilayers in H–H (T–T) configuration at a 0 nN, b 1.4 nN, and c 6.0 nN

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For comparison, the potential energy and its first derivative with respect to the interlayer distance along z direction for H–T configuration are shown in Fig. 4. Meanwhile, the position of the minimum energy also locates at ~ 3.5 Å. The potential energy and Fz also exhibit the same change trend compared to that of H–H (T–T) configuration. Subsequently, the three-dimensional potential energy surfaces for H–T configuration are calculated as shown in Fig. 5, under the normal load of 0 and 0.6 nN, the continuous mountain-like potential energy surfaces are observed in Fig. 5a, b, the low-friction path is obviously detected along the mountain valley (Y direction). However, increasing the normal load to 3.7 nN, the shape of PES is quite different from that in Fig. 5a, b, the low-friction paths along Y direction are completely blocked; however, the low-friction path can be obtained along the edge of X direction (low potential energy region).

Fig. 4
figure 4

Variable of potential energy and its first derivative with respect to the interlayer distance along z direction

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Fig. 5
figure 5

Three-dimensional potential energy surfaces for the sliding motion of h-BN bilayers in H–T configuration at a 0 nN, b 0.6 nN, and c 3.7 nN

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As we know, the tribological behavior is closely related to its electronic structure. Thus, the charge density difference of h-BN bilayer in H–H (T–T), H–T configuration is calculated as shown in Fig. 6. The mixed covalent-ionic bonding characteristics, the valence charges are located around N atoms. A common conclusion can be obtained from Fig. 6a–f that the electron clouds between B atoms in h-BN bilayer overlap each other with increase of the normal load, indicating that low-friction path can be assigned to the interactions between B atoms in h-BN bilayer.

Fig. 6
figure 6

The charge density difference of h-BN bilayer in H–H or T–T configuration at a 0 nN, b 1.4 nN, and c 6.0 nN; in H–T configuration at d 0 nN, e 0.6 nN, and f 3.7 nN

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Zhang et al. reported [22] the kinetic frictional force (fx) equals the predicted \(\left( {\frac{{\partial {\sigma _{\operatorname{int} }}}}{{\partial x}}} \right)\left| {\hbox{max} ,\,{\sigma _{\operatorname{int} }}} \right.\) is interfacial energy of h-BN slabs. And they also pointed out that kinetic frictional force was closely related to the change of potential energy along the sliding direction. Thus, the kinetic frictional forces with respect to sliding distance along x direction are plotted in Fig. 7. We obtain a similar result that the kinetic frictional force is determined by the change of potential energy along x direction. Besides, there is a one-to-one correspondence between frictional force and charge density difference. It has to be noted that the so-called D disorder peak is observed in well-ordered graphene, corresponding to A edge vibration modes on both external and internal edges, which can only exist on single hexagonal ring or for cluster of reduced number of cyclic rings. Differently, we calculated Raman spectra of monolayered h-BN as shown in Fig. 8. A D peak at 1339.5 cm−1 is obviously observed corresponding to the high-frequency intra-layer E2g vibration mode [7, 23] (for detailed vibration mode, see Supplementary Movie), which is well in agreement with the experimental data (1330.2–1367.4 cm−1) [24].

Fig. 7
figure 7

Variation of frictional force along x direction: a 0 nN, b 1.4 nN, and c 6.0 nN for H–H or T–T configuration; d 0 nN, e 0.6 nN, and f 3.7 nN for H–T configuration

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Fig. 8
figure 8

Calculated Raman spectra of monolayered h-BN. Inset is vibration mode of h-BN at 1339.5 cm−1

