Introduction

Strong correlations among electrons arise when their mutual Coulomb interaction is similar or larger than their kinetic energy, and the delicate balance between these two energy scales determines the ground state of the system and its low-energy excitations. The rise of long period moiré patterns, formed by vertically stacking two atomically thin crystals with a lattice mismatch (δ) and/or relative twist angle (θ), provides a unique capability to tune the two critical energy scales (Coulomb interaction and kinetic energies)—and the electron density—over several orders of magnitude, creating a highly versatile solid-state quantum material platform1. The potential to engineer and probe strongly correlated states has been highlighted in several moiré material systems, including twisted bilayer graphene heterostructures near the magic angle2,3,4 and homo-5,6,7 or heterobilayer8,9,10,11 transition metal dichalcogenide (TMD) moiré heterostructures. The physics of these systems is determined by the strong electronic interactions in the landscape defined by the underlying moiré potentials. Further examples of synthetic moiré materials in which new physical properties arise due to interlayer hybridization can be found in moiré trapped excitons12,13, twisted 2D magnets14,15,16, moiré solitons17,18, moiré polaritons19, and ferroelectric moiré materials20,21,22.

The simplest model to understand strongly interacting quantum systems is the Hubbard model, which consists of a kinetic term defined by the hopping parameter t and the on-site Coulomb repulsion U. In the Hubbard picture, when U/t > 1 the kinetic energy is quenched, and strong electronic correlations emerge. Although it is expected that the Hubbard model can provide valuable insight into phenomena such as high-temperature superconductivity or exotic magnetic states23,24, only the one-dimensional (1D) case can readily be solved. The solution to higher dimensional problems depends on a delicate balance between the different model parameters and the lattice geometry1. Hence, quantum simulators based on ultracold atomic optical lattices, where it is possible to preset the lattice geometry and tune U/t by adjusting the laser power and U using Feshbach resonance techniques25, have become an exciting avenue to probe the Hubbard model and explore emerging quantum materials26.

Although 2D moiré materials offer access to different energy scales and a broader range of density and temperature than what is achievable in cold atom optical lattices, the same wide-ranging in situ tunability (of lattice symmetry, U, and t) possible in optical lattices has proven to be elusive for 2D quantum materials. While the interlayer coupling and band alignment can be tuned in situ by displacement fields5,6,22, U and t can only be tuned via the choice of constituent materials, the stacking angle θ or adjusting the dielectric environment of the active moiré region27,28. Since θ and the permittivity of the surrounding layers are set during the fabrication of the moiré heterostructure, direct continuous modification of the Hubbard model parameters is not readily achievable29, restricting the capability to fine-tune parameters near critical points and broadly explore phase diagrams. This limitation constrains the usefulness of moiré materials as quantum simulators.

The effect of heterostrain30,31 in homo- and heterostructures, i.e., the presence of a differential strain between the layers that compose the structure, was found as a path forward to tailor their electronic properties. The heterostrain issue was extensively studied from the theoretical point of view30,32,33,34,35 with the aim to, for instance, design flat bands in bilayer graphene and TMD structures32,34. From the experimental side, heterostrain was mainly studied in unintentionally strained structures or systems without in situ tuning31,36,37,38 and only a few experimental realizations have focused on the implementation of devices to intentionally apply heterostrain on demand17,39. However, the use of heterostrain to directly affect the moiré superlattice with the perspective to tune U/t has not been fully investigated. In this work, we explore the effect of heterostrain on the size and geometry of moiré lattices formed in vertically stacked bilayers of van der Waals materials. Using a generic analytical approach that is independent of the intrinsic crystal structure, we describe the effects of biaxial, uniaxial, and shear heterostrains for realistic experimental conditions. We find that depending on the type of strain, we can tune the moiré wavelength or even smoothly modify the moiré lattice from, e.g., triangular into rectangular lattices. Furthermore, the moiré wavelength can be tuned in situ in different lattice directions to break symmetries in the system, satisfying lattice conditions required for exotic quantum models beyond the standard Hubbard model, such as coupled Luttinger chains40. Finally, we show that the use of heterostrain can be implemented to fine-tune and tailor structures to overcome fabrication variabilities that are typically present, for instance, in magic-angle graphene and other moiré materials41. We thus show that strain control is a promising strategy to provide the critical degree of freedom required to realize reconfigurable quantum materials and achieve a fully tunable on-chip quantum simulator.

