To begin with, we found that in the collinear FM, AFM-s and AFM-t spin configurations, all spins completely point along the c-axis (without considering the spin-orbit coupling) as shown in Table 5.3 (a). The AFM-s state is the most stable state and has a largest gap of 1.472 eV. When considering the spin-orbit coupling, all of them converge to similar configurations (Table 5.3 (b)) that the canting of spins away from the c-axis occur, especially the b-component of spins on Cu1, and the FM state has relatively large canting than AFM-s amd AFM-t states. As is mentioned before, the magnetic structure of Cu3Mo2O9 is still not well understood. Two different guessed noncollinear AFM spin configurations are suggested [38, 39]. We began our calculations without considering the relativistic effect by consulting the results of neutron diffraction experiments [38] which give the guessed spin moments on Cu1 and Cu3. We also considered the suggestion of Hamasaki et al. that spins on Cu2 and Cu3 form singlet dimers. The noncollinear AFM spin configurations we found which are slightly unstable than the collinear AFM-s state (Table 5.3(a)) are similar to the results of neutron diffraction experiments. All spins lie in the ac-plane.
The spins on Cu1 form the AFM chain along the b-axis, and the spins on Cu2 and Cu3 form singlet dimers as shown in Figure 5.5. We rotated the spins on Cu1 around the b-axis, while spins on Cu2 and Cu3 remain almost fixed. The relation between the energy and θ of spins on Cu1 in the system is displayed in Figure 5.6. The sinusoidal relation appears. We found that the most stable noncollinear AFM spin configuration in our calculations is that of angle θ = 107◦ or 287◦approximately between a-axis and direction of spins of Cu1 appear, while spins on Cu2 and Cu3 remain unchanged. As we consider the relativistic effect, all spins still lie nearly in the ac-plane in the most stable noncollinear AFM spin configuration. However, when spins on Cu1 rotating to another angle, the spins cant away from the ac-plane occur, especially on Cu1.
The relation of ratio of spins along b-axis on Cu1 and θ of the spins on Cu1 in the system is pictured in Figure 5.7.
5.4 Exchange Interactions
To discuss the magnetic properties of Cu3Mo2O9, we simplified the system to the effective Heisenberg model neglecting the Zeeman energy, i.e. do not considering the external field,
He f f = −1 2∑
k
Jkei· ej, (5.1)
where Jk is the parameter of the exchange interaction between Cui and Cuj atoms, and ei is an unit vector along the direction of local spin moment on Cui atom. A negative exchange
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Table 5.3: Non-collinear spin-polarized GGA+U calculations. Energy (meV/unit cell), band gap (eV) and spin moments (µB) on Cu ions have been given. Spin-orbit coupling has not been considered in (a), and has been considered in (b).
(a)
Cu1-1 -0.002 0.054 0.697 0 0.013 0.629 0.002 0.037 0.627 Cu1-2 0.002 -0.054 0.697 0 0.013 0.629 -0.002 -0.037 0.627 Cu1-3 -0.002 -0.054 0.697 0 0.013 -0.629 -0.002 0.037 -0.627 Cu1-4 0.002 0.054 0.697 0 0.013 -0.629 0.002 -0.037 -0.627 Cu2-1 -0.008 0 0.663 0.014 -0.019 -0.642 0.015 0.004 -0.655 Cu2-2 0.008 0 0.663 0.014 -0.019 0.642 0.015 -0.004 0.655 Cu2-3 0.008 0 0.663 -0.014 -0.019 -0.642 -0.015 -0.004 -0.655 Cu2-4 -0.008 0 0.663 -0.014 -0.019 0.642 -0.015 0.004 0.655 Cu3-1 0.002 0 0.679 0.008 0.010 0.662 -0.008 0.012 -0.667 Cu3-2 -0.002 0 0.679 0.008 0.010 -0.662 -0.008 -0.012 0.667 Cu3-3 -0.002 0 0.679 -0.008 0.010 0.662 0.008 -0.013 -0.667 Cu3-4 0.002 0 0.679 -0.008 0.010 -0.662 0.008 0.013 0.667
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Figure 5.5: Calculated spin configurations are similar to the results of neutron diffraction ex-periments. All spins lie in the ac-plain almost. The spins on Cu1 form the AFM chain, and the spins on Cu2 and Cu3 form singlet dimers. We rotated the spins on Cu1 around the b-axis, while the spins on Cu2 and Cu3 remain fixed. The most stable noncollinear AFM spin configuration is that of an angle θ = 107◦or 287◦between a-axis and direction of moments on Cu1.
