Ab-Initio Examination of Ni’s Stability, Electrical, Optical, and Magnetic Properties with GGA and GGA+U

Magnetic properties and structure

Using GGA and GGA+U techniques, Figure 1 shows how the total energy (atomic energy) changes with volume of the fcc and bcc structures of Ni metal in the FM, AFM, and NM phases. According to the figure, the Ni atom’s ground state is found to be FM fcc by GGA as well as GGA+U computations. Our findings supported earlier studies.^[3] The outcomes agree with GGA as well as GGA+U computations. We determined the energy of an atom, E, for each phase at the optimal volume of atoms, V_0, and the lattice constant a, as displayed in Table 1 in accordance with results from other investigations.^[35-37] This table confirms the precision of our GGA as well as GGA+U computations, which showed that Ni’s ground state is a FM face-centered cubic structure. Table 1 shows that the addition of strong correlations led to a drop in atomic volume for every state examined. Our results are strikingly similar to those of our earlier work^[4] and other earlier investigations.^[35,38]

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Figure 1:

(Color online) Total energies for Ni were computed as a function of atomic volume with Murnaghan EOS; solid symbols for fcc and open symbols for bcc, square for NM, star for AFM, and triangle for FM states with, respectively, (a) GGA and (b) GGA+U.

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Table 2 presents the outcomes of ground-state characteristics such as magnetic moments (µB), bulk modulus at zero pressure B₀, its derivatives of zero pressure B’₀, volume of equilibrium V₀, alongside relevant data from experiments and earlier all-electron computations. Table 2 demonstrates that when significant correlations are considered, the moments of magnetism rose in every research instance with the exception of the bcc FM state, where they fell, producing an unusual outcome. This data clearly shows that, in comparison to other findings, the GGA yields results that are similarly reasonable.^[37] Because the GGA results are more in line with actual evidence and other theoretical analyses, they perform noticeably better than previous DFT results.^[36,39]

Electronic properties

This work examines the impact of states on the electronic configuration of the bcc and fcc Ni energy levels. We present the partial density of states (PDOS) and total density of states (TDOS) based on GGA and GGA+U computations for the FM, AFM, and NM stages, as shown in Figures 2, 3, and 4. We recorded the computations within the energy range of -5 to 3 eV, respectively. The vertical dotted line in our computed DOS represents the level of Fermi (E_F). According to Figure 2, the DOS profile for NM and AFM states is essentially unchanged. However, it differs considerably for FM states in computations GGA and GGA+U are used. It is supported by Iota’s investigations^[1] and associated theoretical research.^[16]

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Figure 2:

(Color online) Calculated TDOS of FM, AFM, and NM states of Ni (a) bcc and (b) fcc, Blue for FM, Red for AFM, and Green for NM, respectively, for [A] GGA as well as [B] GGA+U.

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Figure 3:

(Color online) Calculated PDOS for states (a) NM, (b) FM and (c) AFM of Ni bcc, Red for total, Blue for d and Green for s DOS, respectively for [A] GGA and [B] GGA+U.

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Figure 4:

(Color online) computed PDOS for states (a) NM, (b) FM and (c) AFM of Ni fcc, Red for total, Blue for d and Green for s DOS, respectively for [A] GGA and [B] GGA+U.

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Additionally, the AFM phase of both bcc and fcc nickel exhibits the highest Fermi level DOS (E_F). The only situation where this is not the case is the bcc NM state, which has the highest value. Once more, for GGA as well as GGA+U computations, the FM state provides lower E_F values, as seen in Table 3. Figure 2 demonstrates that itinerant magnetism’s Stoner condition is satisfied when N(E_F)I>1 for both AFM and NM states in fcc and bcc structures. The DOS at the EF is N(EF), and I is the Stoner parameter. Stability in the FM state is shown by N(E_F)I<1. Furthermore, Figure 2 demonstrates that the Eg level has the most significant impact on the Fermi level of NM and AFM. Moreover, it is possible to understand how the Stoner parameter I is influenced by U and J.

Figures 3 and 4 show the outcomes of GGA and GGA+U computations for the PDOS of total, s, and d orbitals of Ni’s bcc and fcc for NM, FM, and AFM states. The large DOS close to the Fermi energy causes instability, which is lessened by an AFM state. This result is very similar to what was found in a similar investigation with other 3D elements.^[45,46] Table 4 shows how the peak energy levels in the band layers for the PDOS differ in each of our case studies. The Ni:3d level is the primary source of these peaks. Furthermore, Figures 3 and 4 show that orbital d is primarily in charge of the prevailing DOS. VBM, or valence band maximum, is mainly affected through the d-states of Ni, whilst s and f-states have the least effect. This is consistent with what Cococcioni M. et al.^[47] claimed. Furthermore, plots of PDOS display the hybridization of p-d, and it is evident by contrasting the GGA and GGA+U DOS distributions that the Hubbard term (U) modifies the states’ local positions. The hybridization of Ni 3d is weakened when U is added because it causes a downward shift of the Ni states, as indicated in Table 4 and illustrated in Figure 6 for the maximum amount of band layer energy. For NM and AFM, near the Fermi level, the impurity states are almost identical. However, for FM calculations, they appear to differ, GGA as well as GGA+U possess the Fermi level’s lowest energy, respectively. At E_F, the DOS is also the highest for bcc NM and fcc AFM, but it is the lowest for GGA as well as GGA+U of bcc and fcc FM [Figure 2].