Results and discussion

Mathematical description

Moiré lattices are large-scale interference patterns formed when the unit cells of adjacent layers deviate either by a small twist angle θ or a lattice constant mismatch δ42. In this work, we assume that the unit cell of both layers of the homo-/heterostructure are of similar geometry, while the size of the unit cell of one layer may deviate by a scaling factor (1 + δ). The following formulation describes a pure geometrical deformation of moiré lattices under a global 2D strain tensor in which each layer is considered as a rigid object, i.e., atomic reconstruction effects can be neglected. However, to reach an experimental realization, it is important to take atomic reconstruction into account. Possible experimental implementations are presented in Supplementary Note 5. In structures with small stacking angles, atoms within each layer try to adjust the stacking landscape forming commensurate regions17,43,44,45. This effect increases interlayer adhesion and leads to higher friction between layers that hinders the possibility of modifying the atomic registry29,46,47. Conversely, higher stacking angles reduce the reconstruction effects and avoid strain transfer between layers (μ ≠ 0). For MoSe2-WSe2 heterostructures, these critical angles are found to be 2.5° and ~1.0° for 3R and 2H stacking, respectively44,48,49, and increase to ~3.0° in the case of 2H homobilayers45. Hence, it is possible to avoid atomic reconstruction by using a stacking angle above the critical angle of the bilayer. On the other hand, while the atomic reconstruction is always present, heterostrain can also reduce interlayer interaction and prevents atomic reconstruction50, as it has been shown that ϵu ~ 3% can transform a totally commensurate moiré lattice (θ = 0°) into a partially incommensurate structure with reduced atomic reconstruction17.

The most general moiré pattern is formed between two monoclinic lattices with Bravais lattice vectors ai for the lower layer and bi for the upper one, as sketched in Fig. 1a. These primitive lattice vectors can be written as follows:

$$\begin{array}{lll}{{{{\bf{a}}}}}_{{{{\rm{i}}}}}\,=\,{a}_{{{{\rm{i}}}}}{R}_{{\psi }_{{{{\rm{i}}}}}}\cdot \left(\begin{array}{l}1\\ 0\end{array}\right)\\ {{{{\bf{b}}}}}_{{{{\rm{i}}}}}\,=\,{a}_{{{{\rm{i}}}}}(1+\delta ){R}_{{\psi }_{{{{\rm{i}}}}}+\theta }\cdot \left(\begin{array}{l}1\\ 0\end{array}\right),\end{array}$$
(1)

where ai are the lattice constants of a general 2D monoclinic lattice and \({R}_{{\psi }_{{{{\rm{i}}}}}}\) is the 2D rotation matrix with an angle ψi. The angle ψi is

$${\psi }_{{{{\rm{i}}}}}=\left\{\begin{array}{ll}{\theta }_{0}\,;\,{{{\rm{i}}}}=1\\ {\theta }_{0}+\beta \,;\,{{{\rm{i}}}}=2\end{array}\right.$$
(2)

with θ0 being the overall rotation of the lattices compared to the x-axis and β the angle between the primitive lattice vectors a1(b1) and a2(b2). The stacking of these TMDs results in a bilayer system with a moiré pattern as presented in Fig. 1b. Such a lattice is characterized by the moiré lattice vectors A1 and A2 whose magnitudes are A1 and A2, respectively. In addition, we define as α the angle between those vectors.

Fig. 1: Relevant parameters describing the moiré lattice formed by monolayers of hexagonal crystals, such as graphene and TMDs.
figure 1

a Sketch of the different TMD layers that compose the moiré lattice. The lattice parameters of the first (second) layer are a1 (b1) and a2 (b2), the angle between those vectors is β for both layers, and the relative angle between the lattices is θ. b Moiré lattice parameters A1 and A2 and the internal angle α between those vectors in a bilayer formed by the hexagonal monolayer crystals.

