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Anisotropic spin filtering by an altermagnetic barrier in magnetic tunnel junctions

Boyuan Chi, 1 Leina Jiang, 1 Yu Zhu, 1 Guoqiang Yu, 1, 2 Caihua Wan, 1 and Xiufeng Han 1, 2, 3, ∗

arXiv:2409.03415v1 [cond-mat.mtrl-sci]

5

Sep 2024

1 Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100190, China

2 Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China

3 Center of Materials Science and Optoelectronics Engineering, University of Chinese Academy of Sciences, Beijing 100049, China

The spin filtering effect, distinct decaying lengths experienced by oppositely spin-polarized electrons in a magnetic barrier, generally occurs in ferromagnetic (FM) insulators or semiconductors. With the rise of altermagnetic (ALM) materials which exhibit similar capability of spin-polarizing electrons with ferromagnets, it is a nature question whether the ALM insulators or semiconductors can also act as unique barriers for the spin splitting effect. Here, through first-principles calculations, we investigated the complex band structure of the ALM insulator FeF 2 and found that it possesses an anisotropic spin filtering effect: along the [001] direction of FeF 2 , a current remains spin-neutral but has locally nonvanishing spin polarizations in the momentum space; moreover, along the [110] direction of FeF 2 , a current will be globally spin-polarized by different attenuation lengths of oppositely spin-polarized electrons. Leveraging this anisotropic spin filtering effect, we designed two types of MTJs with the ALM barrier: ALM electrode/ALM insulator barrier/non-magnetic (NM) electrode and FM electrode/ALM insulator barrier/NM electrode, using RuO 2 (001)/FeF 2 /IrO 2 and CrO 2 (110)/FeF 2 /IrO 2 as the corresponding prototypes, respectively. We found that these two proposed MTJs exhibited the tunneling magnetoresistance (TMR) ratios of 216% and 3956%, by matching the conduction channels of the electrodes and the spin-resolved lowest decay rate of the barrier in the momentum space. Our work deepens and generalizes understanding toward the spin filtering effect for the rising ALM insulators and semiconductors, and broadens applications of the AFM spintronics.

The magnetic tunnel junction (MTJ), composed of two ferromagnetic (FM) electrodes and a sandwiched insulating barrier, is elementary for spintronic devices [1–3]. The tunneling magnetoresistance (TMR) effect is the core of MTJ, making it suitable for magnetic random-access memories (MRAMs), magnetic sensors, and others spintronic devices [4–8]. To achieve reliable reading and writing, a higher TMR ratio is imperative. Initially, TMR was regarded dominantly determined by the spin polarization of FM electrodes. Therefore, various FM metals with high spin polarizations such as Heusler alloys have been paid plenty of attentions. Thereafter, the importance of symmetry filtering of crystallized barriers for tunneling was gradually realized [9]. By analyzing the evanescent states, MgO was predicted to exhibit weak attenuation effect for electrons with the ∆ 1 symmetry [10, 11]. Consequently, by combining Fe electrode with MgO, one successfully developed MTJs with large TMR ratios [10–13], manifesting the powerfulness of the barrier engineering for MTJs.

Recently, the discovery of the antiferromagnetic (AFM) materials with spin splitting band structures, i.e., altermagnetic (ALM) materials [14–16], has attracted researchers to explore their applications in magnetic devices, twodimensional materials, superconductivity, etc. [16–25]. Focusing on the local spin polarization in momentum space, researchers have designed a novel all-antiferromagnetic tunnel junctions (AFMTJs) utilizing two ALM (or spin splitting AFM) electrodes instead of FM electrodes [17, 26–32]. Moreover, the TMR ratio can be further enhanced by matching the distribution of the lowest decay rates of the barrier with that of the conduction channels of the AFM electrodes in the re-

ciprocal space [33].

