Energy Minimization of Two Dimensional Incommensurate Heterostructures

Abstract

We derive and analyze a novel approach for modeling and computing the mechanical relaxation of incommensurate two dimensional heterostructures. Our approach parametrizes the relaxation pattern by the compact local configuration space rather than real space, thus bypassing the need for the standard supercell approximation and giving a true aperiodic atomistic configuration. Our model extends the computationally accessible regime of weakly coupled bilayers with similar orientations or lattice spacing, for example materials with a small relative twist where the widely studied large-scale moiré patterns arise (Kim et al. in Proc Natl Acad Sci 114:3364–3369, 2017; Yoo et al. in Atomic and electronic reconstruction at van der Waals interface in twisted bilayer graphene, Nat Mater 18:448–453, 2019). Our model also makes possible the simulation of multi-layers for which no inter-layer empirical atomistic potential exists, such as those composed of $\hbox {MoS}_2$ layers, and more generally makes possible the simulation of the relaxation of multi-layer heterostructures for which a planar moiré pattern does not exist.

Appendix A. Proof of Prop. 3.2

In this appendix, we detail the technical proof of Proposition 3.2, which we recall first for ease of reading.

Proposition A.1

Let $\mathbf {u}\in W^{2,q}(X)$, then

$$\begin{aligned} \left\| \widehat{\mathbf {u}}_j - \widetilde{\mathbf {u}}_j \right\| _{L^q(\Omega )} \leqq \left( \frac{\vert \Gamma _j \vert }{2q+1} \right) ^{1/q} \theta ^2 \; \left\| \nabla ^2_\omega \mathbf {u}_j \right\| _{L^q(X_j)}, \end{aligned}$$

(A.1)

where is the subset of the transversal corresponding to lattice sites of layer j, $\nabla ^2_\omega \mathbf {u}_j$ is understood as a 2-linear form for which the norm is defined as $\Vert \ell \Vert := \sup _{\vert \mathbf {h}_1 \vert = \vert \mathbf {h}_2 \vert = 1 } \left| \ell [\mathbf {h}_1, \mathbf {h}_2] \right|$ and

$$\begin{aligned} \theta = \sqrt{p} \sup _{1 \leqq i,j \leqq p} \left\| \mathrm {E}_i - \mathrm {E}_j \right\| , \qquad \text {where } \Vert \cdot \Vert \text { denotes the } \mathbb {R}^2\text {-operator norm}. \end{aligned}$$

(A.2)

Proof

It is natural to relate this result to local finite element interpolant error estimation and follow similar steps. Let $\omega \in \Omega$ be an arbitrary configuration, j be an arbitrary layer number. Consider a Taylor expansion up to degree 1 of $\mathbf {u}_j$ around the point $\Pi _j\omega$:

$$\begin{aligned} T^1_\omega \mathbf {u}_j(\delta \omega ) = \mathbf {u}_j \left( \Pi _j \omega \right) + \nabla _\omega \mathbf {u}_j \left( \Pi _j \omega \right) \cdot \delta \omega \text { where } \delta \omega = (\varvec{\delta \omega }_1, \ldots , \varvec{\delta \omega }_p) \in \mathbb {R}^{2p},\ \varvec{\delta \omega }_j = \varvec{0}. \end{aligned}$$

Using the integral formula for the residual of the Taylor series, we have

$$\begin{aligned} \mathbf {u}_j(\Pi _j \omega + \delta \omega ) - T^1_\omega \mathbf {u}_j(\delta \omega )&= \int \nolimits _0^1 \frac{(1-h)^2}{2}\frac{d^2\mathbf {u}_j}{dh^2} (\Pi _j \omega + h \delta \omega )\,\mathrm{d} h\\&= \int \nolimits _0^1 \frac{(1-h)^2}{2} \nabla _\omega ^2 \mathbf {u}_j (\Pi _j \omega + h \delta \omega ) [\delta \omega , \delta \omega ] \,\mathrm{d} h, \end{aligned}$$

so by Jensen’s inequality we obtain the estimate

$$\begin{aligned} \Vert \mathbf {u}_j(\Pi _j \omega + \delta \omega ) - T^1_\omega \mathbf {u}_j(\delta \omega ) \Vert ^q \leqq \Vert \delta \omega \Vert ^{2q} \int \nolimits _0^1 \frac{(1-h)^{2q}}{2^q} \left\| \nabla _\omega ^2 \mathbf {u}_j (\Pi _j \omega + h \delta \omega ) \right\| ^q \,\mathrm{d} h. \end{aligned}$$

(A.3)

As earlier (see (3.2)), let us write $\varvec{\gamma }_j = \mathrm {E}_j \begin{bmatrix} s \\ t \end{bmatrix}$ with $0 \leqq s, t < 1$. The lattice sites in layer j around the origin are located at the four points $\mathbf {r}_{ab}$ with $a, b \in \{ 0,1 \}$ defined as in (2.12). One checks easily from the definition (3.4) that

$$\begin{aligned} \mathtt {T}_{-\mathbf {r}_{ab}} \omega = \Pi _j \omega + \delta \omega _{ab} \qquad \text {for } a,b \in \{0,1\}, \end{aligned}$$

where the shifts $\delta \omega _{00}$, $\delta \omega _{10}$, $\delta \omega _{01}$, $\delta \omega _{11}$ can be chosen as