Figure 5.6: The energy versus θ of the spins on Cu1 in the system. The most stable noncollinear AFM spin configuration is that of an angle θ = 107◦or 287◦approximately between a-axis and direction of the spins on Cu1.
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Table 5.4: Non-collinear spin-polarized GGA+U calculations without the spin-orbit coupling.
Energy (meV/unit cell), band gap (eV) and spin moments (µB) on Cu ions of spins configurations are given.
θ =20◦ θ =46◦ θ =61◦
energy 12.50 11.37 9.72
gap 1.46 1.46 1.47
Cu1-1 0.596 0 0.212 0.442 0 0.452 0.292 0 0.560 Cu1-2 -0.596 0 0.212 -0.442 0 0.452 -0.292 0 0.560 Cu1-3 -0.596 0 -0.212 -0.442 0 -0.452 -0.292 0 -0.560 Cu1-4 0.596 0 -0.212 0.442 0 -0.452 0.292 0 -0.560 Cu2-1 -0.225 0 -0.606 -0.206 0 -0.613 -0.221 0 -0.608 Cu2-2 0.226 0 -0.606 0.206 0 -0.613 0.222 0 -0.607 Cu2-3 -0.226 0 0.606 -0.206 0 0.613 -0.222 0 0.607 Cu2-4 0.225 0 0.606 0.206 0 0.613 0.221 0 0.608 Cu3-1 0.236 0 0.624 0.211 0 0.633 0.230 0 0.627 Cu3-2 -0.234 0 0.625 -0.212 0 0.633 -0.228 0 0.627 Cu3-3 0.234 0 -0.625 0.212 0 -0.633 0.228 0 -0.627 Cu3-4 -0.236 0 -0.624 -0.212 0 -0.633 -0.230 0 -0.627
θ =73◦ θ =107◦ θ =122◦
energy 8.09 2.68 3.75
gap 1.47 1.47 1.47
Cu1-1 0.188 0 0.603 -0.196 0 0.600 -0.323 0 0.542 Cu1-2 -0.188 0 0.603 0.196 0 0.600 0.323 0 0.542 Cu1-3 -0.188 0 -0.603 0.196 0 -0.600 0.323 0 -0.542 Cu1-4 0.188 0 -0.603 -0.196 0 -0.600 -0.323 0 -0.542 Cu2-1 -0.203 0 -0.614 -0.218 0 -0.609 -0.231 0 -0.604 Cu2-2 0.203 0 -0.614 0.217 0 -0.609 0.231 0 -0.604 Cu2-3 -0.203 0 0.614 -0.217 0 0.609 -0.231 0 0.604 Cu2-4 0.203 0 0.614 0.217 0 0.609 0.231 0 0.604 Cu3-1 0.208 0 0.634 0.239 0 0.623 0.241 0 0.622 Cu3-2 -0.208 0 0.634 -0.238 0 0.624 -0.240 0 0.623 Cu3-3 0.208 0 -0.634 0.237 0 -0.624 0.240 0 -0.623 Cu3-4 -0.208 0 -0.634 -0.239 0 -0.623 -0.241 0 -0.622
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Cu1-1 -0.494 0 0.394 -0.596 0 0.211 -0.632 0 0.006
Cu1-2 0.494 0 0.394 0.596 0 0.211 0.632 0 0.006
