Strain-tunable magnetic and electronic properties of monolayer CrI₃

Abstract

Two-dimensional CrI₃ has attracted much attention as it is reported to be a ferromagnetic semiconductor with a Curie temperature of around 45 K. By performing first-principles calculations, we find that the magnetic ground state of CrI₃ is variable under biaxial strain. Our theoretical investigations show that the ground state of monolayer CrI₃ is ferromagnetic under compression, but becomes antiferromagnetic under tension. Particularly, the transition occurs under a feasible in-plane strain of around 1.8%. Accompanied by the transition of the magnetic ground state, CrI₃ undergoes a transition from magnetic-metal to half-metal to half-semiconductor to spin-relevant semiconductor when the strain varies from −15% to 10%. We attribute these transitions to the variation of the d-orbitals of Cr atoms and the p-orbitals of I atoms. Generally, we report a series of magnetic and electronic phase transitions in strained CrI₃, which will help both theoretical and experimental researchers in further understanding the tunable electronic and magnetic properties of CrI₃ and its analogs.

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Introduction

Two-dimensional (2D) materials are of great concern theoretically and experimentally because of their unique electronic and optoelectronic properties.^1,2 Typically, graphene, which consists of six carbon atoms in a honeycomb lattice, is a zero-gap semimetal with the carrier mobility being reported up to ∼10⁵ cm² V⁻¹ s⁻¹.³ When it is chemically decorated with added atoms or cut into 1D nanoribbons, it will exhibit tunable electronic and magnetic properties.^4–6 When combined with other 2D materials to form van der Waals (vdW) heterostructures,⁷ it would exhibit much more interesting physical properties.^6,8 Besides graphene, other 2D materials from semiconducting black phosphorus⁹ to transition metal dichalcogenides^10–12 to insulating hexagonal boron nitride¹³ have also been identified as important candidates for the post-silicon electronic and optical devices.¹⁴ However, none of them is reported to exhibit intrinsic magnetism, which limits their application in spintronics.¹⁵ Theorists predicted that most 2D materials are nonmagnetic because the thermal fluctuations at finite temperature would break the spontaneous symmetry.¹⁶ However, a composite of monolayer Cr₂Ge₂Te₆ has been reported to be ferromagnetic recently, which is regardless of the restriction.¹⁷ The rise of Cr₂Ge₂Te₆ paves a new way to search for the long-range Ising ferromagnetism in atomically thin 2D materials, where an intrinsic magnetocrystalline material could exist because of the reduction of the crystal symmetry.¹⁸ Very recently, another ferromagnetic semiconductor chromium triiodide (CrI₃) appeared in the research field again¹⁹ because of its high Curie temperature in the monolayer.²⁰ Upon carrier doping, room-temperature magnetism is observed in CrI₃ due to its flat band structure.^21–23 Both experimental and theoretical research shows that monolayer CrI₃ is a ferromagnetic semiconductor.^18,24 When increasing the number of layers, the ferromagnetic order persists within each layer, but the antiferromagnetic coupling dominates different layers.⁸ Moreover, the ferromagnetic and antiferromagnetic states can be switched on and off by changing the external gate voltage.¹⁸ When CrI₃ forms heterostructures with other 2D materials like graphene, it also exhibits some topological insulating properties.²⁵ However, all this is concluded from the fact that CrI₃ is a ferromagnetic semiconductor with an equilibrium lattice constant.²⁰

On the other hand, strain plays an important role in determining the physical properties of 2D materials. Considering the fact that typically CrI₃ is transferred on the substrate of SiO₂ and on other 2D materials after exfoliation,²⁰ the intrinsic physical properties of CrI₃ would be affected by the substrate due to the lattice-mismatch-induced strain.^26,27 There are some studies on the physical properties of CrI₃ under strain, but they are based on the assumption of a robust ferromagnetic ground state.²⁸ Meanwhile, it is challenging for experimental researchers to identify the magnetic order of the monolayer in the atomic resolution. Here, we wish to identify the magnetic ground states of CrI₃ under strain via ab initio first-principles calculations.