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In the next step, both lattices are subject to a geometric deformation due to an applied strain on the bottom layer, expressed by a general 2D strain tensor ϵ. We restrict our discussion to a global strain that is uniform across the whole layer. To account for any slippage between layers, we introduce the strain transfer parameter μ. This strain transforms the Bravais lattice vectors ai and bi of the individual layers in the form

$$\begin{array}{lll}{{{{{\bf{a}}}}}^{{\prime} }}_{{{{\rm{i}}}}}\,=\,{a}_{{{{\rm{i}}}}}\left({\mathbb{I}}+\epsilon \right)\cdot {R}_{{\psi }_{{{{\rm{i}}}}}}\cdot \left(\begin{array}{l}1\\ 0\end{array}\right)\\ {{{{{\bf{b}}}}}^{{\prime} }}_{{{{\rm{i}}}}}\,=\,{a}_{{{{\rm{i}}}}}(1+\delta )\left({\mathbb{I}}+\mu \epsilon \right)\cdot {R}_{{\psi }_{{{{\rm{i}}}}}+\theta }\cdot \left(\begin{array}{l}1\\ 0\end{array}\right).\end{array}$$
(3)

The parameter μ allows us to model different experimental setups (see Supplementary Note 6), for instance, a system where the lower layer is clamped directly to the substrate while the upper one is only in contact with the first one. Since the interlayer friction is high for commensurate structures (small θ or δ) and low for incommensurate structures (high θ or δ), the transferred strain in the second layer is different for each case29,46,47. In the incommensurate case, the strain on the lower layer is not transferred to the upper one, i.e., μ = 0, which we refer to as heterostrain. For small θ or δ, the layers are more commensurate such that 0 < μ < 1. A full transfer of strain μ = 1 is referred to as homostrain.

The strain tensor ϵ is commonly written as a symmetric 2 × 2 matrix with three independent parameters32. Hence, we express the strain matrix as follows:

$$\epsilon =\left(\begin{array}{cc}{\epsilon }_{{{{\rm{xx}}}}}&{\epsilon }_{{{{\rm{xy}}}}}\\ {\epsilon }_{{{{\rm{xy}}}}}&{\epsilon }_{{{{\rm{yy}}}}}\end{array}\right)={\epsilon }_{{{{\rm{c}}}}}{\mathbb{I}}+{\epsilon }_{{{{\rm{s}}}}}{S}_{{\phi }_{{{{\rm{s}}}}}},$$
(4)

where we define

  • \({\epsilon }_{{{{\rm{c}}}}}=\frac{{\epsilon }_{{{{\rm{xx}}}}}+{\epsilon }_{{{{\rm{yy}}}}}}{2}\) as biaxial strain,

  • \({\epsilon }_{{{{\rm{s}}}}}=\sqrt{{\left(\frac{{\epsilon }_{{{{\rm{xx}}}}}-{\epsilon }_{{{{\rm{yy}}}}}}{2}\right)}^{2}+{\epsilon }_{{{{\rm{xy}}}}}^{2}}\) as shear strain,

  • \({S}_{{\phi }_{{{{\rm{s}}}}}}=\cos ({\phi }_{{{{\rm{s}}}}}){\sigma }_{{{{\rm{x}}}}}+\sin ({\phi }_{{{{\rm{s}}}}}){\sigma }_{{{{\rm{z}}}}}\) as shear matrix, and

  • \({\phi }_{{{{\rm{s}}}}}=\arccos \left(\frac{{\epsilon }_{{{{\rm{xy}}}}}}{{\epsilon }_{{{{\rm{s}}}}}}\right)\) as shear strain angle.

Our strain parameterization follows ref. 36, which separates the strain into biaxial strain ϵc (which changes the size of the unit cell) and shear strain ϵs (which alters the shape of the unit cell). Details on the physical meaning of the shear angle and different experimental feasibilities can be found in Supplementary Note 5. In this context, uniaxial strain (ϵxx = ϵu, ϵxy = 0, ϵyy = −νϵu) is a mixture of biaxial and shear strain with a shear angle set to ϕs = 90°. Note that under uniaxial strain, the crystal is elongated along one direction, while in the perpendicular direction, the crystal deforms proportional to the Poisson ratio − ν.

Using the previous definitions, it is possible to calculate the real space moiré lattice under general strain, as presented in Supplementary Note 1. The new moiré lattice vectors \({{{{{\bf{A}}}}}^{{\prime} }}_{{{{\rm{i}}}}}\) take the form