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Figure 6:

(Online color) Calculated DOS at E_F, Fermi energy of Ni for FM, AFM, and NM cases, bcc square and fcc tringle, respectively, solid symbols for GGA and open for GGA+U.

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Ni’s band structure in its fcc FM phase determined by GGA+U approximation is displayed in Figure 5. As seen in the figure, the d-band’s border is crossed by the Fermi level in the region around Γ- point of fcc’s unit cell for initial Brillouin Zone, giving rise to partially-filled d- band, which is indicating metallic nature of Ni. Consequently, no band gap is observed at region of the d-electrons located above the Fermi level. The inception of U correction in the GGA approximation for exchange potential apparently did not give rise to significant changes in Ni band structure in comparison with the mere GGA approximation.

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Figure 5:

(Color online) Calculated band structure of FM phase of fcc Ni for GGA+U (a) spin-up and (b) spin-down, respectively.

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Optical properties

Since crystalline structures can be utilized in thermomagnetic recording media, it’s critical to comprehend the magneto-optical characteristics of materials with magnets.^[48-50] The band structures with transitions between bands are intimately associated with the optic characteristics. ε(ω), the dielectric function, can be utilized to characterize the material’s optical response, which indicates how it reacts to an incident photon. The crystal is considered isotropic in a cubic symmetry structure (ε=ε_xx=ε_yy=ε_zz). A crucial component in determining the physical characteristics of solids is the interband dielectric function, ε(ω), which characterizes how a medium reacts to light at all photon energies. As mentioned in,^[51] it comprises two components: the imaginary component ε₂(ω) and the real component ε₁(ω).

Here, ω represents the imaginary ε₂(ω) and the real ε₁(ω) components of The intricate dielectric function as well as the incident photon frequency. Furthermore, the intraband contribution was considered, since outcomes of band structure computations reveled metallic character for Ni, In the range of high frequencies, the dielectric function for metal is estimated in terms of the plasma frequency $\left({\omega }_p\right)$ by the simple form $\epsilon \left(\omega \right)=1-\frac{{\omega }_p^2}{{\omega }^2}$. Our results show the absorptive behavior as defined by the Kramers-Kronig relation and are inherently connected to the electronic band structure.^[52]

The Cauchy integral’s primary value is indicated by the letter P. All other optical characteristics may be easily obtained once ε₁(ω) and ε₂(ω) are known; for example, calculations were made for the absorption coefficient α(ω), energy loss function L(ω), and refractive indices n(ω).^[45,53] Both the ε(ω)’s real ε₁(ω) and imaginary ε₂(ω) components, the interband dielectric function, which was calculated applying FP-LAPW for Ni bcc and fcc’s NM, FM, and AFM states, are shown in Figures 7 and 8. Utilizing both GGA and GGA+U techniques, ε₁(ω) and ε₂(ω) values are calculated within 0 and 13 eV of energy. The findings show that the behaviors of ε₁(ω) and ε₂(ω) accord well with previous studies.^[54] It predicts new and uncommon behavior, except for the bcc NM GGA computation. With the exception of the fcc ε₁(ω) calculation, which showed an increase, as seen in Figures 7 and 8, the peaks showed a diminishing oscillatory pattern as energy increases in all cases examined. This suggests a novel behavior. Our findings are consistent with previous studies.^[16]

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Figure 7:

(Color online) Calculated the Real component ε₁ (ω) of Ni’s dielectric function for NM, FM, and AFM states (a) bcc and (b) fcc, from 0 to 13 eV [A] GGA [B] GGA+U, respectively.

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Figure 8:

(Color online) Calculated the Imaginary part ε₂(ω) of Ni’s NM, FM, and AFM states’ dielectric functions (a) bcc and (b) fcc, from 0.0 to 13.0 eV, respectively, with [A] GGA and [B] GGA+U.

