Abstract
Large perpendicular magnetic anisotropy (PMA) in transition metal thin films provides a pathway for enabling the intriguing physics of nanomagnetism and developing broad spintronics applications. After decades of searches for promising materials, the energy scale of PMA of transition metal thin films, unfortunately, remains only about 1 meV. This limitation has become a major bottleneck in the development of ultradense storage and memory devices. We discovered unprecedented PMA in Fe thin-film growth on the
INTRODUCTION
Magnetic anisotropy is a relativistic effect originating from spin-orbit coupling (SOC). Perpendicular magnetic anisotropy (PMA) in magnetic thin films has led to rich physics and become a key driving force in the development of magnetic random-access memory (MRAM) devices (1–3). Establishment of PMA in nanostructures and nanopatterned magnetic multilayers paves a new avenue toward nanomagnetism, in which fascinating physics such as spin Hall switching and skyrmions is blooming (4–7). Because the strength of the SOC is a quartic function of the atomic number, it is not surprising to have large magnetic anisotropy in heavy metals such as rare earth materials, which are commonly used for permanent magnets (8, 9). It is, however, a challenge to enable large anisotropy, especially PMA, in commonly used 3d transition metals such as Fe thin films.
Actually, the strength of PMA is determined by the energy correction from the SOC, which couples the orbital angular momentum L to the spin momentum S via Hso = λL ⋅ S. In the atomic limit of a single Fe, six Fe 3d valence electrons could ideally have a total spin of Sz = 2 and angular momentum of Lz = 2.The atomic limit of the SOC energy λL ⋅ S is thus 75 meV, given by the SOC coefficient λ ≈ 19 meV (10). However, in all existing discussions of Fe-based thin films on MgO substrates, such as Fe/MgO-based (11–15) and CoFeB/MgO-based systems (16–18), the size of PMA is only 1 meV, far below the atomic limit. This leaves a vast window to escalate PMA in transition metal thin films unexploited.
Under crystal field, five d orbitals are superposed and form xy, yz, xz, x2 − y2, and 3z2 − r2 orbitals as the eigenstates. All of the new orbitals have zero Lz because of the time reversal symmetry. If these orbitals are nondegenerate, the first-order energy correction from SOC vanishes, leaving the second-order perturbation as the dominant contribution (11, 19). This is the scenario in most thin film systems (11–18, 20). The energy scale of PMA is then λ2/Δ, where Δ is the bandwidth of the state crossing the Fermi level. For a typical 3d magnetic element, Δ ~ 1 eV and λ ~ 0.03 eV (10). It is thus not surprising to achieve 1-meV PMA in most 3d magnet thin films.
To escalate the PMA, one thus would like to find a regime in which the first-order perturbation of SOC is dominant. In this regime, PMA is proportional to λ instead and has the chance to approach the atomic limit of SOC energy. It occurs when partially filled degenerate orbitals exist around the Fermi level. A successful example has already been demonstrated in a single adatom (21–24) or dimer (25, 26) deposited on specific substrates. However, once a thin film is formed, PMA in these systems is greatly reduced and brought back to the second-order perturbation scheme.
Here, we report giant PMA in Fe ultrathin films grown on the wurtzite
(A) Surface structures of one-monolayer (1ML) Fe deposited on
RESULTS
As a central result of this work, total energies with different magnetization directions of 1 monolayer (1ML) Fe on BN, AlN, GaN, and InN, respectively, were obtained. The relative magnetoanisotropy energy as a function of
The values of Ku1, Ku2, total PMA in units of millielectron volt per unit cell and millijoule per square meter, spin moments (ms), orbital moments (ml), and occupation number
DISCUSSION
The giant PMA in Fe/III-V nitrides is considerably beyond the energy scale of the second-order perturbation of SOC. To understand the origin of this giant PMA, we studied the electronic structure of Fe(3d) orbitals. Without loss of generality, the Fe(1ML)/GaN system was analyzed in detail below. Figure 2 (A and B) displays the difference between the total charge density of our Fe(1ML)/GaN system and the sum of charge densities of a suspended 1ML Fe and a pure GaN supercell. Electron density is reduced in blue contours, whereas it is increased in yellow ones. Thus, charge transfer occurs from the blue contour to the yellow contour during the formation of the Fe-GaN interface. The yellow contour indicates the formation of strongly polarized Fe–N bonds and the enhancement of in-plane x2 − y2 and xy orbitals. From blue contours, a significant reduction of Fe’s itinerant and xz/yz electrons was witnessed. The reduction of itinerant electrons is reasonable because Fe electrons saturate the dangling bonds from N atoms on the N-terminated surface, so that Fe atoms lose electrons and become cations. The ionic behaviors of Fe are doubly confirmed by the Bader charge (29–31) results (Table 2), of which the difference corresponds to the charge increasing/decreasing on one atom. About 0.4e− electrons per Fe atom is transferred to N atoms on the interface. These interfacial N atoms thus have almost the same number of valence electrons as that in bulk GaN. In addition, there is no additional interatomic charge transfer when SOC is included, shown on the last column in Table 2.
