Wannier function approach to realistic Coulomb interactions in layered materials and heterostructures

We introduce an approach to derive realistic Coulomb interaction terms in freestanding layered materials and vertical heterostructures from ab initio modeling of the corresponding bulk materials. To this end, we establish a combination of calculations within the framework of the constrained random-phase approximation, Wannier function representation of Coulomb matrix elements within some low-energy Hilbert space, and continuum medium electrostatics, which we call Wannier function continuum electrostatics (WFCE). For monolayer and bilayer graphene we reproduce full ab initio calculations of the Coulomb matrix elements within an accuracy of $0.3$ eV or better. We show that realistic Coulomb interactions in bilayer graphene can be manipulated on the eV scale by different dielectric and metallic environments. A comparison to electronic phase diagrams derived in M. M. Scherer et al. [Phys. Rev. B 85, 235408 (2012)] suggests that the electronic ground state of bilayer graphene is a layered antiferromagnet and remains surprisingly unaffected by these strong changes in the Coulomb interaction.

Figure 1

(left) Three-dimensional layered material. The infinite stack of single layers is numbered by $j$. (right) Corresponding two-dimensional monolayer in a variable dielectric surrounding. The arising electric field of two charges $q_a$ and $q_b$ is indicated by dashed lines. The dielectric properties in the layer are denoted by ${\varepsilon }_1$, which is actually a nonlocal and thus ${\mathbf{q}}_{\parallel }$-dependent function (see main text).

Figure 2

Flowchart of the Wannier function continuum electrostatics (WFCE) algorithm to obtain the screened Coulomb matrix elements of a freestanding monolayer directly from the Coulomb interaction of the corresponding layered bulk material.

Figure 3

(left) Leading eigenvalues of the bare and screened Coulomb matrices of graphite for $q_z=0$ obtained from ab initio calculations together with the analytical description of the unscreened interaction (dashed blue line). The dark gray (light gray) area indicates the interval of all other eigenvalues of the screened (bare) Coulomb matrix. (right) Momentum-dependent leading eigenvalue of dielectric matrix of graphite obtained from cRPA calculations for different values of $q_z$. The red markers for $q=0$ indicate the parallel (circle) and perpendicular (square) limits of the screening.

Figure 4

Leading eigenvalue of the dielectric matrix (outer frame) and bare and screened Coulomb matrices (inner frame) of (left) monolayer and (right) bilayer graphene. Red markers indicate ab initio calculations, and gray markers show the WFCE values.

Figure 5

Leading eigenvalue of the dielectric matrix (outer frame) and bare and screened Coulomb matrices (inner frame) of Gr/Ir. Red markers indicate cRPA calculations, and gray markers show the values from the WFCE approach.

Figure 6

Density-density matrix elements of the screened Coulomb interactions for graphite, monolayer and bilayer graphene, and Gr/Ir in real space. Colored markers show ab initio values, and gray markers connected by dashed lines show the WFCE values.

Figure 7

(left) WFCE density-density matrix elements of the background screened Coulomb interactions for bilayer graphene in real space for different dielectric surroundings $\left({\varepsilon }_2/{\varepsilon }_3\right)$. (right) Corresponding dielectric functions in momentum space.

Figure 8

(a) Top and (b) side view of the Gr/Ir system.