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We have employed the generalized Bloch theorem to evaluate the spin stiffness constants of 3d transition metals (bcc-Fe, fcc-Co, and fcc-Ni) within the linear combination of pseudo-atomic orbitals (LCPAO). The spin stiffness constants were obtained by fitting the spin-wave energy curve, which relates to the total energy difference and the spiral vectors. In order to convince the reliable spin stiffness constants, we also provided the convergences of spin stiffness constants in terms of the cutoff radius and the number of orbitals. After observing the specific cutoff radius and the basis orbital, at which the spin stiffness constant converges, we used those two parameters to compute the Curie temperature by using the mean field approximation and the random phase approximation. For the latter approximation, we applied the so-called Debye approximation, which is intended to reduce very significantly many required wavevectors to evaluate the Curie temperature. We claimed that our results are in good agreement with both other calculations and experiments.
The recent study based on the magnetic structures within the itinerant electron model gives a strong description of why some metals should have a ferromagnetic ground state. This ground state can be well-explained by the so-called Stoner criterion, which is determined by the Coulomb interaction. The Stoner criterion has successfully predicted that some
The excitations in magnetic systems are usually addressed to two different kinds of excitations. The first excitations are called the Stoner excitations. These excitations do exist due to the transfer of electron–hole, namely, the transition of an electron from the filled state to the empty state. Due to the excitations of electron–hole, the continuum region (Stoner continuum) is created along the certain wavevector. The second ones are the spin-wave excitations (magnons), where the configuration of magnetic moments of all atoms gives a spiral form. The clear difference lies in the range of the wavelength spectra. Unlike the Stoner excitations, the spin-wave excitations only hold for the short wavevector (long wavelength) and are then damped when entering the Stoner continuum. Nevertheless, the spin-wave excitations become much more dominant in the nearly lowest excitations spectrum. This means that the Stoner excitations can be excluded up to the critical temperature on this condition. This approach enables us to estimate the Curie temperature by considering the spin-wave excitations.
There are two approaches to consider the spin-wave excitations, i.e., the real space approach and the reciprocal space approach. In the real space approach, one should first calculate the exchange coupling constant of two different atoms to obtain the spin-wave energy. On the contrary, the spin-wave energy can be directly calculated by using the reciprocal space method, which implements the so-called generalized Bloch theorem (GBT). The limitations of these approaches lie in the efficiency of some calculations. According to Padja et al.,1) the more efficient calculations of the exchange coupling constant or the Curie temperature can be performed within the real space approach than the reciprocal space approach. Contrarily, the spin-wave energy or the spin stiffness constant can be computed more efficiently using the reciprocal space approach.
So far, we only note that the use of GBT with an LCPAO was never used by the previous authors to study the spin-wave excitations, especially for calculating the Curie temperature, in the framework of first-principles approach for the ferromagnetic
In this study, we apply the GBT with the constraint scheme method to calculate the magnon energy using the reciprocal space method (frozen magnon method) within the density functional theory (DFT). Our electronic calculation, in which the wavefunction of a single particle is described by an LCPAO with the norm-conserving pseudopotentials, reproduces successfully the spin stiffness constants and Curie temperatures of
We perform the first-principles approach on the magnetic excitations of
The implication of the wavefunction in Eq. (1) is that the magnetic moment of each magnetic atom will be rotated along the chosen spiral vector
Since the Heisenberg model suits to investigate the long wavelength spin-wave excitations, the calculated total energy in DFT calculation should be mapped onto the energy in the Heisenberg Hamiltonian. If the two magnetic moments interact with each other characterized by the exchange coupling constant
The Curie temperature can be computed in two ways, using either the MFA or the RPA. In this case, the Curie temperature can be evaluated by the arithmetic average value of the magnon energies in MFA or by the harmonic average value of the magnon energies in RPA.24) Due to these different treatments, the MFA tends to give a larger value than the RPA if using the same number of discrete