(A) The positive and (B) negative part of the charge difference between the total charge density of Fe(1ML)/GaN and the sum of charge densities of a suspended 1ML Fe and pure GaN supercell. Charge deficiency in xz/yz-like orbitals shown in (A) is transferred to x2 − y2/xy-like orbitals shown in (B). (C to F) Orbital-resolved projected density of state (PDOS) of 3z2 − r2, xz/yz, and x2 − y2/xy orbitals, respectively, in the absence of SOC. The bandwidth Δ of x2 − y2/xy is labeled in (F). (E) PDOS for Fe(3d) orbitals with SOC included. Positive and negative values of PDOS refer to spin majority and spin minority channels, respectively, and the Fermi level is set to zero. (G) Occupation number of each Fe(3d) orbital and (H) PMA as a function of the Hubbard U when SOC is included.
Bulk refers to the bulk Fe and GaN; surface refers to the clean GaN
To explain the charge transfer from xz/yz to x2 − y2/xy orbitals and thereby identify valence states of Fe cations, we investigated the crystal field and orbital-resolved PDOS of Fe(3d) without SOC first. As shown in Fig. 2 (C to F), all five d orbitals in the spin majority channel are far below the Fermi level and fully filled. Rich physics is present in the spin minority channel. Three orbitals predominated by xz, yz, and 3z2 − r2 are far above the Fermi surface and almost unoccupied. Double-degenerate orbitals, labeled as e orbitals, predominated by x2 − y2/xy orbitals are low-lying (Fig. 2F). They are crossing the Fermi level and thus partially occupied. In Fig. 2D, the twofold degeneracy of the xz/yz-predominated orbitals, labeled as e′ orbitals, is explicitly shown. Double degeneracies of e and e′ orbitals are protected by the two-dimensional irreducible representation E of the C3v point group of the crystal field around each Fe cation. In reality, an overlap between xz/yz and x2 − y2/xy orbitals is present but small. According to the density matrix of Fe(3d) orbitals and the corresponding occupation number, e states are mixed with 3% xz/yz orbitals, and e′ orbitals contain 3% x2 − y2/xy components.
In spin minority channels, e states are almost half-filled, with an occupation number of 0.457, whereas occupation number of e′ states is only 0.035. As a comparison, in suspended 1ML Fe, the occupation number of x2 − y2/xy is 0.095, and that of xz/yz is 0.564. Therefore, x2 − y2/xy orbitals have escalated occupation once Fe is deposited on GaN. It is consistent with the charge density contours discussed earlier.
