Liang
Liu
^{ab},
Zezhou
Lin
^{a},
Jifan
Hu
^{c} and
Xi
Zhang
*^{a}
^{a}Institute of Nanosurface Science and Engineering, Guangdong Provincial Key Laboratory of Micro/Nano Optomechatronics Engineering, Shenzhen University, Shenzhen 518060, China. E-mail: zh0005xi@szu.edu.cn
^{b}Key Laboratory of Optoelectronic Devices and Systems of Ministry of Education and Guangdong Province, College of Optoelectronic Engineering, Shenzhen University, Shenzhen, 518060, China
^{c}School of Physics, State key laboratory for crystal materials, Shandong University, Jinan 250100, China
First published on 30th March 2021
Atomic thin two-dimensional (2D) ferromagnetic (FM) semiconductors with high Curie temperatures (T_{c}) are essential for future spintronic applications. However, reliable theoretical searching for 2D FM semiconductors is still hard due to the complexity of strong quantum fluctuations in 2D systems. We have proposed a full quantum search (FulQuanS) method to tackle the difficulty, and finally identified five 2D semiconductors of CrX_{3} (X = I, Br, Cl), CuCl_{3} and FeCl_{2} with FM order at finite temperature from the pool of 3721 potential 2D structures. Via the method of renormalized spin wave theory (SW) and quantum Monte Carlo simulations (QMC), we located the T_{c} for CrX_{3} (X = I, Br, Cl), CuCl_{3} and FeCl_{2} at 48 K, 31 K, 18 K, 74 K and 931 K respectively, which excellently agree with experiments for CrX_{3} and reveal the superior performances of the new predicted structures. Furthermore, our QMC results demonstrated that the systems with low-spin numbers and/or low anisotropies have much higher T_{c} than the estimations of classical models e.g., Monte Carlo simulations based on classical Heisenberg models. Our findings suggest excellent candidates for future room-temperature spintronics, and shed light on the quantum effects inherent in 2D magnetism.
Recently, theoretical searching for 2D ferromagnetism has drawn intense attention, and significant progress has been made, providing essential references for the realization of superior 2D ferromagnets in laboratories.^{18–23} To date, researchers have proposed more than six hundred potential 2D ferromagnets via comparing the energies of ferromagnetic and some antiferromagnetic (AFM) configurations at the density functional theory (DFT) level.^{24,25} However, it is still highly difficult to provide accurate descriptions and predictions for 2D ferromagnets. Firstly, the complexities of AFM configurations e.g., C-type, G-type, A-type AFM and some long-range AFM configurations make it impossible to enumerate all possible AFM configurations and ensure the FM ground states within conventional methods. On the other hand, the Curie temperature which is one of the key properties in 2D magnets is very nontrivial to estimate via usual theoretical methods. In the literature, the locations of T_{c} were accomplished by the mean-field method, spin wave (SW) theory^{26} and Monte Carlo simulations (MC) of Ising model^{27} or classical Heisenberg model.^{19,28} The mean-field method is a linear mapping from exchange interactions J to T_{c}. It did not count the effects of magnetic anisotropy which is crucial in the 2D limits, failing to capture the phase transitions in 2D magnetism.^{19} So did the Ising model. SW and classical Heisenberg MC involve the considerations of magnetic anisotropies, but treat spins as classical sticks in which the approximation integrities were based on the large spin-number limit. To date, there is a surprising paucity of studies that seek to identify the effect of the quantum effect on T_{c} of 2D magnets which is supposed to be crucial in the case of a small spin-number such as in the famous CrX_{3} family. All these signify the imperatives of new and plausible methods that capture the ground state properties i.e. the exchange interactions and anisotropies of 2D FM systems, as well as going deep into the excited states near phase transitions with full considerations of quantum effects, predicting new 2D FM semiconductors with high efficiency and credibility.
