Structural, Electronic, and Nonlinear Optical Characteristics of Europium-Doped Germanium Anion Nanocluster EuGe_n⁻ (n = 7–20): A Theoretical Investigation

Abstract

Doping rare-earth metals into semiconductor germanium clusters can significantly enhance the stability of these clusters while introducing novel and noteworthy optical properties. Herein, a series of EuGe_n⁻ (n = 7–20) clusters and their structural and nonlinear optical properties are investigated via the ABCluster global search technique combined with the double-hybrid density functional theory mPW2PLYP. The structure growth pattern can be divided into two stages: an adsorption structure and a linked structure (when n = 7–10 and n = 11–20, respectively). In addition to simulating the photoelectron spectra of the clusters, their various properties, including their (hyper)polarizability, magnetism, charge transfer, relative stability, and energy gap, are identified. According to our examination, the EuGe₁₃⁻ cluster exhibits a significant nonlinear optical response of the β_tot value of 7.47 × 10⁵ a.u., and is thus considered a promising candidate for outstanding nonlinear optical semiconductor nanomaterials.

1. Introduction

Nanomaterials have emerged as key materials with multidisciplinary potential in various fields, like energy and optics, driven by their unique structure–property relationships [1,2]. In the field of advanced semiconductors, germanium and germanium-based nanomaterials play an essential role [3]. Since 1958—when it was used to develop the world’s first integrated circuit—germanium has gradually been used in various advanced manufacturing fields [4]. For example, germanium has high transparency towards infrared light, allowing it to be used for fabricating infrared lenses and windows [5], narrow-emitting quantum dots for high-color-index displays [6], non-silicon solid-state transistors with high-mobility charge carriers [7], solar cells for high photoelectric conversion efficiency [8], and so on. As next-generation nanomaterials, germanium and germanium-based semiconductors are gradually becoming irreplaceable in cutting-edge technologies and high-precision manufacturing. Additionally, more and more unique physical and chemical properties await further discovery, so it is urgent to develop novel germanium-based semiconductor materials to further satisfy current industrial and societal needs.

Pure germanium nanoclusters provide excellent semiconductor performance and are already widely applied in mainstream advanced optoelectronic devices. However, pure germanium has a large number of dangling bonds, leading to a highly sensitive device service environment [9,10]. To overcome this situation and prolong the devices’ lifetimes, one effective and easy-to-implement strategy is doping transitional metal (TM) into pure germanium nanoclusters. There are several examples of this in the literature. Manganese-doped anionic germanium nanoclusters MnGe₆⁻ and Mn₂Ge₇⁻ were studied by means of joint photoelectron spectroscopy (PES) with theoretical calculations, and the results showed that MnGe₆⁻ and Mn₂Ge₇⁻ have the same symmetry of C_2v and vertical electron detachment energies of 2.58 ± 0.08 and 2.88 ± 0.08 eV, respectively [11]. A series of transitional and lanthanide metals, including Sc, Ti, V, Y, Zr, Nb, Lu, Hf, and Ta, were examined by means of anion PES at a 213 nm pulse using a Nd³⁺:YAG laser, which revealed a Ge₁₆ cage structure with a large cavity, allowing the metal to be encapsulated [12]. In a theoretical investigation, bimetallic Mo₂Ge_n (n = 9–15) clusters were studied via the B3LYP density functional approach, and the Mo₂Ge₉ theoretical predicted structure was analogous to the one that was experimentally observed [13]. Group 4 and 5 elements in germanium clusters TMGe_8–17⁻ were studied by applying a genetic algorithm coupled with density functional theory. The calculated PES highly agreed with the measured spectra, in which the complete closed cage motif showed great stability [14]. Monoanionic Ru₂Ge_n⁻ (n = 3–20) was examined by using a genetic algorithm associated with PBE functionality. The ground-state structure study showed that, when extra electrons were added in, the cluster structure was significantly influenced [15]. Regarding noble metal-doped germanium nanoclusters, AuGe₁₂ clusters with a bicapped pentagonal prism geometry were investigated via density functional theory calculations. The results showed that the hybridization between the transitional metal Au and the germanium cage enhanced the chemical stability of the whole cluster [16]. Such conclusions were also made for IrGe_n⁻ (n = 1–20) clusters [17]. Because of the similar chemical and physical properties of silicon and germanium, Kumar and Kawazoe, using the ab initio pseudopotential plane wave method, calculated the structures of transitional metal Ti-, Zr-, and Hf-doped Ge₁₆, forming the Frank–Kasper structure MGe₁₆ and capped decahedral or cubic MGe₁₆ and MGe₁₄; moreover, they found that the growth behaviors were different from those for the MSi_n nanocluster [18]. Another research study by Han and Hagelberg proved that TMSi_n with n = 12 or n = 16 and TMGe_n with n = 10, n = 12, or n = 16 are the most promising candidates for modular nanomaterials [19]. Besides doping with d-block transition metals, another way to achieve such goals is through doping with rare-earth metals. A scandium atom doped into an anion germanium cluster can vastly enhance the stability of the cluster, owing to the form of high-symmetry Frank–Kasper ScGe₁₆⁻ with 68 valence electron-filled shell configurations [20,21,22,23]. The different cage types of ScGe₁₂ were also theoretically studied [24]. Li et al. systematically explored the structural evolution, relative stability, and electronic properties of LaGe_n⁻ (n = 3–14), and their results indicated that a lanthanum atom doped into a pure germanium cluster increased the interactions between germanium atoms and enhanced the stability of the whole system; such building blocks have potential for the design of molecular chains [25]. A series of germanium anionic clusters ReGe₆⁻ doped with the heavy rare-earth elements Gd, Tb, Dy, Ho, Er, Tm, Yb, and Lu were identified using the PBE0 functional method combined with the def2-TZVP basis set. The calculations showed that rare-earth atoms doped into center pentagonal ring structures have greater stability and aromaticity than other types, making them potential building blocks for novel rare-earth-based semiconductor materials [26]. The lowest energy structures of rare-earth element Yb-, Pm-, and Dy-doped neutral and cationic germanium clusters Ge_nM (n = 9, 10; M = Yb, Pm, and Dy) were determined using a genetic algorithm coupled with the PBE functional and projected augmented wave method. The results showed that rare-earth metal doping into germanium is favorable in terms of the energy direction [27]. With the great number of potential applications in the current semiconductor industry and the huge demand in today’s society, there is an urgent need to discover novel, stable, and ultra-efficient nanomaterials.