Once SOC is included, one can expect the lift of degeneracy between x2 − y2 and xy orbitals due to nonzero off-diagonal matrix elements. It is confirmed by the PDOS shown in Fig. 2E, where almost no states in the spin minority channel are found near the Fermi level. Degenerate e states are split, and a large splitting of about 3.0 eV is present. According to the density matrix, the occupation number of the lower splitting state is 0.904, and the corresponding eigenstate is
In terms of spherical harmonics, this state is quite close to
According to the discussion above, large band splitting and the partial occupation in consequence are the precursor of large PMA. However, one should note that the SOC of Fe(3d) is on the scale of 20 meV, two orders of magnitude smaller than the bandwidth (~ 2.2 eV) of x2 − y2/xy orbitals, labeled as Δ in Fig. 2F. SOC alone can hardly generate such a large splitting of the entire band. This is resonated by the fact that, in simple non–self-consistent calculations, the PMA contributed by SOC alone is only 3.0 meV, a typical value in the second-order correction scheme. Because the spin splitting changes slightly after SOC is included and no structural reconstruction driven by SOC happens, SOC cannot be the major driving force of the band splitting. The only interaction on the scale of electron volts under investigation is the on-site electron-electron correlation interaction described by the Hubbard U. It thus suggests that the correlation interaction between occupied electrons must play a significant role on large PMA here.
The energy contribution from Hubbard U can be given by the single-particle expression under the Dudarev formation of L(S)DA + U (32);
To further confirm this conclusion, we performed the SOC-included self-consistent calculations with multiple values of the Hubbard U. As shown in Fig. 2G, the splitting between
Electronic structures of 1ML Fe on N-terminated (000
(A to C) PDOS of x2 − y2/xy orbitals without SOC. The bandwidth in the spin minority channel is indicated by double arrows. (D to F) PDOS of Fe(3d) with SOC. In (D), the shadow region gives the projection onto x2 − y2/xy orbitals where the magnitude of splitting is indicated by the double arrow.
It is confirmed by the PDOS of Fe(3d) orbitals with SOC, as shown in Fig. 3 (D to F). Relevant magnetic and orbital properties for each Fe(1ML)/III-V nitride are listed in Table 1. Considerable splitting around 3 eV at Fermi level of spin minority is found in Fe/AlN, Fe/GaN, and Fe/InN but is reduced to ~1.5 eV in Fe/BN, as shown in Fig. 3E. According to Table 1, such small splitting in Fe/BN is consistent with the occupation number of the
We further investigated the thickness dependence of PMA by using the Fe/GaN
Fe thin films grown on the (
CONCLUSION
In summary,
In the rapidly developing technology of MRAM, the lack of large PMA becomes a bottleneck in downsizing the binary bits. Giant PMA discovered here suggests that a 2.0 nm × 2.0 nm flake of Fe(1ML)/III-V nitride has a total uniaxial magnetic anisotropy energy of about 1.2 eV, reaching the criteria for 10-year data retention at room temperature (3). Therefore, giant PMA in this thin film can ultimately lead to nanomagnetism and promote revolutionary ultrahigh storage density in the future. Furthermore, large anisotropy energy could lead to large coercivity. Fe/III-V nitride could lead to a new type of permanent magnet without rare earth element potentially.
METHODS
The calculations were carried out in the framework of the noncollinear spin-polarized first-principles calculations with the projector-augmented wave pseudopotential (37) implemented in the Vienna ab initio simulation package (38). We used the generalized gradient approximation (GGA) of Perdew-Burke-Ernzerhof (PBE) formation (39) plus Hubbard U (GGA + U) (32) with U = 4.0 eV on Fe(3d) orbitals.
To build the slab supercell, we used four X-N (X = B, Al, Ga, and In) principal layers as the substrate and deposited 1 to 3 ML of Fe on the N-terminated
Charge density of the SOC-free ground state was used as the initial state. Self-consistent total energy calculations were used to derive the noncollinear calculation with SOC included. Γ-centered 25 × 25 × 1 K-point meshes in the two-dimensional Brillouin zone were used with an energy cutoff of 600 eV for the plane-wave expansion. The accuracy of the total energy is thus guaranteed to be better than 0.1 meV/u.c.
To obtain a reasonable U value for GGA + U calculations, we compared density of state (DOS) and PDOS of 1ML Fe on GaN
DOS (gray dashed line) and PDOS (solid blue line) of Fe(1ML)/GaN without SOC derived by (A) GGA-PBE, (B) GGA + U at U = 4.0 eV and (C) HSE06, respectively. Consistency between (B) and (C) shows that GGA + U at U = 4.0 eV can well describe Fe(3d) orbitals in this system.
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REFERENCES AND NOTES
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