First of all, the spin stiffness constant will be confirmed by observing its convergence based on two parameters, i.e., the cutoff radius and the number of orbitals. To do so, in the OpenMX code we specified these parameters by the symbol, for example, Fe5.0-
To start the calculation, a conical spin spiral configuration was constructed by fixing a 10° cone angle by applying the constraint scheme method. To obtain the magnon energy, we computed the total energy difference from the self-consistent calculations and mapped it onto the effective Heisenberg Hamiltonian, as formulated in Eq. (5). We then varied the cutoff radius and increased the number of orbitals systematically. In this case, we stopped the number of orbitals, at which the overcompleteness appears, i.e., the total energy difference becomes unreliable. The spin stiffness constant D will be then evaluated by a fourth-order fit
Figures 1–3 show the magnon energies [(a)–(e)] and the convergences of spin stiffness constant (f) for bcc-Fe, fcc-Co, and fcc-Ni, respectively. For all figures, the number of orbitals can only be increased at the short cutoff radius, e.g., 3.5 a.u. or 4.0 a.u., while at the long cutoff radius, such as 5.0 a.u., the small number of orbitals can only be applied due to overcompleteness. Note that although the number of orbitals can be increased at the short cutoff radii, the small number of orbitals sometimes cannot be employed due to the insufficient basis set, as shown in Fig. 2(a) for fcc-Co, in which the number of orbitals of
Figure 1. (Color online) Spectra of magnon energy near
Figure 2. (Color online) Spectra of magnon energy near
Figure 3. (Color online) Spectra of magnon energy near
The calculated spin stiffness constants for bcc-Fe, fcc-Co, and fcc-Ni using the cutoff radius of 4.0 a.u. with the orbitals of
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To support our LCPAO, we also provide the spectra of magnon energy on the high symmetry line in the Brillouin zone for bcc-Fe, fcc-Co, and fcc-Ni, as shown in Figs. 4(a), 4(c), and 4(e). The tendencies of our calculated magnon spectra are in good agreement with those using the LMTO method with the real space approach1) and the frozen magnon approach.9) Our results are only different on the peaks of the magnon energy in the special k point. The parabolic curves, which are nearly isotropic, are observed in the long wavelengths near Γ point for fcc-Co and fcc-Ni. For the short wavelengths in bcc-Fe, we also observe the so-called Kohn anomalies shown by two local minima in the interval of Γ–H and H–N, in good agreement with Refs. 1 and 9.
Figure 4. Spectra of magnon energy using the cutoff radius of 4.0 a.u. and the orbitals of
We used two approaches for calculating the Curie temperatures. The first approach is to apply the MFA by taking the average of magnon energies in the Brillouin zone, as formulated in Eq. (6). In general, our calculated results are in good agreement with the previous calculations, as shown in Table II. Our estimation for bcc-Fe is also in good agreement with the experiments, whereas the deviation less than 15% is addressed to fcc-Co and fcc-Ni. Note that, although our spin stiffness constant of fcc-Ni is overestimated from the experiment, as shown in Table I, we obtain a Curie temperature closer to the experiment that those with the different methods in Refs. 1, 5, 6, 9, and 10.
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The second approach to calculate the Curie temperature is addressed to the RPA. Since the number of
To apply the Debye approximation, we initially modify Eq. (7) in the integral formulation39,40)
We would like to give some comments on why the Curie temperature is underestimated for bcc-Fe. By seeing the dashed lines in Figs. 4(b), 4(d), and 4(f), we suppose that the underestimate or the overestimate depends on the tendency of the fitting function. The calculated Curie temperature for fcc-Co gets the error of less than 5%, which is the best result, followed then by fcc-Ni and bcc-Fe. In this case, we see that the dashed line for fcc-Co follows the flow of the solid line, which means that there is almost no deviation, see Fig. 4(d). For fcc-Ni, we see a small deviation, especially at
We have presented a schematic procedure to calculate the spin stiffness constants, as well as the Curie temperatures by using the GBT with an LCPAO as the basis set. The convergences of spin stiffness constant of bcc-Fe, fcc-Co, and fcc-Ni have been determined by two parameters, the cutoff radius and the number of orbitals. These convergences can only be achieved by choosing the cutoff radius of 4.0 a.u. with the orbitals of
The spectra of the magnon energy and the Curie temperatures are derived by setting the cutoff radius of 4.0 a.u. with the orbitals of
Acknowledgments
T.B.P. wishes to thank Dr. M. Ležaíc for the available discussion on the Debye approximation. The computations were carried out using ISSP supercomputers located at the University of Tokyo, while the spectra of magnon energy and calculated Curie temperatures were performed at the Universitas Negeri Jakarta.
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