The element europium (Eu) is a typical and unique luminescent activator, widely used in a variety of luminescent materials. Depending on the coordination environment and electron transition of its 4f-4f and 4f-5d orbitals, it can exhibit different optical properties, and there is high demand for its application in different situations. In view of this, we selected Eu as a dopant in pure germanium clusters to design novel semiconductor building blocks. The ground-state structures of the EuGe_n⁻ (n = 7–20) clusters were systematically determined through stepwise optimization, and their nonlinear optical properties were investigated. Moreover, their photoelectron spectra, magnetic behavior, relative stability, energy gap, and density of states were all studied. The results reveal the large polarizability and hyperpolarizability of the EuGe₁₃⁻ clusters, which may serve as potential nonlinear optical materials for the next-generation semiconductor industry.

2. Results and Discussion

2.1. Ground-State Structure and Evolution Pattern of EuGe_n⁻ (n = 7–20) Nanoclusters

The ground-state structures, symmetry, and energy differences in EuGe_n⁻ (n = 7–20) optimized at the mPW2PLYP level are shown in Figure 1, and the Cartesian coordinates of each cluster are collected in Table S1. To further ensure the ground-state structures, the AIMD results are shown in Figure S1 (Supporting Information), and the total energies at mPW2PLYP with zero-point energy at TPSSh are collected in Table S2. Each cluster is noted as nA-x, where n is the number of Ge atoms, A is the anionic cluster, and x is the sequence number from small to large for increasing energy differences compared to the energy of the ground-state structure. The spin multiplicities of EuGe_n⁻ (n = 7–20) clusters were all predicted for nonets. For EuGe₇⁻, 7A-1 and 7A-2 have the same symmetry of C_s, which can be seen as one Eu atom replacing one Ge atom in the pure Ge₈ cluster [28]. The structure of 7A-3 is two Ge atoms forming a capped tetragonal bi-pyramid, of which the Eu atom is the vertex. The energies of 7A-2 and 7A-3 are higher than that of 7A-1, by 0.29 and 0.33 eV, respectively. For EuGe₈⁻, 8A-1 is one Eu atom adsorbed on the top of the square frustum of Ge_8, and 8A-2 is a typical tricapped triangular prism (TTP) structure where the Eu atom is located on one side of the triangular prism. The structure of 8A-3 can be described as one extra Ge atom adsorbed onto 7A-3. The energy differences for 8A-2 and 8A-3 with regard to 8A-1 are 0.32 and 0.49 eV, respectively. For EuGe₉⁻, all isomers have a similar structure, depicted as one Eu atom and two Ge atoms capped on the TTP Ge₆ basic framework, where the difference is only in the three atoms’ cap location. 9A-1 is more stable than 9A-2 and 9A-3, with energy differences of 0.37 and 0.76 eV, respectively. For EuGe₁₀⁻, 10A-1 and 10A-2 are similar types, with one Eu atom site on the side of two capped anti-prisms Ge₁₀. Due to the similarity of their structures, the energies of 10A-1 and 10A-2 are close, with a difference of only 0.07 eV. 10A-3 is derived by adding one Ge atom to the ground-state structure of EuGe₉⁻. The energy of 10A-3 is higher than that of 10A-1 by about 0.67 eV. For EuGe₁₁⁻, a linked structure emerges. The isomers of 11A-1 and 11A-2 are all depicted with Eu atoms as linkers between two parts of TTP Ge₉ and Ge₂. The structure of 11A-3 is one Eu atom and one Ge atom on the plane of C_s symmetry, co-connected to two trigonal bi-pyramids Ge₅. 11A-4 can be seen as one extra Ge atom added onto the side of 10A-1. The energy differences for 11A-2, 11A-3, and 11A-4 with regard to 11A-1 are 0.20, 0.27, and 0.39 eV, respectively. For EuGe₁₂⁻, the lowest-energy 12A-1 is a linked type with a Eu atom connected to Ge₃ as one part and one capped anti-prism Ge₉ as the other part. The structure of 12A-2 exhibits a center Eu atom joining two tetragonal bi-pyramids Ge₆. 12A-1 is more stable than 12A-2, with an energy difference of 0.18 eV. For EuGe₁₃⁻, the minimum-energy structure is a Eu atom associated with TTP Ge₉ and a four-membered ring Ge₄. 13A-2 and 13A-3 are similar motifs viewed as a four-membered ring Ge₄ sub-cluster adsorbed onto two capped anti-prism basic frameworks; their energies are higher than that of the 13A-1 linked structure by 0.21 and 0.31 eV, respectively. For EuGe₁₄⁻, all selected isomers are of linked forms. 