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. 2026 Oct 5. Online ahead of print. doi: 10.1039/d6ra07688b

Exciton states of HfO2 and Janus HfSO monolayers: first-principles investigations

Nguyen Thi Han a,✉
PMCID: PMC13637422  PMID: 42835699

Abstract

In this work, we perform first-principles calculations to study the electronic and optical properties of HfO2 and its Janus HfSO monolayers, with a particular emphasis on electron–hole interaction (excitonic effects). The electronic properties are investigated using a hierarchy of theoretical approaches, including standard density functional theory DFT (PBE), DFT (PBE)+U, DFT (PBE)+SOC, the hybrid functional DFT (HSE06), and many-body perturbation theory within the GW approximation. Both HfO2 and HfSO monolayers are indirect-gap semiconductors, and the predicted band gaps increase as we apply more advanced levels of theory. We also examine electronic properties through complex orbital hybridization. Furthermore, we observed strongly bound excitonic states in both materials, including both optically active (bright) and inactive (dark) excitons, highlighting the significant role of excitonic effects. The pronounced excitonic effects identified in both materials suggest their potential for next-generation two-dimensional (2D) optoelectronic and photonic applications.

Optical absorption and excitonic states of (a and b) HfO2 and (c and d) HfSO monolayers. Distinct peaks show excitonic contributions and strongly bound excitons.graphic file with name d6ra07688b-ga.webp

1. Introduction

Today, 2D materials have gained significant attention for their extraordinary physical properties and diverse potential applications. Due to their reduced size and strong quantum confinement effects, these atomically thin materials show unique traits that are not found in their bulk forms, such as exceptional mechanical flexibility, high thermal conductivity, and electronic and optical properties.1–6 Among 2D materials, transition metal dichalcogenides (TMDs) are emerging as promising candidates for electronic and optoelectronic applications.5,7,8 TMDs' material features give rise to unique phenomena, including layer-dependent band gaps, intense photoluminescence (PL) from monolayers, and valley-selective circular dichroism with circularly polarized light. Importantly, the optical and electronic properties of TMDs are mainly affected by tightly bound electron–hole pairs (called exciton effects) stabilized by the reduced dielectric screening and increased Coulomb interactions characteristic of 2D systems.9,10 These findings have established TMDs as a foundational platform for exploring exciton physics in reduced dimensions and for the development of electronic and optoelectronic technologies.11

Stimulated by the success of 2D TMD materials, 2D transition-metal dioxides such as ZrO2 and HfO2 monolayers have been synthesized via chemical vapor deposition (CVD).7–10 Among these, 2D HfO2 and Janus HfSO monolayers have been extensively studied through both experimental measurements and theoretical calculations. However, it should be noted that the Janus HfSO monolayer has not yet been experimentally synthesized.

Theoretically, based on first-principles calculations, Weng et al.12 reported that the 1T phase of the HfO2 monolayer is dynamically stable and exhibits a wide indirect band gap of 6.73 eV, as predicted by the Heyd–Scuseria–Ernzerhof hybrid functional (HSE06), highlighting its potential for applications in wide-band-gap semiconductors. In earlier studies, a 2D HfO2 monolayer was also shown to possess band gaps of 4.85 eV and 6.43 eV at the DFT (PBE) and DFT (HSE06) levels, respectively.13 In a separate work, Dat and Vu14 demonstrated that 2D HfO2 and Janus HfSO monolayers exhibit indirect band gaps of 6.50 eV and 2.98 eV, respectively, using the DFT(HSE06) approach.

While our previous work examined the electronic properties of both 2D HfO2 and Janus HfSO monolayers, the quasiparticle band gap was calculated using the more accurate G0W0 method,15 as well as electron–electron and electron–hole interactions (excitonic effects) in optical responses, have not yet been reported.

