# Mexican hat-like valence band dispersion and quantum confinement in rhombohedral ferroelectric α-In2Se3

Geoffroy Kremer, Aymen Mahmoudi, Meryem Bouaziz, Mehrdad Rahimi, François Bertran, Jean-François Dayen, Maria Luisa Della Rocca, Marco Pala, Ahmed Naitabdi, Julien Chaste, et al.

► To cite this version:

Geoffroy Kremer, Aymen Mahmoudi, Meryem Bouaziz, Mehrdad Rahimi, François Bertran, et al. Mexican hat-like valence band dispersion and quantum confinement in rhombohedral ferroelectric α-In2Se3. Physical Review B, 2025, 112, pp.155418. 10.1103/n2l9-7vjl. hal-05318490

HAL Id: hal-05318490

https://hal.science/hal-05318490v1

Submitted on 16 Oct 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
Mexican hat–like valence band dispersion and quantum confinement
in rhombohedral ferroelectric α-In₂Se₃

Geoffroy Kremer,¹ Aymen Mahmoudi,² Meryem Bouaziz,² Mehrdad Rahimi,³ François Bertran,⁴
Jean-Francois Dayen,⁵,⁶ Maria Luisa Della Rocca,³ Marco Pala,⁷ Ahmed Naitabdi,⁸ Julien Chaste,²
Fabrice Oehler,² and Abdelkarim Ouerghi²

¹Université de Lorraine, CNRS, IJL, F-54000 Nancy, France

²Université Paris-Saclay, CNRS, Centre de Nanosciences et de Nanotechnologies, 91120 Palaiseau, France

³Laboratoire Matériaux et Phénomènes Quantiques, Université Paris Cité, CNRS UMR 7162, F-75013 Paris, France

⁴Synchrotron SOLEIL, L'Orme des Merisiers, Départementale 128, 91190 Saint-Aubin, France

⁵Université de Strasbourg, IPCMS-CNRS UMR 7504, 23 Rue du Loess, 67034 Strasbourg, France

⁶Institut Universitaire de France, 1 rue Descartes, 75231 Paris cedex 05, France

⁷DPIA, University of Udine, 33100 Udine, Italy

⁸Laboratoire de Chimie Physique–Matière et Rayonnement, UMR 7614, Sorbonne Université, 4 Place Jussieu, 75005 Paris, France

<figure><img src="image_2.png" /></figure>

(Received 30 April 2025; revised 15 July 2025; accepted 18 August 2025; published 14 October 2025)

Two-dimensional ferroelectric materials offer a large variety of electronic properties depending on chemical composition, number of layers, and stacking order. Among them, α-In₂Se₃ has attracted much attention due to the promise of outstanding electronic properties, attractive quantum physics, in- and out-of-plane ferroelectricity, and high photoresponse. Precise experimental determination of the electronic structure of rhombohedral (3R) α-In₂Se₃ is needed for a better understanding of potential properties and device applications. Here, combining angle-resolved photoemission spectroscopy and density-functional theory calculations, we demonstrate that 3R α-In₂Se₃ phase exhibits a robust inversion of the valence band parabolicity at the Γ point forming a bow-shaped dispersion with a depth of 140 ± 10 meV between the valence band maximum along the ΓK direction of the Brillouin zone. Moreover, we find an indirect band gap of about 1.25 eV, as well as a highly electron doping of approximately 5×10¹² electrons per cm² at the surface. This leads to surface band bending and the formation of a prominent electron accumulation layer. These findings allow a deeper understanding of the rhombohedral α-In₂Se₃ electronic properties underlying the potential of III–VI semiconductors for electronic and photonic technologies.

DOI: 10.1103/n219-7vjl

# I. INTRODUCTION

Two-dimensional (2D) van der Waals (vdW) III–VI semiconductors have drawn intense attention due to their unique electronic properties [1–5]. Among these materials, In₂Se₃ holds different phases and several polytypes, with potential applications in various domains, going from electronics, photonics to thermoelectricity, due to its good electron mobility, excellent photoresponse, exotic ferroelectricity [6], and unique electronic band structure [7]. In₂Se₃ has been extensively investigated in different forms, such as thin films, nanocrystals, and nanostructures [6]. In particular, the investigations of III–VI semiconductors has propelled the study of post-transition-metal 2D semiconductors [8–10]. As In₂Se₃ is integrated into electronic devices, its electronic properties may be further modified by via chemical device [11]. Moreover, it has been also integrated in van der Waals heterostructures with other 2D compounds, allowing to further tune electronic, magnetic, and optical properties of the 2D assembly or to perform band-structure engineering [6]. Therefore, some key aspects of 2D material technological integration directly relate to fundamental surface and interface physics, contacts, and energy band profile control. As a consequence, before transferring from science to technology [12], it is crucial to understand the fundamental properties of

III–VI semiconductors, and to investigate the change of their electronic properties, particularly of the band structure (up to room temperature) [13], with the crystalline phase [3].

In few layer graphene, 2H stacking is generally more stable than 3R stacking [14,15] and also in many vdW layered transition-metal dichalcogenide (TMD) materials [16–18]. The ferroelectric behavior of 3R-stacked MX₂ results from interlayer charge transfer between layer structures lacking inverse symmetry. However, the impact of intralayer- (ionic displacement) and interlayer (charge transfer) effects on ferroelectricity remains largely unexplored in spontaneous ferroelectric van der Waals materials [19,20]. Remarkably, α-In₂Se₃ in its hexagonal structure (2H) hosts a fascinating combination of a robust *n*-type doping and a peculiar band structure, leading to the formation of a quantum confined electron gas (2DEG) directly at its surface [7,21]. The 2H α-In₂Se₃ 2DEG maintains considerably high electron density even at room temperature, comparable to conventional AlGaN/GaN systems, enabling the observations of anomalous optical response and quantum Hall effect [22]. The 2DEG in 2H α-In₂Se₃ represents an alternative platform for achieving high-performance devices not subjected to the constraints of stable stoichiometry and alloy composition [19] required for AlGaN/GaN.
Besides the 2H-hexagonal stacking, the α phase of In₂Se₃ possesses a 3R stacking form, where the basic block of Se-In-Se-In-Se atoms sequence stacks vertically in a rhombohedral structure. It has been recently demonstrated that the ferroelectric properties of α-In₂Se₃ are highly dependent on its stacking form. A distinct ferroelectric polarization switching mechanism in metal/α-In₂Se₃/metal heterostructures has been measured depending on the nature of the embedded polytype [19]. The 2H and 3R structures show a stacking-dependent ferroelectric domain-wall dynamic, implying larger hysteresis windows for the 3R case, and different intralayer dynamics related to the ferro-paraelectric phase transition induced by high electric fields [23]. Thus, exploring and controlling the stacking-dependent physical properties in two-dimensional materials is of fundamental interest, opening routes for interlayer interaction engineering. While fundamental understanding of 2H α-In₂Se₃ electronic structure has recently advanced, the 3R In₂Se₃ polytypes lack such an equivalent knowledge and still require detailed investigation. Experimental band-structure mapping using angle-resolved photoelectron spectroscopy (ARPES), associated with density-functional theory (DFT) simulations, have the potential to answer this question.

