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Figure 5.18: The electronic band structure of the most stable non-collinear AFM state (θ = 107◦ or 287◦approximately) we found from GGA+U calculation with spin-orbit coupling.
the relativistic effect do not make considerable difference in the electronic band structure and densities of states.
5.6 Spontaneous Electric Polarization
The Cu3Mo2O9 has been found by Kuroe et al. [37] to have remarkable magnetoelectric effect, being as strong as that of TbMnO3. A small spin cluster model has been suggested which is different from the magnetic superlattice. Since there has been no ab initio calculation yet, we performed the Berry’s phase calculations to obtain the theoretical spontaneous electric po-larization for Cu3Mo2O9. As is mentioned before, the electric polarization can appear when the spatial inversion symmetry of the system is broken. Such breaking can be caused by crystal structure distortion or the special spin configuration. In the Cu3Mo2O9, the electric polarization is caused by the breaking of the inversion symmetry of spin configuration. We calculated the polarization of the most stable noncollinear AFM state we found (θ = 107◦approximately) with considering the relativistic effect. A theoretical value ('866.58 µC/m2) is in the same order of experimental results. However, the direction is different from the experimental one. The ex-perimental results reveal that the spontaneous polarization appears ('500 µC/m2) along c-axis,
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Figure 5.19: Total and site-projected electronic densities of states of the most stable non-collinear AFM state we found from GGA+U calculation with spin-orbit coupling.
but our theoretical polarization is along b-axis. We look into the most stable spin configuration we found to try to realize the system further. Spins on Cu-sites are listed in Table 5.5. The Cu1 ions are located on the symmetric center of the system, direction of spin on Cu1 does not affect the spatial symmetry of the system. Cu2-1 and Cu2-2 are spatial symmetric sites of Cu2-4 and Cu2-3 respectively, and Cu3-1 and Cu3-2 are spatial symmetric cites of the Cu3-4 and Cu3-3 respectively. We inspected directions of spins in the most stable state we found. Spins on Cu2-1 and Cu2-4 point to oppsite directions in the x and z components. And, spins on Cu2-2 and Cu2-3 (Cu3-1 and Cu3-4, Cu3-2 and Cu3-3) also point to oppsite directions in the x and z components.
The spatial inversion symmetry is broken in the direction lie in the xz-plane. However, the spon-taneous electricpolarization appears along the b-axis, which does not agree with our expectation.
In our opinion, there are some possible explanations for dealing with the disagreement of direc-tion of the electric polarizadirec-tion between experimental results and our calculadirec-tions. First, the direction of polarization found by Kuroe et al. is wrong. Second, the electron-electron Coulomb repulsion has been considered on every Cu sites with the commonly used values of U = 4.5 eV and J = 0.9 eV in our calculations. However, there are three crystallographically inequivalent Cu ions. The Hubbard U on crystallographically different Cu sites may have different values.
Third, since the DFT+U method is still a single particle method, it may be probably insufficient to deal with the strong correlated many-body behavior. Forth, since the magnetic structure is still not well determined, the spin configuration we found is probably not the magntic structure of the ground state.
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Table 5.8: A spin configuration that the theoretical electric polarization of the state agrees with the experimental results. Spin-orbit coupling has been considered. Energy (meV/unit cell), band gap (eV) and spin moments (µB) on Cu ions are listed.
energy 11.98
gap 1.36
Cu1-1 0.105 0.606 0.114 Cu1-2 -0.174 -0.585 0.131 Cu1-3 -0.170 -0.585 -0.137 Cu1-4 0.109 0.606 -0.109 Cu2-1 -0.013 0.650 0.059 Cu2-2 -0.084 0.644 0.057 Cu2-3 0.086 -0.645 -0.022 Cu2-4 0.010 -0.644 -0.096 Cu3-1 0.166 0.626 0.155 Cu3-2 0.176 0.623 -0.154 Cu3-3 -0.155 -0.635 0.131 Cu3-4 -0.146 -0.637 -0.132
In addition to the results above, we have found a spin configuration that the theoretical electric polarization of the state agrees with the experimental results. However, the symmetry of the spin configuration does not satisfy the conditions suggested by results of the neutron diffraction experiment. And, spins on neighboring Cu2 and Cu3 do not form a singlet dimer.
The spin configuration is listed in Table 5.8. We can see that all spins mainly point along b-axis.
The polarization (only '31 µC/m2) appears along c-axis. We hope that our calculated results can contribute to the studies of Cu3Mo2O9.
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We have done a series of ab initio calculations to study the interesting material, Cu3Mo2O9, which reveals strong magnetoelectric effect. There has no ab initio study of it and its magnetic structure is still not well defined. Our ab initio calculations are based on spin density functional theory with generalized gradient approximation (GGA). We used the accurate full-potential pro-jector augmented-wave (PAW) method, as implemented in the VASP code. The core radii of the copper, molybdenum and oxygen atoms are 1.312, 1.455 and 0.8, respectively.
