Piezoelectric Electrostatic Superlattices in Monolayer MoS₂

In this work, we restrict attention to one-dimensional potential modulations, $V(\bm{r})=V_{0}\cos(\bm{q}\cdot\bm{r})$, with the vector $\bm{q}$ directed along either the armchair $(\bm{q}_{AC})$ or the zigzag $(\bm{q}_{ZZ})$ direction: Figure 2A displays an example of a superlattice along the armchair direction with $V_{0}$=0.1 V and period $L$=17.6 nm. The functional form of this potential is motivated by the calculated displacement fields in Ref. 3 and we assume that the field does not vary appreciably across the thickness of the monolayer. In response to this external perturbation, atoms are displaced from regions of higher potential towards regions of lower potential (peak displacement of $\pm 0.6$Å; Fig. 2C), leading to tensile and compressive strains, $\epsilon_{11}$, that range between $\pm 1.8\%$ (Fig. S2). Figure 2B displays the spatial distribution of band edges from which we observe that electronic states, as deep as 0.1-0.2 eV into the valence or conduction band, are predominantly localized within the region of tensile strain. The perturbation reduces the overall band gap by $\sim 30\%$ relative to the unperturbed bandgap of monolayer MoS₂ (from 1.67 eV to 1.13 eV). By comparison, if atoms are held fixed at their original positions (no structural relaxation; Fig. S3), the valence and conduction band edges are spatially separated, localizing within regions of positive and negative potential, respectively, that too becoming more distinct only at larger fields. The decrease in the band gap is now less than 1% ($\sim 15$ meV), over an order of magnitude smaller than the relaxed case at the same applied potential. These observations essentially confirm, at least to lowest order, the simple physical picture presented previously. Of course, the armchair direction is not special in this regard, and a calculation for a cosine potential applied along the zigzag direction shows analogous atomic displacements and localization of band edges within the region of tensile strain, (Fig. S4). However, there is one crucial distinction between the responses of the armchair and zigzag directions: for the armchair case, the VBM and CBM, though localized within the region of tensile strain, have very little spatial overlap (Fig. 2B); in contrast, for the zigzag case, the VBM and CBM overlap spatially within the region of tensile strain (Fig. S4). To understand this markedly anisotropic response, we undertake a more detailed analysis of the structural response in each case.

Monolayer MoS₂ (space group $P\bar{6}m2$; point group $D_{3h}$) lacks a center of inversion and is strongly piezoelectric. [46] The polarization vector ($\bm{P}$) and strain tensor ($\bm{\epsilon}$) are related via the constitutive relation $\bm{P}=\bm{e}{\bm{\epsilon}}$, where $\bm{e}$ is the third-rank piezoelectric tensor. [50] When the external potential $V$ is aligned with the armchair direction ($\bm{q}\parallel\bm{q}_{AC}$), the piezoelectric coupling to the induced strain field induces a polarization field that is aligned with the armchair direction ($\bm{P}\parallel\bm{q}_{AC}$). In contrast, when the external potential $V$ is aligned with the zigzag direction ($\bm{q}\parallel\bm{q}_{ZZ}$), the polarization field is along the orthogonal armchair direction ($\bm{P}\perp\bm{q}_{ZZ}$). Further details can be found in the supplementary text. ¹¹1See Supplemental Material at [URL will be inserted by publisher] for details. Considering the armchair case and focusing on the region of tensile strain $-L/4\leq x\leq L/4$ where the band edges are localized, we calculate a net ionic dipole moment of 6.64 eÅ or $\sim 12\mu\textrm{C/cm}^{2}$ along the direction of quantization ($\bm{q}_{AC}$). ²²2The areal dimensions of the supercell are 176.41 Å$\times$ 3.18 Å while the thickness of the monolayer is 3.13 Å. Note that this degree of polarization is comparable to the remnant polarization in several ferroelectric materials. [53] Thus, separation of the valence and conduction band edges within the tensile region is driven by the internal electric field that arises from piezoelectric coupling to the strain field induced by the external potential. For a larger external potential ($V_{0}$=0.3 V; Fig. S5), both the localization of band edges within the region of tensile strain as well as their spatial separation are more marked, as deep as 0.5 eV into the valence/conduction band. By contrast, for the zigzag case, the dipole moments within the region of tensile strain $-L/4\leq x\leq L/4$ are orthogonal to the direction of quantization and, hence, there is no further spatial separation of band edges within this region of quantum confinement.