Cu1-3 0.494 0 -0.394 0.596 0 -0.211 0.632 0 -0.006 Cu1-4 -0.494 0 -0.394 -0.596 0 -0.211 -0.632 0 -0.006 Cu2-1 -0.231 0 -0.604 -0.230 0 -0.604 -0.227 0 -0.605 Cu2-2 0.232 0 -0.603 0.231 0 -0.604 0.228 0 -0.605 Cu2-3 -0.232 0 0.603 -0.231 0 0.604 -0.228 0 0.605
Cu2-4 0.231 0 0.604 0.230 0 0.604 0.227 0 0.605
Cu3-1 0.243 0 0.622 0.242 0 0.622 0.238 0 0.623
Cu3-2 -0.241 0 0.622 -0.240 0 0.623 -0.237 0 0.624 Cu3-3 0.241 0 -0.622 0.240 0 -0.623 0.237 0 -0.624 Cu3-4 -0.243 0 -0.622 -0.242 0 -0.622 -0.239 0 -0.623
θ =200◦ θ =230◦ θ =265◦
energy 12.50 10.98 6.17
gap 1.46 1.47 1.48
Cu1-1 -0.600 0 -0.199 -0.397 0 -0.491 -0.059 0 -0.629 Cu1-2 0.600 0 -0.199 0.397 0 -0.491 0.003 0 -0.631
Cu1-3 0.600 0 0.199 0.397 0 0.491 0.059 0 0.629
Cu1-4 -0.600 0 0.199 -0.397 0 0.491 -0.003 0 0.631 Cu2-1 -0.226 0 -0.606 -0.222 0 -0.607 -0.237 0 -0.601 Cu2-2 0.226 0 -0.606 0.223 0 -0.607 0.239 0 -0.601 Cu2-3 -0.226 0 0.606 -0.223 0 0.607 -0.239 0 0.601
Cu2-4 0.225 0 0.606 0.222 0 0.607 0.237 0 0.601
Cu3-1 0.236 0 0.624 0.232 0 0.626 0.220 0 0.630
Cu3-2 -0.234 0 0.625 -0.230 0 0.627 -0.217 0 0.631 Cu3-3 0.234 0 -0.625 0.230 0 -0.627 0.217 0 -0.631 Cu3-4 -0.236 0 -0.624 -0.232 0 -0.626 -0.220 0 -0.630
θ =290◦ θ =320◦ θ =350◦
energy 3.49 6.84 10.88
gap 1.47 1.46 1.46
Cu1-1 0.308 0 -0.551 0.475 0 -0.416 0.621 0 -0.120 Cu1-2 -0.308 0 -0.551 -0.475 0 -0.416 -0.621 0 -0.120 Cu1-3 -0.308 0 0.551 -0.475 0 0.416 -0.621 0 0.120
Cu1-4 0.308 0 0.551 0.475 0 0.416 0.621 0 0.120
Cu2-1 -0.231 0 -0.604 -0.231 0 -0.604 -0.229 0 -0.605 Cu2-2 0.231 0 -0.604 0.232 0 -0.603 0.230 0 -0.604 Cu2-3 -0.231 0 0.604 -0.232 0 0.603 -0.230 0 0.604
Cu2-4 0.230 0 0.604 0.231 0 0.604 0.229 0 0.605
Cu3-1 0.241 0 0.622 0.243 0 0.622 0.240 0 0.623
Cu3-2 -0.239 0 0.623 -0.241 0 0.622 -0.239 0 0.623 Cu3-3 0.239 0 -0.623 0.241 0 -0.622 0.239 0 -0.624 Cu3-4 -0.241 0 -0.622 -0.243 0 -0.622 -0.241 0 -0.623
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Table 5.5: Non-collinear spin-polarized GGA+U calculations with the spin-orbit coupling. En-ergy (meV/unit cell), band gap (eV) and spin moments (µB) on Cu ions of spins configurations we found have been given.