$$\begin{aligned} \delta \omega _{00}&= \left( (\mathrm {E}_1 - \mathrm {E}_j) \begin{bmatrix} s\\t \end{bmatrix}, \ldots , (\mathrm {E}_p - \mathrm {E}_j) \begin{bmatrix} s\\t \end{bmatrix} \right) , \\ \delta \omega _{10}&= \left( (\mathrm {E}_1 - \mathrm {E}_j) \begin{bmatrix} s-1\\t \end{bmatrix}, \ldots , (\mathrm {E}_p - \mathrm {E}_j) \begin{bmatrix} s-1\\t \end{bmatrix} \right) , \\ \delta \omega _{01}&= \left( (\mathrm {E}_1 - \mathrm {E}_j) \begin{bmatrix} s\\t-1 \end{bmatrix}, \ldots , (\mathrm {E}_p - \mathrm {E}_j) \begin{bmatrix} s\\t-1 \end{bmatrix} \right) , \\ \delta \omega _{11}&= \left( (\mathrm {E}_1 - \mathrm {E}_j) \begin{bmatrix} s-1\\t-1 \end{bmatrix}, \ldots , (\mathrm {E}_p - \mathrm {E}_j) \begin{bmatrix} s-1\\t-1 \end{bmatrix} \right) , \end{aligned}$$

since $\delta \omega _{ab}$ is defined on $\Omega$ and is thus invariant under lattice shifts. Note that for $a,b \in \{0,1\}$,

$$\begin{aligned} \delta \omega _{ab,j} = \varvec{0}\qquad \text {and}\qquad \Vert \delta \omega _{ab} \Vert \leqq \sqrt{2}\; \theta , \end{aligned}$$

where $\theta$ is defined in (3.6). Furthermore, the weighted average of these shifts is zero:

$$\begin{aligned} \sum _{a,b \in \{0, 1 \}}\alpha _{ab}\; \delta \omega _{ab} = 0, \end{aligned}$$

where we have introduced the bilinear weights

$$\begin{aligned} \alpha _{00} = (1-s)(1-t), \quad \alpha _{10} = s(1-t), \quad \alpha _{01} = (1-s)t, \quad \alpha _{11} = st. \end{aligned}$$

As a consequence, by the affine character of the Taylor approximant $T_\omega ^1 \mathbf {u}_j$ defined above,

$$\begin{aligned} \widehat{\mathbf {u}}_j(\omega ) = T^1_\omega \mathbf {u}_j(0) = \sum _{a,b \in \{ 0, 1 \} } \alpha _{ab} \; T^1_\omega \mathbf {u}_j(\delta \omega _{ab}). \end{aligned}$$

(A.4)

Let us now rewrite the definition (3.2) of the bilinear interpolant $\widetilde{\mathbf {u}}_j$ as

$$\begin{aligned} \widetilde{\mathbf {u}}_j(\omega ) = \sum _{a,b \in \{ 0, 1 \} } \alpha _{ab} \; \mathbf {u}_j(\Pi _j \omega + \delta \omega _{ab}). \end{aligned}$$

(A.5)

Taking the difference of the identities (A.4), (A.5) and using convexity of the norm and the above Taylor estimate (A.3), we find the pointwise estimate

$$\begin{aligned} \begin{aligned} \left\| \widehat{\mathbf {u}}_j(\omega ) - \widetilde{\mathbf {u}}_j(\omega ) \right\| ^q \leqq \theta ^{2q} \int \nolimits _0^1 (1-h)^{2q} \sum _{a,b \in \{ 0, 1 \} } \alpha _{ab} \; \left\| \nabla _\omega ^2 \mathbf {u}_j (\Pi _j \omega + h \delta \omega _{ab}) \right\| ^q \,\mathrm{d} h. \end{aligned} \end{aligned}$$

We may now integrate over the configuration parameter $\omega = (\varvec{\gamma }_1, \ldots , \varvec{\gamma }_p)$. The difference between $\omega$ and $\left( \Pi _j \omega + h \delta \omega _{ab}\right)$ depends only on s, t and h, for example

$$\begin{aligned} \omega - \left( \Pi _j \omega + h \delta \omega _{00}\right) = - \left( \left( h \mathrm {E}_j + (1-h) \mathrm {E}_1 \right) \begin{bmatrix} s \\ t \end{bmatrix} , \ldots , \left( h \mathrm {E}_j + (1-h) \mathrm {E}_p \right) \begin{bmatrix} s \\ t \end{bmatrix} \right) . \end{aligned}$$

Integrating over the variables $\varvec{\gamma }_i$ for $i \ne j$ with fixed values of $\varvec{\gamma }_j= \mathrm {E}_j \begin{bmatrix} s \\ t \end{bmatrix}$ and h we find that by translation invariance of the Lebesgue measure,

This leads to

Since the right-hand side does not depend on the remaining variable $\varvec{\gamma }_j$, one last integration over it yields

$$\begin{aligned} \left\| \widehat{\mathbf {u}}_j(\omega ) - \widetilde{\mathbf {u}}_j(\omega ) \right\| _{L^q(\Omega )}^q \leqq \frac{\vert \Gamma _j \vert }{2q+1} \theta ^{2q} \; \left\| \nabla ^2_\omega \mathbf {u}_j \right\| _{L^q(X_j, \Vert \cdot \Vert _\mathrm {op})}^q, \end{aligned}$$

which proves the desired estimate (3.5). $\quad \square$