To address these issues, we systematically investigate the electronic and optical properties of HfO2 and Janus HfSO monolayers using first-principles calculations based on DFT with the PBE functional, DFT+U, DFT+SOC, HSE06, and the GW approximation. Many-body excitonic effects are further incorporated through the Bethe–Salpeter equation (BSE) to obtain a more accurate description of the optical response beyond the independent-particle picture. We identify and characterize both bright and dark excitonic states in these monolayers, elucidating their distinct optical activities and the underlying electronic transitions. In particular, our comparative analysis reveals how differences in structural symmetry and electronic environments influence the excitonic states and optical properties of HfO2 and HfSO monolayers. These findings provide comprehensive insights into the interplay between electronic structure and excitonic effects in Hf-based 2D materials and offer a theoretical foundation for exploring their potential in optoelectronic applications.

2. Calculation methods

In this study, all first-principles calculations in this work were performed within the framework of density functional theory (DFT)16 as implemented in the Vienna Ab initio Simulation Package (VASP).17 The exchange–correlation interactions were treated using the Perdew–Burke–Ernzerhof (PBE) functional within the generalized gradient approximation (GGA).18 The interactions between the valence electrons and the ionic cores were described using the projector augmented-wave (PAW) method.19 A plane-wave energy cutoff of 500 eV was employed throughout all calculations. The atomic structures of both the pristine HfO2 monolayer and its Janus derivative, 2D HfSO, monolayer were fully relaxed until the residual forces on each atom were less than 0.01 eV Å−1. The electronic self-consistency criterion was set to 10−7 eV for the total energy convergence. For Brillouin zone integration, a 20 × 20 × 1 Monkhorst–Pack k-point mesh was used for the unit cell calculations. A vacuum spacing of at least 15.0 Å was introduced along the out-of-plane direction to eliminate spurious interactions between periodic images. To obtain a more accurate estimation of the electronic bandgap, DFT (HSE06)20 was employed in addition to the standard GGA-PBE calculations, incorporating 25% of the exact Hartree–Fock exchange. The dynamic stability of both materials was assessed by computing the phonon dispersion relations within the harmonic approximation, using density functional perturbation theory (DFPT) as implemented in the PHONOPY package.21 The 5 × 5 × 1 supercells were constructed to include interactions up to at least the fifth nearest neighbors, ensuring the convergence and reliability of the phonon spectra. Furthermore, due to the relatively large atomic number of hafnium (Hf), spin–orbit coupling (SOC)22 effects were included in selected calculations to capture the relativistic contributions to the electronic structure. In addition, the Hubbard U (U = 4) correction was applied to the Hf-d orbitals to describe localized electron correlation effects, as the Hf element is a partially filled transition metal element.23 These additional corrections are expected to enhance the accuracy of predictions for electronic and optical properties.

Following the DFT calculations, quasiparticle corrections were evaluated within the G0W0 framework,24 employing the single-particle Green's function and the dynamically screened Coulomb interaction. A response-function cutoff energy of 200 eV was adopted, and the Brillouin zone was sampled using a 12 × 12 × 1 Γ-centered k-point mesh. These parameters were subsequently employed to calculate optical responses with enhanced accuracy.

For the optical response, we show the interaction between the incident electromagnetic field and the charge carriers. Upon photon excitation, the charge carriers undergo a dynamic response to the external perturbation, generating an induced current density that determines the resulting dielectric response. In the long-wavelength limit ε(q → 0,ω), the screening behavior of the charge carriers is characterized by the transverse dielectric function. Within the quantum-mechanical framework, optical transition probabilities are evaluated using Fermi's golden rule:2.In which the strength of excitation peaks and the available transition channels are related to the square of the velocity matrix element, |e·〈vk|v|ck〉|2 and the joint density of states, δ(ω − Eck − Evk), respectively. Regarding the optical response beyond the independent particle approach, the excitonic effects were included. The connection of the exciton energies ΩS and the corresponding electron–hole amplitude |e·〈0|v|S〉|2 of the correlated electron–hole excitations, S, is obtained by solving the Bethe–Salpeter equation (BSE):252.

