The quantum size effect is a well-known fundamental scientific phenomenon. Due to quantum confinement, downscaling a system to small sizes should increase the bandgap of a solid state material. However, in this work, we present an exception: monolayers of nickel hydroxide have smaller bandgaps than their bulk analogues, due to the surface states appearing at energies within the bandgap region. Our findings are obtained by several state-of-the-art first principles calculations.

The quantum size effect is a remarkable phenomenon that happens at small sizes of semiconducting materials. As a material reduces in size, typically at nanometer scale, then the material can absorb less of the optical spectrum since the bandgap increases.1 This is a basic quantum effect that can be explained by the most elemental model called the particle-in-a-box. Essentially, a particle confined in a box will have available states with energies that depend inversely on the size of the box. Therefore, the energy differences as well as the bandgap are larger for smaller nano-materials.

There is almost no hint in the literature to the possibility of deviation from this fundamental phenomenon of the quantum size effect. One rare example is the theoretical work of Peelaers and Van de Walle2 on 2D Ga2O3, where the calculated bandgap of the Ga2O3 monolayer was not found to be higher than that of the bulk material. In fact, this deviation was not quantitatively significant since the bandgap of the monolayer was almost identical to the bandgap of bulk Ga2O3. In addition, this deviation has not been verified experimentally or beyond the Density Functional Theory (DFT) framework for Ga2O3. Since most of the literature points to consistency with the quantum size effect, this effect has been used to guide technological development.

The ability to control the bandgap through reducing the size of a material has been vastly utilized for developing novel electronic, optical, and chemically active devices. Some examples include superconductivity modulated by the quantum size effect,3 the rise of catalytic activity of titania on dendrimer templates4 or gold clusters on titania with the appearance of nonmetallic properties,5 chaotic Dirac billiard in graphene quantum dots,6 and electrical tuning of quantum plasmonic resonance.7 

Smaller sized systems and in particular 2D materials have attracted great interest in recent years.8 2D materials pose intriguing properties, such as a confined intrinsic size, a large surface area, sensitivity to defects, and chemical reactivity. Recent advances in nanotechnology and development of new low and mixed dimensional materials continue to provide novel solutions for energy harvesting and storage fields, electronic and optical devices, chemical catalysis, and more.8–11 The most commonly studied example is graphene whose discovery led to a Nobel Prize recognition and numerous subsequent studies.12 

A much less studied family of 2D materials that have great potential are transition metal hydroxides [TM(OH)2; TM = Transition Metal].13 2D materials based on layered double hydroxides (LDHs) are an emerging class of 2D materials with potential use in water splitting, as supercapacitor electrodes, in solar cells, and in electronic devices.10,11,14,15 While some of the 2D LDHs were obtained in a laboratory environment,14,16,17 the existence of others is still hypothetical. Fabricating hydroxides in 2D through exfoliation is possible,18 and the possibility to design new properties lies in the ability to change transition metal centers.

One example that has attracted attention in the field of developing renewable energy is Ni(OH)2, whose chemical activity in the NiOOH phase sharply rises when doped with iron, positioning this material as the best-known water oxidation catalyst at alkaline conditions.19 There are several papers on ultrathin/monolayered Ni(OH)2. Harvey et al.14 demonstrated the possibility of producing Ni(OH)2 nanosheets through liquid phase exfoliation. The authors used the material as an effective and stable oxygen evolution reaction (OER) catalyst and as a supercapacitor electrode with high capacitance. Cui et al.20 produced the monolayers of Ni(OH)2 by employing the bottom-up sol-gel method and measured the capacitance of the material. Boettcher et al.17 investigated the behavior of monolayered Ni(OH)2 during an electrochemical reaction by employing atomic force microscopy. According to the results, the Ni(OH)2 nanosheets might be unstable during the reaction or if contaminated with iron, making it difficult to design nanoscale Ni(OH)2-based electrochemical systems.

Due to the rising interest in this material, this study focuses on Ni(OH)2. We discover that Ni(OH)2 has a smaller bandgap when reduced from 3D to 2D. We use a combination of state-of-the-art first principles calculations to evaluate the bandgap of monolayered and bulk Ni(OH)2. We find that Ni(OH)2 has low-energy surface states that lie at the conduction band edge and reduce the bandgap, in contrast to the quantum size effect.