θ =20◦ θ =46◦ θ =61◦
energy 10.58 9.84 8.31
gap 1.44 1.45 1.46
Cu1-1 0.591 0.043 0.212 0.437 0.038 0.451 0.290 0.031 0.558 Cu1-2 -0.591 0.043 0.212 -0.437 0.039 0.451 -0.290 0.031 0.558 Cu1-3 -0.591 0.043 -0.212 -0.437 0.039 -0.451 -0.290 0.031 -0.558 Cu1-4 0.591 0.043 -0.212 0.437 0.038 -0.451 0.290 0.031 -0.558 Cu2-1 -0.213 -0.010 -0.606 -0.195 -0.014 -0.611 -0.209 -0.016 -0.607 Cu2-2 0.213 -0.009 -0.606 0.195 -0.014 -0.611 0.210 -0.016 -0.607 Cu2-3 -0.213 -0.009 0.606 -0.195 -0.014 0.612 -0.210 -0.016 0.607 Cu2-4 0.213 -0.009 0.606 0.195 -0.014 0.612 0.209 -0.016 0.607 Cu3-1 0.245 -0.008 0.614 0.220 -0.002 0.624 0.240 0.003 0.616 Cu3-2 -0.244 -0.008 0.615 -0.220 -0.002 0.624 -0.238 0.003 0.617 Cu3-3 0.243 -0.008 -0.615 0.220 -0.002 -0.624 0.238 0.003 -0.617 Cu3-4 -0.245 -0.008 -0.614 -0.220 -0.002 -0.624 -0.240 0.003 -0.616
θ =73◦ θ =107◦ θ =122◦
energy 7.02 1.87 2.51
gap 1.46 1.47 1.46
Cu1-1 0.189 0.025 0.600 -0.191 0.001 0.599 -0.320 -0.008 0.541 Cu1-2 -0.189 0.025 0.600 0.191 0.002 0.599 0.320 -0.008 0.541 Cu1-3 -0.189 0.025 -0.600 0.191 0.002 -0.599 0.320 -0.008 -0.541 Cu1-4 0.189 0.025 -0.600 -0.191 0.001 -0.599 -0.320 -0.008 -0.541 Cu2-1 -0.193 -0.016 -0.612 -0.212 -0.017 -0.606 -0.217 -0.012 -0.604 Cu2-2 0.193 -0.016 -0.612 0.212 -0.017 -0.606 0.218 -0.012 -0.604 Cu2-3 -0.193 -0.016 0.612 -0.212 -0.017 0.606 -0.218 -0.012 0.604 Cu2-4 0.193 -0.016 0.612 0.212 -0.017 0.606 0.217 -0.012 0.604 Cu3-1 0.217 0.005 0.625 0.242 0.015 0.616 0.249 0.017 0.613 Cu3-2 -0.217 0.005 0.625 -0.240 0.015 0.616 -0.247 0.017 0.613 Cu3-3 0.217 0.005 -0.625 0.240 0.015 -0.616 0.247 0.017 -0.613 Cu3-4 -0.217 0.005 -0.625 -0.242 0.015 -0.616 -0.249 0.017 -0.613
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Cu1-1 -0.490 -0.022 0.393 -0.592 -0.033 0.210 -0.628 -0.041 0.006 Cu1-2 0.490 -0.022 0.393 0.592 -0.033 0.210 0.628 -0.041 0.006 Cu1-3 0.490 -0.022 -0.393 0.592 -0.033 -0.210 0.628 -0.040 -0.005 Cu1-4 -0.490 -0.022 -0.393 -0.592 -0.033 -0.210 -0.628 -0.041 -0.006 Cu2-1 -0.218 -0.006 -0.604 -0.216 -0.001 -0.605 -0.215 0.004 -0.605 Cu2-2 0.219 -0.006 -0.604 0.217 -0.001 -0.604 0.215 0.005 -0.605 Cu2-3 -0.218 -0.006 0.604 -0.217 0 0.604 -0.215 0.005 0.605
Cu2-4 0.218 -0.006 0.604 0.216 0 0.605 0.215 0.005 0.605
Cu3-1 0.250 0.016 0.612 0.249 0.014 0.612 0.248 0.012 0.613 Cu3-2 -0.248 0.016 0.613 -0.247 0.014 0.613 -0.246 0.012 0.614 Cu3-3 0.248 0.016 -0.613 0.247 0.014 -0.613 0.246 0.012 -0.614 Cu3-4 -0.250 0.016 -0.612 -0.249 0.014 -0.612 -0.248 0.012 -0.613
θ =200◦ θ =230◦ θ =265◦
energy 10.57 9.41 5.15
gap 1.44 1.45 1.47
