Communication: Generalization of Koopmans’ theorem to optical transitions in the Hubbard model of graphene nanodots

Many-body effect in a finite system such as molecules and their artificial counterparts, semiconductor quantum dots usually involve direct, exchange, and correlation interactions among tens to hundreds of electrons. Quasiparticle gap, as a direct result of many-body effect, can be generally established by photoemission or inverse photoemission spectroscopy¹ and are usually compared with the single-particle description of the interacting system. For an example, Koopmans’ theorem states that the first ionization energy of a molecular system within the closed-shell Hartree-Fock (HF) theory is equal to the negative of the orbital energy of the highest occupied molecular orbital (HOMO).² As a similar conclusion also applies to the electron affinity while the quasiparticle gap measures just its difference with the ionization energy, it is straightforward to extend the original theorem to that the HF quasiparticle gap in a closed-shell system is equal to its single-particle energy gap.

The other important property of an semiconductor is the optical gap, which measures the edge of its absorption spectrum. For a closed-shell system with the ground state of total spin zero, the first few excited states are often triplets and thus optically inactive. Therefore, the optical gap is usually larger than the first energy gap of the many-electron system, i.e., the energy difference between the ground triplet and singlet. For this reason, there has been no simple way to relate the optical gap to the single-particle energy gap like what Koopmans’ theory does to the quasiparticle gap. In the work, we will generalize the theory to the optical transitions in graphene nanodots and propose that the optical gap in the Hubbard model of graphene nanodots is equal to its single-particle gap in the absence of electron correlation. Moreover, we will show that the generalized rule is so robust that it would hold even in the presence of strong electric field applied to the system.

Due to the extraordinary electronic³ and optical properties,⁴ graphene is believed to be a promising candidate in future microelectronic devices.⁵ For optoelectronic applications of graphene nanostructures, quantitative understanding and control of their spectroscopic properties in the case of various substrates like SiO₂,⁶ diamond,⁷ SiC,⁸ or metal materials⁹ are of paramount importance. Bulk graphene is known to have an almost zero band-gap; however, a finite gap can be opened and even engineered by quantum confinement effect in graphene nanoribbons and nanodots.¹⁰ Moreover, the peculiar dimensional dependence of electron-electron interactions¹¹ makes graphene nanodots a perfect model system of strongly correlated electrons.

The quasiparticle gap is modified into the optical gap by the electron-electron interactions through the excitonic effect.^12–15 Exciton binding energies in quasi-one-dimensional graphene nanoribbons^16,17 and excitonic absorption in triangular graphene quantum dots with zigzag edges^18,19 have been studied by solving the Bethe-Salpeter from first principles or diagonalizing the configuration interaction Hamiltonian. In this work, we will show that the quasiparticle correction to the optical gap and exciton binding energy in graphene nanodots are dominated by the long-range Coulomb interaction and thus both become small when only on-site Hubbard interactions are present in the case of metal substrates. More interestingly, our systematic configuration interaction calculations will reveal that the two many-body contributions to the optical gap would nearly cancel each other, which results in an unexpected overlap of the optical and single-particle gaps in the graphene systems.

Figure 1 provides an illustration of quasiparticle energies, optical gaps, and their relations with the exciton binding energy for a close-shell system. When an electron is optically excited from fully occupied valence states (denoted by shaded area in the figure), the N-electron system has a higher energy of $E N 1$ than its ground state energy E_N. The optically excited system now consists of two components, the (N − 1) valence electrons and the one in the conduction state. The energy $E N 1$ can then be considered as a sum of the energies of the two individual components and their interaction, i.e.,

Here, G_SP is the single-particle gap, i.e., the kinetic energy of the excited electron if the energy of HOMO is set to zero. If one puts an electron back into the vacant state from which the electron is excited, the interaction I_N−1,e can be written as

where E_ex denotes the interaction between the electron replaced to the vacant state and the one excited to the conduction state, commonly known as the exciton binding energy. Here, I_N,e represents the sum of all the pair interactions between the electron excited to lowest unoccupied molecular orbital (LUMO) and those occupying the valence states, i.e.,

where V_ijkl is the Coulomb matrix element.²⁰ 

Furthermore, considering a (N + 1)-electron system with N electrons fully occupying the valence states and one electron in the lowest conduction state, similarly one has

Combining these equations together and after eliminating I_N−1,e and I_N,e, one has

It is then straightforward to derive that

where the optical and quasiparticle gaps are defined by $G OP = E N 1 − E N$ and G_QP = μ_N+1 − μ_N, respectively, and μ_N = E_N − E_N−1 is the chemical potentials of the system. Here, the optical gap G_OP is seen to have two contributions, one is the quasiparticle correction G_QP − G_SP and the other is exciton binding energy E_ex.