$$\begin{array}{lll}{{{{{\bf{A}}}}}^{{\prime} }}_{{{{\rm{i}}}}}\,=\,\frac{{a}_{{{{\rm{i}}}}}(1+\delta )}{{{\Delta }}}\\ \qquad\quad\,\cdot\, \left[(1+\delta ){c}_{\mu }\left((1+{\epsilon }_{{{{\rm{c}}}}}){\mathbb{I}}+{\epsilon }_{{{{\rm{s}}}}}{S}_{{\phi }_{{{{\rm{s}}}}}}\right)\right.\\ \qquad\quad\,-\,\left.{c}_{1}\left((1+\mu {\epsilon }_{{{{\rm{c}}}}}){R}_{\theta }+\mu {\epsilon }_{{{{\rm{s}}}}}{S}_{{\phi }_{{{{\rm{s}}}}}+\theta }\right)\right]\cdot {R}_{{\psi }_{{{{\rm{i}}}}}}\cdot \left(\begin{array}{l}1\\ 0\end{array}\right)\end{array}$$
(5)

where the denominator Δ is the expression

$$\begin{array}{lll}{{\Delta }}\,=\,{c}_{\mu }{(1+\delta )}^{2}+{c}_{1}\\ \qquad\,\,-\,2(1+\delta )\left[(1+{\epsilon }_{{{{\rm{c}}}}}+\mu {\epsilon }_{{{{\rm{c}}}}})+\mu ({\epsilon }_{{{{\rm{c}}}}}^{2}-{\epsilon }_{{{{\rm{s}}}}}^{2})\right]\cos (\theta ),\end{array}$$
(6)

\({c}_{1}={(1+{\epsilon }_{{{{\rm{c}}}}})}^{2}-{\epsilon }_{{{{\rm{s}}}}}^{2}\), and \({c}_{\mu }={(1+\mu {\epsilon }_{{{{\rm{c}}}}})}^{2}-{\mu }^{2}{\epsilon }_{{{{\rm{s}}}}}^{2}\). Finally, we calculate the moiré unit cell area M as follows:

$$M=\parallel {{{{{\bf{A}}}}}^{{\prime} }}_{1}\times {{{{{\bf{A}}}}}^{{\prime} }}_{2}\parallel ={a}_{1}{a}_{2}\parallel \sin (\beta )\parallel \frac{{\left(1+\delta \right)}^{2}}{\parallel {{\Delta }}\parallel }{c}_{1}{c}_{\mu }.$$
(7)

The full calculation is provided in Supplementary Note 2.

We emphasize that all manifestations of 2D strain are covered by these calculations and no approximations are made, in contrast to previous descriptions that focus on hexagonal lattices only and use approximations that only apply to specific strain configurations30,32,33. Note that expression (5) defines the moiré lattice vectors and can be employed for any homo- and heterobilayer structure with a similarly shaped unit cell.

In the particular case of homostrain, expression (5) leads to \({{{{{\bf{A}}}}}^{{\prime} }}_{{{{\rm{i}}}}}=\left({\mathbb{I}}+\epsilon \right)\cdot {{{{\bf{A}}}}}_{{{{\rm{i}}}}}\) which means that the strain is applied to the moiré vectors as if the moiré lattice is strained itself. Even though 2D materials can withstand very high strain levels51, the maximum applicable strain is ϵc, ϵs < < 1. Hence, the homostrain effect on the moiré lattice is very small and can be neglected in comparison to the effect of tuning the twist angle θ. However, in the case of imperfect strain transduction between the layers (μ < 1) we find that strain has a substantial effect on the moiré size and shape, since the magnitude of the moiré vectors is dominated by the denominator Δ in expression (5). For example, in the homobilayer case, increasing biaxial strain ϵc decreases the moiré lattice parameters with the approximate dependence of \({A}^{{\prime} }{{{\rm{i}}}}\propto 1/\parallel {\epsilon }_{{{{\rm{c}}}}}\parallel\). In the context of the Hubbard model, the hopping parameter t decreases exponentially with \({A}_{{{{\rm{i}}}}}^{{\prime} }\) while U scales inversely with \({A}_{{{{\rm{i}}}}}^{{\prime} }\)27,28 and, therefore, the ratio U/t can be modified by tuning ϵc. In contrast to θ, in situ strain tuning is experimentally viable, which allows direct tuning of U/t (see Supplementary Note 5).

In contrast to biaxial strain, shear strain can make the denominator Δ identically zero, leading to the divergence of the moiré vectors. This condition, referred to as a 1D moiré pattern33, has already been experimentally observed52. This effect occurs when the deformation of one layer due to shear strain aligns the lattice sites in one direction while leaving a mismatch in the perpendicular direction. We emphasize that this effect is not dependent on the orientation of the lattice θ0, the shear angle ϕs or the shape of the layer lattices.

