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Article

Prediction of Intriguing Valley Properties in Two-Dimensional Hf2TeIX (X = I, Br) Monolayers

by
Kaiyuan He
* and
Peiji Wang
*
School of Physics and Technology, Institute of Spintronics, University of Jinan, Jinan 250022, China
*
Authors to whom correspondence should be addressed.
Crystals 2024, 14(9), 794; https://doi.org/10.3390/cryst14090794
Submission received: 29 July 2024 / Revised: 2 September 2024 / Accepted: 5 September 2024 / Published: 9 September 2024
(This article belongs to the Section Crystal Engineering)
Browse Figures
Figure 1
<p>(<b>a</b>) Top and side views of geometric structure of Hf<sub>2</sub>TeIX (X = I, Br) monolayer. The green, blue, yellow, and gold balls denote the I, Hf, Te, and I (Br) atoms, respectively. (<b>b</b>) Excluding the Hf<sub>2</sub>TeI<sub>2</sub> electronic band structure of the SOC. (<b>c</b>) Consider the electronic band structure of the SOC. (<b>d</b>) Orbital projection component analysis of electronic structure under SOC.</p> "> Figure 2
<p>Band structures of Hf<sub>2</sub>TeIBr (<b>a</b>) without SOC and (<b>b</b>) with SOC; blue is spin down, and red is spin up.</p> "> Figure 3
<p>(<b>a</b>) Berry curvature of Hf<sub>2</sub>TeIBr as a contour map over the 2D Brillion zone. (<b>b</b>) Schematic diagram of the three-dimensional Berry curvature of the Brillouin area. (<b>c</b>) Schematic diagram of the spin-valley Hall effect.</p> "> Figure 4
<p>Energy band diagrams of Hf<sub>2</sub>TeIBr lattice under biaxial strain at different stresses (−6–6%); blue is spin down, and red is spin up.</p> "> Figure 5
<p>(<b>a</b>) Hf<sub>2</sub>TeIBr total energy of the monolayer as a function of the biaxial strain. (<b>b</b>) The change in the band gap under the biaxial strain, and the properties of the band gap. (<b>c</b>) Change in the split value (<math display="inline"><semantics> <mrow> <msub> <mrow> <mo>∆</mo> <mi mathvariant="normal">E</mi> </mrow> <mrow> <mi mathvariant="normal">C</mi> </mrow> </msub> </mrow> </semantics></math>) (<math display="inline"><semantics> <mrow> <msub> <mrow> <mo>∆</mo> <mi mathvariant="normal">E</mi> </mrow> <mrow> <mi mathvariant="normal">V</mi> </mrow> </msub> </mrow> </semantics></math>) under biaxial strain.</p> "> Figure 6
<p>(<b>a</b>–<b>c</b>) The Berry curvature of a single-layer Hf<sub>2</sub>TeIBr under biaxial strains of ε = −6%, 0%, and 6%, respectively.</p> ">
Versions Notes

Abstract

:
The valley degree of freedom, as a new information carrier, is important for basic physical research and the development of advanced devices. Herein, using first-principle calculations, we predict that two-dimensional Hf2TeIX (X = I, Br) monolayers harbor intriguing valley properties. Without considering spin–orbit coupling (SOC), the Hf2TeI2 monolayer has a semi-metallic nature, with Dirac cones located at the high-symmetry point K, and feature, with considerable Fermi velocity. When the SOC is taken into account, a band gap opening of 271 meV can be observed at the Dirac cones. More interestingly, the Hf2TeIBr monolayer exhibits intrinsic spatial inversion symmetry breaking, which leads to the emergence of valley-contrasting physics under SOC. This is demonstrated by the presence of spin–valley splitting and opposite Berry curvature at adjacent K points. Besides, the spin–valley splitting, the band gap and magnitude of the Berry curvature of the Hf2TeIBr monolayer can be effectively tuned by strain engineering. These findings contribute significantly to the design of valleytronic devices and extend opportunities for exploring two-dimensional valley materials.