In this work, we present a systematic study on the tunable electronic and magnetic properties of monolayer CrI₃ under strain. Our results show that the Cr atoms in the unit cell are ferromagnetically aligned under compression strain, and CrI₃ retains the magnetic order up to a maximum tension strain of around 2%, then it dramatically becomes an antiferromagnetic half-semiconductor when the tension strain is further increased. During this transition, the magnetic moment on Cr atoms increases, and CrI₃ undergoes a transition from magnetic-metal to half-metal to half-semiconductor, owing to the variation of the d-orbitals of Cr atoms and the p-orbitals of I atoms. Our results will provide a new way to understand the magnetic ground state in monolayer CrI₃ and its analogs, which is useful for the design of spintronic devices²⁹ based on ferromagnetic semiconductors.^17,30–32

Results and discussion

The atomic structure of monolayer CrI₃ is shown in Fig. 1a, which can be simply imaged as a super cell of 1T SnS₂ with one point vacancy of Sn atoms.^33–35 It belongs to space group C2/m containing two formula units.³⁶ Its optimized lattice parameters are calculated to be a = b = 6.978 Å and c = 21.476 Å, which are in good agreement with the X-ray diffraction data³⁷ and previous DFT results.²⁴ Both spin-unpolarized and spin-polarized calculations are performed to get an overview of the ground state of monolayer CrI₃. Our result shows that the ferromagnetic (FM) state is more favorable in energy, which is 63 meV lower than the antiferromagnetic (AFM) state, indicating stable ferromagnetism at room-temperature. Thus, the intrinsic electronic and magnetic properties of CrI₃ can be represented by the FM state as shown in Fig. 1b and c. The spin-unpolarized band structure shows that nonmagnetic CrI₃ is a metal with several bands crossing through the Fermi level. Upon considering the spin-polarization, the degenerated bands get split, resulting in an indirect energy gap of 1.124 eV and 2.169 eV for the spin-up (solid) and -down (dashed) electrons, respectively. It is found that the spin-polarized electrons in monolayer CrI₃ exhibit anisotropic transport properties. For the spin-up electrons, the conduction band minimum (CBM) and valence band maximum (VBM) are located at the Gamma point and in the line from gamma to M; for the spin-down electrons, the indirect gap originates from M and gamma points, respectively. It is also noted that the conduction and valence band edges around the Fermi level are fully spin-polarized and exclusively occupied by electrons with the same spin component, rendering a typical half-semiconductor character. Moreover, the CBM and VBM of the spin-up electrons are contributed by both Cr and I atoms, while the CBM and VBM of the spin-down electrons are contributed by Cr atoms and I atoms, respectively. Our conclusion of the intrinsic electronic structure of monolayer CrI₃ is further confirmed by the density of state (DOS) calculations, where sharp peaks composed of hybrid states appear around the Fermi level, suggesting strongly localized states. As a result, electrons are bound in these states and the carrier mobility of CrI₃ is very slow, which can be concluded directly from the nearly flat bands around the Fermi level. Thus, in some vdW heterostructures CrI₃ is usually used as a FM substrate to generate spin-polarized electrons.⁷

Fig. 1 Atomic and electronic structure of unstrained monolayer CrI₃. (a) Structural schematic of CrI₃ with top, front and right views. The green dashed line presents the unit cell and the red vectors indicate the spin configuration. (b) Band structure of ferromagnetic CrI₃. The Fermi level is set to zero, and the color indicates the attribution of Cr and I atoms. (c) Density of states of CrI₃ obtained by spin-polarized calculations.

We have shown above that pristine CrI₃ is a ferromagnetic semiconductor with the magnetic moment on Cr atoms being 3.106 μ_B.^19,38 However, these 2D materials are usually supported by the substrate in device design. Besides the interlayer interaction, strain induced by lattice mismatch and lattice orientation is the most common case in these 2D materials. We noted in a very recent work that when a non-collinear spin configuration is introduced by considering the spin orbital coupling effect, monolayer CrI₃ will undergo a transition from the FM to AFM state under compression.³⁹ Here, we would like to show novel electronic and magnetic properties of monolayer CrI₃ under biaxial strain with a collinear spin configuration. The total energies are calculated using spin-polarized calculations for both FM and AFM configurations. It shows typically parabolic characters as in Fig. 2a when biaxial strain is applied. For the AFM configuration, its equilibrium lattice constant is slightly larger than that of the FM configuration. Under compression strain, the FM configuration prefers a much lower energy. To clearly show the variation of the total energy, we further plot the energy difference ΔE = E_AFM − E_FM as a function of biaxial strain in Fig. 2b, where E_AFM and E_FM are the total free energies of the monolayer with AFM and FM configurations, respectively. Within a reasonable range from −10% to 10%, ΔE decreases monotonously and drops down to zero around 1.8%, indicating a possible transition from FM to AFM. Taking the tensile strain of 3% as an example, the corresponding ΔE is 64 meV, which is twice larger than that calculated from the fluctuation of 300 K, suggesting stable AFM states at room temperature. When tension strain is further increased, the AFM configuration becomes much more stable. To confirm our conclusion, we performed further calculations with DFT+U, which usually give better results for transition metals with d orbitals.^40,41 The parameters J and U are chosen to be 0.7 eV and 2.7 eV, respectively, which have shown great success in predicting the magnetic anisotropic properties of monolayer CrI₃.^39,42 We listed the total energies in Table S1 of the ESI.† One can clearly see that though the total energy of monolayer CrI₃ is higher than that of the standard DFT results, the relative variation trends of ΔE, E_AFM and E_FM are the same. As a result, the transition from the FM to AFM state occurs when the tension strain is applied.