In addition to the size of the moiré pattern, shear strain is also able to change the moiré lattice geometry since ϵs changes the shape of the individual lattice depending on the orientation of ϕs and θ0. The main defining parameter for the geometry of the moiré pattern is the angle α between the moiré vectors (see Fig. 1b). This angle takes the form

$$\sin (\alpha )=\left\Vert \frac{{{{{{\bf{A}}}}}^{{\prime} }}_{1}\times {{{{{\bf{A}}}}}^{{\prime} }}_{2}}{{{A}^{{\prime} }}_{1}{{A}^{{\prime} }}_{2}}\right\Vert .$$
(8)

In the case of zero shear strain (ϵs = 0), it can be shown that \(\frac{{a}_{1}}{{a}_{2}}\mathop{=}\limits^{{\epsilon }_{{{{\rm{s}}}}}=0}\frac{{{A}^{{\prime} }}_{1}}{{{A}^{{\prime} }}_{2}}\) and that \(\alpha \mathop{=}\limits^{{\epsilon }_{{{{\rm{s}}}}}=0}\beta\), proving that the shape of the moiré lattice is equal to the shape of the underlying lattice defined by ai, as presented in detail in Supplementary Note 3.

In contrast to in situ tuning of θ53,54,55, 2D strain has the potential to tune the size and shape of the moiré pattern via the three independent strain parameters ϵc, ϵs and ϕs. In the following section, we will focus on the case of pure heterostrain (μ = 0) on homo- and heterobilayer structures and show how the different types of strain can affect the moiré lattice geometry.

Biaxial heterostrain on hexagonal homo- and heterobilayers

In this section, we center our analysis on moiré patterns generated by monolayers with hexagonal lattices; in particular, we take WSe2 homobilayers and MoSe2-WSe2 heterobilayers as paradigmatic cases. The lattice parameters used to perform the calculations are \({a}_{{{{{\rm{MoSe}}}}}_{2}}=0.329\) nm for MoSe256 and 0.4% smaller for the WSe256,57,58. As shown in Supplementary Note 6, strategies can be proposed to avoid strain transfer for most bilayer structures. In the following, we analyze the pure heterostrain case, in which the deformation is applied to the lower layer, which, in our specific heterobilayer example, corresponds to the WSe2 monolayer. The Poisson ratios are \({\nu }_{{{{{\rm{MoSe}}}}}_{2}}=0.23\) and \({\nu }_{{{{{\rm{WSe}}}}}_{2}}=0.19\) for MoSe2 and WSe2, respectively56.

Figure 2a shows a sketch of the bilayer formation in which ϵc produces an in-plane elongation/contraction of the lower layer unit cell and the upper layer is stacked at an angle θ. The effect of ϵc and θ on the moiré pattern is depicted in Fig. 2b for homobilayer (left) and heterobilayer (right) structures. The vertical lines at ϵc = 0 correspond to the well-known behavior of homo- and heterobilayers described in ref. 59, i.e., in the homobilayer case, the structure presents a divergence of \({A}_{1}^{{\prime} }\) and \({A}_{2}^{{\prime} }\) at θ = 0 that evidences the absence of moiré patterns in naturally stacked bilayers (2H and 3R stacking order). However, the stacking of layers with θ ≠ 0 and/or ϵc ≠ 0 gives rise to moiré patterns whose periodicity (\({A}_{1}^{{\prime} }\) and \({A}_{2}^{{\prime} }\)) decreases by increasing ϵc and/or θ. In the heterobilayer case, as the lattice parameter of WSe2 is ~0.4% smaller than MoSe2, the stretching of the WSe2 layer by ~0.4% recovers the divergence observed at the point ϵc = 0, θ = 0 for homobilayers. Therefore the use of biaxial heterostrain in heterobilayers enables the formation of arbitrarily large moiré lattices as observed in homobilayer structures. Another important point to note is that due to the hexagonal symmetry of each layer, neither ϵc nor θ can modify the moiré lattice geometry that is also hexagonal, i.e., \({A}_{1}^{{\prime} }={A}_{2}^{{\prime} }\) and α = 60°, as demonstrated in Supplementary Note 3. Consequently, different combinations of strain and stacking angle can lead to the same moiré lattice (but with a relative rotation). For instance, the dotted lines in both panels of Fig. 2b describe a unique moiré lattice with \({A}_{1}^{{\prime} }={A}_{2}^{{\prime} }=10\) nm. Supplementary Note 4 provides details regarding the direction of the moiré lattice with respect to the underlying lattice in the absence of shear strain.