14A-1, with C_s symmetry, consists of a Eu atom connected to one unit of TTP Ge₉ and one unit of trigonal bi-pyramid Ge₅. 14A-2 is an energy-degenerate form of 14A-1 showing one sub-structure of TTP Ge₉ and another of square-pyramid Ge₅. The structure of 14A-3 shows mirror symmetry with that of 14A-2, but with a slight difference. The energy difference between 14A-1 and 14A-2 is only 0.04 eV, and that between 14A-1 and 14A-3 is 0.16 eV. For EuGe₁₅⁻, the lowest-energy structure of 15A-1 with C_2V symmetry can be divided into one Eu atom combined with three Ge atoms forming a co-planar quaternary ring that connects two mirror-symmetric pentagonal pyramids Ge₆. 15A-2 and 15A-3 can both be described as a Eu atom binding one sub-unit of TTP Ge₉ and one sub-unit of Ge₆. The energies of 15A-2 and 15A-3 are higher than that of 15A-1 by 0.22 and 0.48 eV. For EuGe₁₆⁻, the architectures of 16A-1, 16A-2, and 16A-3 can all be noted as a Eu atom as a connector joining one sub-cluster of TTP Ge₉ and another sub-cluster of Ge₇. The energy of 16A-1 is lower than those of 16A-2 and 16A-3 by 0.34 and 0.46 eV, respectively. For EuGe₁₇⁻, the skeleton of 17A-1 can be described as a center Eu atom bridging one capped anti-prism Ge₉ and one anti-prism Ge₈. 17A-2 is of a chain type with two parts of two Ge atoms capping an anti-prism Ge₁₀ and two Ge atoms capping a tetragonal bi-pyramid where the Eu atom is not located at the middle position as a linker. The energy of 17A-2 is 0.26 eV higher than that of 17A-1. The cage structure of 17A-3 has a larger energy difference relative to the linked type of 17A-1 of about 1.02 eV. For EuGe₁₈⁻, 18A-1 and 18A-2 are analogous structures portrayed as a Eu atom anchoring two TTP Ge₉ sub-clusters. 18A-3 is an endohedral cage structure where a Eu atom is encapsulated in a Ge₁₈ cage skeleton. The energies of 18A-2 and 18A-3 are 0.16 and 1.54 eV higher than the energy of 18A-1. For EuGe₁₉⁻, 19A-1 and 19A-2, with the same C_s symmetry, can be viewed as a Eu atom bridging one Ge atom capping a Ge₉ anti-prism on one side and two Ge atoms capping a Ge₁₀ anti-prism on the other. The energies of 19A-1 and 19A-2 are degenerative with a small energy difference of 0.04 eV. The 19A-3 cage structure originates from the cage motif of 17A-3 with the addition of two extra Ge atoms into the Ge bones. The energy of 19A-3 is 1.67 eV larger than that of 19A-1. For EuGe₂₀⁻, the energy-minimum pattern 20A-1 can be deconstructed as a middle triangle composed of one Eu and two Ge atoms that connects symmetrical bi-lateral TTP Ge₉ blocks. The geometry of 20A-2 is a Eu atom located at the center, bridging two capped anti-prisms. The energies of 20A-2 and the cage structure 20A-3 are 0.24 and 1.06 eV larger than that of 20A-1.

To sum up, the structural evolution of EuGe_n⁻ (n = 7–20) can be classified into two stages: (i) for small-sized clusters with n = 7–10, the configurations mainly present an adsorbed mode, where a Eu atom is adsorbed onto the Ge_n⁻ cluster; (ii) for medium-sized clusters with n = 11–20, the structures exhibit a pronounced tendency to form elongated link structures. Specifically, Eu atoms serve as connection points, bridging sub-clusters with TTP structures of Ge₉ and Ge_n-9. Consequently, the EuGe₉ sub-cluster, which is formed via linkage between a Eu atom and a TTP-structured Ge₉ unit, constitutes the fundamental structural unit for the growth of medium-sized clusters.

