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[Y. Miura](https://orcid.org/0000-0002-5605-5452)

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[First-Principles Studies on Spin-Dependent Transport of Magnetic Junctions with Half-Metallic Heusler Compounds](https://mdr.nims.go.jp/datasets/3ea7c9cd-1a08-4735-b6bd-b179c7df4621)

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JMSJ47_390 Journal of the Magnetics Society of Japan Vol.47, No.4, 2023INDEXINDEXCopyright ©2023 by the Magnetics Society of Japan. This article is licensed under the Creative Commons Attribution International License (CC BY 4.0)  http://creativecommons.org/licenses/by/4.0/J. Magn. Soc. Jpn., 47, 90-102 (2023) <Review>FFiirrsstt--pprriinncciipplleess  SSttuuddiieess  oonn  SSppiinn--ddeeppeennddeenntt  TTrraannssppoorrtt  ooff  MMaaggnneettiicc  JJuunnccttiioonnss  wwiitthh  HHaallff--mmeettaalllliicc  HHeeuusslleerr  CCoommppoouunnddss   Y. Miura*,** *Research Center for Magnetic and Spintronic Materials (CMSM), National Institute for Materials Science (NIMS),  1-2-1 Sengen, Tsukuba, Ibaraki 305-0032, Japan  **Center for Spintronics Research Network (CSRN), Graduate School of Engineering Science, Osaka University,  Machikaneyama 1-3, Toyonaka, Osaka 560-8531, Japan      Two types of magnetoresistance (MR), i.e., tunnel magnetoresistance (TMR) and current perpendicular to plane giant magnetoresistance (CPP-GMR), are theoretically discussed for systems with half-metallic (HFM) Co-based full Heusler compounds. In TMR, the non-collinear spin structure at the Co2MnSi(CMS)/MgO(001) interface that results from thermal spin fluctuations significantly reduced the TMR ratio at room temperature (RT). Enhancement of the exchange stiffness of CMS/MgO was essential to suppress the reduction in TMR at RT. Furthermore, it is proposed that inserting of a B2-CoFe layer between CMS and MgO is a promising way to enhance the interface exchange stiffness constant. In CPP-GMR, the interface spin asymmetry γ in the Valet-Fert model was essential to enhance GMR at RT, because the temperature dependence of γ in experiments was much larger than that of the bulk spin asymmetry β. I showed that the experimental CPP-GMR ratio increases with the decrease in the theoretical resistance-area product RPA, which is roughly consistent with the RP dependence of γ. This means that matching of the conductive channel between HMF and a non-magnetic metallic spacer is a key parameter to enhance γ. These findings will be very important for room temperature spintronics devices applied in future artificial intelligence hardware.  KKeeyy  wwoorrddss:: magnetoresistance, first-principles calculation, Heusler compound, TMR, CPP-GMR, ballistic transport, exchange stiffness    11..  IInnttrroodduuccttiioonn        I introduce ”Theoretical Study on Spin-dependent Transport Properties in Magnetic Junctions” in this review paper, which was a research subject of MSJ Outstanding Research Award 2022.  The spin-dependent tunneling between two ferromagnetic (FM) electrodes separated by a non-magnetic (NM) layer provides magnetoresistance (MR) that depends on the relative magnetization directions of the ferromagnetic electrodes.1–3) The effect is applied in magnetoresistive random access memory (MRAM) and ultra-sensitive magnetoresistive sensors, which are essential to realize artificial intelligence (AI) hardware and neuromorphic devices in the next-generation information society.4,5) There are two types of MR devices according to the type of nonmagnetic layer, which are tunneling magnetoresistance (TMR) devices with an insulating barrier layer, and current-perpendicular-to-plane giant magnetoresistance (CPP-GMR) devices with a NM metal spacer. To realize AI-dedicated and neuromorphic spintronics devices, a large MR ratio and low resistance are required.  High TMR ratios over 300% at room temperature have been realized in magnetic tunnel junctions (MTJs) with single-crystalline MgO and bcc-FeCo6–8) from coherent tunneling through the Δ1 evanescent state in MgO and the highly spin-polarized Δ1 band around the Fermi level in bcc-FeCo9,10). This means that in TMR devices, the spin dependence of the current can be characterized by the spin polarization of the Δ1 state of the ferromagnetic electrodes at in-plane wave vector kk║ = 00 rather than that of the total density of states (DOSs) around the Fermi level. In MgO-based MTJs, it is difficult to reduce the resistance while maintaining high TMR. Reducing the barrier thickness of MTJs decreases the TMR as well as the