Fig. 2: Moiré lattice deformation in hexagonal bilayers under biaxial heterostrain.
figure 2

a Sketch of the moiré lattice formation. To perform the calculations, the biaxial strain ϵc, indicated through blue arrows, is applied to the lower layer. The upper layer is stacked with an angle θ respect to the lower one. b Moiré lattice parameters \({A}_{1}^{{\prime} }\) and \({A}_{2}^{{\prime} }\) as function of biaxial strain and stacking angle for homo- (left) and heterobilayer (right) structures.

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Uniaxial heterostrain on hexagonal homo- and heterobilayers

We continue by describing the effect of uniaxial heterostrain on the moiré lattice generated in homo- and heterobilayers using the same material parameters as in the previous section. Figure 3a shows a sketch of this situation in which the lower layer is stretched/contracted along the zigzag direction by applying a uniaxial strain ϵu. Due to the Poisson effect, the direction perpendicular to ϵu is deformed by −νϵu, where ν is the Poisson ratio. In the case where the layer is stretched in the zigzag direction, the armchair direction is contracted (stretched) if ν > 0 (ν < 0). The upper layer is stacked at an angle θ with respect to the lower layer. Figure 3b depicts the moiré lattice parameters \({A}_{1}^{{\prime} }\) (upper panels) and \({A}_{2}^{{\prime} }\) (lower panels) for homobilayer- and heterobilayer structures (left and right panels, respectively). Uniaxial strain has a stronger influence on the moiré lattice than biaxial strain: (i) For uniaxial strain, the moiré lattice parameters \({A}_{1}^{{\prime} }\) and \({A}_{2}^{{\prime} }\) diverge along curves in the θ − ϵu space instead of a single point (see red curves in Fig. 3b). (ii) ϵu breaks the degeneracy between \({A}_{1}^{{\prime} }\) and \({A}_{2}^{{\prime} }\). (iii) In heterobilayers, uniaxial strain cannot retrieve the moiré lattice geometry found in homobilayers.

Fig. 3: Deformation of the moiré lattice parameters in hexagonal bilayers under uniaxial heterostrain.
figure 3

a Schematic of moiré lattice formation under uniaxial heterostrain ϵu. The deformation of the lower layer due to strain along a zigzag direction and Poisson ratio ν are depicted by the arrows. The upper layer was stacked with an angle θ respect to the lower layer. b Moiré lattice parameters \({A}_{1}^{{\prime} }\) (upper panels) and \({A}_{2}^{{\prime} }\) (lower panels) for the homobilayer- and heterobilayer structures (left and right panels, respectively) as a function of uniaxial strain and stacking angle. Spots labeled as I, II, and III correspond to the different moiré lattices presented in real space in Fig. 4b and those marked as v1 and v2 identify the vertexes of the divergence curves in the heterobilayer case.

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In homobilayers, \({A}_{1}^{{\prime} }\) and \({A}_{2}^{{\prime} }\) diverge along lines that intersect at the origin (ϵu = 0, θ = 0). For heterobilayers, these curves do not intersect but show an avoided crossing behavior instead with vertices located on the axis θ = 0° (points marked as v1 and v2 on Fig. 3b). The position of v2 is given by the mismatch of the lattice parameters between the two layers, which in our case of a MoSe2/WSe2 heterostructure, is ϵu = 0.4%. The position of v1 is weighted by \({\nu }_{{{{{\rm{WSe}}}}}_{2}}\), as the deformation on the armchair axis is 19% of that of the zigzag axis. Now, the vertex is located at −0.4/0.19 ≃ −2.1%. It can easily be seen that a vanishing Poisson ratio can eliminate the divergence of the lattice parameters and lattice area in the negative semi-space. Further information about the effect of the Poisson ratio on the moiré lattice can be found in Supplementary Note 7.