In this work, we report the electronic properties of 3R-α-In₂Se₃. Complementary micro-Raman spectroscopy analysis and scanning transmission electron microscopy (STEM) was conducted to study the structural properties of the crystal exploring the vibration frequencies of phonons corresponding to the characteristic vibrational modes of the sample. ARPES was used to investigate the 3R α-In₂Se₃ band structure, electronic band gap, and the position of the valence band maximum relative to the Fermi level. Such result is further supported by the excellent agreement with the density-functional theory calculated band structure with the experimental data. Our results show an indirect band gap of ~1.25 eV, with the bottom of the conduction band localized at the center of the Brillouin zone, just below the Fermi level. Furthermore, we demonstrate the presence of Mexican hat-like valence band dispersion in 3R α-In₂Se₃ at the Γ point using ARPES measurements and DFT calculations. Such a peculiar electronic structure results in density of state (DOS) singularities, opening exciting opportunities for engineering electronic devices and for exploring material solutions with enhanced thermoelectric properties. We show that thin flakes exfoliated from the same 3R α-In₂Se₃ crystals display ferroelectric (FE) properties when integrated into field-effect transistor (FET)-like devices. By performing thermoelectric experiments, we confirm an *n*-type doping of the flakes with a large Seebeck coefficient of the order of 200 µV/K in absolute value. The extracted electron density of approximately 5×10¹² electrons per cm² matches correctly the value determined from ARPES and DFT calculation. The ensemble of these observations confirms the great potential of α-In₂Se₃ for future applications where stacking engineering can play a major role.

## II. STRUCTURAL AND CHEMICAL PROPERTIES

III–VI materials crystallize in a common layered van der Waals configuration, with well-defined covalently bond

monolayers in the plane, weakly coupled to each other along the out-of-plane axis by vdW forces [Fig. 1(a)]. For α-In₂Se₃ each monolayer consists of two different indium atoms with different (covalent) coordination polyhedra: In¹ is coordinated tetrahedrally by four Se atoms while In² is octahedrally coordinated by six Se [1]. The inner structure of an individual monolayer thus decomposes into five sublayers, in a Se-In¹-Se-In²-Se repetition along the vertical direction. Figure 1(a) shows a schematic of three vertical stackings of such a quin-tuple sublayer (QL) unit. In addition, α-In₂Se₃ possesses several polytypes depending on the relative position and in-plane orientation of each QL. In Fig. 1(a), the three QLs show a small in-plane translation, implying an ABC stacking order characteristic of the rhombohedral configuration of the 3R α-In₂Se₃ polytype. Figure 1(b) shows a schematic of the bulk and surface-projected hexagonal Brillouin zone of α-phase In₂Se₃. The crystal unit cell being hexagonal, the important high-symmetry points are Γ, M, and K in the three-dimensional bulk Brillouin zone (3D BZ), and their respective planar projection (2D BZ). In Fig. 1(c), we show the electronic structure of bulk 3R α-In₂Se₃, as calculated from the Heyd-Scuseria-Ernzerhof (HSE) hybrid functionals [24]. The minimum of the conduction band appears in the Γ point, while the valence band maximum (VBM) is located near the Γ point, thus resulting in a negative effective mass for the highest valence band at the Γ point. Most interestingly, the energy difference between the VBM and the highest valence band state near the Γ is higher, with the Mexican hat-like landscape extending over a significant fraction of the Brillouin zone. This unusual character of the band structure gives rise to a high DOS and an almost one-dimensional (1D)-like van Hove singularity near the VBM [25]. A large DOS near the Fermi level (*E*<sub>F</sub>), in general, would lead to instabilities and transitions to different phases such as magnetism, superconductivity, and other phenomena [23]. Also, sharp features in the DOS near *E*<sub>F</sub> are often indicative of a material with a large thermoelectric Seebeck coefficient [26].

The specific crystalline phase of In₂Se₃ is generally difficult to identify. There are several different phases (α, β, γ, δ, and κ), some of which exhibit similar properties. For example, the γ- and δ phases have hexagonal and trigonal crystal structures, respectively, whereas the α- and β phases have highly similar rhombohedral structures. In order to probe the chemical and structural properties of 3R α-In₂Se₃ single crystal (HQ Graphene), micro-Raman and high-resolution x-ray photoemission spectroscopy investigations were performed on the same sample. The 3R α-In₂Se₃ micro-Raman spectrum measured at room temperature is reported in Fig. 1(d). We can distinguish from this spectrum the typical vibrational modes reported previously for bulk α-In₂Se₃ [6,13]. As shown in Fig. 1(d), multiple micro-Raman peaks are observed near 88, 104, 159, 179, 193, and 250 cm⁻¹ in 3R α-In₂Se₃ crystals, which can be assigned to the *E* and *A* modes of α-In₂Se₃, respectively [1,2,27]. We can see that the *E* mode of 3R α-In₂Se₃ is stronger than that of *E* mode. The intensity ratio between *E* and *A* modes can therefore be used to identify the 3R α-In₂Se₃ polytypes. High-resolution x-ray photoemission spectroscopy (XPS) measurements were carried out on the 3R α-In₂Se₃ crystals at the CASSIOPEE beamline of the synchrotron SOLEIL (France) [28–30].
<figure><img src="image_3.png" /><figcaption>FIG. 1. (a) Side view of the surface crystal structure of 3R α-In₂Se₃. The black line shows the conventional unit cell. (b) Corresponding three-dimensional (3D) Brillouin zone (BZ) and its 2D projection in the (001) plane. (c) Calculated band structure. (d) Micro-Raman spectroscopy spectra for 3R α-In₂Se₃ crystals using 532- and 633-nm laser. (e), (f) XPS of 3R α-In₂Se₃ at $h\nu = 110$ eV: In 4d and Se 3d.</figcaption></figure>