2.2. Predicted Photoelectron Spectroscopy of EuGe_n⁻ (n = 7–20)

Photoelectron spectroscopy (PES) is a common technique for indirectly testing the geometry and electronic structure of nanoclusters. Because nanoclusters have multiple isomers and states that are difficult to accurately determine, PES characterization of nanoclusters is absolutely crucial. In view of this, the theoretical predicted PES of EuGe_n⁻ (n = 7–20) was calculated via the method of Koopmans’ theorem and the first peak equivalent to vertical detachment energy (VDE) calculated under the mPW2PLYP level. The first ionization potential (IP) is defined as E(anion) − E(neutral). The first peak is set to the first IP + E(HOMO); then, the second peak is obtained by adding the absolute value of the energy difference between the occupied orbital below the HOMO and the HOMO orbital as the second IP, and so on. The number of peaks and their position on the PES reveal the energy differences between diverse N-1 states and the original N-electron state. By default, all orbitals have a strength equal to 1.0, where peak broadening can be plotted by means of strength stacking and Gaussian broadening. A plot for each cluster is shown in Figure 2, with a Gaussian broadening of 0.30 eV for the energy region smaller than 5 eV. The peaks of each EuGe_n⁻ (n = 7–20) cluster with energy from low to high are marked as X and A to E in alphabetical order. For EuGe₇⁻, the PES results exhibit four peaks at 2.00 (X), 2.47 (A), 3.55 (B), and 4.71 (C) eV. For EuGe₈⁻, in the energy region smaller than 5 eV, there are only two peaks located at 3.43 (X) and 4.87 (A) eV. For EuGe₉⁻, there are four peaks (X, A to C) located at 2.33, 3.28, 3.69, and 4.29 eV (respectively). For EuGe₁₀⁻, the PES results for the ground-state structure 10A-1 and its energy-degenerate isomer 10A-2 are shown. They have similar shapes and positions of their X, A, and B peaks due to the similarity of their structures. The PES results for 10A-1 show five peaks in the energy region from 0 to 5 eV, sited at 2.10, 3.49, 3.85, 4.51, and 4.87 eV, compared with those for 10A-2 showing only four peaks at 2.42, 3.60, 4.06, and 4.71 eV; thus, they can be distinguished by the number of peaks. For EuGe₁₁⁻, the six peaks X and A to E are located at 2.63, 2.99, 3.77, 4.28, 4.62, and 4.92 eV, respectively. For EuGe₁₂⁻, three obvious peaks, X, A, and B, are located at 2.61, 3.21, and 4.23 eV, with peaks C and D as tail peaks of B at 4.64 and 4.98 eV. For EuGe₁₃⁻, three clear spikes, X, B, and D, are located at 2.05, 3.92, and 4.66 eV. A and C are shoulder peaks of B at 3.50 and 4.22 eV. For EuGe₁₄⁻, the PES curves of the two near-energy structures 14A-1 and 14A-2 have different numbers of peaks and shapes, making them easy to distinguish. The peaks X, A, and B of 14A-1 are positioned at 3.21, 4.23, and 4.76 eV. The PES results for 14A-2 show two more peaks than those for 14A-1, at 2.76 eV for A, 3.20 eV for B, 3.77 eV for C, 4.10 eV for D, and 4.55 eV for E. For EuGe₁₅⁻, there are three independent peaks at 2.07 eV (A), 4.09 eV (B), and 4.83 eV (C). For EuGe₁₆⁻, four consecutive peaks, X and A to C, appear at positions of 3.20, 3.78, 4.43, and 4.74 eV. For EuGe₁₇⁻, five peaks, X and A to D, are located at 3.02, 3.51, 3.79, 4.28, and 4.75. Among them, the peaks A and B are very close to each other. For EuGe₁₈⁻, three evident peaks, X, B, and C, are sited at 3.19, 4.48, and 4.87 eV. A is a satellite peak in the middle of X and B at 3.84 eV. For EuGe₁₉⁻, the PES results for 19A-1 present two clear peaks, X at 3.30 eV and A at 4.26, with tailed peaks at the end of the energy region denoted as C at 4.52 and D at 4.84 eV. The PES results for 19A-2, compared with those for 19A-1, only show three peaks at energy positions of 3.05, 3.72, and 4.73 eV, so they can be discerned by the curve shape and number of peaks. For EuGe₂₀⁻, in the selected energy region, there are three peaks at 2.77, 4.09, and 4.58 eV. Because there are no comparable experimental results, to some extent, these predicted PES results can provide a reference and instructions for future experimental investigations.

2.3. Magnetic Moments and Charge Transfer of EuGe_n⁻ (n = 7–20)

The Eu atom possesses remarkable magnetism as a result of the spin magnetic moment of electrons in its 4f orbitals. When a Eu atom is doped into a pure germanium cluster, it brings this novel magnetic property to the nanocluster. Considering this, a natural population analysis (NPA) in the NBO 3.1 software was performed to discover the charge, valence electron configuration, and magnetic moments of each orbital and the total structure of EuGe_n⁻ (n = 7–20). The data are collected in

Table 1. The NPA valence electron configuration, each valence orbital’s magnetic moment, the charge of the Eu atom in EuGe_n⁻ (n = 7–20), and the total magnetic moment of the whole cluster.

Cluster	Charge (a.u.)	Electron Configuration	Magnetic Moment of Eu Atom					Molecule (μB)
			6s	4f	5d	6p	Total
EuGe7−	0.57	[core]6s0.964f6.985d0.326p0.18	0.75	6.97	0.05	0.12	7.89	8
EuGe8−	0.51	[core]6s0.894f6.985d0.406p0.23	0.71	6.97	0.05	0.13	7.86	8
EuGe9−	0.69	[core]6s0.804f6.985d0.416p0.14	0.64	6.97	0.06	0.07	7.74	8
EuGe10−	0.67	[core]6s1.004f6.985d0.186p0.17	0.70	6.98	0.03	0.13	7.84	8
EuGe11−	0.81	[core]6s0.404f6.975d0.676p0.17	0.00	6.97	0.09	0.01	7.07	8
EuGe12−	0.80	[core]6s0.414f6.975d0.696p0.15	0.01	6.97	0.13	0.01	7.12	8
EuGe13−	0.70	[core]6s0.514f6.975d0.636p0.21	0.28	6.97	0.11	0.07	7.43	8
EuGe14−	0.69	[core]6s0.274f6.975d0.796p0.31	0.01	6.97	0.09	0.01	7.08	8
EuGe15−	0.66	[core]6s0.344f6.975d0.826p0.24	0.10	6.97	0.14	0.02	7.23	8
EuGe16−	0.74	[core]6s0.274f6.975d0.756p0.30	0.01	6.97	0.09	0.00	7.07	8
EuGe17−	0.68	[core]6s0.244f6.975d0.856p0.29	0.00	6.97	0.16	0.01	7.14	8
EuGe18−	0.73	[core]6s0.244f6.975d0.826p0.29	0.04	6.96	0.12	0.01	7.13	8
EuGe19−	0.77	[core]6s0.264f6.975d0.756p0.28	0.00	6.97	0.09	0.00	7.06	8
EuGe20−	0.66	[core]6s0.264f6.975d0.836p0.32	0.02	6.96	0.09	0.01	7.08	8