In fact, the momentum-dependent spin splitting of ALM materials is not only in the real part as reported in [14, 34– 36] but also in the imaginary part of their band structures as shown below. Here, by analyzing the complex band structure of the ALM insulator FeF 2 , we found a noticeable splitting in the decay rate κ of the evanescent states for the two spinchannels. This feature is distinguished from that observed in conventional non-magnetic (NM) or trivial AFM insulators with the same decay rate for opposite spins. The splitting decay rates of evanescent states thereby result in the spinresolved effective barrier for tunneling electrons with opposite spins, thus allowing ALM insulators to act as a spin filter with zero stray field. The generation of spin filtering effect without producing stray fields is a distinctive property of ALM materials, which makes them applicable in spin- and angleresolved photoemission spectroscopy (SARPES) and enhancing the TMR effect, etc.

Utilizing the spin filtering effect of ALM insulators, it is viable to design non-volatile MTJs with a magnetic compensated barrier. In this study, through first-principles calculations, we demonstrated the anisotropic spin filtering effect of ALM material FeF 2 . Based on this effect, we designed two kinds of MTJ: ALM/ALM insulator barrier/NM and FM/ALM insulator barrier/NM MTJs. Specifically, we used RuO 2 (001)/FeF 2 /IrO 2 and CrO 2 (110)/FeF 2 /IrO 2 as the corresponding prototypes to study these two kinds of MTJs. The quantum transport calculations revealed that the two MTJs exhibit the TMR ratios of 216% and 3956%, respectively. The TMR effect in both arises from the concerted play of the spin filtering effect of FeF 2 and the spin-polarization of the FM and ALM electrodes, CrO 2 and RuO 2 . The computational details are presented in the Supplementary Materials [37]. Due to the 2 usage of NM electrode and ALM barrier with compensated magnetization, both MTJs do not require additional pinning structures, simplifying the device structures. Furthermore, the former MTJ benefits from zero stray field; the latter can be applied in magnetic sensor applications.

In general, the transmission coefficient T σ (k ∥ ) is usually factorized as [7, 38, 39]:

(a) Transport along [001] of FeF 2 (1)

where t σL (k ∥ ) and t σR (k ∥ ) are the probabilities for an electron with spin σ to transmit through the left and right electrode/barrier interfaces, respectively. κ(k ∥ ) is the lowest decay rate in the barrier and d is the barrier width.

This equation applies to NM or trivial AFM barriers with spin degeneracy, such as MgO, NiO, and CoO, etc. However, for FM insulators, the lowest decay rate κ is dependent on spin. Thus, Eq.(2) should be rewritten as:

The spin-dependent κ σ (k ∥ ) indicates that electrons with opposite spins experience different effective barrier heights during tunneling, and the barrier acts as a spin filter.

An ALM material is a nontrivial AFM material that exhibits spin splitting in the absence of spin-orbit coupling. Therefore, for ALM insulators, like FeF 2 , LaMnO 3 , MnTe, et al., spin splitting occurs along certain high-symmetry directions. This indicates that the spin splitting can also manifest itself in the complex band along these directions as shown below, resulting in a spin-dependent κ σ (k ∥ ), analogous to the case of FM insulators, as described by Eq. (2). As a result, when electrons tunnel along these spin-split directions of the ALM insulators, the current will be spin-polarized, and the ALM insulators can thus exhibit the spin filtering effect.

The properties of ALM materials are closely related to its crystalline orientation [40, 41]. Electrons undergo different spin filtering effect when tunneling along different crystalline orientations of ALM insulators. Figs. 1(a) and 1(b) show the crystal-orientation-dependent spin filtering effect when electrons tunnel through the ALM FeF 2 . Dashed circles denote areas with small decay rates. As shown in Fig. 1(a), when a current flows along the [001] direction of FeF 2 , due to the same shape of decay rates for the two spin channels, the total spin polarization remains zero, though the current is locally polarized in the momentum space. When a current flows along the [110] direction of FeF 2 , the current is then globally spin-polarized [Fig. 1(b)]. Therefore, the ALM insulators can polarize currents as FM materials, while still retaining advantages of zero stray field and robustness against external magnetic fields.

