國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
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.
5.5 Electronic Structure
We ploted the electronic band structure and densities of states of our NM calculations in Figure 5.8 and 5.9. The lowest valence band at the Γ point is dominant Cu1-dxy hybridized with O1-py orbitals in the vicinity of −7.1 eV with bandwidth of about 0.5 eV. In the vicinity of −5.75 eV and −5.5 eV, bands are dominated by the Cu2-dz2 orbital hybridized with O6-pz orbital and Cu3-dx2−y2 orbital hybridized with O5-px orbital. Above −5.5 eV and below −0.5 eV, bands are too complex to be recognized. In the vicinity of −0.4 eV, bands are dominated by the Cu3-dx2−y2 and the Cu3-dzx orbitals hybridized with O1-px orbital. The NM state is
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
Figure 5.9: Total and site-projected electronic densities of states of the NM state.
metallic, and the bands crossing the Fermi level are dominated by Cu2-dz2 orbital hybridized with O1-pzand O2-pyorbitals, and by Cu3-dx2−y2 and Cu3-dzxorbitals hybridized with O1-px, O1-pz and O3-py orbitals. In the vicinity of 0.4 eV, bands are dominated by Cu1-dx2−y2 and Cu1-dxyorbitals hybridized with O1-pyorbital. Above 2.5 eV, bands are dominated by orbitals of Mo atoms hybridized with p-orbitals of the surrounding O atoms.
Here we look into orbital- and site-projected DOS of Cu atoms more closely to examine the crystal field theory. The Cu1 ion and its surrounding O ions (two O1 and two O3 ions) form a parallelogram, and the nearest ligands are O1 ions (1.86 ˚A) which must affect the splitting of d-orbitals of Cu1 the most, while O2 (2.30 ˚A) ions are too far away from the Cu1 ion to cause considerable splitting. There is an angle of about 21◦ between y-axis and the direction that O1 ion attached to the central Cu1 ion along, and an angle of about 25◦ between x-axis and the direction that O3 ion attached to the central Cu1 ion along. We infered that dx2−y2 and dxy orbitals must correspond to the high energy levels. Since O3 ions have longer bond length, dz2 and dzxorbitals must correspond to low energy levels. And the dyz orbital must correspond to the lowest energy level. As can be seen in Figure 5.10 of Cu1, in the vicinity of about 0.4 eV, dx2−y2 and dxy orbitals have almost same DOS, due to locations of O1 ions that dx2−y2 and dxy orbitals experience almost same strength Coulomb repulsion. In the vicinity of about −0.6 eV, dz2 and dzx orbitals dominate and must correspond to higher energy level than dyz orbital.
In the same way, the Cu2 ion and its surrounding O ions (one O1, two O2 and one O6 ions) form a quadrangle-like structure, the nearest ligands are O6 ions which must lead to the largest effect of splitting of the d-orbitals, while O7 ions are too far away from the Cu2 ion to cause
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
Figure 5.10: NM orbital- and site-projected electronic densities of states of Cu1 atoms.
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
Figure 5.11: NM orbital- and site-projected electronic densities of states of Cu2 and Cu3 atoms.
the considerable effect. The quadrangle-like structure almost lie in yz-plane. Since O1, O2 and O6 ions almost attached to the central Cu2 ion along the y and z axes, dz2 and dx2−y2 orbitals must correspond to higher energy levels than remaining dxy, dyz and dzx orbitals. As can be seen in Figure 5.11 of Cu2, the results agree with our expectation generally. The Cu3 and its surrounding O ions (one O1, two O3 and one O5 ions) also form a quadrangle-like structure, The nearest ligands are O5 ions which can lead to the largest effect of splitting of the d-orbitals, while the surrounding O4 and O7 ions are too far away from the Cu3 ion to cause considerable effects. The two O3 and one O5 ions attached to the central Cu3 ion along y-axis and x-axis almost. Therefore, the dx2−y2 orbital must correspond to high energy levels. The O1 ion lie in-between the x and z axes, pushing that dzx orbital also to higher energy level. The remaining orbitals experience smaller electron-electron Coulomb repulsion so that they stay in the lower energy levels. As can be seen in Figure 5.11 of Cu3, the results agree with our expectation.
By the way, those bands across the Fermi level signify that ions near corresponding d-orbitals experience less Coulomb repulsion, causing shorter bond length. Our results agree with crystal field theory well.
Then, we ploted the electronic band structure and densities of states of the collinear FM state from our collinear spin-polarized GGA+U calculations in Figure 5.12 and 5.13. As mentioned before, the lowest valence band at the Γ point is the dominant Cu1-dxyband hybridized with O1-pyorbital near −7.1 eV in the NM state. When considering the spin-polarization, in the collinear
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
Figure 5.12: The electronic band structure of the collinear FM state from the collinear spin-polarized GGA+U calculations.
Figure 5.13: Total and site-projected electronic densities of states of the collinear FM state from the collinear spin-polarized GGA+U calculations.
‧
FM state which is listed in Table 5.3 (b), the band split into two bands (spin up and down) between −7.26 eV and −6.3 eV. Besides, considerable states dominated by Cu1-dx2−y2 orbital hybridized with O1-py orbital appear in the lowest valence band at about −7.26 eV. Near −6.6 eV, spin-up bands are Cu2-dz2 orbital hybridized with O1-pzand O6-pzorbitals. Here it should be mentioned that bands are dominated by Cu2-dz2 orbital hybridized with mainly O6-pzorbital in the vicinity of about −5.75 eV in the NM state. There must be some transferring of electrons after considering spin-polarization. At about −6.4 eV, spin-up bands are Cu3-dx2−y2 orbital hybridized with O5-pxorbital and Cu3-dzxorbital hybridized with O1-pxorbital. Likewise, there must be some transferring of electrons infered from the appearance of the band dominated by corresponding Cu3-dzxorbital hybridized with O1-pxorbital after considering spin-polarization.