In this work, we organized comprehensive investigations to find a series of 2D FM semiconductors with relatively high T_{c}. We proposed high throughput full quantum search (FulQuanS) to discover promising 2D ferromagnetic semiconductors and locate their T_{c}. The magnetic properties and effective parameters in the Heisenberg model were extracted from relativistic density functional theory (DFT) and we finally screened five candidates from 3712 structures in a computational 2D material database (C2DB), including the well-known CrX_{3} (X = Cl, Br, I), and the other two new structures of CuCl_{3} and FeCl_{2}. We employed renormalized SW and a quantum Monte Carlo (QMC) method to locate the phase transitions. For CrX_{3} (X = Cl, Br, I), CuCl_{3} and FeCl_{2}, T_{c} values predicted by FulQuanS were 48 K, 31 K, 18 K, 74 K and 931 K respectively, showing excellent agreement with experiments in the CrX_{3} family. More crucially, the results obtained using FulQuanS clarified the key roles of quantum effects in the phase transitions of 2D magnetism, including the frustrations in low spin systems with trigonal lattices, and the stabilizations of magnetic orders in systems with low anisotropies and low spin numbers. All these findings suggest promising candidates for future 2D magnetism study and spintronic application as well as clarifying the implications of quantum effects inherent in 2D magnetism.
Stage 1 is the data mining and high-throughput search for 2D FM semiconductors. In this stage we screened out all the potential high-T_{c} FM semiconductors by means of high throughput searching. We examined totally 3721 2D structures from C2DB,^{24} which is one of the largest 2D material databases. The considerations of stabilities (with formation energy above hull less than 0.2 eV) and band gaps (>0) screened 772 structures out of whole C2DB. In Fig. 1(b) we summarized their band gaps and formation energy above hull. Since all of them have relatively low formation energy, these structures thus are promising to be easily synthesized via bottom-up chemical methods. And the finite band gaps reveal the semiconducting nature, ensuring the integrities of utilizing Heisenberg models to capture the magnetic interactions in these structures.
Afterwards, we screened out structures with open d- or f-shells, consisting of 454 structures and denoted by orange dots in Fig. 1(b). The magnetism of s- and p-orbitals rarely exists in perfect crystals and nor does that of those with only closed shells, they are thus safely skipped. Then, the remaining structures were optimized and their ground electronic structures were computed with colinear-DFT. It was shown that only 128 of them converged to ferromagnetic ground states (orange dots in Fig. 1(b)).
Finally, we studied the magnetic anisotropies of the remaining structures based on noncollinear-DFT to screen out the FM structures with out-of-plane anisotropies, resulting in eight structures as shown by the red dots in Fig. 1(b). According to Mermin–Wagner theorem, breaking the continual symmetries via anisotropy is crucial for a 2D system to sustain magnetic order at finite temperature.^{29} Furthermore, most of the experimentally confirmed 2D magnetic semiconductors are with out-of-plane preferred magnetic anisotropy. Therefore, we only leave those with out-of-plane anisotropies for further considerations.
Stage 2 is the construction of the anisotropic Heisenberg model:
(1) |
Stage 3 is to solve the anisotropic Heisenberg model eqn (1) for the determinations of T_{c} based on Hatree–Fock renormalized spin wave theory (HF-SW) and path-integral QMC. In HF-SW, the low-energy behaviors of Heisenberg model were captured with boson (each spin wave is a boson) systems, via operator substitutions known as Dyson–Maleev transformation:^{30}
(2) |
It may be noticed that the exact value of Δ^{SW} depends on the occupations and the temperature once the interactions were taken into consideration. Fortunately, the property of Δ^{SW} > 0 always remains in stable ferromagnetic systems until the phase transition (see the ESI† for more discussions on this point). Therefore, utilizing the 0 K Δ^{SW} as the criteria for screening is sufficient.
Besides, the averaged magnetic moment per site at finite temperature T is related to the quantity of spin wave excitons: M(T) = 2(S − n(T)), in which n(T) satisfies the standard Boson–Einstein distribution. T^{SW}_{c} is obtained at the temperature where magnetism vanishes, i.e. M(T_{c}) = 0.
The HF-SW method is analytical and costless, providing good descriptions for the large-scale magnetic configurations in the ground state and coarse description for the magnetic states near T_{c}. Meanwhile QMC is supposed to be numerically accurate near T_{c} but expensive. Therefore, FulQuanS was designed to integrate the advantages of both. We utilized HF-SW to have a coarse estimation for T_{c}, which is treated as the initial guess for QMC. Afterwards, we performed QMC^{31,32} on 16 × 16 and 32 × 32 supercells, using the scaling behavior of 4^{th} order Binder cumulate B(T) = m^{22}/m^{4} to extrapolate the numerically accurate T_{c} of infinitely sized systems.^{31,32} To investigate how the quantum effects impact phase transition and thermal dynamics of 2D magnetism, we also conducted the classical Monte Carlo (CMC)^{33,34} simulation in this stage for comparisons.