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The computation of the dielectric function’s real and imaginary components, shown in Figures 7 and 8, the energy loss function, and the coefficient of optical absorption were computed. Using GGA as well as GGA+U computations, Figure 9 displays the spectrum of electron energy loss L(ω) in relation to photon energy for NM, FM, and AFM states. A crucial parameter for describing events involving electron elastically scattering and non-scattering, particularly at a 0% loss of energy, is the energy loss function. According to ^[55], energy losses at moderate energies 0.0 to 13.0 eV are mainly caused by electron excitations:

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Figure 9:

(Color online) Determined L(ω), the energy loss function of Ni’s FM, AFM, and NM states (a) bcc and (b) fcc from 0.0 to 13.0 eV, respectively, with [A] GGA and [B] GGA+U.

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For computations involving GGA as well as GGA+U, the two notable peak groups are found in the range of energy 3.0 to 13.0 eV. The initial and subsequent peak groups for bcc are located at 3.00 to 4.50 eV and 6.00 to 9.00 eV, respectively. The peaks for fcc are located between 3.00 and 4.50 eV and between 9.00 and 12.50 eV. For FM, AFM, and NM states under GGA and GGA+U calculations, we noticed peculiar behavior in the bcc NM with GGA. This behavior was typified by the existence of only a second peak at (9.00 – 10.00) eV. Table 5 shows the main peak points for the first and second groups, GGA and GGA+U. How the graph behaves, and observed values are consistent with previous research findings.^[56] Electron scattering within that energy range is indicated by the energy loss function’s peaks, which match the valleys in the optical conductivity. Our results are consistent with previous research.^[55]

Figure 10 shows the absorption coefficient α(ω) for bcc and fcc Ni in FM, AFM, and NM states using GGA and GGA+U approaches, over an energy range from 0 to 13 eV, as determined by Equation (5) from the reference.^[57] The optical absorption activation spots are shown in this figure and are described in Table 6 for FM, AFM, and NM based on GGA and GGA+U computations. When significant correlations are taken into consideration, the active points for both bcc and fcc Ni are raised in GGA+U calculations, as shown in Figure 10 and Table 6. In line with results from other studies,^[45,54] Figure 10 shows that both GGA and GGA+U estimates show an increase in absorption with growing energy.

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Figure 10:

(Color online) Determined α(ω), the absorption coefficient of Ni’s FM, AFM, and NM states (a) bcc and (b) fcc from 0.0 to 13.0 eV, respectively, with [A] GGA and [B] GGA+U.

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A glance at Figure 8, together with Figures 9 and 10, shows a first peak in ε₂(ω) spectrum at ∼ 0.5 eV, while the onsets of energy loss function and absorption occur at even lower energy (∼0.25 eV). Taken together, these phenomena provide a strong clue for the interband transition in the narrow d-band around the Fermi level.

Figure 11 demonstrates the refractive index n(ω) for the Ni’s fcc and bcc for NM, FM, and AFM states by GGA as well as GGA+U, as determined by Eq. (6), fluctuates as a function of energy over the range of 0 to 13 eV. According to Eq. (6), the dimensionless refractive index, or n(ω), describes how a beam travels through an optical material. According to Mark Fox,^[58] the connection $n\left(\omega \right)=\sqrt{{\epsilon }_r}$ indicates that the spectral line shape of n(ω) is identical to that of ɛ ₁(ω). The behavior of our data is consistent with findings from other studies, as this figure illustrates.^[46]

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Figure 11:

(Color online) Determined n(ω), the refractive index of Ni’s FM, AFM, and NM states (a) bcc and (b) fcc from 0.0 to 13.0 eV, respectively, by [A] GGA as well as [B] GGA+U.

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Using GGA beside GGA+U, the reflectivity R(ω) for the Ni’s bcc and fcc of FM, AFM, and NM states computations varies from 0.0 to 13.0 eV, respectively, as shown in Figure 12. This figure demonstrated how our findings behaved in a way that was in line with previous research.^[45]

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Figure 12:

(Color online) Determined R(ω), the reflectivity of Ni’s FM, AFM, and NM states (a) bcc and (b) fcc, from 0.0 to 13.0 eV, respectively, with [A] GGA and [B] GGA+U.

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Figures 13 and 14 display, respectively, the computed σ(ω) optical conductivity’s real and imaginary components in relation to photon energy from 0 to 13 eV for FM, AFM, and NM states of bcc and fcc Ni computations, respectively, with GGA beside GGA+U. In all frequencies, the frequency-dependent dielectric function ɛ(ω) is associated with the optical conductivity’s real portion, Re σ(ω). Our findings are consistent with earlier research.^[56]

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Figure 13:

(Color online) Determined the variation of the real component σ₁(ω) for optical conductivity for FM, AFM, and NM states (a) bcc and (b) fcc from 0.0 to 13.0 eV, respectively, with [A] GGA beside [B] GGA+U.

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Figure 14:

(Color online) Determined the variation of σ₂(ω), the imaginary component for optical conductivity for FM, AFM, and NM states (a) bcc and (b) fcc from 0.0 to 13.0 eV, respectively, by [A] GGA as well as [B] GGA+U.

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