1. Introduction

After the successful exfoliation of graphene [1], two-dimensional materials have attracted much attention due to their rich physical properties. At first, researchers focused on the development and application of spin degrees of freedom [2]. Later, researchers found that there was an energy valley at the K point of some Dirac materials, such as graphene, with a new degree of freedom; the term “valley” denotes the points of local energy extrema found within the valence or conduction bands [3,4,5,6]. The valley degree of freedom is robust against low-energy phonons and smooth deformations [7,8,9]. The valley can also serve as a carrier of information storage and processing. Compared with traditional electronics, valleytronics has the advantages of rapid computing speed, high integration, minimal information distortion, and low energy consumption. Therefore, valley electronics have become the focus of research in the field of condensed matter physics in recent years.
In the field of valleytronics, two-dimensional (2D) transition metal disulfide (TMDC) with inversion symmetry breaking, which harbors valley characteristics, provide excellent platforms for the research on valleytronics and are regarded as excellent candidate materials for valleytronics. Due to the strong spin–orbit coupling (SOC) effect, the properties of large spin splitting, spin–valley coupling, orbital magnetic moment, and physical effect related to the Berry curvature [10] in TMDC, have aroused strong interest in researchers [11,12,13,14]. Therefore, most of the research objects on valleytronics are focused on these compounds. So far, molybdenum- and tungsten-based TMDCs have been widely used in valleytronics and spintronics by using valley degrees of freedom and spin degrees of freedom [15,16,17,18,19,20,21,22,23,24,25].
Despite extensive research on valley-contrasting physics, these properties are mainly limited to TMDCs and graphene. Therefore, the exploration of valley materials beyond TMDC and graphene is necessary for the further development of valleytronics. However, for materials with spatial inversion symmetry, the Berry curvature is zero. Therefore, for valleytronics, the destruction of inversion symmetry is a prerequisite. In fact, in addition to TMDC, there are many materials with good properties, such as graphene, silicene, and other Dirac materials, with weak SOC effects. In contrast, Dirac materials have two developmental advantages: one is that there is an energy valley in the structure; the other is that the coherence length is relatively large, the thermal conductivity and electron mobility are high, and the carrier transport channel loss of trajectory is very low. However, these materials also have some disadvantages in application. Like silicene with energy valleys at K and K’, it has limitations that cannot be comparable with transition metal chalcogenides in energy valley electronics. It is mainly because the band gap of this kind of material is close to zero under SOC, which makes the electronic devices based on this structure unable to be effectively adjusted on and off; Secondly, under the condition that the structure has the symmetry of spatial inversion and time inversion at the same time, the minimal SOC effect, such as that in silicene, is not conducive to the coupling of energy valley and spin, which also leads to the failure to realize the locking of energy valley and spin. Therefore, if the strong SOC effect can be introduced into this Dirac material and the ideal large band gap can be opened, it is very ideal for the development of energy valley electronics. Based on the above, we urgently need to develop an excellent alternative material for TMDC, and considering the advantages of the Dirac material, such as ultra-high electron mobility, simple structure, and easy preparation, we can overcome the disadvantages, introduce a strong SOC effect by introducing heavy metal transition elements, and produce excellent valley electronic properties by strain means.
Recently, the Ln2TeI2 (Ln = La, Gd) crystal was successfully prepared in an experiment [26,27]. The corresponding monolayer was expected to be exfoliated. In view of this, we predicted the 2D Hf2TeIX (X = I, Br) monolayers through first-principle calculations. Based on these considerations, we propose a novel two-dimensional hexagonal Hf2TeIX (X = I, Br) film in this study. Drawing inspiration from valley electronics and based on the energy valley characteristics of K and K’, we propose replacing the I atoms on one side of the structure with the Br atoms of the same main family to break the spatial inversion symmetry of the structure. This results in significant valley splitting in the Hf2TeIBr monolayer, enabling energy valley–spin coupling in the structure. Additionally, the opposite Barry curvature is protected by the structure’s C3 symmetrical biaxial strain, allowing for flexible adjustment and greatly improving the lateral velocity of the carrier and the efficiency of the valley Hall device. By utilizing the optical selection rules of spin and valley coupling, we can manipulate the valley degree of freedom in Hf2TeIBr. These results are essential for realizing the design of valley electronics-related devices, providing a novel semiconductor material, and broadening the channel for exploring two-dimensional valley materials.