Further insight into how uniaxial strain can affect the moiré lattice geometry can be observed in Fig. 4a, b. Figure 4a shows α (upper panels) and M (lower panels) as function of ϵu and θ for homo- and heterobilayers. α is a measure of the strong geometrical deformation of the moiré lattice. For instance, by combining θ and ϵu, it is possible to generate a rectangular lattice (α = 90°), identified by gray dotted lines. On the other hand, for a 1D moiré lattice, indicated by diverging moiré lattice parameters (red lines in Fig. 3b), \({{{{\bf{A}}}}}_{{{{\bf{1}}}}}^{{\prime} }\) and \({{{{\bf{A}}}}}_{{{{\bf{2}}}}}^{{\prime} }\) tend to be collinear, i.e., α → 0° or α → 180°. Note that the divergence of \({A}_{1}^{{\prime} }\) and \({A}_{2}^{{\prime} }\) also causes a divergence of the moiré unit cell area, even though these lattice vectors tend to be collinear. Figure 4b depicts real space images of some of the accessible moiré lattices by combining ϵu and θ. The different geometries correspond to the three points marked as I, II, and III in Figs. 3b and 4a. Point I describes a 1D moiré lattice, point II a rectangular geometry, and III a hexagonal arrangement.

Fig. 4: Effects of uniaxial heterostrain on the geometry and size of moiré lattices formed from hexagonal bilayers.
figure 4

a Relative angle α (upper panels) and moiré lattice area M (lower panels) as function of ϵu and θ for the homobilayer (left panels) and heterobilayer structure (right panels). Gray dotted lines denote configurations in which the moiré lattice is rectangular (α = 90°). The spots labeled as I, II, and III correspond to the moiré lattices presented in panel b. b Illustration of moiré lattices corresponding to the points labeled as I, II, and III in panel a and in Fig. 3b. The scale bar in II has a length of 30 nm and is valid for all real space moiré patterns I–III. Case I shows a 1D moiré lattice, II a rectangular one, and III a hexagonal one.

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Shear heterostrain on hexagonal homo- and heterobilayers

We finish our discussion on the effect of heterostrain on moiré lattices by presenting the case of shear strain (ϵs), once again employing the same material parameters as in previous sections. In Fig. 5a, we sketch the different layers and their deformation/rotation under the influence of ϵs. The case presented corresponds to a sideways load applied along the zigzag direction on the lower layer (violet arrows) while the upper layer is stacked at an angle θ. For simplicity, we will focus our analysis on α and M, shown in Fig. 5b for homobilayer (left) and heterobilayer structures (right). As in the case of uniaxial strain, the moiré lattice parameters and the moiré area diverge along curves in the θ − ϵs space (red lines). Once again, such divergence is accompanied by the formation of 1D moiré lattices, as the internal angle α tends to 0° or 180°. Again, the main difference between homo- and heterobilayers is in the vicinity of zero strain and θ = 0, where for heterostructures, the divergence curves do not cross (see red lines in the moiré lattice area). The positions of the vertexes of the curves are located at ±0.4%, since this strain is necessary to fulfill the mismatch between the lattice parameters of the TMDs. As a confirmation of the universality of our mathematical description of moiré lattices, we show similar behavior for moiré patterns formed by rectangular unit cells, such as a homobilayer of WTe2 in Supplementary Note 8.

Fig. 5: Moiré lattice deformation in hexagonal bilayers under shear heterostrain.
figure 5

a Scheme of the moiré lattice formation under shear heterostrain ϵs. The deformation of the lower layer due to shear strain along the zigzag direction is depicted through arrows. The upper layer was stacked with an angle θ respect to the lower one. b Relative angle α (upper panels) and moiré lattice area M (lower panels) as a function of shear strain ϵs and stacking angle θ for homobilayer (left panels) and heterobilayer (right panels) systems. Gray dotted lines denote configurations in which the moiré lattice is rectangular (α = 90°).

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In summary, we have presented a general geometrical description of the effect of strain in homo- and heterobilayer systems. We show that heterostrain can be used to form a vast variety of moiré lattice geometries, independent of the underlying lattice size and shape. We demonstrate how an initial moiré lattice geometry can be tuned into a variety of particular moiré patterns, e.g., a hexagonal lattice can be made rectangular or even 1D.

Outlook

Our results are experimentally feasible and can be realized with state-of-the-art strain tuning setups60,61,62 (see the Supplementary Material for further details). Although heterostrain is beginning to be explored experimentally17,63, most strain-tuning experiments focus on homostrain. However, most existing homostrain setups can also achieve heterostrain by clamping one layer to the substrate while leaving the other one mechanically decoupled from the substrate64,65. A thorough presentation of the different experimental setups and clamping configurations is given in Supplementary Note 5, where we also display how to apply pure biaxial, uniaxial, or shear strain as well as their possible combinations.