. From Table 1, it can be seen that the Eu atoms in all ground-state structure clusters have a positive charge of from 0.51 a.u. to 0.81 a.u., which means that the Eu atoms in the clusters serve as electron donors and the Ge frameworks serve as electron acceptors. Between the Eu and Ge atoms, ionic bonds are formed. When n = 7–10, the ground-state structure is replaced by a structure in which the Eu atom has a valence electron configuration of 6s^0.80–1.004f^6.985d^0.18–0.416p^0.14–0.23. When the ground-state structure changes to a linked type (n = 11–20), the valence electron configuration of Eu becomes 6s^0.24–0.514f^6.975d^0.63–0.856p^0.15–0.32. From the valence electron configuration of the Eu atom of Xe [6s²4f⁷], it is known that the 4f electrons of the Eu atom in EuGe_n⁻ (n = 7–20) do not participate in bonds, mainly providing magnetic moments for the cluster. Without bonding involving the 4f electrons of Eu, the bonding between Eu and Ge is mainly attributed to electrons in Eu’s 5d and 6s orbitals hybridizing with electrons in Ge’s 4s and 4p orbitals.

To further explore the magnetism of EuGe_n⁻ (n = 7–20), the spin density and spin population were examined. The spin density three-dimensional contour maps of EuGe_n⁻ (n = 7–20) shown in Figure 3 were obtained using the following equation:

$${\rho }^s(r)={\rho }^{\alpha }(r)-{\rho }^{\beta }(r)$$

(1)

The equation definition is the electron density of Alpha minus the electron density of Beta in three-dimensional space. It can be easily seen that the iso-surfaces of spin density are all located in the Eu atom, and the values of ${\rho }^s(r)$ are all close to or equal to 8, indicating that the magnetism of EuGe_n⁻ (n = 7–20) originates from the Eu atom. A further analysis was carried out via the spin population method, as shown in Table 1. For n = 7–10, the adsorption structure stage, the total magnetic moment contribution by the Eu atom is 7.86 to 7.95 μ_B, with a contribution ratio exceeding 99.10%. For n = 11–20, the linked stage, the total magnetic moment provided by the Eu atom is 7.11 to 7.48 μ_B, with a contribution ratio exceeding 88.96%—a slight decrease compared to that of the adsorption structure.

2.4. Relative Stability of EuGe_n⁻ (n = 7–20)

The average binding energy (ABE) is an important parameter used to describe the relative stability with the change in the size of a cluster. The second-order energy difference (Δ²E) is another critical parameter used to evaluate the stability within one cluster of its left and right sides. The ABE and Δ²E are calculated through following Equations (2) and (3):

$$ABE\left({\mathrm{EuGe}}_n^{-}\right)={\displaystyle \frac{\left[\left(n-1\right)E\left(\mathrm{Ge}\right)+E({\mathrm{Ge}}^{-})+E(\mathrm{Eu})-E\left({\mathrm{EuGe}}_n^{-}\right)\right]}{n+1}}$$

(2)

$${\mathsf{\Delta }}^{2}E{(\mathrm{EuGe}}_n^{-})=E({\mathrm{EuGe}}_{n-1}^{-})+E({\mathrm{EuGe}}_{n+1}^{-})-2E({\mathrm{EuGe}}_n^{-})$$

(3)

The values of ABE and Δ²E are plotted in Figure 4. From Figure 4a, in the adsorption pattern phase, the ABE rises sharply, with a peak observed for EuGe₁₀⁻ of 4.16 eV. At the linked structure stage, two obvious sections can be seen. When n = 11–17, the increasing trend is smaller than that for the adsorption stage, tending toward a flat curve for n = 18–20. The maximum and minimum values of the ABE within the cluster size were found for EuGe₁₉⁻ at 4.35 eV and EuGe₇⁻ at 3.90 eV. In Figure 4b, the Δ²E curve presents four peaks at 0.81, 0.48, 0.45, and 0.37 eV, belonging to EuGe₉⁻, EuGe₁₃⁻, EuGe₁₅⁻, and EuGe₁₈⁻; this means that these clusters are more stable than their adjacent ones.

The energy gap (E_g) is a very important physical quantity that denotes the energy difference between the lowest unoccupied molecular orbital and the highest occupied molecular orbital, described in Equation (4):

$$E_g=E(\mathrm{LUMO})-E(\mathrm{HOMO})$$

(4)

The E_g values in Figure 4c were calculated using the mPW2PLYP and PBE0 methods. In general, the number of Hartree–Fock components in a functional strongly affects the value of E_g. The mPW2PLYP functional has more Hartree–Fock components and usually over-estimates E_g. However, having fewer Hartree–Fock components in a functional mostly leads to under-estimation of E_g, such as with the pure PBE functional. In view of this, the hybrid density functional PBE0 was implemented to calculate E_g. The trends in E_g values calculated using mPW2PLYP and PBE0 are generally similar, being only slightly different at n ≤ 11; when n > 11, the two E_g curves present the same trend. The maximum E_g values pertain to EuGe₁₄⁻, with 3.67 eV at the mPW2PLYP level and 2.40 eV at the PBE0 level. The minimum E_g values belong to EuGe₁₅⁻, with 2.00 eV at the mPW2PLYP level and 1.24 eV at the PBE0 level. The above results show that the EuGe_n⁻ (n = 7–20) clusters are all semiconductors and potential semiconductor building block candidates for next-generation optoelectronic materials.