High-resolution XPS spectra for In (In 4d) and Se (Se 3d) are recorded at 110 eV [Figs. 1(f) and 1(e), respectively]. The In spectrum presents two doublets, which can be attributed to the In 4d$_{5/2}$ and In 4d$_{3/2}$ with a spin-orbit (SO) splitting of 0.8 eV. The high-intensity doublet corresponds to the In atoms embedded in the 3R α-In₂Se₃, while the second In 4d at higher binding energy is the signature of a defective/substoichiometric 3R α-In₂Se₃ with Se vacancy or In₂O₃ [31]. Similarly, two components are also present for the Se 3d peak, corresponding to the Se 3d$_{3/2}$ and 3d$_{1/2}$ at 54.4 and 55.2 eV, respectively. The obtained binding energy values agree with previously reported values [6,32]. The crystalline phase of the sample was demonstrated by energy-dispersive x-ray spectroscopy (EDX) and x-ray photoelectron spectroscopy (XPS), which mapped the elemental distribution of Se and In in a selected area of the In₂Se₃. The EDX images (see Fig. 7) and XPS spectra demonstrated a continuous film in which the atomic range of Se and In could be identified. The intensity of the In and Se maps across the sample also indicates that the Se defect stoichiometry is lower than 1%.

In order to confirm the nature of our polytype, a part of our commercial 3R α-In₂Se₃ bulk crystal was analyzed by transmission electron microscopy (Titan THEMIS, Thermofisher). Figure 2(a) shows a corresponding low-resolution scanning-TEM (STEM) image. The stacking structure along the vertical axis is already apparent. For 3R α-In₂Se₃ (symmetry *R*3*m*) [1], all atoms in the unit cell can be grouped in triplets (In₁,

In₂, Se₁, Se₂, Se₃), which are all located in Wyckoff position 3a. Each triplet comprises equivalent positions (0, 0, z) (2/3, 1/3, 1/3+z) (1/3, 2/3, 2/3+z) with arbitrary z. Consequently, there are many systematic extinctions in the structure factor and *HKL* reflections are only allowed when *H*-*K*-*L*=3*n*, with *n* an arbitrary integer. We retrieve this particular rule in the experimental diffraction pattern presented in Fig. 2(b). We

<figure><img src="image_4.png" /><figcaption>FIG. 2. Transmission electron microscopy (TEM) characterization of the In₂Se₃: (a) Large bright-field scanning transmission electron microscopy (STEM) image of 3R α-In₂Se₃, (b) Selective area diffraction patterns of In₂Se₃, and (c) high-resolution STEM image of 3R α-In₂Se₃.</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>FIG. 3. Constant ARPES energy surfaces for 3R α-In₂Se₃ (*hv* = 90 eV) with 2D projections taken at *E*-*E*<sub>F</sub> equals (a) 0 eV, (b) -1.7 eV, and (c) -6 eV. In panel (a), red line corresponds to the 2D hexagonal BZ obtained using the in-plane cell parameter of α-In₂Se₃ (a = 4.0 Å). Corresponding ARPES spectra along (d) **M** - **Γ** - **M** - **Γ** and (f) **Γ** - **K** - **M** high-symmetry directions corresponding to green and pink dashed lines in panel (a), respectively. (e), (g) Corresponding DFT calculations with HSE functional for two distinct values of *k*<sub>*z*</sub> in the 3D BZ, namely *k*<sub>*z*</sub> = 0 (dark red) and *k*<sub>*z*</sub> = π/c (light red). A rigid energy shift of +1 eV has been applied to fit the ARPES dispersion. (h) Normal emission energy distribution curve extracted from ARPES [red dashed line in panel (d) showing the amplitude of the band gap. (i) Corresponding value extracted from density of state (DOS), as extracted from DFT calculations at *k*<sub>*z*</sub> = 0 and at the center of the 2D BZ [black dashed lines in panel (e)].</figcaption></figure>

also determine an experimental ratio c/a = 0.1408 for the
crystal. The matching c/a ratio and the observed systematic
extinction pattern are a strong indication for the 3R α-In₂Se₃
nature of our sample [1]. To confirm the stacking order of the
sample, we show in Fig. 2(c) an atomically resolved STEM
image of the In₂Se₃ crystal. We distinguish easily the van der
Waals gap between each consecutive lamella along the vertical
direction. Inside each lamella, we observe a clear asymmetry
between the bottom In1 atom coordinated tetrahedrally by
four Se atoms (blue ellipse) and the top In2 atom coordinated
octahedrally by six Se atoms (red ellipse). We clearly recog-
nize the α-In₂Se₃ monolayer structure [see also Fig. 1(a)]. As
there is no rotation between consecutive layers, but a simple
in-plane translation with a period of three layers along the
vertical direction [1], we confirm the polytype agrees with the
3R α-In₂Se₃ configuration.

### III. ELECTRONIC BAND STRUCTURE: ARPES VS DFT

We now focus on the electronic properties of 3R α-In₂Se₃
as measured by ARPES. First, we show in Figs. 3(a)–3(c) a
series of isoenergetic contours in the (*k*<sub>x</sub>, *k*<sub>y</sub>) plane obtained
between binding energy (BE) = 0 eV (*E*<sub>F</sub>) and BE = 6 eV at a
photon energy of *hv* = 90 eV. The use of this photon energy is
sufficient to reach the Brillouin zone (BZ) edges. In particular,
we observe in Fig. 3(a) the evidence of the periodicity of the
electronic band structure at this energy between the **Γ** point of
the first Brillouin zone, and the **Γ** points of the second BZ. At
this energy, a single intensity point is observed at the **Γ** points
and no photoemission intensity at other places in the BZ. By
superimposing the projected 2D hexagonal BZ (red line) on
the isoenergetic contour, we show an excellent agreement us-
ing the in-plane lattice parameter of the crystal (*a* = 4.026 Å)
[1]. We have consequently located the other high-symmetry points of the BZ, namely $\bar{\Gamma}$, $\bar{K}$, and $\bar{M}$. Other isoenergetics contours are plotted in Figs. 3(b) and 3(c) corresponding to BE equals to 1.7 and 6 eV, respectively. We experimentally resolve a lack of photoemission intensity between approximately 1.7 eV BE [Fig. 3(b)] and $E_F$ [Fig. 3(a)], which we ascribe to the electronic band gap of the material, reflecting its semiconducting character. At 6 eV BE [Fig. 3(c)], we can resolve some spectral weight at the $\bar{K}$ point, no photoemission intensity at the $\bar{M}$ point, and a circular structure around the $\bar{\Gamma}$ point. These behaviors are very similar to what we observed for the 2H stacking [1], and we did not resolve at this point different spectroscopic signatures between the 3R and 2H $\alpha$-In$_2$Se$_3$.