Our first principles calculations include DFT, the most commonly used method for solid-state materials. In order to properly describe the electronic interactions in open shell transition metals, a U correction term is added (DFT+U, Dudarev’s approach21). We also perform calculations beyond DFT, based on many-body perturbation theory (GW approximation).22 Several GW approaches are used, which differ by the starting point as well as the self-consistency level of the calculation, including G0W0, scGW0, and QPGW.23 All the DFT+U and GW calculations were performed with the VASP24,25 5.4.4 software package, employing the Projector Augmented Wave (PAW) formalism26 to describe the electron-ion interactions. Further computational details are provided in the supplementary material.

We consider three model systems to demonstrate and explain the deviation of transition metal hydroxides from the quantum size effect: (1) Monolayers of Ni(OH)2. We show a systematic comparison between theories for calculating the bandgap and a deep analysis of the band diagram. (2) Two, three, and four monolayers of Ni(OH)2. For the bilayer, we show bandgap dependence on inter-sheet distance and orientation. (3) A “sandwich” of Co(OH)2 layers inside a bulk Ni(OH)2 is to demonstrate that the surface states of Co(OH)2 are eliminated by bonding to outer Ni(OH)2 layers.

The calculated DFT+U bandgap in three-dimensions is 3.00 eV for β-Ni(OH)2. However, the calculated bandgap of a monolayer of Ni(OH)2 is 2.17 eV, much smaller than in the bulk material—in contrast to what is expected from the quantum size effect. These results agree with previous experimental literature reports that measure a bandgap of 2.3827 eV for thin film Ni(OH)2 and 3-3.528 eV for bulk β-Ni(OH)2. We note that there is difficulty in extracting an accurate bandgap from the reported experimental data due to the shallow increase of the absorption spectra.

To account for electron screening effects not captured with DFT, the monolayer and bulk bandgaps of Ni(OH)2 were calculated with several GW approaches. However, this improvement in the theory did not change the deviation from the quantum size effect: as seen in Fig. 1, the calculated ratio between the 3D and 2D bandgaps is above one for all the calculation approaches. Please see more information in Table S1 in the supplementary material.

FIG. 1.

Calculated ratio between two-dimensional (single monolayer) and three-dimensional bandgaps (EgML and Egbulk, respectively) for Ni(OH)2 using several state-of-the-art approaches based on DFT, hybrid DFT, and GW. The inset shows the structure of a single monolayer (side and top views).

FIG. 1.

Calculated ratio between two-dimensional (single monolayer) and three-dimensional bandgaps (EgML and Egbulk, respectively) for Ni(OH)2 using several state-of-the-art approaches based on DFT, hybrid DFT, and GW. The inset shows the structure of a single monolayer (side and top views).

Close modal

In the remainder of this paper, we deeply analyze the source of the deviation from the quantum size effect for Ni(OH)2. The band structure of ultra-thin Ni(OH)2 shown in Fig. 2(a) indicates that the two lowest unoccupied bands are lower in energy for thicker slabs. The bandgaps for the monolayer, bilayer, three-layer, and four layers are 2.17 eV, 2.27 eV, 2.31 eV, and 2.32 eV, respectively. In contrast, the behavior of the higher energy bands is consistent with the quantum size effect (see Table S2 in the supplementary material). Hence, the lowest lying states of the conduction band are responsible for the deviation from the quantum size effect. We can plot the spatial distribution of the electronic states in the three lowest energy bands above the valence band [see Fig. 2(b) for the electron density in the bi-layer], and see that the states are located at the outer surface of the slab, on the O–H groups. Similar charge density images for 1-4 layers are in the supplementary material (Figs. S1-S4). Higher energy states at the Γ point are located inside the slab far from the surface [Fig. 2(b) and Figs. S1 and S2]. Therefore, the surface states act as “defect” states with low energies located inside the bandgap.

FIG. 2.

Electronic information for Ni(OH)2: (a) Band diagram for monolayer, bilayer, three-layers, and four-layers (nML = n monolayers). (b) Spatial distribution of electron density of states in the three lowest unoccupied bands (total and at Γ point) in the bilayer. (c) Bandgap values of a Ni(OH)2 bilayer as a function of interlayer distance. (d) Charge density of Ni(OH)2 bilayers for increasing interlayer distances (from left to right); the dashed line corresponds to the middle point separating the layers.

FIG. 2.