2.5. Nonlinear Optical Properties of EuGe_n⁻ (n = 7–20)

The Eu atom is a typical optical activator. In previous works [29,30], lone-pair electrons helped to enhance the nonlinear optical effects of optical materials. The Eu atom has seven single electrons in its 4f orbitals; when an extra electron is brought into the cluster, the whole cluster changes to have eight single electrons. In view of this, the dipole moment (μ₀), static polarizability (α₀), static first hyperpolarizability (β_tot), components of β_tot in the x, y, and z directions, and projection of β_tot in the direction of the dipole moment (β_prj) for EuGe_n⁻ (n = 7–20) clusters were calculated at the level of long-range corrected density functional theory CAM-B3LYP, which is commonly used to evaluate nonlinear optical responses [31,32]. The results are summarized in

Table 2. The dipole moment (μ₀, in a.u.); static polarizability (α₀, in a.u.); static first hyperpolarizability (β_tot, in a.u.); components of hyperpolarizability in the x, y, and z directions (β_xxx β_yyy β_zzz, in a.u.); and projection of hyperpolarizability in the direction of the dipole moment (β_prj, in a.u.) of EuGe_n⁻ (n = 7–20).

n.	μ0	α0	βprj	βtot	βxxx	βyyy	βzzz
7	1.37	454.46	−2242.88	2281.64	1179.93	1952.85	0.00
8	0.91	488.37	−3911.63	3911.63	0.00	0.00	−3911.63
9	2.02	548.01	8681.47	8681.48	107.48	8680.81	0.00
10	1.40	553.61	−10,775.99	10,801.04	−6283.05	−8785.54	0.00
11	3.94	527.99	−20,833.66	20,895.13	17,745.60	10,984.98	−1014.91
12	3.47	552.37	−12,056.43	12,066.61	5700.40	10,634.59	118.72
13	3.34	803.99	−726,411.06	747,032.61	373,853.70	−630,754.60	142,967.70
14	2.27	630.10	−2737.34	3113.67	1204.37	2871.31	0.00
15	3.48	680.11	−96,087.04	96,087.04	0.00	0.00	−96,087.04
16	2.98	682.23	−2359.59	2551.03	2538.74	−250.11	0.00
17	2.84	738.93	−2682.55	2776.80	137.43	−2771.25	−109.35
18	2.47	750.25	−2526.95	2557.41	−1881.68	−1731.94	0.00
19	2.70	788.58	−2690.12	2736.34	2618.65	793.88	0.00
20	2.53	860.61	−3405.02	3442.56	−3111.18	1473.71	0.00

. For clusters with n = 7–10, the dipole moment varies between 0.91 and 2.02 a.u. When n increases to the range of 11 to 20, the dipole moment exhibits a significant increase, ranging from 2.27 to 3.94 a.u., which can be attributed to changes in the cluster configuration. The static polarizability of the ground-state structures of EuGe_n⁻ (n = 7–20) demonstrates an overall increasing trend, with particularly notable values at n = 13 and 20, reaching 803.99 and 860.61 a.u., respectively. The second-order nonlinear optical coefficient of the clusters, i.e., the first hyperpolarizability, can reflect the nonlinear optical response of the materials. Thus, the static hyperpolarizability and its components along the three directions of the 14 ground-state clusters were simulated and compared. It was observed that the β_tot value for EuGe₁₃⁻ was markedly higher than those of the other clusters, reaching 747,032.61 a.u., with β_yyy being the predominant contributing component. These findings suggest that EuGe₁₃⁻ holds significant potential for applications in the development of nonlinear optical materials.

To further elucidate the nonlinear optical properties of EuGe_n⁻ (n = 7–20), the electronic spatial extent R² of the clusters serves as an effective descriptor quantifying their polarizability and hyperpolarizability [33]. Figure 5a,b, respectively, illustrate the relative relationship between α₀ and R² and that between β_tot and R² as the number of Ge atoms increases. From Figure 5a, it can be seen that the α₀ value increases as the cluster size and R² increase. However, for β_tot, there is a linear trend only for small-sized clusters with n < 12 (shown in Figure 5b). Notably, the polarizability reaches a significant peak when n = 13.

Besides the descriptor R², the relative relationships between the polarizability, hyperpolarizability, and energy gap are illustrated in Figure 6a,b. In the figure, the HOMO–LUMO gap exhibits a negative correlation with α₀ and β_tot. Smaller E_g values could account for the larger α₀ and β_tot observed for EuGe₁₃⁻ and EuGe₁₅⁻. Given that the symmetry of EuGe₁₅⁻clusters is higher than that of EuGe₁₃⁻ clusters, the electron cloud distribution in the EuGe₁₅⁻ clusters is more uniform, rendering them less susceptible to polarization by external electric fields. Consequently, the α₀ and β_tot values for EuGe₁₅⁻ clusters are lower than those for EuGe₁₃⁻ clusters. Additionally, the calculated β_prj values can serve as a reference and comparison for hyperpolarizability values obtained from future experiments on electric field-induced second harmonic generation (EFISH).