For the excited-state calculations, the k-point mesh, plane-wave cutoff energy, and number of bands were kept consistent with those employed in the calculations. For the Janus HfSO monolayer, convergence tests of the plane-wave cutoff energy and k-point mesh were performed to ensure the reliability of the GW quasiparticle band gap, as shown in Fig. S1 of the SI.

3. Results and discussions

As plotted in Fig. 1a and b, we show the top and side views of the optimized structures of the 2D HfO2 monolayer. The structure consists of a single layer of Hf atoms sandwiched between two layers of oxygen atoms (O), forming a trigonal prismatic coordination. Following full structural relaxation, the optimized lattice constant is determined to be 3.24 Å, while the Hf–O bond length is calculated to be 2.11 Å. The obtained structural parameters agree well with previously reported results, further validating the reliability of the present computational approach. The phonon band structure of a 2D HfO2 monolayer is shown in Fig. 1c, which exhibits a total of nine distinct phonon branches, consistent with the presence of three atoms in the unit cell. Based on the vibrational characteristics in the long-wavelength limit, these branches can be categorized into three acoustic and six optical modes. The acoustic modes comprise the in-plane longitudinal acoustic (LA) and transverse acoustic (TA) branches, which exhibit linear dispersion near the Γ point, as predicted by the Debye model. In contrast, the out-of-plane acoustic (ZA) branch exhibits a quadratic dispersion, a key feature of 2D materials.26

Fig. 1. (a) Top and (b) side views of the fully optimized atomic structure of the HfO2 monolayer. (c) Phonon band structure and (d) corresponding phonon density of states (PhDOS), both verifying the dynamic stability of the system. Hafnium (Hf) and oxygen (O) atoms are represented by brown and red spheres, respectively.

Fig. 1

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An important characteristic of the phonon spectrum is the separation between the acoustic and optical branches, leading in a prominent phonon band gap of approximately 1.5 THz. This acoustic–optical phonon gap arises primarily from the mass disparity between Hf and O atoms in the unit cell. The gap suppresses phonon–phonon scattering processes between the acoustic modes and most optical modes, except for the zone-center out-of-plane optical (ZO) mode. Such phonon band separation is expected to play a critical role in thermal transport, as it reduces anharmonic scattering and may enhance lattice thermal conductivity. Moreover, the observed phonon dispersion does not exhibit any imaginary frequencies across the entire Brillouin zone, confirming the dynamic stability of the 2D HfO2 monolayer. This finding further highlights the potential of this material for practical applications. The corresponding phonon density of states (PhDOS) provides additional insight into the vibrational character of the system. As depicted in Fig. 1d, the low-frequency acoustic branches are dominated by the heavier Hf atoms, whereas the high-frequency optical modes arise predominantly from the vibrations of the lighter O atoms. This element-resolved phonon behavior aligns with expectations based on mass and bonding characteristics.

After fully optimizing the atomic positions and lattice parameters of the pristine HfO2 monolayer (Fig. 2a and b), a Janus-type monolayer, HfSO, was constructed by substituting one of the two oxygen atoms with a sulfur atom (S) on one side of the layer, thereby breaking the out-of-plane mirror symmetry. As a result, the chemical environments on the two surfaces become distinct, leading to different bond lengths for Hf–O and Hf–S interactions. The optimized Hf–O bond length is about 2.14 Å, while the Hf–S bond length increases to 2.52 Å, reflecting the larger atomic radius and lower electronegativity of sulfur (S) compared to oxygen (O).

Fig. 2. (a) Top and (b) side views of the fully relaxed atomic structure of the HfSO monolayer. (c) Phonon band structure and (d) corresponding phonon density of states (PhDOS), both confirm the dynamic stability of the system. Hafnium (Hf), sulfur (S), and oxygen (O) atoms are depicted as brown, cyan, and red spheres, respectively.

