Van der Waals Density Functional for Layered Structures

H. Rydberg, M. Dion, N. Jacobson, E. Schröder, P. Hyldgaard, S. I. Simak, D. C. Langreth, and B. I. Lundqvist

Phys. Rev. Lett. 91, 126402 – Published 18 September 2003

Abstract

To understand sparse systems, we must account for both strong local atom bonds and weak nonlocal van der Waals forces between atoms separated by empty space. A fully nonlocal functional form [H. Rydberg, B. I. Lundqvist, D. C. Langreth, and M. Dion, Phys. Rev. B 62, 6997 (2000)] of density-functional theory (DFT) is applied here to the layered systems graphite, boron nitride, and molybdenum sulfide to compute bond lengths, binding energies, and compressibilities. These key examples show that the DFT with the generalized-gradient approximation does not apply for calculating properties of sparse matter, while use of the fully nonlocal version appears to be one way to proceed.

Received 30 May 2003

DOI:https://doi.org/10.1103/PhysRevLett.91.126402

Authors & Affiliations

H. Rydberg¹, M. Dion², N. Jacobson¹, E. Schröder¹, P. Hyldgaard¹, S. I. Simak¹, D. C. Langreth², and B. I. Lundqvist^1,*

¹Department of Applied Physics, Chalmers University of Technology and Göteborg University, SE-412 96 Göteborg, Sweden
²Center for Materials Theory, Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08854-8019, USA

^*To whom correspondence should be addressed. Electronic address: lundqvist@fy.chalmers.se.

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Vol. 91, Iss. 12 — 19 September 2003

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Figure 1
Schematics of the vdW forces in sparse, layered materials. Top panel: Laterally averaged electron-density profile of two graphene layers at equilibrium separation $d \approx 3.5 Å$ . The dotted ellipse indicates the region determining the interlayer interaction within LDA or GGA. It is unable to describe the true long range interaction (horizontal arrow) between correlated charge fluctuations (extrema illustrated by those of the finite-frequency charge response, and labeled by vertical arrows). Bottom panel: Full $E_{c}^{n l}$ (solid curve) vs the asymptotic form of $E_{c}^{n l} \to - c_{4} / d^{4}$ (dashed line). Also shown: Full ground-state interaction energy (dash-dotted line). Insets: Sketches of the correlated charge fluctuations for different layer separations, explaining why $| E_{c}^{n l} | > c_{4} / d^{4}$ . In the asymptotic region (top inset), attractive interactions $B C$ and $A D$ are nearly canceled by the repulsive interactions $A C$ and $B D$ , leaving little more than the weak dipolar interaction. Near equilibrium separation (bottom inset), the repulsive interactions $A C$ and $B D$ remain specified by the layer separation $d$ , but the relative strengthening of attractive interaction $B C$ substantially overcompensates the relative weakening of the attractive interaction $A D$ . At still smaller $d$ , $E_{c}^{n l}$ becomes weaker and saturates (Fig. 2, inset).Reuse & Permissions
Figure 2
Calculated vdW-DF results for the interlayer binding (adhesion) between two sheets of graphene (upper panel) and ${M o S}_{2}$ (lower panel) as functions of the interlayer separation $d$ . In each panel it is compared (solid curves) with the underlying GGA (dashed curves), and with the adhesion in the absence of the vdW-DF term $E_{c}^{n l}$ (dash-dotted curves) calculated using only the first two terms of Eq. (5) for $E_{x c}$ . The nonlocal-correlation vdW contribution $E_{c}^{n l}$ alone is shown separately (dotted curves). The figure shows that no binding results from exchange or local correlation; the vdW interactions provide adhesion in fair agreement with experiment (Table ). The inset shows that our calculated value of $E_{c}^{n l}$ saturates for small $d$ values.Reuse & Permissions
Figure 3
Comparison of the vdW-DF layer-to-layer binding in the staggered bulk hexagonal-BN system (solid lines) and adhesion between two sheets of BN (broken lines) shown as functions of the interlayer separation $d = c / 2$ . In both descriptions, the GGA gives unphysical results for bond and equilibrium separation. The total adhesion is dominated by the contribution of the vdW interaction $E_{c}^{n l}$ . The nearest-neighbor interlayer binding (broken lines) is seen to strongly dominate the bulk adhesion (solid lines) for layered materials.Reuse & Permissions

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