Cu1-1 -0.595 -0.043 -0.200 -0.394 -0.037 -0.490 -0.058 -0.020 -0.626 Cu1-2 0.595 -0.043 -0.200 0.394 -0.037 -0.490 0.002 -0.016 -0.629 Cu1-3 0.595 -0.043 0.200 0.394 -0.037 0.490 0.058 -0.019 0.626 Cu1-4 -0.595 -0.043 0.200 -0.394 -0.037 0.490 -0.002 -0.015 0.629 Cu2-1 -0.213 0.009 -0.606 -0.210 0.015 -0.607 -0.222 0.020 -0.602 Cu2-2 0.214 0.009 -0.606 0.210 0.015 -0.606 0.223 0.021 -0.602 Cu2-3 -0.213 0.009 0.606 -0.210 0.015 0.606 -0.223 0.021 0.602 Cu2-4 0.213 0.009 0.606 0.210 0.015 0.607 0.222 0.020 0.602
Cu3-1 0.246 0.008 0.614 0.242 0 0.616 0.233 -0.011 0.619
Cu3-2 -0.244 0.008 0.615 -0.240 0 0.617 -0.231 -0.009 0.620 Cu3-3 0.244 0.008 -0.615 0.240 0 -0.617 0.231 -0.009 -0.620 Cu3-4 -0.246 0.008 -0.614 -0.242 0 -0.616 -0.233 -0.011 -0.619
θ =290◦ θ =320◦ θ =350◦
energy 2.32 5.28 8.98
gap 1.46 1.46 1.44
Cu1-1 0.306 0.007 -0.549 0.472 0.020 -0.415 0.617 0.037 -0.119 Cu1-2 -0.306 0.007 -0.549 -0.472 0.020 -0.415 -0.617 0.037 -0.119 Cu1-3 -0.306 0.007 0.549 -0.472 0.021 0.415 -0.617 0.037 0.119 Cu1-4 0.306 0.007 0.549 0.472 0.020 0.415 0.617 0.037 0.119 Cu2-1 -0.217 0.013 -0.604 -0.218 0.006 -0.604 -0.216 -0.002 -0.605 Cu2-2 0.218 0.013 -0.604 0.219 0.006 -0.604 0.217 -0.002 -0.605 Cu2-3 -0.218 0.013 0.604 -0.219 0.006 0.604 -0.216 -0.002 0.605 Cu2-4 0.217 0.013 0.604 0.218 0.006 0.604 0.216 -0.002 0.605 Cu3-1 0.248 -0.017 0.613 0.250 -0.016 0.612 0.249 -0.013 0.613 Cu3-2 -0.247 -0.017 0.614 -0.248 -0.016 0.613 -0.247 -0.013 0.613 Cu3-3 0.247 -0.017 -0.614 0.248 -0.016 -0.613 0.247 -0.013 -0.614 Cu3-4 -0.248 -0.017 -0.613 -0.250 -0.016 -0.612 -0.249 -0.013 -0.613
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Figure 5.7: All spins lie in the ac-plane in the most stable noncollinear AFM spin configuration, i.e. an angle θ = 107◦or 287◦between a-axis and direction of spins of Cu1. When the spins on Cu1 rotate to another angle, the spins cant away from the ac-plane occur, especially on Cu1.
parameter here means that the two corresponding spins tend to point along the oppsite directions.
For simplicity, here we only consider the nearest interaction between Cui and Cuj atoms, and the second-nearest interaction between Cu1 atoms. The four exchange parameters (J1, J2, J3 and J4) correspond to nearest Cu-Cu interactions (Cu1-Cu2, Cu1-Cu3, Cu2-Cu3and Cu1-Cu1) in the system. And the second-nearest interaction between Cu1 atoms is labeled as J5. To obtain these five parameters, we performed a series of collinear spin-polarized calculations. The results are listed in Table 5.6. Through solving the equations,
H = H0+ Hmag+ He f f, (5.2)
where H0is the internal energy of the system, Hmagis the magnetization energy and He f f is the Hamiltonian of the effective Heisenberg model. we can obtain the exchange parameters.