To verify the influence of strain on the moiré superlattice in a physical experiment, we propose the use of piezo response force microscopy, which measures polarizations arising due to strain gradients in the moiré52,66. The method is very well established and allows the local visualization of the moiré lattice for a variety of different 2D heterostructures. Since piezo response force microscopy only measures the local moiré geometry, it is an ideal tool to verify the deformation of the moiré pattern under strain, as proposed in this paper. The information gained about the size and shape of the moiré at different strain levels can then be used to explain electronic and optical observables, like the change in the photoluminescence emission of moiré excitons from circular polarization in a hexagonal moiré to linear polarization in a 1D moiré pattern35,52 or the change in carrier density for strongly correlated states in the fractional filling of the moiré lattice5,6,7,8,9,10,11.

The combination of shear and biaxial strains can become powerful knobs to modify, on demand, the moiré pattern size and shape. For example, Fig. 6 presents the combined effect of biaxial and shear strain in a generic hexagonal homobilayer stacked with θ = 1°. From the condition ϵc = ϵs = 0, it is possible to change U/t, in situ, by tuning the size of the moiré pattern through biaxial strain (point ϵc = ϵs = 0 to point III or to point IV). On the other hand, light green lines in the plot depict the condition \({A}_{1}^{{\prime} }={A}_{2}^{{\prime} }\). Varying the shear strain along the line that connects point I with II modifies the ratio \({A}_{1}^{{\prime} }/{A}_{2}^{{\prime} }\) which realizes a triangular Hubbard model with two tunable hopping parameters t1 and t2. Finally, the point labeled as V in Fig. 6 presents a lattice in which the effect described in ϵc = ϵs = 0 can be equivalently applied to a perfect square lattice. The fact that all these moiré patterns are accessible within one experiment highlights the potential of this novel approach. Moiré straintronics offers a powerful new avenue to explore highly correlated quantum systems. As commonly known, once the sample is fabricated, it is usually not possible to tune the stacking angle. Furthermore, achieving precise rotational alignment within the necessary accuracy required near critical points, such as in magic-angle graphene, is very challenging with state-of-the-art fabrication procedures41. In contrast to twist angle tuning29, strain can be applied much more precisely with an accuracy corresponding to 0.0001° in the twist angle (as shown in Supplementary Note 5). In ref. 67, it is shown that flat bands in twisted bilayer graphene can be achieved at an angle of 1.25° when a compressive shear strain of 0.36% is applied to a single layer. Hence, the twist angle set during the fabrication can provide a rough alignment knob, while heterostrain can be used to fine-tune the moiré lattice. Furthermore, heterostrain offers an additional mechanism to generate flat bands in graphene31,38,63,67,68 or tune highly correlated moiré quantum materials around critical points in the phase diagram28. Especially in twisted bilayer graphene, shear heterostrain not only alters the moiré miniband bandwidth but also lifts the particle-hole symmetry and breaks the valley degeneracy of the system32,67. A recent experiment in an ex-situ shear heterostrained bilayer graphene near the magic angle showed a zero-energy flat band indicating the possibilities of in situ tunable heterostrained systems67,69.

Fig. 6: Geometrical deformation of a general homobilayer stacked at θ = 1° under biaxial and shear heterostrain.
figure 6

Light green lines denote moiré lattices in which \({A}_{1}^{{\prime} }={A}_{2}^{{\prime} }\). Points labeled as I, II, III, IV, V, VI, and VII correspond to the different illustrations of the moiré pattern in real space. The scale bar in IV has a length of 30 nm in case of a MoSe2 bilayer and is valid for all real space moiré patterns I–VII. An experimental setup for the simultaneous control of biaxial and shear strain is shown in Supplementary Note 5. The computational script to reproduce real space images and strain animations can be found in https://github.com/QuantumPhotonicsLab/Strained-Moire-Visualization.

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On the other hand, our mathematical framework presents an important starting point for exploring reconstructed moiré lattices17,18,19,20,21,22. Finally, the capability to in situ tune the geometry and the interaction strength in highly correlated moiré quantum systems is, to the best of our knowledge, unprecedented. We, therefore, expect strain tuning of moiré materials will have a major impact on the exploration of highly interacting quantum systems, from fine-tuning magic-angle graphene to the realization of moiré quantum simulators for Luttinger liquids, the Hubbard model, and beyond.