Large phosphorene in-plane contraction induced by interlayer interactions in graphene-phosphorene heterostructures

Intralayer deformation in van der Waals (vdW) heterostructures is generally assumed to be negligible due to the weak nature of the interactions between the layers, especially when the interfaces are found incoherent. In the present work, graphene-phosphorene vdW heterostructures are investigated with the density functional theory (DFT). The challenge of treating a nearly incommensurate (very large) supercell in DFT is bypassed by considering different energetic quantities in the grand canonical ensemble, alternative to the formation energy, in order to take into account the mismatch elastic contribution of the different layers. In the investigated heterostructures, it is found that phosphorene contracts by $\sim 4\%$ in the armchair direction when compared to its free-standing form. This large contraction leads to important changes in terms of electronic properties, with the direct electronic optical transition of phosphorene becoming indirect in specific vdW heterostructures. More generally, such a contraction indicates strong substrate effects in supported or encapsulated phosphorene—neglected hitherto—and paves the way to substrate-controlled stresstronic in such 2D crystal. In addition, the stability of these vdW heterostructures is investigated as a function of the rotation angle between the layers and as a function of the stacking composition. The alignment of the specific crystalline directions of graphene and phosphorene is found energetically favored. In parallel, several models based on DFT-estimated quantities are presented; they allow notably a better understanding of the global mutual accommodation of 2D materials in their corresponding interfaces, that is predicted to be non-negligible even in the case of incommensurate interfaces.

Figure 1

Schematic construction and representation of the graphene-phosphorene vdW heterostructures studied in this paper. (a) Top view of the primitive cells of graphene and phosphorene, the two constituent building blocks of the heterostructure. (b) Superlattices of graphene and phosphorene are built and superposed for a given zigzag rotation angle ${\theta }_z$ between the layers. For that given angle, commensurate structures can be found with a corresponding strain tensor. From these commensurates, periodic graphene-phosphorene vdW heterostructures are constructed with different composition stackings as illustrated in (c)–(f). These structures are denoted ${\mathrm{G}}_i{\mathrm{P}}_j$ where $i$ and $j$ correspond to the number of adjacent layers of graphene and phosphorene, respectively.

Figure 2

The five commensurate structures investigated in this paper, characterized here by the zigzag $\left({\theta }_z\right)$ and armchair $\left({\theta }_a\right)$ rotation angles between the graphene and phosphorene layers. The primary zigzag $\left({\vec{a}}^{\text{C}}\right)$ and armchair $\left({\vec{b}}^{\text{C}}\right)$ vectors of graphene are represented in dashed red. Similarly, the primary zigzag $\left({\vec{a}}^{\text{P}}\right)$ and armchair $\left({\vec{b}}^{\text{P}}\right)$ vectors of phosphorene are shown in solid blue. The average strain tensors of graphene $\left({\underline{\underline{\overline{\epsilon }}}}^C\right)$ and of phosphorene $\left({\underline{\underline{\overline{\epsilon }}}}^P\right)$ are then defined based on their respective primary lattice vectors.

Figure 3

Schematic representation of the different definitions of energies: cohesive, mixing, stacking, and substitution energies. The scheme is split into three planes, corresponding each to a different system (upper plane: black phosphorus, middle plane: graphene-phosphorene vdW heterostructure, bottom plane: graphite). For sake of clarity, the intralayer and interlayer energies of each system are the plane axes. G and P abbreviate graphene and phosphorene, respectively, Gr and BP their bulk counterparts (graphite and black phosphorus), while GP stands for the graphene-phosphorene vdW heterostructure. The cohesive energy $E_{\text{coh}}$ and mixing energy $E_{\text{mix}}$ are trivially defined by the scheme. The stacking energy $E_{\text{stack}}$ is defined from the phosphorene and graphene geometries in the vdW heterostructure, themselves characterized by the average strain tensors ${\underline{\underline{\overline{\epsilon }}}}^{\text{P}}$ and ${\underline{\underline{\overline{\epsilon }}}}^{\text{C}}$ (the tensor and average notations are dismissed here for sake of clarity), respectively. Finally, the substitution energy $E_{\text{sub}}$ is defined from the same strained phosphorene and graphene in-plane geometries, but now stacked in the optimal stacking sequence on top of each other (interlayer distance optimized). By construction, the intralayer energies computed in the vdW heterostructure and in strained graphite and black phosphorus cancel out. Only remains the difference between interlayer interactions. If neglected, the substitution energy is independent of the commensurate structure in use $\left(E_{\text{sub}}[{\epsilon }_1^{\text{P}},{\epsilon }_1^{\text{C}}\right]=E_{\text{sub}}[{\epsilon }_2^{\text{P}},{\epsilon }_2^{\text{C}}\left]\right)$.

Figure 4

(Left) Mixing energy $E_{\text{mix}}$ and (right) substitution energy $E_{\text{sub}}$ for the different ${\mathrm{G}}_i{\mathrm{P}}_j$ vdW heterostructures as a function of the zigzag and armchair rotation angles ${\theta }_z$ and ${\theta }_a$. The red square and blue circle symbols correspond to the mixing and substitution energies, respectively, and the angle intervals $[{\theta }_z,{\theta }_a]$ are delimited thanks to short horizontal lines with upward and downward arrows. Using the model presented in Sec. 4, the differences between a given commensurate structure and the estimated ground state for that specific rotation angle are shown using error bars for both mixing and substitution energies.

Figure 5

Variation of (dashed red) elastic energy, (dotted blue) vdW energy, (solid green) energy sum as defined in Eq. (9) as a function of (left) zigzag lattice parameter and (right) armchair lattice parameter. The top panels correspond to the case of graphite, the middle panels to black phosphorus, while the bottom panels correspond to the case of the graphene-phosphorene vdW heterostructures. For comparison, in panel (a), the lattice parameters of graphene and graphite along the considered direction are displayed in solid black and in dotted black, respectively. In panel (b), the lattice parameters of phosphorene and black phosphorus along the considered direction are displayed in solid black and in dotted black, respectively. For the left panel (c), the zigzag lattice parameters of phosphorene and black phosphorus are displayed in solid black and in dotted black, respectively. For the right panel (c), the armchair lattice parameters of phosphorene and black phosphorus are displayed in solid black and in dotted black, respectively, while the armchair lattice parameter of graphene is displayed in dotted black. Stars correspond to the computed values, and circles correspond to the minima of the total energy.

Figure 6

Electronic band structures, unfolded on the Brillouin zone of phosphorene shown in (a), of (b) phosphorene (in white) and graphene (in red) monolayers superposed on top of each other, supposing that the graphene and phosphorene armchair lattices match and that the graphene zigzag lattice parameter is exactly 3/4 of its phosphorene counterpart. (b) Graphene and phosphorene strained such as the in-plane crystalline structure corresponds to the ${\mathrm{G}}_1{\mathrm{P}}_1(0^{\circ },0^{\circ },b_P=b_C)$'s one. The layers are spaced by a 15-Å-thick vacuum layer to avoid in that case the interactions between the layers. (c) ${\mathrm{G}}_1{\mathrm{P}}_1(0^{\circ },0^{\circ },b_P=b_C)$ graphene-phosphorene vdW heterostructure. The weight associated with the unfolding technique is given by the color bar and the marker size.