To go into deeper detail, we plot in Figs. 3(d) and 3(f) the ARPES dispersion along the $\bar{M}-\bar{\Gamma}-\bar{M}-\bar{\Gamma}$ and $\bar{\Gamma}-\bar{K}-\bar{M}$ high-symmetry directions of the BZ, respectively. These directions correspond to the green and pink dashed lines in Fig. 3(a). We directly visualize the semiconducting character of 3R $\alpha$-In$_2$Se$_3$ with the bottom of the conduction band and the top of the valence band localized close to the center of the BZ. The Fermi level in 3R $\alpha$-In$_2$Se$_3$, is ~150 ± 15 meV above the conduction band minimum, resulting in intrinsic n-type charge-carrier doping of the “as-grown” material. By comparing our ARPES results to DFT calculations, we find an excellent agreement of the electronic dispersion along both high-symmetry directions except for the complex variation of photoemission spectral weight, which cannot be captured by this type of calculation. The DFT-calculated dispersions are shown in Figs. 3(e) and 3(g) for two values of $k_z$ in the 3D BZ, namely $k_z = 0$ (dark red) and $k_z = \pi/c$ (light red), along the $\bar{M}-\bar{\Gamma}-\bar{M}-\bar{\Gamma}$ [Fig. 3(e)] and $\bar{\Gamma}-\bar{K}-\bar{M}$ directions [Fig. 3(g)]. From this comparison between ARPES and DFT, it is difficult to disambiguate between these two $k_z$ values in the 3D BZ, i.e., what is the vertical momentum probed using this particular photon energy ($h\nu = 90$ eV)? Nevertheless, by looking at the dispersion of the CB, we are likely probing the band structure at the $\bar{\Gamma}$ point (see Fig. 6 in the Appendix for larger energy scale DFT calculations). A striking point of this comparison is the dispersion of the band at -5.5 eV at $\bar{\Gamma}$ as measured by ARPES. Experimentally, we observe a broadening that we can ascribe to the continuous superposition of the bands (with different $k_z$) between the two high-symmetry points in the 3D BZ, i.e., at $k_z = 0$ and $k_z = \pi/c$, as illustrated by the superposition of the DFT electronic dispersions at these two points. This effect is known as $k_z$ broadening and is typical of quasi-2D systems [32]. It is particularly pronounced for electronic states with out-of-plane character such as bands with $p_z$ derived state, as we have discussed in our recent publication on 2H $\alpha$-In$_2$Se$_3$ [4]. Another noticeable agreement between ARPES and DFT is the amplitude of the band gap, which we determine experimentally equal to 1.25 ± 0.1 eV with ARPES in Fig. 3(h) by plotting a normal emission energy distribution curve, and theoretically equal to 1.3 eV using HSE hybrid functional as extracted from the DOS in Fig. 3(i). Typically, HSE-type functionals are known to give quantitative estimates of the band gap of semiconducting materials, which is also in good agreement with the GW calculations [33] or photoluminescence measurements [2].

#### IV. INVERTED VALENCE BAND PARABOLICITY AND 2D ELECTRON GAS FORMATION

Figure 3(f) shows a large-scale ARPES scan along the $\bar{\Gamma}-\bar{K}-\bar{M}$ direction for which we focus on two pertinent regions in energy space, namely the green rectangle in the vicinity of the VB maximum and the pink rectangle close to the Fermi energy, both close to the center of the BZ. In Fig. 4 we present further ARPES data at $h\nu = 108$ eV for which the out-of-plane momentum is again close to 0, i.e., to the $\Gamma$ point in the 3D BZ. The ARPES spectra close to the VB maximum reveal a parabolicity inversion around $\Gamma$, resulting in a Mexican hat-like energy surface around the $\Gamma$ point. The change of the sign of the Hole Valence Band (HVB) parabolicity can be explained by two simultaneous effects: the strong interactions with the many underlying valence bands, which tend to decrease the HVB effective mass, and the negligible interaction with the upper conduction band, which would otherwise increase it [34]. Figure 4(b) shows the corresponding energy distribution curves (EDCs) taken at different values of $k_x$. The experimental data, allowing to determine the position of the VBM and the depth of the local minimum arising from the band parabolicity inversion, are $\Delta k = 0.22 \pm 0.1$ Å and $\Delta E = 140 \pm 10$ meV, respectively. These values agree within the uncertainty of our DFT predictions.