Fig. 2

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On the other hand, the phonon band structures and the corresponding PhDOS of the 2D HfSO monolayer are presented in Fig. 2c and d, respectively. The absence of imaginary frequencies throughout the Brillouin zone confirms the dynamic stability of the material, which contains three atoms per unit cell. Notably, the Janus HfSO monolayer exhibits three acoustic modes, including LA, TA, and ZA, and six optical phonon modes, like the HfO2 monolayer. However, unlike the 2D HfO2 monolayer, the acoustic and optical branches in janus HfSO monolayer exhibit overlap. The low-frequency acoustic modes are primarily governed by the vibrations of the heavier Hf atoms, whereas the high-frequency optical modes originate mainly from the lighter S atoms, as shown in Fig. 2d.

The electronic band structures of the 2D HfO2 monolayer were computed using various levels of exchange–correlation approximations, including standard DFT (PBE), DFT with spin–orbit coupling (DFT+SOC), DFT with on-site Coulomb interactions (DFT+U), and DFT (HSE06) levels. While the PBE functional provides a qualitatively correct description of the band dispersion and the nature of the band gap, it is well known to underestimate the magnitude of the energy band gap. To address this limitation, the DFT (HSE06) hybrid functional was employed. Specifically, to obtain a more accurate bandgap value consistent with experimental data, the GW level was included. As illustrated in Fig. 3a and d, the 2D HfO2 and its Janus HfSO monolayers consistently exhibit an indirect bandgap across all computational approaches employed. Specifically, the valence band maximum sub-bands are located along the Γ–K path, whereas the conduction band minimum is situated at the M point. The calculated band gap of the HfO2 monolayer is 4.84 eV at the DFT(PBE) level and 4.79 eV upon inclusion of SOC. The band gap increases to 5.70 eV with DFT (PBE)+U, 6.48 eV with DFT (HSE06), and 8.64 eV at the (GW) level, reflecting the progressively improved treatment of electron–electron interactions beyond DFT (PBE). Additionally, the energy band gaps of the Janus HfSO monolayer are evaluated in detail, as presented in Table 1. These results are consistent with previous theoretical work.14 The large band gaps indicate the insulating nature of 2D HfO2 and HfSO monolayers and suggest their potential for optoelectronic applications.

Fig. 3. (a)–(d) Electronic band structures of the 2D HfO2 monolayers with various calculations: DFT (PBE), DFT (PBE)+SOC, DFT (PBE)+U, DFT (HSE06) standards.

Fig. 3

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Table 1. Calculated electronic band gaps of the pristine HfO2 and Janus HfSO monolayers using various computational approaches, including standard DFT (PBE), DFT (PBE) with spin–orbit coupling (DFT+SOC), DFT with Hubbard U correction (DFT+U), DFT (HSE06), and the many-body GW approximations.

Materials Energy band gap (eV)
DFT (PBE) DFT (PBE)+SOC DFT (PBE)+U DFT (HSE06) GW
HfO2 4.84 4.79 5.70 6.48 8.64
HfSO 1.95 0.26 1.91 2.85 4.07
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As plotted in Fig. 4a, b, 5a and b, the ionic character of the Hf–O bonds in the HfO2 monolayer is confirmed by Bader charge analysis, which reveals substantial electron transfer from Hf to O atoms. Specifically, each Hf atom donates approximately 2.62 electrons, resulting in a net charge gain of about 1.31e per O atom. In comparison, the 2D HfSO monolayer also exhibits strong ionic bonding characteristics. To further elucidate the asymmetric electronic environments, the three-dimensional (3D) electron localization function (ELF) distribution was calculated, as shown in Fig. 5c.

Fig. 4. (a) Charge density distributions and (b) charge density differences of the HfO2 monolayer, showing Hf–O chemical bonds, (c) the weighted band structures, and (d) projected density of states (PDOS) illustrating the orbital contributions to the electronic structure of the 2D HfO2 monolayer.