Fig. 2 Top and side views of eight 2D FM semiconductors with short-range ferromagnetism and out-of-plane anisotropies including three with experimental confirmations and five new predicted by DFT. |
S | J1 (B1) | J2 (B2) | J3 (B3) | A _{ z } | |
---|---|---|---|---|---|
CrI_{3} | 1.5 | −1.69 (−0.08) | −0.51 (0.02) | 0.46 (0.01) | −0.272 |
CrBr_{3} | 1.5 | −1.41 (−0.03) | −0.34 (0.01) | 0.44 (0.00) | −0.08 |
CrCl_{3} | 1.5 | −1.00 (−0.01) | −0.18 (0.00) | 0.29 (0.00) | −0.031 |
CuCl_{3} | 1.0 | −9.02 (−0.01) | −1.84 (0.00) | 3.20 (0.00) | 0.01 |
TiBr_{3} | 0.5 | −120.66 (−0.02) | 39.43 (−0.03) | −26.52 (0.00) | 0.014 |
TiCl_{3} | 0.5 | −114.60 (−0.02) | 42.71 (−0.02) | −29.96 (−0.01) | −0.011 |
FeCl_{2} | 2.0 | −15.40 (−0.02) | −0.45 (−0.01) | 1.52 (0.03) | −0.049 |
Fe_{2}O_{2}Br_{2} | 0.5 | −28.78 (−0.19) | −8.88 (−0.63) | −24.89 (0.36) | −2.47 |
The calculated leading exchange couplings J_{1} in these nine structures are negative, indicating that the ferromagnetic order is favoured at least in the 1NN range, in agreement with the super-exchange mechanism discussed before. The 2NN exchange couplings J_{2} are much smaller than J_{1}. And J_{2} values in all structures are also of the ferromagnetic type except TiCl_{3} and TiBr_{3}. Most importantly, except FeCl_{2}, exchange couplings of our candidates have non-negligible strengths up to the 3NN range, revealing the long-range nature of super-exchange. For instance, J_{3} values are shown to be 0.46 meV, 0.44 meV and 0.29 meV for CrI_{3}, CrBr_{3} and CrCl_{3} respectively, nearly one third the amount of J_{1}.
The strongest single-ion anisotropy was found in Fe_{2}O_{2}Br_{2} with A_{z} = −2.47 meV, about nine times larger than that of CrI_{3}. This stems from the symmetry-breaking in local octahedra crystal field caused by the large electrostatic differences between oxygen and bromine anions.
One of the particular HF-SW eigenvector of TiBr_{3} with negative eigenvalue and wave vector k = (2π/15a, 2π/15a) was illustrated in Fig. 3(i), in which part of the lattice (top view) was shown in the left-down corner. The arrows denote spins of Ti^{3+} ions and the colours represent the phases of spins. This kind of AFM configuration has energy lower than the FM states and may be missed by conventional considerations within small supercells due to its long-range nature (with a period of 15 times lattice constant).
Due to the out-of-plane magnetic anisotropies, finite energy gaps Δ^{SW} at the Γ point are presented for the remaining six spectra. These energy gaps Δ^{SW} are dependent on the temperature, and the evolutions of Δ^{SW} are governed by the anisotropic properties including the exchange and single-ion parts. However, until the phase transition, all Δ^{SW} values are preserved as positive to protect the magnetic orders (see the ESI† for more details).
To describe the whole anisotropy quantitatively, including both effects from anisotropies and exchange interactions, we define a dimensionless quantity D = 10^{3}Δ^{SW}/E_{m}, where E_{m} is the largest eigenvalue in SW spectra, reflecting the strengths of exchange interactions inherent in the system. In the CrX_{3} family, the largest anisotropy with D = 49.92 was found in CrI_{3} while the anisotropies in CrBr_{3} and CrCl_{3} are weaker with D = 22.66 and 12.22 respectively, in good agreement with experiments.^{5,13,14} The anisotropies of CuCl_{3}, FeCl_{2} and Fe_{2}O_{2}Br_{2} were found to be 0.2387, 0.8565 and 24.29, respectively. Therefore, the 2D magnetism in Fe_{2}O_{2}Br_{2} has anisotropy comparative to CrBr_{3} while the other 2 systems show nearly isotropic behaviours.