A more detailed inspection of the CB dispersion we measured in panel (a), and previously observed at $h\nu = 90$ eV in Fig. 3, is presented in Fig. 4(c). It shows that the CB consists of a single dispersive parabolic band. We noticed that the sample revealed lower doping inhomogeneity on the surface. DFT calculations show that the conduction band structure differs in the KGK and HAH directions. However, ARPES measurements reveal the same structure in both directions. Based on these observations, we can attribute this band to the surface states [7]. This feature is typical of surface charge accumulation layers, which is a key signature of quantum confinement, and it is interpreted as the fingerprint of the occurrence of a 2D electron gas (2DEG). The formation of the 2DEG is due to a surface band bending and the formation of a prominent electron accumulation layer. The origin of the band bending is the presence of donor surface states that can donate their electrons to the conduction band of the In$_2$Se$_3$, therefore inducing a significant electron accumulation at the surface and a consequent downward band bending. Based on the XPS measurement the doping can be attributed to the intrinsic Se vacancies. As in the 2H stacking case, the experimentally observed position of $E_F$ inside the CB can be explained by a strong electron doping of the crystal [4]. We can quantify the dispersive nature of these specific bands using a parabolic dispersion. We obtain an effective mass ($m^*$) of 0.1 $m_0$ with energy minimum located at 110 ± 15 meV BE. The discrepancy in energy position observed in the parabolic dispersion using ARPES at 90 and 108 eV is attributed to inhomogeneous doping on the surface. This 2DEG exhibits one confined state which corresponds to an electron density of $5 \times 10^{12}$ electrons per cm$^2$ using the 2D Luttinger volume $n_{2D} = k_F^2/2\pi$ formula, also confirmed by thermoelectric measurements (see next discussion). As shown in Fig. 4(d), we also performed a 1D self-consistent Poisson-Schrödinger simulation of the potential well and of the accumulation charge
<figure><img src="image_6.png" /><figcaption>FIG. 4. ARPES spectra for 3R α-In₂Se₃ (ħν = 108 eV) along (a) the Ī – K – M̄ high-symmetry direction. It corresponds to the zoom-in close to the VB maximum, as represented by the green rectangle in Fig. 3(f). It shows an inversion of the band parabolicity. (b) Corresponding energy distribution curves (EDCs) taken at different values of kₓ and showing a 140-meV energy difference between the VB maximum and its position at the center of the BZ. (c) ARPES spectrum obtained in the vicinity of the Fermi energy as represented by the pink rectangle in Fig. 3(f) (slightly different spatial region on the sample), showing one parabolic band associated with a two-dimensional electron gas (2DEG). Blue dashed line corresponds to a fit with a nearly free electron dispersion. (d) Conduction band profile along the confinement direction (black line) and first wave function (blue line) extracted using the Quantum Well (QW) model and experimental parameters obtained from (c) (*E*<sub>c</sub>: Conduction band, *E*<sub>1</sub>: quantum-well state, Z(nm): the thickness of the sample).</figcaption></figure>

at the surface of 3R α-In₂Se₃. To fit the experiments, we assumed an effective mass of 0.1 *m*₀, a relative dielectric permittivity of 17 *ε*₀, a temperature of 300 K, and a donor sheet density of 4×10¹³ cm⁻² distributed along the first 3 nm. We found only one confined state below the Fermi level at the energy of *E*₁ = -110 meV and thus contributing to an electron density of about 5×10¹² cm⁻², which remains compatible with the ARPES data. For clarity, the shape of the absolute value of this confined wave function is also plotted.

V. TRANSPORT PROPERTIES

We have fabricated FET-like devices based on multilayer
3R α-In₂Se₃ flakes exfoliated from the same single crystals
used for the ARPES investigation. The 3R α-In₂Se₃ flake
(~100-nm thick) is coupled to a Au local gate through a
hBN flake (~30–40-nm thick), used as a dielectric spacer.
Source and drain electrodes are fabricated on the hBN flake
by standard microfabrication and the 3R α-In₂Se₃ flake is
dry transferred over it just after mechanical exfoliation. As

detailed in Ref. [35], this fabrication approach allows probing
the 3R α-In₂Se₃ flake surface in contact with the electrodes
that remain fresh and protected from oxidization under air ex-
posure. Figure 5(a) shows an atomic force microscopy (AFM)
image of a representative 3R α-In₂Se₃-based FET; an optical
image of the same device is also shown in Fig. 5(d) allowing
to better visualize its detailed structure. The thicknesses of
the 3R α-In₂Se₃ and hBN flakes are typically evaluated by
averaging 50 lines profiles taken on zoomed parts of the AFM
images [Figs. 5(b) and 5(c)].

The electric and thermoelectric characterizations of the
3R α-In₂Se₃-based FET are performed under high vacuum
(~10⁻⁷ mbar) at 35 °C. Figure 5(f) shows the two-point I-V
measurements obtained by voltage biasing the device in
Figs. 5(a) and 5(d) at different temperatures, from 35 to
200 °C. The source-drain voltage is swept first from negative
to positive values (solid dots) and then from positive to neg-
ative values (open dots). A nonlinear and nonsymmetric I-V
trend is visible, which can be attributed to the different injec-
tion barriers at the source and drain contacts. Moreover, the
<figure><img src="image_7.png" /><figcaption>FIG. 5. (a) AFM image of a representative 3R α-In₂Se₃ based FET. Thicknesses estimation of the 3R α-In₂Se₃ (b) and hBN (c) layers: line cuts of the flakes indicated by the dotted line in (a); the extracted thicknesses are $d_{In_2Se_3}$ = 114.0 ± 0.6 nm and $d_{hBN}$ = 35 ± 1 nm. (d) Optical image of device in (a). (e) IV characteristics of the 3R α-In₂Se₃-based device in (a) at different temperatures. Measurements are acquired by sweeping the voltage bias from negative to positive values (filled dots) and from positive to negative ones (open dots). (f) Seebeck coefficient measured as a function of the gate voltage of another analogous device; row data are in red and smoothed data in black.</figcaption></figure>

I-V measurement shows an evident hysteretic behavior, with a current jump around ±1.5 V and a small switching ON/OFF ratio. The I-V hysteresis in the two-point measurement is characteristic of memristive behavior and can be explained by the asymmetric modulation of the Schottky barriers by residual in-plane ferroelectric (FE) polarization. Following the work of Xue *et al.*, residual polarization at the contact interface causes either negative or positive charge accumulation, thereby inducing a change in the effective Schottky barrier. The difference in the measured current at the same VDS can be attributed to different residual ferroelectric polarization depending on the poling history [36]. Figure 5(f) show that the observed hysteresis is robust with the temperature, at least up to 200 °C (curves are vertically shifted for the sake of clarity). Note that at temperature higher than 200 °C, a phase transition from the α to the β phase could occur, so we generally avoid temperature higher than that during the electrical characterization [21].