2. Computational Details

First-principle calculations have been performed using the Vienna ab initio simulation package (VASP), which is based on density functional theory (DFT) [28,29]. To examine the exchange–correlation interaction between electrons, we employ the Perdew–Burke–Ernzerhof (PBE)-type generalized gradient approximation (GGA) [30]. A cutoff energy of 400 eV is applied for the plane basis. The Brillouin zone is sampled using a Γ-centered k mesh with a size of 15 × 15 × 1. The crystal structure is optimized until the force applied to each atom is less than 0.001 eV/Å. The energy convergence criterion is set to 10−7 eV. To prevent interactions between periodically repeated layers, we include a vacuum layer with a thickness of 25 Å. Furthermore, the correlation effects for the Hf-5d electron is handled using the DFT+U method, U = 3 eV [31,32,33]. The Berry curvature of the occupied bands is calculated by the maximally localized Wannier function method implemented in the WANNIER90 package [34,35].

3. Results and Discussion

The structural and electronic properties of the Hf2TeI2 monolayer are first investigated. Based on lattice constant scanning, we find that the optimal lattice constant of Hf2TeI2 is a = b = 4.83 Å. The Hf2TeI2 monolayer exhibits a hexagonal lattice, and its unit cell contains one Te, two Hf, and two I atoms, belonging to the P3m1 space group. Hf2TeI2 monolayers are arranged in the order of I-Hf-Te-Hf-I, where the I-Hf and Hf-Te bond lengths are found to be 3.18 and 2.88 Å, respectively. The thickness of the Hf2TeI2 monolayer is 4.69 Å, as shown in Figure 1a.
To assess the feasibility of the experimental fabrication of Hf2TeI2, we calculate its cohesive energy using the following formula:
E c o h = ( E H f 2 T e I 2 − 2 E H f − E T e − 2 E I ) / 5 .
Here, E H f 2 T e I 2 represents the total energy of the Hf2TeI2 unit cell, while E H f , E T e , and E I , are the energies of Hf, Te, and I atoms, respectively. The calculated E c o h is −6.42 eV/atom, which is slightly smaller than that of graphene (−7.85 eV/atom), indicating that the Hf2TeI2 monolayer is expected to be fabricated in the experiment.
Next, we focus on the electronic properties of the Hf2TeI2 monolayer. Figure 1b presents its band structure without considering SOC. Notably, a semi-metallic nature can be observed, with Dirac cones located at the high-symmetry K point on the Fermi level. The Fermi velocities on both sides at K and K’ are defined as follows:
ℏ v F ≈ d E ( k ) / d k .
Our calculations find that the Fermi velocity of Hf2TeI2 is similar to that of the graphene ν f = 2.9 × 10 5 m/s, further highlighting the potential application of this structure in electronic devices. When the SOC effect is considered, a band gap opening of 271 meV is generated at the Dirac cone. Thus, the Hf2TeI2 transforms into a semiconductor. In this case, both the extremes of the valence band and the extremes of the conduction band are located at the K point, and thus the system shows an energy valley behavior. The orbital component analysis reveals that dxy and dx2−y2 primarily contribute to the energy bands near the Fermi level. Since the system has spatial inversion symmetry, the two energy valleys are equivalent and indistinguishable. To exploit the valley property of the Hf2TeI2 monolayer, breaking its spatial inversion symmetry is essential, and we will discuss this later on.
Next, we investigate the effect of biaxial strain on the electronic properties of the Hf2TeI2 monolayer as shown in Figure S2. The magnitude of the biaxial strain was defined as follows:
ε = a − a 0 / a 0
where a 0 and a represent the in-plane lattice constants of the Hf2TeI2 monolayer with and without strain, respectively. Under biaxial strain, the monolayer maintains its direct band gap semiconductor properties over the −4–6% strain range and the semiconductor- metal phase transition in the ground state at compression to −6%. Because the structure has a spatial inversion symmetry, both the ground state or the external strain, even strong SOC action cannot destroy the band degenerate at K and K’. Although strain can control the band gap of single layer, it does not destroy the spatial inversion symmetry of the system, so it is difficult to apply the valley behavior of the system. In order to realize the valley properties of a material, it is necessary to break its spatial inversion symmetry. Based on the above considerations, the I atoms on one side of the Hf2TeI2 structure are substituted by the Br atoms to break the spatial inversion symmetry and achieve excellent energy valley-related characteristics.