Fig. 4 shows the net magnetic moment evolutions at finite temperatures for the six structures. To figure out the effects of quantum nature, both the results from classical Monte Carlo simulations and QMC were collected here. In the low temperature region, the quantum fluctuations have heavily suppressed magnetism in the low-spin system and even lead to the frustration in Fe_{2}O_{2}Br_{2}. More importantly, quantum effects also play an essential role in high temperature regions and decide the actual positions of phase transitions.
Fig. 4 Mean magnetic moment (per ion) vs. temperature, given by CMC (blue) and QMC (red) for each structure. Green stars correspond to experimentally observed T_{c}. |
The exact T_{c} inherent in quantum models and classical models were obtained by scaling methods and marked in Fig. 4. Detailed scaling behaviours of these systems are shown in Fig. S1–S5 in the ESI.† The experimental T_{c} of the CrX_{3} family (45 K, 28 K and 17 K for CrI_{3}, CrBr_{3} and CrCl_{3} as reported in ref. 5, 13 and 14) were marked with green stars for comparison. The T^{QMC}_{c} predicted by QMC for CrI_{3}, CrBr_{3} and CrCl_{3} are 48 K, 31 K and 18 K, respectively, in excellent agreement with experiments, demonstrating the high accuracy of FulQuamS. However T^{CMC}_{c} values predicted by classical Monte Carlo simulations for CrI_{3}, CrBr_{3} and CrCl_{3} are 33 K, 20 K and 11 K, respectively, significantly below experimental observations, revealing the deficiencies of classical methods for the descriptions of 2D magnetism, also indicating the essential roles of quantum effects in the persisting of 2D magnetic orders near T_{c}.
Notably, the most robust ferromagnetism was found in FeCl_{2} with T_{c} = 931 K. Such high T_{c} is worthwhile for experimental verifications in future works, and the theoretical reasons for this result are two fold: (i) As shown in Table 1, 1NN ferromagnetic exchange couplings of FeCl_{2} are ∼60 meV (equals |J_{1} × S^{2}|), almost 10 times larger than that in CrBr_{3} and (ii) the magnetic lattice of FeCl_{2} is trigonal, supporting six 1NN ferromagnetic couplings for each magnetic moment, which double the case of CrBr_{3}. Therefore, it is straight to understand why FeCl_{2} has T_{c} of ∼30 times higher than CrBr_{3}. Amongst the remaining candidates, Fe_{2}O_{2}Br_{2} was found to be frustrated in QMC due to the spin-1/2 nature and triangular lattice, and the T_{c} of new predicted structures CuCl_{3} turned out to be 74 K, well beyond the T_{c} in CrI_{3}, worth future experimental investigations, too.
Another crucial factor related to quantum effects is the spin number. To figure out this, we performed QMC on a series of systems in which all the magnetic related energies except the spin numbers are the same. Fig. 5(b) shows the averaged spin structure factor (thermally averaged net spin divided by spin number S) of honeycomb ferromagnetic systems which share the same SW spectra with CrCl_{3} (their behaviours in CMC are the same, too) but different spin numbers of 1/2, 3/2 and 9/2, corresponding to the reduced spin, original and enlarged spin models. At temperatures much lower than T_{c}, the spin structure factor of the spin-1/2 system is slightly smaller than that of the spin-3/2 system due to the suppressions from quantum fluctuations embedded in spin-1/2. More importantly, systems with lower spin numbers always reveal higher T_{c}. As a result, at high temperatures, the spin structure factors of spin-1/2, 3/2, 9/2 and infinite (corresponding to classical case) systems are in descending order.
In triangular ferromagnetic systems which share the same SW spectra with FeCl_{2}, we found similar behaviours, as shown in Fig. 5(c). In the low temperature region, the averaged spin structure factors are positively related to the spin numbers due to the strong quantum fluctuations. On the other hand, at high temperatures where phase transition occurs, the spin structure factors become negatively related to the temperature. Hence, the quantum effects from a low spin number also play important roles in the phase transition and always lead to the enhancements of T_{c} in trigonal systems, too.
Footnote |
† Electronic supplementary information (ESI) available: The detailed energy mapping method to extract Heisenberg parameters from DFT calculations, the spin wave theory with interactions solved by Hartree–Fock approximations, the electronic and magnetic structures of five candidates and the impacts of Hubbard U on the d-shell, and verifications on the GGA-level were presented. See DOI: 10.1039/d0nr08687h |
This journal is © The Royal Society of Chemistry 2021 |