The thermoelectric analysis is performed soon after the electrical characterization of the devices by a dc approach. A direct current is applied to a microheater to create a temperature gradient in the longitudinal direction of the 3R α-In₂Se₃ flakes. Such a gradient is measured by monitoring the four-point resistance of the source and drain electrodes that, shaped in a nanowire geometry, act simultaneously also as local thermometers. The thermoelectric voltage between these two electrodes is then measured using a Keithley 2182A nano-voltmeter and the resulting Seebeck coefficient extracted. An

example of Seebeck coefficient measured as a function of the gate voltage of another device is shown in Fig. 5(f); analogous results have been obtained on a total of six devices. The 3R α-In₂Se₃ flake shows systematically a noisy Seebeck signal (red data) with an average absolute value in the 100-200-µV/K range (black data), depending on the measured device [35]. These values are in good agreement with theoretical calculations for the case of monolayer α-In₂Se₃ [37]. This noisy character is possibly related to a robust in-plane residual polarization in the flake induced by the applied source-drain voltage [26]. The Seebeck coefficient is negative and shows no dependence on the gate voltage in the whole explored range. Its negative sign indicates that electrons are the majority charge carriers with a charge density extracted by the Mott formula equal to $10^{12}-10^{13}$ elec/cm². All these conclusions agree well with the ARPES measurements, revealing the bottom of the conduction band located just below the Fermi level.

In this work, we have conducted an extensive study exploring the band structure and transport properties of 3R α-In₂Se₃, demonstrating its remarkable electronic properties. The electronic band structure was studied using both ARPES and density-functional theory, highlighting a Mexican hat-band configuration, relevant for applications in multiple domains. We performed a detailed analysis of the top valence band, which exhibits an inversion of the effective mass sign near the zone center with a local minimum having depth of 140 ± 10 meV. Also, we have shown the occurrence of a highly
<figure><img src="image_8.png" /><figcaption>FIG. 6. (a) Large-scale DFT calculations along the $\bar{M}-\bar{\Gamma}-\bar{M}-\bar{\Gamma}$ high-symmetry direction in the HSE approximation for two distinct values of $k_z$ in the 3D BZ, namely $k_z = 0$ (dark red) and $k_z = \pi/c$ (light red). (b) Corresponding DOS extracted at the center of the 2D BZ [black dashed lines in panel (a)] showing a 1.3-eV band gap between the maximum of the VB and the minimum of the CB at $k_z = 0$.</figcaption></figure>

metallic states at the surface (2DEG) of vacuum-cleaved 3R $\alpha$-In$_2$Se$_3$ single crystals. This study provides experimental access to robust 2DEGs based on ferroelectric materials, which are scarce in literature. It is of pivotal interest to continue this study by correlating the 2DEG properties and the In$_2$Se$_3$ particular phase.

# VI. MATERIALS AND METHODS

## A. Experiments

The micro-Raman measurements were conducted at room temperature, using a commercial confocal Horiba micro-Raman microscope with a 100× objective and a 532-/633-nm laser excitation. XPS/ARPES experiments were performed at the CASSIOPEE beamline of the SOLEIL synchrotron light source at T = 80 K. The CASSIOPEE beamline was equipped with a Scienta R4000 hemispherical electron analyzer and the incident photon beam was focused into a <100>-µm spot on the sample surface with linear horizontal polarization. High-quality samples from the HQ Graphene Co. were cleaved at room temperature with Scotch tape at a base pressure better than $5 \times 10^{-9}$ mbar. The experiment was performed at room temperature with an energy resolution better than 15 meV.

## B. Computational details

The theoretical calculations were carried out by using first-principles calculations based on density-functional theory (DFT) by means of the Quantum espresso suite [38]. We used a fully relativistic pseudopotential and noncollinear simulations to consider the SO interaction and adopted norm-conserving pseudopotentials, and we used the hybrid Heyd-Scuseria-Ernzerhof (HSE) functional to describe the [24] exchange-correlation term. van der Waals interactions were considered in the calculations with the semiempirical Grimme-D3 correction. The self-consistent solution was obtained by adopting an 8×8×8 Monkhorst-Pack grid and a cutoff energy of 80 Ry. Cell parameters and atomic positions

were relaxed according to a convergence threshold for forces and energy of $10^{-4}$ and $10^{-8}$ atomic units, respectively.

# ACKNOWLEDGMENTS

We acknowledge the financial support by Optitaste (Grant No. ANR21-CE24-0002), DEEP2D (Grant No. ANR-22-CE09-0013), MixDferro (Grant No. ANR-21-CE09-0029), and FastNano (Grant No. ANR-22-PEXD-0006), as well as the French technological network RENATECH.

# DATA AVAILABILITY

The data are not publicly available. The data are available from the authors upon reasonable request.

# APPENDIX

Figure 6 displays large-scale band-structure calculations obtained in the HSE approximation [Fig. 6(a)] for $k_z = 0$ (dark red) and $k_z = \pi/c$ (light red), as well as corresponding DOS obtained at the center of the BZ [Fig. 6(b)]. It shows the negligible dispersion of the VB and on the contrary the strong dispersion of the CB as a function of $k_z$. The 1.3-eV band gap is reported between the maximum of the VB and the minimum of the CB at $k_z = 0$ and at the center of the BZ. The stacking order was analysed using STEM cross-section measurements (Fig. 7). The quality of the sample was also demonstrated by the energy-dispersive

<figure><img src="image_9.png" /><figcaption>FIG. 7. EDX elemental maps showing the spatial distribution of In and Se, respectively.</figcaption></figure>
x-ray (EDX) spectroscopy study, which mapped the elemental distribution of indium (In) and selenium (Se) (Fig. 7). The

EDX images show a continuous film of the sample, in which the atomic range of In and Se could be identified.

[1] L. Liu, J. Dong, J. Huang, A. Nie, K. Zhai, J. Xiang, B. Wang, F. Wen, C. Mu, Z. Zhao, Y. Gong, Y. Tian, and Z. Liu, Atomically resolving polymorphs and crystal structures of In₂Se₃, *Chem. Mater.* **31**, 10143 (2019).

[2] N. Balakrishnan, E. D. Steer, E. F. Smith, Z. R. Kudrynskyi, Z. D. Kovalyuk, L. Eaves, A. Patanè, and P. H. Beton, Epitaxial growth of γ-InSe and α, β, and γ-In₂Se₃ on ε-GaSe, 2D Mater. **5**, 035026 (2018).

[3] J. Felton, J. Harknett, J. Page, Z. Yang, N. Alghofaili, J. N. O'Shea, L. Eaves, Y. Kohama, M. T. Greenaway, and A. Patanè, Probing and manipulating the Mexican hat-shaped valence band of In₂Se₃, *Nat. Commun.* **16**, 922 (2025).