Fig. 2 Ground state of monolayer CrI₃ under biaxial strain. (a) Strain effect on the total energy of magnetic CrI₃. (b) Energy difference and magnetic moment of FM and AFM CrI₃ as a function of strain. Red and green symbols present the data of AFM and FM configurations, respectively.

AFM CrI₃ is reported to be stable under tensile strain in previous work, but their physical properties are less studied. The band structure of AFM CrI₃ plotted in Fig. 3 shows that tensed CrI₃ is an indirect gap spin-relevant semiconductor with the VBM and CBM at the K and M points, respectively. Different from that of the FM state, spin-polarized electrons degenerate in AFM CrI₃ as the inversion symmetry is preserved. As a result, spin-polarized electrons are strongly localized as seen from the nearly flat bands both around 1.0 and 2.0 eV, which disperse with an energy window of up to 0.5 eV in the FM configuration. The high degeneracy of the spin-polarized electrons is confirmed by the DOS as well, where the spin-up and -down electrons show identical distribution with mirror symmetry. Detailed analysis shows that though the contribution of the band structure is from the same atoms as FM CrI₃, the non-degenerated bands at some highly symmetric K points get split, indicating the symmetry breaking of the p_x and p_y orbitals in I atoms. It is also noted that when biaxial strain is applied, the magnetic moment on the Cr atoms can be tuned by biaxial strain. When in-plane strain is applied from the compression to the tension region, the magnetic moment on Cr atoms increases for both FM and AFM configurations. Remarkably, the magnetic moment on the Cr atoms increases from 2.966 μ_B to 3.364 μ_B for FM CrI₃ when the strain varies from −10% to 10%. But the slope of the magnetic moment variation is much smaller in the AFM case.

Fig. 3 Electronic structure of stretched CrI₃. Band structure and DOS of CrI₃ with a tension strain of 4%. The inset shows the spin configuration with the antiferromagnetic order. The Fermi level is set to zero.

In addition to the transition and modulation of the magnetic state, the electronic properties of CrI₃ show an interesting response to the external strain. We show the band gap of CrI₃ for both spin-up and -down electrons in Fig. 4. The left and right panels show the band gap modulation in the FM and AFM states, respectively. For the masked section, we will neglect the variation of the band gaps as the magnetic ground state has changed. It is obvious that the FM state is more sensitive to the biaxial strain as the slope of the modulated band gap is sharper than that of the AFM state. In the non-strained case (left panel), both spin-up and -down electrons open a gap showing the character of a half-semiconductor (HS); when a biaxial compression smaller than 13% is applied, the band gaps of both the spin-up and -down electrons decrease, as the band gap of spin-down electrons is larger than that of the spin-up electrons, at a critical value around 14%, the spin-up gap becomes zero, while the spin-down gap remains open with 0.103 eV, making CrI₃ a half-metal (HM); upon further compressing the monolayer, both the spin-up and -down bands close, and CrI₃ becomes a magnetic-metal (MM). In contrast, when tension strain is applied in a reasonable range (right panel), AFM CrI₃ shows spin-relevant semiconductor (SS) characters. Both spin-up and -down electrons occupy the same band and the band gap drops down in a slow slope around 0.015 eV/1%. Even when CrI₃ is stretched by 10%, it remains open with a large band gap of 1.189 eV, suggesting that CrI₃ is a robust AFM-SS when it is under tensile strain. The transition is further confirmed by our benchmark calculations of DFT+U as shown in Fig. S1 in the ESI.†

Fig. 4 Strain-dependent band gaps in CrI₃. (a and b) spin-relevant band gaps as a function of biaxial strain. Blue and green represent spin-up and -down electrons, respectively. The shadow indicates the artificial band gap modulation in the FM or AFM configuration.