To provide a clearer explanation on the spin filtering effect of ALM insulators, we chose FeF 2 as an example. FeF 2 is an ALM insulator with collinear AFM magnetic configuration and a rutile structure as plotted in Fig. 2(a). Thanks to its magnetic space group P 4 ′ 2 /mnm ′ , FeF 2 exhibits a momentumdependent spin splitting band structure. From the density of states (DOS) of FeF 2 (001) depicted in Fig. 2(c), FeF 2 has zero net magnetization and a 3.5 eV band gap, which is consistent

(b) Transport along [110] of FeF 2 electrode barrier electrode M or Low resistance electrode barrier electrode M or High resistance Low resistance High resistance FIG. 1. Schematics of the spin filtering effect of ALM insulator FeF 2 along (a) the [001] and (b) [110] directions, respectively. Schematics of the TMR mechanism for (c) ALM/ALM insulator barrier/NM tunnel junction with spin-neutral current, (d) FM/ALM insulator barrier/NM tunnel junction with spin-polarized current. Blue and Red circles indicate the spin-up and spin-down channels, respectively. Solid and dashed circles indicate the conduction channels of the electrodes and the area of the evanescent states with low decay rates in the barriers, respectively.

with the experimental study [42]. The spin-up and spin-down bands of FeF 2 (001) in Fig. 2(d) degenerate on most highsymmetry lines, but they split along the Γ-M and Z-A directions, reflecting the altermagnetism. Therefore, when electrons tunnel along the Γ-M (Z-A) direction, i.e. the [110] direction of FeF 2 , the band gap of the spin-up and spin-down electrons at this high-symmetry line is diverse [see Supplementary Materials [37] Fig. S1], which leads to a different barrier height felt by oppositely polarized electrons and thus results in the spin filtering effect.

The complex band structure can more accurately illustrate the decay of tunneling electrons inside barrier. In the elastic scattering, the transverse wave vector is conserved, and the longitudinal wave vector k z can be expressed as k z = q + iκ, where the real part q corresponds to the real band, and the imaginary part κ corresponds to the wave functions that decay into the bulk [9]. As a result, tunneling electrons exhibits exponential decay [exp(−2κd)] in the barrier. Figs. 3(a) and 3(b) show the case of tunneling along the [001] direction of FeF 2 . The three complex bands in Fig. 3(b) correspond to the Γ(0, 0), C(0.15, 0.15) and C ′ (0.15, -0.15) point. On one hand, for the spin degeneracy at the Γ point, the spin-up and spin-down electrons share a similar complex band, thus they have the same lowest decay rate κ at the Fermi energy. On the other hand, the spin splitting along Γ-M leads to a different lowest decay rates for both spins in the complex band structure. However, for FeF 2 with altermagnetism, the spin of the Γ-M ′ band is opposite to that of the Γ-M band, resulting in

the same shape of k ∥ -resolved lowest decay rates, producing a 3 M X

FIG. 2. Atomic and magnetic structures of FeF 2 with (a) (001) and (b) (110) crystal faces. (c) The density of states and (d) the band structure of FeF 2 (001).

C (0.15, 0.15) Γ Tunneling along [001] of FeF 2 (b) spin-up C' (0.15,-0.15)

M' spin-down 0.25 Γ 0.50 0.25 C'

0.29 (c) spin-up

Y(d)

D (0.15, 0) Z Γ 0.51 2 -2 0 E-E F (eV)

2 0.8 0.4 κ (2π/c') 0.4 D D -4 0.57 -2 0 E-E F (eV)

Γ 0.8 spin-down Γ -4 0.22 Tunneling along [110] of FeF 2 0.36 spin-down C 0.50 C' spin-neutral current, but with nonzero local spin polarization in the momentum space.