[4] M. Y. Ma, D. Wang, Y. T. Huang, D. Han, N. K. Chen, H. B. Sun, S. Zhang, and X. Bin Li, Vacancy defects in 2D ferroelectric In₂Se₃ and the conductivity modulation by polarization-defect coupling, *Nano Lett.* **25**, 3726 (2025).

[5] G. Rodrigues-Fontenele, G. Fontenele, M. R. Valentim, L. V. C. Freitas, G. Rodrigues-Junior, R. Magalhães-Paniago, and A. Malachias, Structural and electronic response of multigap n-doped In₂Se₃: A prototypical material for broad spectral optical devices, ACS Appl. Mater. Interfaces **16**, 49902 (2024).

[6] F. Xue, W. Hu, K. C. Lee, L. S. Lu, J. Zhang, H. L. Tang, A. Han, W. T. Hsu, S. Tu, W. H. Chang, C. H. Lien, J. H. He, Z. Zhang, L. J. Li, and X. Zhang, Room-temperature ferroelectricity in hexagonally layered α-In₂Se₃ nanoflakes down to the monolayer limit, *Adv. Funct. Mater.* **28**, 1 (2018).

[7] G. Kremer, A. Mahmoudi, A. M'Foukh, M. Bouaziz, M. Rahimi, M. L. Della Rocca, P. Le Fèvre, J. F. Dayen, F. Bertran, S. Matzen, M. Pala, J. Chaste, F. Oehler, and A. Ouerghi, Quantum confinement and electronic structure at the surface of van der Waals ferroelectric α-In₂Se₃, ACS Nano **17**, 18924 (2023).

[8] U. N. Noumbé, C. Gréboval, C. Livache, T. Brule, B. Doudin, A. Ouerghi, E. Lhuillier, and J. F. Dayen, Ionic glass-gated 2d material-based phototransistor: MoSe₂ over LaF₃ as case study, *Adv. Funct. Mater.* **29**, 1902723 (2019).

[9] C. Gréboval, U. Noumbe, N. Goubet, C. Livache, J. Ramade, J. Qu, A. Chu, B. Martinez, Y. Prado, S. Ithurria, A. Ouerghi, H. Aubin, J.-F. Dayen, and E. Lhuillier, Field-effect transistor and photo-transistor of narrow-band-gap nanocrystal arrays using ionic glasses, *Nano Lett.* **19**, 3981 (2019).

[10] M. Tangi, P. Mishra, C.-C. Tseng, T. K. Ng, M. N. Hedhili, D. H. Anjum, M. S. Alias, N. Wei, L.-J. Li, and B. S. Ooi, Band alignment at gan/single-layer WSe₂ Interface, ACS Appl. Mater. Interfaces **9**, 9110 (2017).

[11] W. Shi, S. Yu, P. Liu, W. Fan, H. Luo, and S. Song, Near-infrared photoluminescent flowerlike α-In₂Se₃ nanostructures from a solvothermal treatment, *Chem. Eng. J.* **225**, 474 (2013).

[12] D. A. Bandurin, A. V. Tyurnina, G. L. Yu, A. Mishchenko, V. Zólyomi, S. V. Morozov, R. K. Kumar, R. V. Gorbachev, Z. R. Kudrynskyi, S. Pezzini, Z. D. Kovalyuk, U. Zeitler, K. S. Novoselov, A. Patanè, L. Eaves, I. V. Grigorieva, V. I. Fal'ko, A. K. Geim, and Y. Cao, High electron mobility, quantum Hall

effect and anomalous optical response in atomically thin InSe, *Nat. Nanotechnol.* **12**, 223 (2016).

[13] F. Zhang, Z. Wang, L. Liu, A. Nie, Y. Gong, W. Zhu, and C. Tao, Atomic-scale visualization of polar domain boundaries in ferroelectric In₂Se₃ at the monolayer limit, *J. Phys. Chem. Lett.* **12**, 11902 (2021).

[14] H. Henck, J. Avila, Z. Ben Aziza, D. Pierucci, J. Baima, B. Pamuk, J. Chaste, D. Utt, M. Bartos, K. Nogajewski, B. A. Piot, M. Orlita, M. Potemski, M. Calandra, M. C. Asensio, F. Mauri, C. Faugeras, and A. Ouerghi, Flat electronic bands in long sequences of rhombohedral-stacked graphene, *Phys. Rev. B* **97**, 245421 (2018).

[15] D. Pierucci, H. Sediri, M. Hajlaoui, J.-C. Girard, T. Brumme, M. Calandra, E. Velez-Fort, G. Patriarche, M. G. Silly, G. Ferro, V. Soulière, M. Marangolo, F. Sirotti, F. Mauri, and A. Ouerghi, Evidence for flat bands near the Fermi level in epitaxial rhombohedral multilayer graphene, ACS Nano **9**, 5432 (2015).

[16] X. Wang, K. Yasuda, Y. Zhang, S. Liu, K. Watanabe, T. Taniguchi, J. Hone, L. Fu, and P. Jarillo-Herrero, Interfacial ferroelectricity in rhombohedral-stacked bilayer transition metal dichalcogenides, *Nat. Nanotechnol.* **17**, 367 (2022).

[17] I. Paradisanos, S. Shree, A. George, N. Leisgang, C. Robert, K. Watanabe, T. Taniguchi, R. J. Warburton, A. Turchanin, X. Marie, I. C. Gerber, and B. Urbaszek, Controlling interlayer excitons in MoS₂ layers grown by chemical vapor deposition, *Nat. Commun.* **11**, 2391 (2020).

[18] A. Mahmoudi, M. Bouaziz, A. Chiout, G. Di Berardino, N. Ullberg, G. Kremer, P. Dudin, J. Avila, M. Silly, V. Derycke, D. Romanin, M. Pala, I. C. Gerber, J. Chaste, F. Oehler, and A. Ouerghi, Electronic properties of rhombohedrally stacked bilayer WSe₂ obtained by chemical vapor deposition, *Phys. Rev. B* **108**, 045417 (2023).

[19] Y. Wu, T. Zhang, D. Guo, B. Li, K. Pei, W. You, Y. Du, W. Xing, Y. Lai, W. Ji, Y. Zhao, and R. Che, Stacking selected polarization switching and phase transition in vdW ferroelectric α-In₂Se₃ junction devices, *Nat. Commun.* **15**, 10481 (2024).