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4 Discussion

Generally, the anti-wear of substrate and coating system closely depends on the adhesion between friction pairs and coating. The anti-wear results from lower friction and anti-adherence between coating and the gliding counterpart [25]. Based on Neuville’s views, we find that low friction should correspond to low adhesion between coating and the gliding counterpart. On the other hand, a D disorder peak is observed in Fig. 8, it indicates that h-BN is a similar structure compared to carbon materials. And Neuville et al. believed that h-BN could be transformed into harder c-BN with the incidence of quantum electronic activation, meanwhile c-BN could be heat-degraded to h-BN during sliding [23, 26]. Thus, the quantum electronic atomic rearrangement model has also been applied in h-BN. Thus, we systematically investigate the tribological properties of h-BN bilayer under variable normal load. We obtain a quite different conclusion from the previous work [15]. Based on Tomanek’s theory [27], the kinetic friction coefficient could be calculated by the formula \(\mu \,=\,\frac{{{F_{\text{f}}}}}{{{F_{\text{N}}}}}\), combination of Figs. 7 and 8, FN equals FvdW at 0 nN, thus, the \({\mu _{{k_{\hbox{max} }}}}\) is estimated to be about 0.75 (H–H or T–T) and 0.37 (H–T), the rest FN equals Fz minus FvdW, \({\mu _{{{\text{k}}_{\hbox{max} }}}}\) is estimated to be 1.1 and 2.4 for 1.4 and 6.0 nN (H–H or T–T), and 0.15 and 1.7 for 0.6 and 3.7 nN (H–T), respectively. It is well in agreement with the change trend of friction coefficient as reported by the previous experimental researches [14, 28, 29]. In order to well understand this strength phenomenon, we should pay more attention to its electronic structure. As we known, h-BN has a similar hexagonal structure to graphene that applied in many fields. However, h-BN is a semiconductor with wide band gap, thus it exhibits good electrical insulation. An et al. reported pressure-induced insulator-semiconductor transition in h-BN bilayer [30]. We calculate the band gap for H–H (T–T) and H–T configurations under different normal loads. The band gap is 4.23 (0 nN), 3.79 (1.4 nN), and 2.26 eV (6.0 nN) for H–H (T–T) configurations, and 4.68 (0 nN), 4.57 (0.6 nN), and 4.19 eV (3.7 nN) for H–T configurations. It indicates that the band gap of H–H (T–T) configurations sensitively decreases with increase of the normal load compared to that of H–T configurations. And the band gap has a significant effect on the sliding properties of semiconductor materials as reported in our previous work [31]. We point out that the low friction can retain under higher band gap. Thus, high band gap should be one of key factors to construct the low-friction path in 3D-PES. On the other hand, it is reported that h-BN has strong bonding in xy plane but weak bonding along z direction [20]. It implies the strong covalent interlayer bonding and weak interlayer van der Waals bonding. As increasing the normal load, the van der Waals interaction is enhanced as shown in Fig. 9, the positive values of FvdW indicate a repulsion [32] in h-BN bilayer. The strong repulsive FvdW can maintain the low friction as reported in most of previous investigations [33,34,35]. In this work, the Fz is almost balanced by Fvdw at 0 and 0.6 nN (H–T), as shown in Figs. 2, 4, and 9, ensuring its low friction. Thus, the change of charge density difference with the augmenter of the normal load leading to enhancing repulsive FvdW should be another key factor for the low-friction path in 3D-PES.

Fig. 9
figure 9

The repulsive force FvdW as a function of interlayer distance along z direction for H–H (T–T) and H–T configurations

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Of course, a detailed tribological investigation should take into account numerous variables such as hardness, the sliding velocity, the compressive stress, tensile stress, and so on. Given that studying the effect of these factors on low friction beyond the scope of this work, here, we simply aim at probing the possible relationship between low friction and potential energy surfaces.

5 Conclusions

By means of first-principles method, we construct the potential energy surfaces for the sliding motion of h-BN bilayers to probe their tribological properties depending on the applied the normal load or lateral orientation. Two h-BN bilayers as H–H (T–T) and H–T configuration are established. The load-induced low-friction paths are obviously observed for two models, which indicates its better lubricant properties under high load.

We also detect the delicated interplay between electrostatic, band gap, and van der Waals contributions to the h-BN bilayer. We find that high band gap of h-BN bilayer is conducive to constructing low-friction paths in 3D-PES and electron interactions of B atoms in between monolayer h-BN accelerate the increase of the repulsive Van der Waals force maintaining low-friction paths. We believe that this investigation on h-BN provides positive guidance for further experimental investigations.