The highest occupied spin-up and spin-down bands are dominated by corresponding Cu1-dx2−y2
and Cu1-dxyorbitals hybridized with O1-pyorbital. An energy gap (0.92 eV) has been opened here. Those spin-down bands in the interval between about 1.0 eV and 2.0 eV are effects of considering Coulomb repulsion of electrons. A spin-down electron can occupy those orbitals only if it overcomes the Hubbard U . In the energy range between 1.0 eV and 2.0 eV, bands are the dominantly Cu1-dx2−y2 and Cu1-dxyorbitals. In the vicinity of 1.2 eV, bands are mainly Cu3-dx2−y2 and Cu3-dzx orbitals. And near 1.5 eV, bands are dominant Cu2-dz2 orbital slightly hybridized with O1-pz orbital. Above 3.0 eV, bands are dominated by orbitals of Mo atoms hybridized with p-orbitals of the surrounding O atoms.
We also ploted the electronic band structure and densities of states of the collinear AFM-t and AFM-s states (Table 5.3 (b)) in Figure 5.14, 5.15, 5.16 and 5.17. The electronic structure of the two states are similar, the main difference between them that we can recognize are located in the vicinity of 1.5 eV. The lowest valence band is also the dominant Cu1-dxyband hybridized with O1-pyorbital at −6.6 eV. However, the bandwidth is narrowed to about 0.2 eV. The bands dominated by Cu2-dz2 orbital hybridized with O1-pz and O6-pz orbitals in the collinear FM state rise to about −6.4 eV in the collinear AFM state. And, the bands dominated by Cu3-dx2−y2
orbital hybridized with O5-px orbital and Cu3-dzx orbital hybridized with O1-px orbital rise to about −6.05 eV. Likewise, bands above −6 eV and below −0.5 eV are too complex to be recognized. The highest occupied band with narrow bandwidth of about 0.1 eV is dominated by Cu1-dzx orbital hybridized with O2-pz and O3-px orbitals, and Cu1-dz2 orbital hybridized with O2-pz orbital. A reasonable energy gap of 1.41 eV was obtained in the AFM-t state, and increases to 1.472 eV in the AFM-s state. In the intervals between 1.35 eV and 1.8 eV, there is a series of narrow conduction bands. In the AFM-t state, the lowest unoccupied bands in the vicinity of 1.35 eV are dominated by Cu3-dx2−y2 and Cu3-dzx orbitals. The bands pushed up to about 1.5 eV in the AFM-s state. Near 1.63 eV is mainly Cu2-dz2 orbital slightly hybridized with O1-pzorbital in the AFM-t state, and it rises slightly in the AFM-s state. The highest bands of the series of the bands in the vicinity of 1.75 eV is dominated by Cu1-dx2−y2 and Cu1-dxy orbitals in the AFM-t state. The bands do not move apparently in the AFM-s state. Above 3 eV, bands are dominated by orbitals of Mo atoms hybridized with p-orbitals of the surrounding O
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
Figure 5.14: The electronic band structure of the collinear AFM-t state from the collinear spin-polarized GGA+U calculations.
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
Figure 5.15: Total and site-projected electronic densities of states of the collinear AFM-t state from the collinear spin-polarized GGA+U calculations.
Figure 5.16: The electronic band structure of the collinear AFM-s state from the collinear spin-polarized GGA+U calculations.
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
Figure 5.17: Total and site-projected electronic densities of states of the collinear AFM-s state from the collinear spin-polarized GGA+U calculations.
atoms.
Finally, we ploted the electronic band structure and densities of states of the most stable non-collinear AFM state (θ =107◦in Table 5.4) we found in Figure 5.18 and 5.19. The lowest valence band at the Γ point is Cu1-dx2−y2 and Cu1-dxy dominant bands hybridized with O1-py orbitals at −6.6 eV with a narrow bandwidth of about 0.2 eV. Bands dominated by Cu2-dz2 orbital hybridized with O1-pz and O6-pz orbitals near −6.4 eV in the collinear AFM state slightly rise to about −6.3 eV here. And, bands dominated by Cu3-dx2−y2 orbital hybridized with O5-pxorbital and Cu3-dzxorbital hybridized with O1-pxorbital slightly drop to about −6.1 eV. Above −6 eV and below −0.5 eV, bands are too complex to be recognized. Likewise, the highest occupied band with narrow bandwidth of about 0.1 eV is dominated by Cu1-dzx orbital hybridized with O2-pz and O3-px orbitals, and Cu1-dz2 orbital hybridized with O2-pz orbital.
The energy gap increase further to 1.47 eV. In the intervals between 1.4 eV and 1.8 eV, there is a series of narrow conduction bands. The lowest bands in the vicinity of 1.43 eV are dominated by Cu3-dx2−y2 and Cu3-dzx orbitals. The bands dominated by Cu2-dz2 orbital slightly hybridized with O1-pz orbital near 1.63 eV in the collinear AFM state drop to 1.61 eV here. The highest band in the vicinity of 1.75 eV is dominated by Cu1-dx2−y2 and Cu1-dxyorbitals. Above 3 eV, bands are dominated by orbitals of Mo atoms hybridized with p-orbitals of the surrounding O atoms. As the angle is rotated away from θ = 107◦and 287◦, the energy gap decrease gradually to 1.44 eV which correspond to θ = 197◦and 17◦approximately. With and without considering
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
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.