Electronic information for Ni(OH)2: (a) Band diagram for monolayer, bilayer, three-layers, and four-layers (nML = n monolayers). (b) Spatial distribution of electron density of states in the three lowest unoccupied bands (total and at Γ point) in the bilayer. (c) Bandgap values of a Ni(OH)2 bilayer as a function of interlayer distance. (d) Charge density of Ni(OH)2 bilayers for increasing interlayer distances (from left to right); the dashed line corresponds to the middle point separating the layers.

Close modal

The bandgap does not depend significantly on geometrical effects related to the stacking of the layers (see Figs. S5–S7, Tables S3 and S4). However, the bandgap does depend on the number of layers and the distance between them. As seen in Figs. 2(c) and 2(d), the surface charge density builds up between the layers at larger interlayer distances of a bilayer and causes dramatic changes in the bandgap value. As we increase the interlayer distance, the bandgap decreases and the interlayer electron density increases as a result of new interlayer states. As the distance continues to increase, the bandgap reaches its minimum and the interlayer density reaches its maximum. Eventually, the electron buildup between the two layers reduces to the regular surface density, and the bandgap stabilizes at the bandgap of the monolayer. A movie showing changes in charge density between bilayers as a function of distance is provided in the supplementary material.

An interesting architecture that demonstrates the deviation from the quantum size effect is a “sandwich” of Co(OH)2 layers inside a bulk of Ni(OH)2. Interestingly, an isolated Co(OH)2 bilayer also has a deviation from the quantum size effect—it has a smaller calculated bandgap of 1.47 eV, compared to the bulk bandgap of 2.23 eV. This deviation vanishes when the bilayer is sandwiched by Ni(OH)2 since the surface states are not present anymore. The model and the results are presented in Fig. 3. The bandgap for this assembly is 2.30 eV, which is converged at eight layers of Ni(OH)2. This model has no deviation from the quantum size effect: the bandgap of the sandwiched four-layer (2.25 eV) is smaller than the bandgap of the sandwiched bilayer (2.30 eV; see Fig. 3).

FIG. 3.

Atomic models of Co(OH)2 layers “sandwiched” inside a Ni(OH)2 bulk matrix. The bandgap values are indicated for each unit cell. Created with Jmol29 visualization software.

FIG. 3.

Atomic models of Co(OH)2 layers “sandwiched” inside a Ni(OH)2 bulk matrix. The bandgap values are indicated for each unit cell. Created with Jmol29 visualization software.

Close modal

In summary, we have demonstrated a deviation of the quantum size effect in 2D Ni(OH)2, which is a material that attracted great interest lately due to the high performance of the NiOOH phase in oxidizing water. The exception from the quantum size effect was demonstrated and analyzed using several theoretical model systems. The calculated bandgap of a monolayer (2.17 eV, PBE+U) is smaller than the bandgap of the bulk (3.00 eV, PBE+U). We ruled out that the deviation from the quantum size effect observed through DFT calculations arises from electronic interactions that are accounted for in several many-body GW approaches. A band diagram analysis reveals that the sources for reducing the bandgap are unoccupied surface states located on O–H bonds. Furthermore, our models show that geometrical effects related to stacking between the layers do not significantly affect the bandgap, but the amount of layers and distance between them do play a role. Finally, we modelled a Co(OH)2 [which shows a similar to Ni(OH)2 behavior] bilayer “sandwiched” between many Ni(OH)2 layers that eliminate the surface states and retain consistency with the quantum size effect. All this evidence supports the existence of an exception to the quantum size effect for Ni(OH)2. We anticipate that the quantum size effect will not appear in other systems that have mid-gap surface states.

Further details on the calculations relaxed geometries are provided as supplementary material.

This research was supported by the Nancy and Stephen Grand Technion Energy Program (GTEP), the I-CORE Program of the Planning and Budgeting Committee, the Israel Science Foundation (Grant No. 152/11), and a grant from the Ministry of Science and Technology (MOST), Israel. This work was supported by the post LinkSCEEM-2 project, funded by the European Commission under the 7th Framework Programme through Capacities Research Infrastructure, INFRA-2010-1.2.3 Virtual Research Communities, Combination of Collaborative Project and Coordination and Support Actions (CP-CSA) under Grant Agreement No. RI-261600. We wish to thank Professor Richard Robinson from Cornell University and Professor Shannon Boettcher from the University of Oregon for most helpful discussions. M.N. acknowledges the Rosen Award for excellence given by the Department of Materials Science and Engineering at the Technion.

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Supplementary Material