To obtain a more comprehensive insight into the relationships among the parameters, we performed additional data processing and analysis. Figure 7a reflects the relationship between lg(β_tot) and the cluster size. Figure 7b shows the relative relationship between α₀ and R², as well as the fitting curve. In Figure 7a, the hyperpolarizability of EuGe₁₃⁻ is clearly higher than that of any other type. The second peak is observed for EuGe₁₅⁻, which is consistent with the discussion above. In Figure 7b, α₀ and R² exhibit a better linear relationship, with a fitting R-squared value equal to 0.78. Finally, Figure 7c shows a volcano plot to further identify the best hyperpolarizability, which belongs to EuGe₁₃⁻. Overall, the static polarizability exhibits a distinct positive correlation with R², while the hyperpolarizability shows a trend of first increasing and then decreasing with R². This suggests that as the electron cloud becomes more extended, the electrons become more responsive to external electric fields, thereby enhancing the likelihood of polarization. Furthermore, the volume and morphology of the clusters influence the outcome.

To avoid basis set effects, the hyperpolarizability was also calculated with aug-cc-pVTZ for Ge and ma-def2-TZVP for Eu [34]. The outcomes were consistent with the discussion above (seen Table S3, Supporting Information).

In summary, the nonlinear optical properties of EuGe_n⁻ (n = 7–20) clusters are strongly correlated with their growth patterns and structural symmetry. As the cluster size increases, the (hyper)polarizability of the cluster exhibits a positive correlation with the overall electron space R² and a negative correlation with the energy gap. The smaller E_g values might be the reason why the clusters have larger (hyper)polarizability values. Among the investigated clusters, EuGe₁₃⁻ had the highest (hyper)polarizability value of 7.47 × 10⁵ a.u.; that is, R² and E_g are two important descriptors for evaluating the nonlinear optical properties of EuGe_n⁻ (n = 7–20).

2.6. Nonlinear Optical Properties of EuGe₁₃⁻ from TD-DFT

Whether compared within the same system or with some other materials [31,32,35], EuGe₁₃⁻ displays significant β_tot values; thus, its (hyper)polarizability density was targeted for further examination. Figure 8 presents the local contributions of EuGe₁₃⁻ polarizability and hyperpolarizability along the y direction in a static electric field. Positive contributions are represented by blue isosurfaces, whereas negative contributions are depicted by white isosurfaces. This visualization method facilitates a clearer understanding of the three-dimensional spatial contributions to the (hyper)polarizability. From Figure 8a, it is evident that the region of positive contribution substantially exceeds that of negative contribution. This discrepancy in areas provides a clear rationale for the cluster’s relatively large positive polarizability. For the hyperpolarizability, despite the presence of distinct positive and negative contribution areas in Figure 8b, the negative contribution regions are more extensive. This is consistent with the significantly negative hyperpolarizability of EuGe₁₃⁻. In general, both the polarizability and hyperpolarizability exhibit a significant contribution region surrounding the Eu atom, which indicates that Eu atoms in the system have a stronger response to the external electric field, enhancing the nonlinear optical properties of the cluster.

The two-level expression (5) is widely employed to qualitatively investigate the NLO property of a system in depth [36,37]. Utilizing this formula, the crucial excitation energies of the EuGe₁₃⁻ cluster were calculated within the framework of TD-DFT:

$${\beta }_{\mathrm{tot}}\propto {\displaystyle \frac{\mathsf{\Delta }\mu \cdot f_0}{\mathsf{\Delta }{E}^3}}$$

(5)

where Δμ is the difference in dipole moment between the ground state and the crucial excited state, f₀ is the oscillator strength, and ΔE is the transition energy of the key transition. Figure 9 shows the simulated UV–vis spectrum and the molecular orbital diagram of the main transitions of the EuGe₁₃⁻ cluster. The strongest absorption peak at 2710.63 nm is dominated by N₀ → N₁, where N stands for a nonet. ΔE is 0.457a.u. The dominant electronic excitation mode in this transition is HOMO(α) → LUMO(α) with a contribution rate of 97%. N₀ → N₃₇ mainly contributes to the absorption peak at 474 nm. The f₀ value of this crucial excitation is 0.0214 a.u., ΔE is 2.51 eV, and Δμ is 0.348 a.u. Multiple electronic excitation modes are observed in this transition, with significant contributions from HOMO(α) → LUMO+12(α) (35%) and HOMO-3(α) → LUMO+1(α) (14%). The relatively large β_tot value of the cluster can be attributed to factors such as the large Δμ and small ΔE values associated with the transitions. The relevant parameters for other clusters are collected in Table S4.

2.7. Density of States of EuGe₁₃⁻

The density of states (DOSs) of a cluster can provide a deep understanding of the bonding characteristics within it. Figure 10 displays the DOS of the EuGe₁₃⁻ cluster, where the black curve is the total DOS and the colored curves are the partial DOS curves of different orbits. As illustrated in the figure, the HOMO and LUMO are predominantly contributed by the Ge-s, p, and Eu-s orbitals; for the HOMO level, these contributions account for 25.91%, 35.13%, and 36.83%, respectively, and for the LUMO level, the corresponding contributions are 46.02%, 25.96%, and 27.11%. In the energy region below the HOMO, the most prominent peak in the DOS is observed at approximately −7.05 eV, which is predominantly attributed to the 4f orbitals of the Eu atom (89.18%). The remainder is mainly contributed by the Ge-s and p orbitals. It is evident that there is a pronounced asymmetry between the spin-up Alpha and spin-down Beta, especially the spin-up peak at −7.05 eV. Based on the NPA, it is once again demonstrated that the magnetic property of the cluster is mainly provided by the 4f orbitals of Eu and that the cluster exhibits spin polarization effects and magnetism.