Esinglet = E0+ 8J3+ 8J4− 4J5= 0 Etriplet = E0− 8J3+ 8J4− 4J5= 23.36 E(a)= E0− 8J1− 8J2− 8J3− 4J5= 306.92
E(b)= E0− 8J1− 16J2− 4J3− 8J4− 4J5= 531.02 E(c)= E0− 8J1− 8J2− 8J3− 8J4− 4J5= 521.60 E(d)= E0− 8J1− 8J2− 8J3− 4J4= 424.24 .
(5.3)
The results are listed in Table 5.7. The two of parameters (J1 and J5) are positive, and the remaining are negative. It is impossible to simultaneously minimize the energy of all exchange interactions. As is mentioned before, geometric frustration possibly plays an important role in this system. The two strongest exchange interaction, J4 and J5, would lead to the Cu1 AFM
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Table 5.6: A series of collinear spin-polarized GGA+U calculations. To obtain the four parame-ters (J1, J2, J3and J4), we performed a series of collinear spin-polarized calculations concluding ferromagnetic configuration and some ferrimagnetic configurations which randomly flip spins in the unit cell. Energy (meV/2 unit cell) and spin moments (µB) on Cu ions are listed.
AFM-s AFM-t (a) (b) (c) (d)
energy 0 23.36 306.92 531.02 521.60 424.24
Cu1-1(5) 0.631 0.630 0.701 0.692 0.687 −0.643 0.669 Cu1-2(6) 0.631 0.630 0.642 0.702 0.702 0.701 0.701 Cu1-3(7) −0.631 −0.630 0.701 0.692 0.687 −0.642 0.669 Cu1-4(8) −0.631 −0.630 −0.618 0.701 0.701 0.701 0.701 Cu2-1(5) −0.648 −0.657 0.659 0.664 0.664 0.659 0.659 Cu2-2(6) 0.648 0.657 0.665 0.664 0.664 0.665 0.665 Cu2-3(7) −0.648 −0.657 0.665 −0.643 −0.654 0.665 0.665 Cu2-4(8) 0.648 0.657 0.659 0.665 0.664 0.657 0.663 Cu3-1(5) 0.667 −0.674 0.679 0.685 0.684 0.685 0.686 Cu3-2(6) −0.667 0.674 0.685 0.685 0.681 0.679 0.679 Cu3-3(7) 0.667 −0.674 0.685 0.675 −0.668 0.677 0.684 Cu3-4(8) −0.667 0.674 0.679 0.685 0.685 0.685 0.685
Table 5.7: The distance D between Cu ions is listed and the exchange parameters (meV) in (a) our results qualitatively agree with (b) the experimental results (only J3 and J4 obtained in the experiments).
J1 J2 J3 J4 J5
(Cu1-Cu2) (Cu1-Cu3) (Cu2-Cu3) (Cu1-Cu1) (Cu1-Cu1) D 2.97 ˚A 2.98 ˚A 3.17 ˚A 3.44 ˚A 6.88 ˚A
(a) 0.4375 -1.9075 -1.460 -26.835 2.50
(b) -4.0 -22.0
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Figure 5.8: The NM electronic band structure.
chain. The exchange parameter J3 which is larger than J1 and J2 would lead to singlet dimer formed by Cu2 and Cu3. The singlet dimers weakly interact with the Cu1 chain. Our results agree with the experimental results well (only J3 and J4 obtained in the experiments). The strength of J4we obtained is slightly larger than result of experiments. And J3is slightly smaller.
When considering the relativistic effect, the spins cant away from the ac-plane, especially on Cu1, as can be seen in Table 5.5. We think that the exchange interactions tend to make spins lie in the ac-plane, and the relativistic effect tends to make spins cant along the b-axis. When spins on Cu1 rotate away from the more stable pointings (θ = 107◦ or 287◦ between a-axis and direction of the spins on Cu1), the dominance of exchange interactions decrease and b-component of spins increase. Therefore, the spins on Cu1 cant the most in the FM state.