Solution of the mean-field Hubbard model of graphene rectangulenes

TL;DR

This work provides a complete analytical mean-field Hubbard solution for undoped and doped graphene rectangulenes of arbitrary size by transforming the interaction into the eigen-state basis built from bulk and edge states. It derives explicit bulk, edge, and edge-bulk Coulomb integrals, and decomposes the Hamiltonian into bulk, edge, and coupling parts, enabling closed-form mean-field solutions for paramagnetic, ferromagnetic, and antiferromagnetic phases. The authors compute eigen-energies, occupations, spin densities, addition energies, and phase-energy differences, and show how to reformulate the MF Hamiltonian back into a real-space tight-binding form suitable for transport and optical-model calculations. This framework captures edge magnetism, edge-state doping effects, and finite-size scaling from nanometers to structures approaching the bulk limit, and provides a scalable bridge to beyond-mean-field methods such as GW. The approach thus offers a powerful, analytical tool to predict electronic, magnetic, and optoelectronic properties of experimental graphene rectangulenes and related graphene flakes.

Abstract

We present a complete analytical solution of the mean-field Hubbard model of undoped and doped graphene rectangulenes. These are non-chiral ribbons of arbitrary length and width, whose dimensions range from simple short acene molecules all the way up to the bulk limit. We rewrite the Hubbard model in the basis of bulk and edge non-interacting eigen-states, and provide explicit expressions for the Coulomb matrix elements. We present a general mean-field decoupling of the Hamiltonian, and discuss in detail the paramagnetic, ferromagnetic and antiferromagnetic mean-field solutions. We calculate the eigen-energies, occupations, spin densities and addition energies of rectangulenes with lengths and widths ranging from a nanometer to several hundreds of them. We rewrite the exact mean-field tight-binding Hamiltonian back in the site-occupation basis, that can be used to model electronic, thermo-electric, transport and optical properties of experimental-size graphene flakes.

Figures (8)

• Figure 1: (a) Rectangulene with dimensions $M_x \times M_y$. ${\cal A}$/${\cal B}$ atoms are indicated by dark/bright red circles. Fake atoms, where wave-function coefficients are set to zero, are indicated by blue circles. Each unit cell is surrounded with a grey dotted box. (b) Each unit cell contains two ${\cal A}$ and two ${\cal B}$ atoms, whose internal coordinates are written in Eq. (\ref{['eqn:internalcoordinates']}). The black arrow indicates the lattice constant.
• Figure 2: Two-dimensional plot of the mesh of allowed ${\bar{\bf k}}$-vectors for a rectangulene with dimensions $(\,M_x,\,M_y\,)\,=\,(\,10,\,11\,)$. The red lines correspond to the $\bar{k}_y$ quantized values. Blue lines correspond to solving Eq. (\ref{['Equation:kx_quantization']}) for $\bar{k}_y$ as a function of $\bar{k}_x$. Black dots at the intersections between blue and red lines correspond to bulk states. Green dots correspond to edge states.
• Figure 3: Coulomb integrals ${\cal U}^B_{m\alpha}$, ${\cal U}^{EB}_{m\alpha}$, ${\cal U}^{BE}_m$ and ${\cal U}^{E}_m$ in units of $U$ (black, blue, red and green dots, respectively) as a function of the wave-number $\bar{k}_m$ for a rectangulene with dimensions $(M_x,\,M_y) = (30,\,41)$.
• Figure 4: Zero-temperature AFM order parameter $P_m$ of an undoped rectangulene of width $N=1501$, that corresponds to 184,7 nm and hosts 500 states. Black, red and green dots correspond to lengths $M_x=10,\,100$ and $1000$ (4.26, 42,6 and 426 nm, respectively). Dashed lines show the fitting of the results to $\tanh^4{\left ( {\cal M}_x \left ( \frac{\bar{k}_m}{\pi}-\frac{2}{3} \right ) \right ) }$
• Figure 5: Electronic structure of a $(\,M_x,\,M_y\,) = (10,15)$, $N=9$ rectangulene as a function of the $k_y$ wave-number at an edge filling (a) $\delta^E=0$ and (b) $\delta^E=4$ electrons. This rectangulene has dimensions 4.3 nm $\times$ 3.7 nm, and hosts 8 edge states. The left column plots the non-interacting (top) and mean-field PM electronic structure. The central/right columns plot the mean-field FM/AFM electronic structure for spin-up (top) and spin-down (bottom). Blue dots correspond to bulk states; red/green dots correspond to $\tau=+1\,/\,-1$ edge states.