Resonating valence bond pairing energy in graphene by quantum Monte Carlo

Abstract

We determine the resonating-valence-bond (RVB) state in graphene using real-space quantum Monte Carlo with correlated variational wave functions. Variational and diffusion quantum Monte Carlo (DMC) calculations with Jastrow-Slater-determinant and Jastrow-antisymmetrized-geminal-power ansatze are employed to evaluate the RVB pairing energy. Using a rectangular graphene sample that lacks $π/3$ rotational symmetry, we found that the single-particle energy gap near the Fermi level depends on the system size along the $x$-direction. The gap vanishes when the length satisfies $L_x=3n\sqrt{3}d$, where $n$ is an integer and $d$ is the carbon-carbon bond length, otherwise, the system, exhibits a finite gap. Our DMC results show no stable RVB pairing in the zero-gap case, whereas the opening of a finite gap near the Fermi level stabilizes the electron pairing. The DMC predicted absolute value of pairing energy at the thermodynamic limit for a finite-gap system is $\sim 0.48(1)$ mHa/atom. Our results reveal a feometry-driven electron pairing mechanism in the confined graphene nanostructure.

Figures (4)

• Figure 1: A spin-singlet dimer configuration (blue color) on graphene lattice. The resonating valence bond (RVB) state is composed of a superposition of such configurations. Three $sp^2$ orbitals form the $\sigma$ band containing three localized electrons, while the bonding among the $p_z$ orbitals of different lattice sites generate a valence band ($\pi$ band) having one electron. The antibonding configuration generates the conduction $\pi^*$ band.
• Figure 2: VMC energy as a function of the number of optimization steps computed using JSD and JAGP wave functions for finite-gap system with number of atoms in simulation cell $N_{at}=80$ (left panel) and zero-gap system with $N_{at}=72$ (right panel).
• Figure 3: (a) VMC and DMC energies for finite gap (Ins) and zero gap (Met) systems obtained with JSD and JAGP WFs as a function of system size. (b) VMC and DMC pairing energies for finite gap (Ins) and zero gap (Met) systems obtained with JSD and JAGP WFs as a function of system size. The error bars are smaller than data points size.
• Figure 4: Single particle energy band as a function of momentum computed by DFT. The valence and conduction band along $\Gamma(0,0)\rightarrow X(\frac{1}{2},0)$ obtained for different $\mathbf{k}$-meshes. The horizontal and vertical lines at $E=0$ and $k=\frac{1}{3}$ represent the Fermi energy and Dirac point, respectively. (a) and (b) show bands for non-zero $E_g\neq0$ and zero gap $E_g=0$ systems for which the RVB pairing energy is obtained by VMC and DMC. (c) and (d) show bands for larger system sizes with non-zero and zero gaps. Comparing the bands obtained by $\mathbf{k}$-mesh $13\times13$ and $12\times13$ demonstrate that vanishing gap is only depend on $k_x$. (e) and (f) show bands near the Dirac point obtained using $\mathbf{k}$-meshes of $100\times100$ with $E_g=0.093$ eV and $99\times99$ with $E_g=0$.