We also focus on the valley characteristics of the 2D hexagonal Hf2TeIBr lattice. As mentioned above, the key to achieving nonequivalent valleys in the Hf2TeIBr monolayer is to eliminate the spatial inversion symmetry. In view of this, we further propose the Janus Hf2TeIBr monolayer, in which the lower I atomic layer is substituted by the Br atom, as shown in Figure 1a. The spatial inversion symmetry in the Hf2TeIBr monolayer is intrinsically broken. After full structural optimization, the lattice constant of the Hf2TeIBr monolayer is found to be a = b = 4.81 Å. We also calculated the cohesive energy of the Hf2TeIBr monolayer and obtained the result of −3.83eV/atom. This indicates that the Hf2TeIBr monolayer is equally feasible for experimental preparation, which proves that the Hf2TeIBr monolayer has good dynamic stability.
Figure 2a shows the band structure of the Hf2TeIBr monolayer without considering the SOC. It can be seen that the characteristics of the Dirac cone are preserved, and there is a band gap of 18 meV, which is caused by the spatial inversion of symmetry breaking. The calculation of Fermi velocities at K and K’ exhibit anisotropic characteristics, where v f L e f t = 4.8 × 10 5 m/s, v f R i g h t = 5.3 × 10 5 m/s. Such considerable carrier velocity makes the Hf2TeIBr monolayer an excellent candidate for high-speed spintronic devices.
Then, the band structure of the Hf2TeIBr monolayer with SOC is calculated, as shown in Figure 2b. The band gap of the Dirac cone at the Fermi level is enhanced to 240 meV. More interestingly, the energy dispersion shows spin splitting because of the spatial inversion symmetry breaking. However, due to the time inversion symmetry, the inequivalent K and K’ valleys are determined by the spin-down and spin-up states, respectively, revealing the emergence of valley–spin coupling phenomenon. The spin splitting of the valence and conduction bands are 8 meV and 27 meV, respectively.
The valley-related properties of the Hf2TeIBr monolayer are investigated. The Berry curvature Ω n z ( k ) of the Hf2TeIBr monolayer is calculated using the Kubo formula [36], which is expressed as follows:
Ω z ( k ) = ∑ n f n Ω n z k
fn is the Fermi–Dirac distribution, where Ω n z ( k ) is defined as follows:
Ω n z ( k ) = − 2 I m φ n ′ k v x φ n ′ k φ n ′ k v y φ n k E n ′ − E n 2
In the formula mentioned above, v x ( y ) and φ n k represent the Bloch states of the velocity operator with the eigenvalue E n . Figure 3a shows the calculated Berry curvature across the entire Brillouin zone. We can observe that non-zero Berry curvatures are distributed around the K and K’ points, exhibiting opposite signs. The magnitude of Berry curvatures is approximately 60 Bohr2, which is comparable to that of MoS2. The three-dimensional diagram of the Berry curvature is also presented in Figure 3b, in which the opposite peak distribution of Berry curvatures at K and K’ and the C3v symmetry can be observed intuitively. This indicates that the monolayer achieves valley-contrasting physics. In this case, the Dirac fermions in two valleys cannot be distinguished energetically, but they can be distinguished by the opposite Berry curvature at K and K’ and the out-of-plane spin moment.
For a valley electronic material with a two-dimensional hexagonal structure of Hf2TeIBr, the Berry curvature of the nonequivalent valley is non-zero Ω z ( k ) , and the sign is opposite. When the in-plane electric field E is applied (B = 0), considering the influence of Berry curvature, the motion of the carrier is defined as follows:
x ˙ = 1 h ∂ E n ( k ) ∂ k + k × Ω z ( k )