[20] D. Huo, Y. Bai, X. Lin, J. Deng, Z. Pan, C. Zhu, C. Liu, H. Yu, and C. Zhang, Tuning of the valley structures in monolayer In₂Se₃/WSe₂ heterostructures via ferroelectricity, *Nano Lett.* **22**, 7261 (2022).

[21] S. M. K. Walker, F. Y. Bruno, Z. Wang, A. De La Torre, S. Riccó, A. Tamai, T. K. Kim, M. Hoesch, M. Shi, M. S. Bahramy, P. D. C. King, and F. Baumberger, Carrier-density control of the SrTiO₃ (001) surface 2D electron gas studied by ARPES, *Adv. Mater.* **27**, 3894 (2015).

[22] G. J. Syaranamual, W. A. Sasangka, R. I. Made, S. Arulkumaran, G. I. Ng, S. C. Foo, C. L. Gan, and C. V. Thompson, Role of two-dimensional electron gas (2DEG) in AlGaN/GaN High Electron Mobility Transistor (HEMT) on-state degradation, *Microelectron. Reliab.* **64**, 589 (2016).

[23] Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi, E. Kaxiras, and P. Jarillo-Herrero, Unconventional superconduc-
tivity in magic-angle graphene superlattices, *Nature* (London) **556**, 43 (2018).

[24] J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, *Phys. Rev. Lett.* **77**, 3865 (1996).

[25] T. Cao, Z. Li, and S. G. Louie, Tunable magnetism and half-metallicity in hole-doped monolayer GaSe, *Phys. Rev. Lett.* **114**, 236602 (2015).

[26] K. Fukutani, T. Sato, P. V. Galiy, K. Sugawara, and T. Takahashi, Tunable two-dimensional electron gas at the surface of thermoelectric material In₄Se₃, *Phys. Rev. B* **93**, 205156 (2016).

[27] A. Pant, Z. Mutlu, D. Wickramaratne, H. Cai, R. K. Lake, C. Ozkan, and S. Tongay, Fundamentals of lateral and vertical heterojunctions of atomically thin materials, *Nanoscale* **8**, 3870 (2016).

[28] G. Kremer, A. Mahmoudi, M. Bouaziz, C. Brochard-Richard, L. Khalil, D. Pierucci, F. Bertran, P. Le Fèvre, M. G. Silly, J. Chaste, F. Oehler, M. Pala, F. Bisti, and A. Ouerghi, Direct observation of highly anisotropic electronic and optical nature in indium telluride, *Phys. Rev. Mater.* **7**, 074601 (2023).

[29] A. Mahmoudi, M. Bouaziz, N. Chapuis, G. Kremer, J. Chaste, D. Romanin, M. Pala, F. Bertran, P. Le Fèvre, I. C. Gerber, G. Patriarche, F. Oehler, X. Wallart, and A. Ouerghi, Quasi van Der Waals epitaxy of rhombohedral-stacked bilayer WSe₂ on GaP(111) heterostructure, *ACS Nano* **17**, 21307 (2023).

[30] L. Khalil, P. M. Forcella, G. Kremer, F. Bisti, J. Chaste, J. C. Girard, F. Oehler, M. Pala, J. F. Dayen, D. Logoteta, M. Goerbig, F. Bertran, P. Le Fèvre, E. Lhuillier, J. Rault, D. Pierucci, G. Profeta, and A. Ouerghi, α-As₂Te₃ as a platform for the exploration of the electronic band structure of single layer β-tellurene, *Phys. Rev. B* **106**, 125152 (2022).

[31] N. Balakrishnan, Z. R. Kudrynskyi, E. F. Smith, M. W. Fay, O. Makarovsky, Z. D. Kovalyuk, L. Eaves, P. H. Beton, and A. Patanè, Engineering *p*-n junctions and bandgap tuning of

InSe nanolayers by controlled oxidation, *2D Mater.* **4**, 025043 (2017).

[32] H. Henck, D. Pierucci, J. Zribi, F. Bisti, E. Papalazarou, J.-C. Girard, J. Chaste, F. Bertran, P. Le Fèvre, F. Sirotti, L. Perfetti, C. Giorgetti, A. Shukla, J. E. Rault, and A. Ouerghi, Evidence of direct electronic band gap in two-dimensional van der Waals indium selenide crystals, *Phys. Rev. Mater.* **3**, 034004 (2019).

[33] P. Scherpelz, M. Govoni, I. Hamada, and G. Galli, Implementation and validation of fully relativistic GW calculations: Spin-orbit coupling in molecules, nanocrystals, and solids, *J. Chem. Theory Comput.* **12**, 3523 (2016).

[34] D. V. Rybkovskiy, A. V. Osadchy, and E. D. Obraztsova, Transition from parabolic to ring-shaped valence band maximum in few-layer GaS, GaSe, and InSe, *Phys. Rev. B* **90**, 235302 (2014).

[35] M. Rahimi, K. Sobnath, L. Chen, F. Mallet, P. Lafarge, C. Barraud, A. Alekhin, C. Brochard-Richard, M. Bouaziz, J. Chaste, A. Ouerghi, and M. L. Della Rocca, Probing the electric and thermoelectric response of ferroelectric 2H and 3R α-In₂Se₃, *Appl. Phys. Lett.* **124**, 251101 (2024).

[36] F. Xue, X. He, J. R. D. Retamal, A. Han, J. Zhang, Z. Liu, J. K. Huang, W. Hu, V. Tung, J. H. He, L. J. Li, and X. Zhang, Gate-tunable and multidirection-switchable memristive phenomena in a Van Der Waals ferroelectric, *Adv. Mater.* **31**, 1901300 (2019).

[37] T. Nian, Z. Wang, and B. Dong, Thermoelectric properties of α-In₂Se₃ monolayer, *Appl. Phys. Lett.* **118**, 126222 (2021).

[38] P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni, I. Dabo, A. Dal Corso, S. De Gironcoli, S. Fabris, G. Fratesi, R. Gebauer, U. Gerstmann, C. Gougoussis, A. Kokalj, M. Lazzeri *et al.*, QUANTUM ESPRESSO: A modular and open-source software project for quantum simulations of materials, *J. Phys.: Condens. Matter* **21**, 395502 (2009).