We now investigate the influence of structural relaxation and strain-induced internal electric fields on the electronic bandstructure of monolayer MoS₂ and the emergence of superlattice minibands. As the external potential and structural relaxation can be viewed as perturbations to the pristine MoS₂ monolayer, unfolding the superlattice bandstructure [54, 55, 56] from its reduced Brillouin zone to the full (extended) Brillouin zone of monolayer MoS₂ (Appendix B; Fig. 6) allows for a more transparent analysis. Figures 3(A, B) display the unfolded bandstructures for the unrelaxed and relaxed armchair superlattices, from which we note several important distinctions. In the unrelaxed case (Fig. 3A), the change in band gap is small ($<1\%$;  15 meV) for this magnitude of external potential ($V_{0}$=0.1V), and the gap remains direct at the K valleys. In the extended zone picture, we clearly observe the formation of superlattice minibands that are gapped due to the quantum confinement effect of the external potential. When the lattice is allowed to relax, these gapped minibands are further dispersed and the decrease in band gap is more substantial ($\sim 30\%$; 0.54 eV for $V_{0}$=0.1 V), the gap now being indirect between the $\Gamma$ valley at the valence band edge and the (degenerate) K valleys at the conduction band edge; the direct gap at the K valleys is $\sim 40$ meV larger. Similar results are seen in the zigzag case (Fig. S6). For a larger external potential ($V_{0}$=0.3 V; Fig. S7), the miniband dispersion becomes even more distinct with substantial miniband gaps (tens of meV). To demonstrate conclusively the structural basis of our findings, we recalculated the bandstructures of fully relaxed superlattices by artificially fixing atoms at their (field-dependent) positions and turning off the external potential (Figs. S8, S9) and found relatively small changes ($<10\%$) in the band gap with no qualitative changes in the miniband dispersion. We therefore conclude that the miniband dispersion occurs primarily due to Stark shifts driven by the internal polarization field that arises from piezoelectric coupling to the induced strain in the superlattice. Such effects are well known in quantum wells heterostructures composed of piezoelectric materials (e.g., zincblende-structure III-V superlattices [2]) in which strains at interfaces produce built-in electric fields that lead to Stark shifts and consequent dispersion of minibands that are accompanied by a decrease in the band gap. The important distinction is that, unlike quantum-well heterostructures involving dissimilar materials, we have here an electrostatic superlattice in a pristine monolayer of a single-phase 2D material that displays analogous physics.

To complete our analysis of bandgap tuning of monolayer MoS₂ electrostatic superlattices, we sampled the parameter space of such superlattices, including wavelengths and amplitudes of potential modulations as well as the effects of structural relaxation. The outcome of these studies is displayed in Figure 4, from which we see that: (i) for a given superlattice length, $L$, the band gap, $E_{g}$, decreases nearly linearly with the amplitude, $V_{0}$, of the applied potential; [32] (ii) for a fixed value of $V_{0}$, a longer supercell experiences a greater reduction in band gap; and (iii) the reduction in band gap is substantially greater when structural relaxation is allowed. These observations hold irrespective of the direction – armchair or zigzag – of the potential modulation. Based on these observations, we make a scaling ansatz