The second term in the above equation indicates that the reverse phase Ω z ( k ) causes the carrier to produce two reverse abnormal velocities [37] perpendicular to the electric field. Figure 3c shows a schematic diagram of the valley and spin Hall effect. At this time, the carrier of unequal energy valley moves backwards under the action of the surface electric field, and then spatial separation occurs, producing the polarization of valley and spin, and its transverse velocity can be described as follows:
v ⊥ = − e ℏ E × Ω ( k )  
The reverse transverse current generated is called the energy valley Hall current. Large transverse velocities are important for reducing the excited carrier recombination since the valence band maximum (VBM) and conduction band minimum (CBM) show opposite Berry curvatures, namely Ω c ( k ) = − Ω v ( k ) . Thus, when there is an in-plane electric field E, excited electrons and holes in the same valley will also have opposite transverse velocity; in fact, large Ω ( k ) can increase the transverse velocity and thus accelerate the excited carriers moving along the direction perpendicular to E and reduce their recombination. On the other hand, if the Berry curvature is large, a smaller outfield E can be imposed for a specific lateral velocity. In general, the magnitude of the carrier velocity can be regarded as the carrier mobility, which can be calculated by the Boltzmann transport equation, and generating the valley polarization in a controlled way is crucial for the use of the valley degrees of freedom. However, to control the lateral transport speed of charge carriers and thus achieve the controllable valley electronics performance of Hf2TeIBr films, it is necessary to gain insights into the physical mechanisms of adjustable Berry curvature.
To achieve a robust valley Hall effect, a larger valley–spin splitting is required. Therefore, the spin-splitting magnitude of valence and conduction bands of the Hf2TeIBr monolayer was investigated by the biaxial strain. In contrast to the uniaxial strain, which disrupts the D3h symmetry of monolayer MoS2, biaxial strain engineering is friendly to the hexagonal structure with C3 symmetry and has been demonstrated as an effective avenue to tune the magnitude of the Berry curvature of a 2D valley material. Here, the biaxial strain does not disrupt the C3 symmetry of Hf2TeIBr and avoids introducing a Berry curvature dipole.
Here, we apply a stress of reasonable magnitude (−6–6%) and calculate the structural energy and corresponding energy bands, the results are shown in Figure 4 and we observe that the Hf2TeIBr monolayer maintained direct band gap semiconductor properties under the tensile strain. Under compressive strain, it manifests as an indirect band gap semiconductor.
In Figure 5b, the variation in band gap as a function of strain is presented. It can be observed that both tensile and compressive strains enhance the degree of band splitting. We define the spin-splitting values of the CBM and VBM at K as ∆ E C and ∆ E V , and the evolution of both with strain is shown in Figure 5c. It is found that the splitting values of CB and VB increase with the increase in tensile strain, and the splitting value of the structure reaches its maximum at a tensile stress of 4%, ∆ E C = 122   m e V and ∆ E V = 128   m e V . For valleytronic materials, the effective tunability of band gap and spin splitting under strain c enables them to be well applied in valleytronic devices.
Furthermore, the Berry curvature at different strains is calculated. Figure 6 shows the Berry curvature of the Hf2TeIBr monolayer under −6~6% biaxial strain. In contrast to the ground state, the compression strain not only reverses the sign of the Berry curvature in K and K’ valleys but also makes the Berry curvature increase from 60 Bohr2 to 80 Bohr2. And the appearance of this phenomenon can be attributed to the strain-induced exchange of the K’ and K valley spin index. For the tensile strain, the main effect is to increase the Berry curvature to 200 Bohr2 while keeping the sign unchanged. These results demonstrate that biaxial strain is an effective way to modulate the Berry curvature, implying that lattice parameters are the main factors affecting the Berry curvature. Therefore, by applying biaxial strain to enhance the Berry curvature, the efficiency of valleytronic devices can be improved.

