Nature of the excited states of layered systems and molecular excimers: Exciplex states and their dependence on structure

Weakly bound systems, like noble-gas dimers or two-dimensional layered materials (graphite, hexagonal boron nitride, or transition-metal dichalcogenides), exhibit excited electronic states of a particular nature. These so-called exciplex states combine on-site (or intralayer) and charge-transfer (or interlayer) configurations in a well-balanced way. We show by ab initio many-body perturbation theory that the energy and composition of the exciplex states depend sensitively on the bond length or interlayer distance of the material. When the constituents approach each other, the charge-transfer contribution increases and the excitation is redshifted to lower energy. If the system is excited into the exciplex state, then a covalent-like bond results. In consequence, noble-gas dimers form excimer complexes, while layered materials exhibit interlayer contraction.

Figure 1

(a) Electronic ground state of two radicals (A and B), which form a doubly occupied bonding orbital ($\sigma$). (b) Excited electronic state of (a). Depopulation of the bonding $\sigma$ HOMO and population of the antibonding ${\sigma }^{*}$ LUMO weakens the bond and causes repulsion of the two constituents. (c) Electronic ground state of two chemically saturated objects (A and B). Each object has a doubly occupied electronic level (o), from which a bonding and an antibonding molecular orbital ($\sigma$ and ${\sigma }^{*}$) are formed. The double occupation of both $\sigma$ and ${\sigma }^{*}$ does not constitute any significant covalent bond. The sketch includes unoccupied higher-energy levels (u) on both A and B, which also hybridize into a bonding ($\sigma$) and an antibonding (${\sigma }^{*}$) MO. (d) Excited electronic state of (c). Depopulation of the antibonding ${\sigma }^{*}$ HOMO and population of the higher-energy bonding $\sigma$ LUMO constitutes twofold “covalent” bonding, causing attraction and bonding of the formerly weakly interacting objects into an excimer.

Figure 2

Total energy of the ${\mathrm{H}}_2$ molecule as a function of the interatomic distance. The ground-state data are taken from DFT-LDA calculations, which yields an equilibrium distance of 0.764 Å in good agreement with the experimental value of 0.741 Å [35]. Among the excited states we focus on the lowest-energy spin-triplet and spin-singlet excitation (mostly $\mathrm{HOMO}\to \mathrm{LUMO}$) and their respective total energies result from adding the $GW$/BSE excitation energy to the DFT-LDA ground-state energy at each interatomic distance. For the singlet state ${}^1{\mathrm{\Sigma }}_u^{+}$ the equilibrium at $d\sim 1.21\AA$ (11.5 eV above the ground state) agrees well with measured data of $d=1.29\AA$ at $E=11.4\mathrm{eV}$ [35]. The underlying thin dotted lines show data from highly accurate quantum-chemical calculations [36, 37, 38].

Figure 3

MO diagram of a dimer of two Ar atoms. The $3p_x$ (with $x$ referring to the dimer axis) orbitals show much stronger hybridization (into ${\sigma }_x$ and ${\sigma }_x^{*}$) than $3p_y$ and $3p_z$ (into ${\pi }_{y,z}$ and ${\pi }_{y,z}^{*}$). Analysis of our computed spatial structure of the orbitals confirms the antibonding nature of the HOMO (${\sigma }_x^{*}$) and the bonding nature of the LUMO ($\sigma$), in analogy to Fig. 1.

Figure 4

Excited-state total energy of ${\mathrm{Ar}}_2$ as a function of the interatomic distance, relative to the ground-state total-energy minimum (at 0 eV). The excited state is given by the lowest-energy spin-triplet and spin-singlet excitation (mostly $\mathrm{HOMO}\to \mathrm{LUMO}$), and its respective total energy results from adding the excitation energy to the DFT-LDA ground-state energy at each interatomic distance. Both for the singlet and the triplet state the equilibrium is at $d=2.3\AA$ with minima 10.5 eV (10.3 eV) above the ground state.

Figure 5

Composition of the singlet excitation of ${\mathrm{Ar}}_2$ from on-site configurations on each atom A/B (${\left|\lambda \right|}^2$, upper curve) and charge-transfer configurations $\mathrm{A}\to \mathrm{B}$ and $\mathrm{B}\to \mathrm{A}$ (${\left|\mu \right|}^2$, lower curve).

Figure 6

Schematic illustration of the composition of an exciplex state from on-site configurations [vertical arrows, with prefactor $\lambda$ in Eq. (7)] and charge-transfer configurations [slanted arrows, with prefactor $\mu$ in Eq. (7)]. At small interatomic distance (a) on-site and charge-transfer configurations are of similar excitation energy, facilitating mixing and leading to prefactors $\lambda$ and $\mu$ of similar magnitude. At large interatomic distance (b) the charge-transfer configurations are of much higher energy (see text) and do not contribute significantly ($\mu \sim 0$), as indicated by the thin slanted arrows.

Figure 7

Calculated optical spectrum of a monolayer of ${\mathrm{MoS}}_2$ at its equilibrium lattice constant $a_0$ and for modified lattice constants.

Figure 8

Spatial density of the excited electron of the $A$ exciton in bulk 2H-${\mathrm{MoS}}_2$, as a side view of the crystal. The excited hole is located in the center of the plot. On the right-hand side the same quantity is shown after integration over the lateral coordinates. About 89% of the electron is located in the same layer as the hole (intralayer or on-site contribution), while 11% of the electron is located in other layers (interlayer or charge-transfer contribution).

Figure 9

Probability distribution of the excited electron relative to the hole for the lowest-energy exciton in bulk $h$-BN. (a) Shown as a function of distance from the layer on which the hole is located. (b) Same data as in (a) but shown as a function of perpendicular lattice constant $c$. The ground-state equilibrium is given by $c=6.6\AA$.

Figure 10

Excitation energy of the lowest exciton of $h$-BN as a function of the perpendicular lattice constant $c$. Open squares refer to a single layer of $h$-BN, treated in the same unit cell as bulk $h$-BN (see text). Blue triangles refer to bulk $h$-BN with dielectric-screening interaction between the layers but no interlayer interaction of their electronic degrees of freedom (i.e., neglecting charge-transfer configurations between the layers, see text). Red circles denote the final BSE calculation for bulk $h$-BN (within LDA+$GdW$/ BSE).

Figure 11

Imaginary part of the dielectric function of graphite, calculated for various values for $a$ and $c$. Experimental data (for $a_0=2.45\AA$ and $c_0=6.71\AA$) are taken from Refs. [48] and [49] (see text).

Figure 12

Dependence of the excitation energy of graphite exciton states on the perpendicular lattice constant $c$ (upper panel) and on the lateral lattice constant $a$ (lower panel). Each exciton state is characterized by its excitation energy ${\mathrm{\Omega }}_S$ (horizontal axis) and is weighted by its optical oscillator strength (quantifying its significance for the perpendicular lattice contraction studied in this paper), thus generating a spectrum. Note the different vertical scales in the two panels.

Figure 13

Relative lateral lattice expansion (upper panel) and perpendicular lattice contraction (lower panel) of hexagonal boron nitride (when excited at its lowest-energy exciton at 5.87 eV) and of graphite (when excited by light of 1.5- and 4.5-eV photon energy, respectively), as a function of exciton density $x$.