In our GGA calculations, the NM and the FM states are matallic. Though a gap (∼0.2 eV) opens in collinear AFM state, it is too small to agree with the crystal field theory. Since Cu3Mo2O9was prepared as a brown-orange powder, the absorbed energy corresponding to the complementary color of visible light must be at least 1.0 eV, i.e. the energy gap must be at least 1.0 eV. Though results of neutron diffraction experiments reveal that the magnetic structure of Cu3Mo2O9 probably be non-collinear, rather than our collinear configurations, the effects of exchange interaction are insufficient to open a wide enough gap. Furthermore, the spin mo-ments on Cu ions are much smaller than the theoretical value (1.0 µB). Therefore, we can have a conclusion that the electron-electron Coulomb repulsion in the system must be considered.
The GGA+U method consists in Hubbard-like correction gives a good description of electronic correlation with the commonly used values of U = 4.5 eV and J = 0.9 eV so that a reasonable band gap could be obtained.
In our GGA+U calculations, a reasonable gap was obtained. The most stable state we found is the AFM-s state with a largest gap of 1.472 eV in our calculations. The noncollinear AFM spin configurations we found are similar to results of neutron diffraction experiments. When we do not consider the relativistic effect, all spins lie in the ac-plane. Spins on Cu1 form an AFM chain. Spins on Cu2 and Cu3 form a singlet dimer weakly interacting with the AFM chain formed by Cu1 ions. The most stable noncollinear AFM state we found is that an angle θ = 107◦ and 287◦ approximately appear between a-axis and direction of spin of Cu1. The energy gap is 1.47 eV. As spins on Cu1 are rotated away from θ = 107◦ and 287◦, the energy gap decrease gradually to 1.44 eV when θ = 197◦and 17◦. As we consider the relativistic effect, all
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spins still lie in the ac-plane almost in the most stable spins configuration. However, when spins on Cu1 rotating to another angle, the spins cant away from the ac-plane, especially on Cu1. The b-component of spin moments on Cu1 increase gradually to the upper limit (about 7% of spin moment) when θ = 197◦and 17◦.
To obtain the theoretical exchange parameters (J1, J2, J3, J4 and J5) of the system, a series of collinear spin-polarized calculations with different spin configurations have been performed.
Two of exchange parameters (J1 and J5) in our results are positive, and the remaining (J2, J3 and J4) are negative. Our results agree with the experimental results well. We think that the exchange interactions tend to make spins lie in the ac-plane, and the relativistic effect tends to make spins point along the b-axis. Since the relativistic effect is fixed, when spins on Cu1 are rotated away from the most stable pointings, the dominance of exchange interactions decrease and b-component of spins increase.
We analysed the electronic band structure and densities of states of NM, collinear FM, AFM-s and AFM-t, and the moAFM-st AFM-stable noncollinear AFM AFM-stateAFM-s (GGA+U). The NM AFM-state iAFM-s metallic.
When considering the spin-polarization, some transfering of electrons occur, and a series of con-duction band appear in the vicinity of 1.5 eV. These concon-duction bands are effects of considering Coulomb repulsion of electrons. An electron can occupy one these bands only if it overcomes the Hubbard U from another electron in the band. We can see that the bandwidth in AFM states (both collinear and noncollinear) are narrow than the collinear FM state. And, the apparent shifts occur between collinear and noncollinear states in the vicinity of 1.5 eV.
We also have done the Berry’s phase calculations with considering the relativistic effect.
The strength of theoretical spontaneous electric polarization of the most stable state we found ('866.58 µC/m2) agrees with the results of experiments ('500 µC/m2). However, the direc-tions are different. The polarization measured by Kuroe et al. is along the c-axis, but along the b-axis in our calculations. In our opinion, there are some possible explanations for dealing with the different directions of the electric polarization between experimental and our results.
First, the direction of polarization measured by Kuroe et al. is wrong. Second, the electron-electron Coulomb repulsion has been considered on every Cu sites with the commonly used values of U = 4.5 eV and J = 0.9 eV in our calculations. However, there are three crystallo-graphically inequivalent Cu ions. The Hubbard U on crystallocrystallo-graphically different Cu sites may have different values. Third, since the DFT+U method is still a single particle method, it may be probably insufficient to deal with the strong correlated many-body behavior. Forth, since the magnetic structure is still not well determined, the spin configuration we found is probably not the magntic structure of the ground state.
In addition to the results above, we have found a spin configuration that the theoretical electric polarization ('31 µC/m2along c-axis) of the state agrees with the experimental results.
However, the symmetry of the spin configuration does not satisfy the conditions suggested by results of the neutron diffraction experiment. And, spins on neighboring Cu2 and Cu3 do not form a singlet dimer.
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The theoretical electric polarization in our calculations does not completely agree with ex-perimental results. Nevertheless, the spin configurations, theoretical exchange parameters have been obtained in our calculations. And, the electronic structure has been analysed. We hope that our results can help those who wants to study this interesting material further.
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