3. Computational Details

All density functional theory calculations were performed using the Gaussian 09 quantum chemistry software [38]. The initial structures were obtained (1) using the ABCluster global searching software, version 3.1 and (2) from previously reported structures [39,40,41]. By way of (1), each cluster generated at least 400 isomers to ensure that the global minimum structure was found. The initial structures were optimized with TPSSh [42] combined with BS-1 basis sets (ECP28MWB [43] for Ge and ECP53MWB [44,45] for Eu). Then, the structures with an energy difference of within 0.8 eV were selected and further optimized using TPSSh combined with BS-2 basis sets (cc-pVTZ-PP [46,47] for Ge and ECP28MWB [48] for Eu). At the same time, frequency calculations were carried out to rule out imaginary frequencies, indicating that the structure is a true local minimum. Then, the above structures were further optimized at the mPW2PLYP [49] level with BS-2 basis sets, but frequency calculations were not carried out in order to reduce the calculation time. Finally, the single-point energy was calculated at the mPW2PLYP level with BS-3 basis sets (aug-cc-pVTZ [50] for Ge and ECP28MWB for Eu).

The PES spectra of EuGe_n⁻ (n = 7–20) nanoclusters were obtained by applying Koopmans’ theorem [51,52] at the level of mPW2PLYP, and ultraviolet–visible (UV–vis) spectra were simulated via the time-dependent density functional theory method of the PBE0 functional with BS-3 basis sets [53,54]. The dipole moment, static polarizability, and first hyperpolarizability of each EuGe_n⁻ (n = 7–20) cluster were calculated by means of the CAM-B3LYP functional [55] with BS-3 basis sets. The results were visualized using Multifwn [56,57] and VMD software, VMD 1.9.4 [58].

The formulas for the dipole moment (μ₀), static polarizability (α₀), and static first hyperpolarizability (β_tot) are presented as follows:

$${\mu }_0={\left({\mu }_{\mathrm{x}}^2+{\mu }_y^2+{\mu }_z^2\right)}^{1/2}$$

(6)

$${\alpha }_0=1/3\left({\alpha }_{xx}+{\alpha }_{yy}+{\alpha }_{zz}\right)$$

(7)

$${\beta }_{tot}={\left({\beta }_x^2+{\beta }_y^2+{\beta }_z^2\right)}^{1/2}$$

(8)

where ${\beta }_i=\left(1/3\right){\displaystyle \sum _j\left({\beta }_{ijj}^2+{\beta }_{jji}^2+{\beta }_{jij}^2\right)}i,j=\left\{x,y,z\right\}$.

The ab initio molecular dynamics (AIMD) of EuGe_n⁻ (n = 7–20) were obtained using Orca quantum chemistry code [59] at the level of the B97-3c functional [60] with def2-mTZVP for Ge and def2-mTZVP for Eu combined with Def2-ECP basis sets [48]. The B97-3c functional contains the Becke–Johnson damping scheme (D3BJ) [61,62].

mPW2PLYP displays high reliability in predicting the ground-state structures of rare-earth elements doped into silicon or germanium clusters. The different density functionals mPW2PLYP, TPSSh, PBE, wB97X, and B3LYP were compared with the ROCCSD(T) method for ScSi_n^0/− (n = 4–9) in a previous work [63], and the results showed that all the ground-state structures predicted using mPW2PLYP were in line with those obtained using ROCCSD(T). Besides those of rare-earth-element-doped silicon clusters, the ground-state structures of ScGe_n⁻ (n = 6–17) and CeGe_n⁻ (n = 5–17) predicted using the mPW2PLYP approach agreed with those predicted via closed-shell DLPNO-CCSD(T1) and open-shell DLPNO-CCSD(T) methods [23,64], which further proved that the mPW2PLYP functional is suitable for such systems, combining accuracy and speed. The optimization scheme and TD-DFT calculation were also based on the above previous works.

4. Conclusions

This study systematically investigated the structural evolution and nonlinear optical properties of europium-doped germanium anionic clusters EuGe_n⁻ (n = 7–20), utilizing an approach that combined global search techniques and double-hybrid density functional theory mPW2PLYP. The growth patterns were categorized into two distinct stages: (i) for n = 7–10, the structures were mainly characterized by an adsorption configuration with a Eu atom adsorbed onto a pure Ge cluster, and (ii) for n = 11–20, the clusters exhibited linked structures wherein a Eu atom was individually linked to a Ge₉ sub-cluster with a TTP configuration and a Ge_n-9 sub-cluster. The PES spectra, charge transfer, magnetic moments, relative stability, energy gaps, and (hyper)polarizability of the clusters were simulated and analyzed. EuGe₁₃⁻, which exhibits notable NLO properties and a β_tot value of 7.47 × 10⁵ a.u., was subjected to a further investigation. The findings indicate that EuGe₁₃⁻ holds potential as a nonlinear optical semiconductor material for application in multifunctional nanomaterials.