The model system is chosen to be the triangular graphene nanodot as shown in the inset of Fig. 2. Recently, a graphene nanostructure of similar shape and size has been successfully fabricated.²¹ The nanodot has armchair edges and the number of carbon rings along each edge is set to be n = 4, which corresponds to a total number of atoms N = 60.

For the case of graphene nanostructures on a substrate of semi-insulating material with a large dielectric constant or even metal, long-range Coulomb interaction is effectively suppressed, and therefore, Hubbard model is a good choice to describe the interacting electron system. The Hamiltonian reads

where i and j are the nearest neighbors, the hopping energy t is set to be 2.7 eV and the Hubbard U is chosen to be between 1.0t and 3.5t.^22,23 A single-particle gap, which separates the occupied and unoccupied states, is defined by the difference between the energies of HOMO and LUMO.^24,25 For n = 4, we find that G_SP is about 2.2454 eV.

To obtain optical and quasiparticle gaps, one needs to calculate not only the ground state energy of the system with various number of electrons, i.e., E_N+1, E_N, and E_N−1, but also a few excited states to determine the optically active final state. Moreover, the electron correlation is known to play an important role in graphene nanostructures and thus cannot be neglected. For these reasons, one has to go beyond the mean-field approximation which is commonly used to solve the Hubbard model. Here, many-particle wave functions are first expanded on the basis of single-particle states obtained by the tight-binding method and then solved by using the configuration interaction approach.²⁰ 

For the triangular model system, we choose N_s valence states from HOMO down and the same number of conduction states from LUMO up, as our basis. All the occupied states in the closed-shell system form a single reference configuration. For a given N_s, one can choose to move m(≤N_s) electrons from the occupied states to the unoccupied states, usually referred as a mth excitation, to construct a many-particle configuration. As a variational formalism, configuration interaction approach is used to determine upper bounds of the energy levels for an interacting electron system. In principle, one may systematically lower the upper bounds by increasing the number of states N_s or the order of excitations m. For the model systems considered in this work, we find that it is necessary to have m ≥ 5 in order for the low-lying levels to be fully converged. Here, we choose (N_s,  m) = (16,  5), and the total number of configurations N_c is 6 689 001 for the neutral half-filling system (N = N_s). To obtain E_N+1 and E_N−1, N_c almost doubles and would become 11 919 336. PARPACK is used for the diagonalization of the resulting sparse matrix to obtain the energy levels of the many-electron system.

Figure 2 plots the optical gap G_OP calculated for the triangular nanodot model as a function of the Hubbard U together with its two contributions, the quasiparticle correction G_QP − G_SP and exciton binding energy E_ex. The single-particle gap G_SP, which does not depend on the Hubbard interaction, is shown for comparison.

A convergence check has been performed before the calculation. For U/t = 3.5, the optical gap G_OP is obtained as 2.2398 eV for (N_s,  m) = (16,  6), larger than 2.2384 eV for (N_s,  m) = (16,  5) by less than 0.1%. For this model system, the first excited singlet is found to be separated from the ground state by two triplets.

One of the most interesting phenomenon to be seen from the result is that the optical gap of the model system is nearly independent on the Hubbard interaction. In fact, as U/t increases from 1.0 to 3.5, G_OP is hardly found to decrease from 2.2433 eV to 2.2384 eV by only about 0.2%, which is apart from the corresponding single-particle gap G_SP by only 0.3%. The other striking result, which is closely related to the unexpected overlap between the optical and single-particle gaps, is that the quasiparticle correction G_GP − G_SP and exciton binding energy E_ex exhibit almost the same dependence on the Hubbard interaction U/t. Both are seen to start from a small value of about 10 meV when U/t = 1.0 and steadily increase to about 130 meV when U/t = 3.5.