Fig. 4

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Fig. 5. (a) Charge density distributions and (b) charge density differences of the HfSO monolayer, covering Hf–S and Hf–O chemical bonds, (c) three-dimensional electron localization function (ELF) distribution of the Janus HfSO monolayer, plotted at an isovalue of 0.6, (d) the weighted band structures, and (e) projected density of states (PDOS) illustrating the orbital contributions to the electronic structure of the HfSO monolayer.

Fig. 5

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The ELF isosurface at 0.6 reveals distinct electron-localization characteristics around the inequivalent Hf–O and Hf–S bonds. Here, both S and O atoms serve as electron acceptors, with Bader analysis indicating a total charge transfer of approximately 3.91 electrons from each Hf atom, distributed as 1.85e to S atoms and 2.05e to O atoms. This substantial charge redistribution highlights the pronounced ionic nature of both Hf–S and Hf–O chemical bonds in the 2D HfSO monolayer.

To gain a deeper understanding of the orbital contributions to the electronic states, we analyze the charge density distributions, charge density difference, and projected density of states (PDOS) for 2D HfO2 and its Janus counterpart, the 2D HfSO monolayer, as shown in Fig. 4c, d, 5d and e. The projected band structure reveals that, in both systems, the unoccupied conduction states are primarily composed of Hf 5d orbitals, indicating that the conduction process is primarily controlled by the Hf sublattice. However, the nature of the hole valence states shows differences between the two materials. In a pure 2D HfO2 monolayer, the hole valence states are mainly derived from O 2pz orbitals (red subbands and red curves), suggesting that hole transport is dominated by the O atoms. This contribution remains dominant across the entire valence band states, especially near the Fermi level.

In contrast, for the 2D Janus HfSO monolayer, the replacement of one O layer by S significantly modifies the electronic structure near the Fermi level. The hole valence states in the 2D HfSO monolayer are governed by S 3py (cyan subbands and black curves) orbitals, highlighting a shift in the hole localization from O to S atoms. This change in orbital character near the Fermi level reflects the broken mirror symmetry of the Janus configuration. It can influence charge transport, optical transitions, and excitonic behavior in both materials.

In this work, we investigate the optical properties of the 2D HfO2 monolayer and its Janus counterpart, the 2D HfSO monolayer, both with and without the inclusion of excitonic effects. To accurately capture many-body interactions, we employ the single-shot G0W0 approximation in conjunction with BSE, which enables us to account for both electron–electron and electron–hole correlations. The presence of excitonic effects can induce a redshift of the optical gap, alter the absorption profile, and enhance the oscillator strength of optical transitions compared to the exciton-free scenario.

The real and imaginary parts of the frequency-dependent dielectric functions are presented in Fig. 6a and b for both HfO2 and HfSO monolayers, respectively. For the 2D HfO2 monolayer (Fig. 6a), the imaginary part of the dielectric function reveals a weakly bound excitonic peak, indicative of a relatively low exciton binding energy. In the absence of excitonic effects, represented by the blue curves calculated at the G0W0 + RPA level, the onset of optical absorption occurs at approximately 9.00 eV for HfO2 and 4.25 eV for the 2D HfSO monolayer. These values are notably higher than the GW band gaps obtained from G0W0 calculations, which are 8.64 eV and 4.07 eV for both 2D HfO2 and HfSO monolayers, respectively. This discrepancy can be attributed to momentum conservation constraints that suppress direct transitions near the band edges.

Fig. 6. Real and imaginary parts of the dielectric function for (a) HfO2 and (b) HfSO monolayers. The red and blue lines correspond to calculations with and without electron–hole interactions, respectively, highlighting the influence of excitonic effects on the optical response.

Fig. 6

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Several pronounced peaks are observed in the imaginary part of the dielectric functions, and the overall spectrum remains qualitatively similar across different computational levels. Nonetheless, a noticeable blue shift in the absorption edge is observed when comparing spectra obtained at the G0W0 + RPA level with those including excitonic effects, emphasizing the significance of excitonic effects.