Based on this local spin filtering effect in the momentum space, we proposed an MTJ with an ALM metal and an NM metal as electrodes, i.e., ALM/ALM insulator barrier/NM MTJ. As shown in Fig. 1(c), for each spin channel (blue and red), its conductance is proportional to the overlapping between the distribution of the conduction channels in the ALM electrode and the distribution of the lowest decay rates in the barrier. Switching the Néel vector of the ALM electrode, the spin polarization at each k ∥ point in the momentum space will be reversed, and the overlapping area between the conduction channels and the lowest decay rates is also changed, resulting in a magnetoresistance between the parallel (P) and antiparallel (AP) states.

We calculated RuO 2 (001)/FeF 2 /IrO 2 MTJ as a prototype to verify our design. RuO 2 is an ALM metal, and IrO 2 is an NM metal, whose conduction channels are shown in Fig. S2. Fig. S3 shows the atomic structure and the transmission coefficient distributions for the P and AP states of this MTJ, it can be seen that the barrier suppresses the transmission coefficients outside the Γ point in the AP state, which leads to a TMR ratio of 216%. This ALM/ALM insulator barrier/NM MTJ can produce the TMR effect with a spin-neutral current and benefit from zero stray field and fast spin dynamics, similar to the AFMTJs [17, 26–29, 32]. This [001] transporting MTJ also has the same disadvantages as AFMTJs, which have high demand for single crystals and are hard to be switched by a magnetic field. To overcome these drawbacks, we shift our focus on the transport configuration along the [110] direction.

Fig. 2(b) is the atomic and magnetic structures of FeF 2 cleaved from the (110) crystal face. When electrons tunnel along the [110] direction of FeF 2 , the distribution of the lowest decay rates satisfies Fig. 3(c). Since the band splits along the Γ-M direction, the two spins have completely different complex band structures at the Γ̄ point, as shown in Fig. 3(d), and this is also true at other k ∥ points, resulting in entirely diverse spin-resolved distribution of the lowest decay rates. Consequently, as long as the current flows along the [110] direction, it will be globally polarized, and the spin polarization will rapidly increase with the thickness of FeF 2 . As a result, the ALM insulator FeF 2 (110) has the spin filtering effect just like FM insulators.

Since the spin filtering effect of the FeF 2 (110) can polarspin-up 0.25 C Γ 0.50 κ (2π/c)

κ (2π/c)

κ (2π/c)

κ (2π/c')

0.45 FIG. 3. The spin- and k ∥ -resolved lowest decay rates of the evanescent states in FeF 2 at the Fermi energy when electrons tunneling along (a) the [001] and (c) the [110] directions. (b) The complex band structures of FeF 2 (001) at Γ, C, C ′ points. (d) The complex band structures of FeF 2 (110) at Γ̄, D̄ points. The black and red hollow circles indicate the lowest decay rates for spin-up and spin-down electrons at the Fermi energy, respectively.

ize current globally, the FM electrode is a matched choice of detecting the spin polarization of the current flowing through FeF 2 (110) by the TMR effect. As depicted in Fig. 1(d), we designed a kind of MTJ composed of FM and NM electrodes, i.e., FM/ALM insulator barrier/NM MTJ. The TMR effect can be generated by switching the magnetization of FM electrode. In this MTJ, only the ALM barrier needs to be singlecrystallized for the spin filtering effect. Thus, it is probably easier for experimentalists to grow than the MTJ with both ALM electrode and barrier as discussed before. Furthermore, this MTJ is free from the additional complex pinning structures thanks to the robustness of the ALM reference layer against external fields, and is easily switched by both electrical and magnetic methods, etc.

Hence, we constructed CrO 2 (110)/FeF 2 /IrO 2 MTJ as an example to exhibit the transport properties of the FM/ALM insulator barrier/NM MTJ. CrO 2 is a half-metallic material with 100% spin polarization, whose conduction channel is shown in Fig. S2(c) of the Supplementary Material [37]. In fact, CrO 2 and IrO 2 are only chosen for their same atomic structure with FeF 2 . We recommend replacing CrO 2 and IrO 2 with 4