where $E_{g,0}$ is the band gap of an unperturbed MoS₂ monolayer, $a_{0}$ is the lattice constant of the monolayer, $q$ is the elementary charge, and $\alpha$ and $\gamma$ are fitting parameters. Given the vast difference in behavior between unrelaxed and relaxed cases, as well as smaller but noticeable differences between the armchair and zigzag cases, we find it necessary to use different prefactors, $\alpha_{i}$ for each case; however, it is sufficient to use a single exponent $\gamma$ to fit the data accurately. For a specific direction (armchair/zigzag) of applied potential, a comparison of the prefactors, $\alpha_{i}$, for the relaxed and unrelaxed cases (Fig. 4C) indicates that the band gap is reduced by over an order of magnitude ($\sim$15 times) when superlattice relaxation is permitted. This unambiguously underscores the need to account for structural relaxation, which has been neglected in theoretical modeling of 2D electrostatic superlattices. ³³3In a practical device, one would expect the MoS2 monolayer to be partially constrained by the substrate, gate dielectric, contacts, etc., that preclude complete structural relaxation, the unrelaxed and (free-standing) fully relaxed cases serving as limiting cases. In contrast, the direction of the applied potential modulation is of less importance for bandgap tuning although it is significant for charge-carrier separation, as noted before. Finally, it is worth noting that the quantitative values of band gaps are underestimated due to the well-known limitations of semilocal exchange-correlation functionals, but the qualitative behavior presented here is expected to be robust (see Appendix C).

One of the most interesting properties of monolayer MoS₂ is the coupling between valley and spin degrees of freedom, due the combination of strong spin-orbit splitting at the valence band edge and lack of centrosymmetry. [48, 49] Specifically, in the absence of magnetic fields, the $K$ and $-K$ valleys of monolayer MoS₂ are related only by time-reversal symmetry and associated with distinct and opposite magnetic moments. Thus, it is natural to inquire whether these desirable valley-selective properties are still preserved in electrostatic superlattices, in spite of large changes in electronic band gaps and, more importantly, significant Stark shifts at the band edges. Figures 3(C, D) display the spin-orbit-split band edges at the $\pm K_{2}$ valleys of the corresponding bandstructures displayed in Figures 3(A, B) respectively. In the unrelaxed case (Fig. reffig3C), which we recall presents a very small change in band gap and no significant miniband dispersion, the superlattice states essentially follow the dispersion of the primitive cell eigenstates and preserve the valley polarization. The situation is more interesting for the relaxed case (Fig. 3(D)), which shows the formation of distinct spin-polarized minibands that preserve the strong spin-polarization ($m_{z}\to\pm 1$) of the $\pm K_{2}$ valleys. At the valence band edge, in particular, minibands of exclusively one spin-polarization populate the spin-orbit gap ($\sim$150 meV [58, 59]) and, importantly, these minibands are gapped by several tens of meVs. At the conduction band edge, the spin-orbit splitting is very small [60, 61] and, while the miniband gaps are appreciable, the up/down spin states are nearly degenerate. Increasing the potential amplitude ($V_{0}$=0.3V; Fig. S9) further increases the miniband splitting to $\sim$100 meV but there still remains a single spin-polarized miniband within the spin-orbit gap, suggesting a degree of tunability of these miniband energies. The zigzag superlattice, in contrast, does not show significant miniband dispersion at the valence band edge of the $K$ valleys (Fig. S6) and thus, there are no miniband states within the spin-orbit gap.

While valley-polarization and the consequent valley-selective properties of monolayer MoS₂ (and several other 2D materials) are well known, the possibility of engineering valley-polarized minibands within a single-phase monolayer is of particular significance. Not only are minibands of importance for optical emission/detection in the mid-infrared and THz spectral ranges, but the fact that these minibands are valley-polarized implies that the such emission/detection could also be spin-selective, enabling future spintronic and quantum electronics applications. Furthermore, while MoS₂ already presents an appreciable spin-orbit gap ($\sim$150 meV), opening up this gap further could provide an even larger energy window within which miniband splitting can be tuned. To this end, we display in Figure 5, an example of the bandstructure of a relaxed superlattice of monolayer WS₂ – also a piezoelectric material [46] – subjected to a cosine potential of amplitude $V_{0}$=0.1V along the armchair direction (in exact correspondence with the MoS₂ case). Spin-orbit splitting of the valence band edge at the K valleys in monolayer WS₂ is nearly three times larger than in MoS₂ [58, 59], and we now observe several spin-polarized minibands; increasing the potential to $V_{0}$=0.3V further enhances the miniband splitting (Fig. S10). Thus, we suggest that electrostatic superlattices in piezoelectric 2D materials [62] with strong spin-orbit coupling can add a new dimension of miniband engineering to ongoing efforts in quantum electronics.