When excitonic effects are included (red curves), two strong excitonic resonances, denoted as E1 and E2, are identified. For the 2D HfO2 monolayer, these peaks appear at 6.50 eV and 7.00 eV, while in the 2D HfSO monolayer, they are located at 3.50 eV and 4.00 eV, respectively (Fig. 6a and b). These excitonic features originate from direct interband transitions at high-symmetry points in the Brillouin zone. The emergence of such bound excitonic states below the quasiparticle band gaps highlights the crucial role of many-body interactions in governing the optical response of 2D materials.

Bright excitons,27,28 which are optically active, contribute directly to light emission and absorption processes, making them vital for applications in light-emitting diodes (LEDs),29 photodetectors, and solar cells. Dark excitons,27,28 on the other hand, are optically inactive due to spin or momentum mismatch but play an important role in exciton dynamics and non-radiative recombination, and can influence the overall efficiency of optical devices. Here, after applying the BSE approach (red curves), which incorporates excitonic effects, we observe strongly bound bright (optically active) excitonic states in the 2D HfO2 monolayer, as evidenced by the discrete exciton energy levels shown in Fig. 7b. In contrast, both bright and dark (optically inactive) exciton states are identified in the 2D HfSO monolayer (Fig. 7d).

Fig. 7. Optical absorption spectra and excitonic states of (a and b) HfO2 and (c and d) HfSO monolayers. The figures illustrate the excitonic contributions to the absorption profiles, where distinct peaks indicate strongly bound exciton states in both materials.

Fig. 7

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The exciton binding energies, defined as the difference between the quasiparticle band gap and the energy of the lowest-energy exciton, are calculated to be 1.14 eV for the HfO2 monolayer and 0.57 eV for the HfSO monolayer. Compared with the reported values for MoS2 (0.54 eV)30 and WS2 (0.79 eV),31 the HfO2 monolayer exhibits a substantially larger exciton binding energy, whereas that of HfSO is comparable to that of MoS2 and lower than that of WS2. These results highlight the distinct excitonic properties of the two Hf-based monolayers, which may arise from differences in dielectric screening and electronic band structures.

4. Conclusions

In summary, we explored the structural, electronic, and optical properties of both HfO2 and HfSO monolayers, using first-principles calculations. Both monolayers exhibit indirect-gap semiconductor behavior, based on theoretical frameworks, including DFT (PBE), DFT (PBE)+SOC, DFT (PBE)+U, DFT (HSE06), and GW standards. Crucially, excitonic effects were also investigated, revealing significant exciton binding energies. The 2D HfO2 monolayer exhibits strongly bound bright excitons, whereas the Janus HfSO monolayer supports both bright and dark excitonic states due to its reduced structural symmetry and modified electronic structure. Overall, these findings demonstrate the insulating nature of 2D monolayers, highlight their potential for electronic and optoelectronic applications, and suggest opportunities for developing Janus structures through suitable material engineering strategies.

Conflicts of interest

The authors declare no competing interests.

Supplementary Material

RA-OLF-D6RA07688B-s001

Data availability

The data that support the findings of this study are available upon reasonable request from the corresponding author.

Supplementary information (SI): additional computational details, convergence tests, and supporting results for the electronic and optical properties of the HfO2 and Janus HfSO monolayers, including GW convergence with respect to the response-function cutoff energy and k-point sampling. See DOI: https://doi.org/10.1039/d6ra07688b.

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Associated Data

This section collects any data citations, data availability statements, or supplementary materials included in this article.

Supplementary Materials

RA-OLF-D6RA07688B-s001
RA-OLF-D6RA07688B-s001.pdf (340.5KB, pdf)

Data Availability Statement

The data that support the findings of this study are available upon reasonable request from the corresponding author.

Supplementary information (SI): additional computational details, convergence tests, and supporting results for the electronic and optical properties of the HfO2 and Janus HfSO monolayers, including GW convergence with respect to the response-function cutoff energy and k-point sampling. See DOI: https://doi.org/10.1039/d6ra07688b.

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