4. Conclusions

In summary, we propose novel 2D hexagonal Hf2TeIX (X = I, Br) monolayers. Based on its good stability, we further found that the energy band of Hf2TeI2 has a Dirac cone at the high-symmetry K point near the Fermi surface, and the SOC opens a band gap of 271 meV and shows the potential energy valley characteristics. It can change the size of the band gap under the external biaxial strain, producing a phase transition from the semiconductor state to the metallic state. Based on its band characteristics, inspired by valley electronics, we want to find a two-dimensional material with controlled valley characteristics and successfully observe the valley-splitting effect in the Hf2TeIBr lattice-breaking spatial inversion symmetry. Importantly, we found that the structure achieves energy valley spin locking. Finally, it is found that the Berry curvature of the system can be improved by external stress, which improves the lateral velocity of the carrier to improve the efficiency of the valley Hall device.

Supplementary Materials

The following supporting information can be downloaded at: https://www.mdpi.com/article/10.3390/cryst14090794/s1, Figure S1: Phonon band dispersions of SL Hf2TeIBr; Figure S2: Band structures of SL Hf2TeI2 with considering SOC under various strains. The Fermi level is set to 0 eV.

Author Contributions

Data, drawing, writing, editing, K.H.; review, P.W. All authors have read and agreed to the published version of the manuscript.

Funding

This work was supported by National Natural Science Foundation of China (Grant No. 52173283 and No. 62071200), Taishan Scholar Program of Shandong Province (Grant No. ts20190939), and Independent Cultivation Program of Innovation Team of Jinan City (Grant No. 2021GXRC043).

Data Availability Statement

The data used in this study are available upon reasonable request due to privacy. The data were collected from the physics group of University of Jinan. For access to the data, please contact the corresponding author at [email protected].

Acknowledgments

Thanks to Peiji Wang, Shengsi Li and the physics group of University of Jinan for their help.

Conflicts of Interest

The authors declare no conflict of interest.

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Figure 1. (a) Top and side views of geometric structure of Hf2TeIX (X = I, Br) monolayer. The green, blue, yellow, and gold balls denote the I, Hf, Te, and I (Br) atoms, respectively. (b) Excluding the Hf2TeI2 electronic band structure of the SOC. (c) Consider the electronic band structure of the SOC. (d) Orbital projection component analysis of electronic structure under SOC.
Figure 1. (a) Top and side views of geometric structure of Hf2TeIX (X = I, Br) monolayer. The green, blue, yellow, and gold balls denote the I, Hf, Te, and I (Br) atoms, respectively. (b) Excluding the Hf2TeI2 electronic band structure of the SOC. (c) Consider the electronic band structure of the SOC. (d) Orbital projection component analysis of electronic structure under SOC.
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Figure 2. Band structures of Hf2TeIBr (a) without SOC and (b) with SOC; blue is spin down, and red is spin up.
Figure 2. Band structures of Hf2TeIBr (a) without SOC and (b) with SOC; blue is spin down, and red is spin up.
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Figure 3. (a) Berry curvature of Hf2TeIBr as a contour map over the 2D Brillion zone. (b) Schematic diagram of the three-dimensional Berry curvature of the Brillouin area. (c) Schematic diagram of the spin-valley Hall effect.
Figure 3. (a) Berry curvature of Hf2TeIBr as a contour map over the 2D Brillion zone. (b) Schematic diagram of the three-dimensional Berry curvature of the Brillouin area. (c) Schematic diagram of the spin-valley Hall effect.
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Figure 4. Energy band diagrams of Hf2TeIBr lattice under biaxial strain at different stresses (−6–6%); blue is spin down, and red is spin up.
Figure 4. Energy band diagrams of Hf2TeIBr lattice under biaxial strain at different stresses (−6–6%); blue is spin down, and red is spin up.
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Figure 5. (a) Hf2TeIBr total energy of the monolayer as a function of the biaxial strain. (b) The change in the band gap under the biaxial strain, and the properties of the band gap. (c) Change in the split value ( ∆ E C ) ( ∆ E V ) under biaxial strain.
Figure 5. (a) Hf2TeIBr total energy of the monolayer as a function of the biaxial strain. (b) The change in the band gap under the biaxial strain, and the properties of the band gap. (c) Change in the split value ( ∆ E C ) ( ∆ E V ) under biaxial strain.
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Figure 6. (a–c) The Berry curvature of a single-layer Hf2TeIBr under biaxial strains of ε = −6%, 0%, and 6%, respectively.
Figure 6. (a–c) The Berry curvature of a single-layer Hf2TeIBr under biaxial strains of ε = −6%, 0%, and 6%, respectively.
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He, K.; Wang, P. Prediction of Intriguing Valley Properties in Two-Dimensional Hf2TeIX (X = I, Br) Monolayers. Crystals 2024, 14, 794. https://doi.org/10.3390/cryst14090794

AMA Style

He K, Wang P. Prediction of Intriguing Valley Properties in Two-Dimensional Hf2TeIX (X = I, Br) Monolayers. Crystals. 2024; 14(9):794. https://doi.org/10.3390/cryst14090794

Chicago/Turabian Style

He, Kaiyuan, and Peiji Wang. 2024. "Prediction of Intriguing Valley Properties in Two-Dimensional Hf2TeIX (X = I, Br) Monolayers" Crystals 14, no. 9: 794. https://doi.org/10.3390/cryst14090794

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