To understand why G_QP − G_SP and E_ex exhibit almost the same dependence on U/t, let us remind that both quasiparticle and excitonic effects origin from the interactions among the same electrons close to the Fermi level. If this argument is true, these two many-body effects shall not only have similar dependence on U/t but also on other factors like the dielectric constant. A calculation including the long-range Coulomb interaction for the same model system on a SiC substrate shows that G_QP − G_SP = 625.1 meV and E_ex = 608.3 meV. Again, it is seen that both quantities have very similar values even in the presence of long-range Coulomb interaction which is absent from the Hubbard model. Further computation reveals that the difference between the quasiparticle and excitonic effects increases slowly but steadily with 1/ϵ^⋆. In the meantime, it is also noted that both effects are dominated by the long-range Coulomb interaction and become quite small as 1/ϵ^⋆ → 0.

The difference between the quasiparticle correction and the exciton binding energy can also be written as (see Fig. 1)

The last term E_N − E_N−1 can be regarded as the interaction between the electron which is excited to the conduction state during the optical absorption and the rest (N − 1)-electron system, i.e., E_N − E_N−1 = I_N−1,h. One thus has

If only the nearest neighbor hopping is taken into account in the tight-binding model, it can be shown that there is a symmetry between the valence (occupied) and conduction (unoccupied) states. For the doubly degenerate HOMO and LUMO states, one has²⁶ 

Furthermore, it is reminded that the Hubbard interaction depends on only the electron density instead of the wave function. Hence, if electron correlation is negligible, we would have I_N−1,e = I_N−1,h and thus G_OP = G_SP. The slight difference seen between I_N−1,e and I_N−1,h or between G_OP and G_SP arises from either the electron correlation or other factor like an applied electric field which breaks the symmetry of electrons and holes.

Figure 3 plots the optical and single-particle gaps calculated for the triangular model system with various sizes. The quasiparticle correction and exciton binding energy are shown together in the same figure. As the nanodot size increases, the single-particle gap is found to decrease steadily due to the weakening quantum confinement effect. The two many-body effects, i.e., the quasiparticle correction and exciton binding energy, are seen to also decrease but more rapidly. If the long-range Coulomb interaction were present, G_QP and E_ex would have dropped more slowly with the increasing dimension.²⁰ In the meantime, it is found that G_QP − G_SP and E_ex stay very close to each other during the size increasing. Because the symmetry between HOMO and LUMO remains as the nanodot size increases, I_N−1,e keeps to be identical to I_N−1,h and only a small difference could be seen between the optical and single-particle gaps due to the electron correlation.

Combining the results presented in Figs. 2 and 3, we now can generalize Koopmans’ theory to the optical transitions in the Hubbard model of graphene nanodots. Here, we propose that the optical gap of a closed-shell graphene system within the Hubbard model is equal to its tight-binding single-particle energy gap in the absence of electron correlation. When the correlation interaction is taken into account, this generalization becomes an empirical rule with an accuracy better than 99.0%. For other nanodots of different geometry like the hexagonal ones, our calculations show very similar results. Finally, it would be interesting to see how this empirical rule for the optical gap holds in the presence of externally applied electric field which breaks the symmetry between HOMO and LUMO. For a smaller nanodot of n = 3 with an electric field applied along the apex-bottom direction, we find that the discrepancy between G_OP and G_SP falls well below 1.0%, and thus, the empirical rule holds well.

To conclude, we have generalized Koopmans’ theorem to the optical transitions in the Hubbard model of graphene nanodots. Based on systematic configuration interaction calculations, we have proposed that the optical gap of a closed-shell graphene system within the Hubbard model is equal to its tight-binding single-particle energy gap in the absence of electron correlation. In these systems, we have shown that the quasiparticle correction to the optical gap and exciton binding energy is dominated by the long-range Coulomb interaction, and thus, both become small when only on-site Hubbard interactions are present. Moreover, we have revealed that the contributions of the quasiparticle and excitonic effects to the optical gap nearly cancel each other and have attributed the unexpected overlap of the optical and single-particle gaps to the symmetry between the electron and hole states in the graphene nanodots.

This work is supported by National Natural Science Foundation of China (Project No. 11474063) and National Basic Research Program of China (973 Program No. 2011CB925602). W. Sheng would like to thank Professor Pawel Hawrylak for fruitful discussion and valuable comments on the manuscript.