To understand the mechanism of the biaxial-strain-induced electronic phase transition from MM to HM to HS to SS, we examined the projected density of states (PDOS) of CrI₃ under various strains as shown in Fig. 5. Our result shows that the low-energy electronic properties of CrI₃ are mainly attributed to the in-plane components of the d orbitals of Cr atoms and the p orbitals of I atoms. And the p_x and p_y orbitals of I atoms are degenerated, which explains the degenerated VBs at some highly symmetric K points in FM CrI₃. On the other hand, the PDOS of Cr atoms is much higher than that of I atoms above the Fermi level, suggesting that the transport properties of electrons are dominated by the variation of the d-orbitals of Cr atoms, while the electronic properties of holes are determined by the p_x and p_y orbitals of I atoms. When the compression strain increases, the PDOS of all d-orbitals of Cr atoms shift downwards with a slightly decreasing gap for both the spin-up and spin-down electrons. However, the p_x and p_y orbitals of I atoms are much more sensitive to the biaxial strain, and the p orbitals at the conduction and valence region become hybridized, resulting in a closed gap. Specifically, at a compression strain state of −15%, all the d and p orbitals show peaks at the Fermi level; when the compression strain decreases to −14%, only the p_y orbitals of the spin-up electrons occupy the Fermi level, resulting in a MM–HM transition. For stretched CrI₃ with a tensile strain of 10%, the peaks of the PDOS from the p_x orbital of I atoms are enhanced and the conduction region is contributed by two Cr atoms with mirror symmetry. We also show the spin density of CrI₃ at the bottom of Fig. 5. As the compression strain decreases from −15% to −10%, the spin density represented by the iso-surface increases slightly, which is in good agreement with the magnetic moment modulation in Fig. 2. To this end, we have shown that the electronic and magnetic properties of monolayer CrI₃ can be effectively tuned by biaxial strain, which is dominated by the d_xy, d_yz, d_z² and d_xz orbitals of Cr atoms and the p_x and p_y orbitals of I atoms.

Fig. 5 Projected density of states and spin density at different strains for CrI₃ showing the MM–HM–HS transition. The scale bar in the slice of the spin density is set from −0.04 e Å⁻³ to 0.04 e Å⁻³.

Conclusions

In summary, we have systematically investigated the electronic and magnetic modulations of monolayer CrI₃ under biaxial strain by first-principles calculations. The applied strain yields a pronounced transition of the magnetic ground state between FM and AFM. When compression strain is applied, CrI₃ remains ferromagnetic. As the strain increases from −15% to 2%, a series of electronic phase transitions of MM–HM–HS–SS occur. In contrast, it becomes antiferromagnetic under tensile strain and the band gap of AFM CrI₃ is robust against external strain. These modulations of electronic and magnetic properties stem from the shift of the d-orbitals in Cr atoms and the p-orbitals in I atoms under strain. The tunable electronic and magnetic properties of monolayer CrI₃ investigated in this work are helpful in understanding the magnetism in CrI₃ and its analogs observed by experimental researchers and would inspire extensive research interest in modulation of the electronic and magnetic properties in ferromagnetic semiconductors.

Computational methods

All our simulations were carried out by performing spin-polarized density functional theory (DFT) calculations as implemented in the Vienna ab initio Simulation Package (VASP).⁴³ The Perdew–Burke–Ernzerhof (PBE) pseudopotentials⁴⁴ within the general gradient approximation (GGA)⁴⁵ were used to describe the electron exchange and correlation interactions and the energy cutoff was set at 520 eV. The Brillouin zone was represented by a 12 × 12 × 1 mesh for geometry optimization and the total energy calculation. And for the DOS calculation, a much denser grid of 24 × 24 × 1 was used. The atomic structure was fully relaxed with the energy convergence being 10⁻⁵ eV. To avoid the interlayer interaction between adjacent images, the vacuum was set to be 21 Å normal to the monolayer.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This work is supported by the National Key R&D Program of China (Grant No. 2018FYA0305800) and the Dutch Science Foundation NWO/FOM 16PR1024. Yu acknowledges financial support from the NSFC grant (No. U1530401) and MOST 2017YFA0303404 from the Beijing Computational Science Research Center. Numerical calculations presented in this paper have been performed on a supercomputing system in the Supercomputing Center of Wuhan University.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c8cp07067a

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