resistance, because the tunneling of non-spin-polarized Δ5 and Δ2 states is not negligible in a MgO layer with a thickness of a few monolayers. Thus, half-metallic ferromagnets (HMFs), which are metallic for the majority-spin band and insulating for the minority-spin band with complete (100%) spin polarization at the Fermi level, will be important to realize both high TMR and low resistance for MTJs with few-monolayer MgO thickness. CPP-GMR devices composed of all-metallic layers have lower resistance than TMR devices, but their MR ratios are also typically more than an order of magnitude smaller. This is due to the presence of many scattering processes in all-metallic systems, because in a metallic system, not only electrons with kk║ = 00 but also those with kk║ ≠ 00 contribute equally to the current. Thus, the spin-dependent transport in GMR devices can be characterized by the spin polarization of the total DOSs of the ferromagnetic electrodes rather than the spin polarization at the specific kk║, and HMFs again are                                              Corresponding author: Y. Miura  (e-mail: MIURA.Yoshio@nims.go.jp) 91Journal of the Magnetics Society of Japan Vol.47, No.4, 2023INDEXINDEXthe Co-based full Heusler compounds Co2YZ (Y = Cr, spin-polarization of the sp density of states (DOS) of Mn, Fe, Z = Al, Si, Ga, Ge) are the most promising Co2FeGa0.5Ge0.5 (CFGG) is weaker than that of candidates for use in spintronics devices because of CMS.23,24) Furthermore, several new Co-based Heusler their high Curie temperatures above room temperature  compounds such as Co2MnGa1−xAsx and Co2FeAl1−xSnx (RT). A characteristic feature of Co-based full Heusler were proposed by the machine learning method with compounds is that their electronic structure hardly finite temperature first-principles calculations using the changes with respect to the specific atomic disorder,  DLM.25)  such as B2-type disorder, where Y and Z atom sites are Table 1 summarizes experimental studies on TMR randomly occupied.17–19) The robustness of high spin ratios at low temperature (LT) and RT for Co-based full polarization against the atomic disorder of Co-based Heusler MTJs26-42). Half-metallic Co2YZ compounds Heusler compounds is important in terms of the have been applied in TMR devices with an amorphous relaxation of experimental requirements, because it is alumina barrier26–30) and then in MTJs with a MgO expected that some degree of atomic disorder will occur barrier, and large TMR ratios were during thin-film fabrication. The valence-band demonstrated.31–38,40) Furthermore, new barrier electronic structures of typical halfmetallic Co-based materials such as MgAl2O439) and Cu(In1−xGax)Se241,42) full Heusler compounds Co2MnSi (CMS) were have been applied as the barrier layer in Heusler-based investigated by photoelectron spectroscopy to confirm MTJs. Particularly, MTJs with Mn-rich CMS and the temperature dependence of the electronic Co2(Mn,Fe)αSi with MgO barriers provided huge TMR structures.20) The experimental valence band exceeding 2000% at very low temperature,38,40) photoemission spectra at 30 K were in good agreement indicating the potential of the halfmetallic electronic with density functional theory (DFT) calculations at structures. However, the TMR also rapidly decreases zero temperature, and no distinct temperature with increasing temperature.  dependence was observed in the experimental valence As is shown in Table 1, experiments on the TMR effect band spectra. Furthermore, the temperature of MTJs with half-metallic Co-based full Heusler dependence of the spin-polarization of CMS was shown compounds have not been reported after 2019. There are to be small by spin-resolved hard x-ray photoelectron two reasons for the decrease in experimental studies on spectroscopy.21) Recently, we calculated the temperature TMR in Heusler-based MTJs. One is the large dependence of the bulk electronic structures of temperature dependence of the TMR ratios of MTJs half-metallic Co-based Heusler compounds using the with half-metallic Co-based full Heusler compounds.  important for obtaining large GMR. Among the many theoretically predicted HMFs,11–16) that the temperature dependence of the disordered local moment (DLM) method,22) and found Bottom electrode Barrier Upper electrode TMR % @ LT TMR % @ RT Year [Reference] Co2(Cr0.6Fe0.4)Al Al-O Co2(Cr0.6Fe0.4)Al 26.5 16 200326) Co2MnAl Al-O Co0.5Fe0.5 40 27 200427) Co2FeAl Al-O Co0.75Fe0.25  47.3 200528) Co2MnSi Al-O Co2MnSi 570 67 200629) Co2Fe(Al0.5Si0.5) Al-O Co0.75Fe0.25 106 76 200630) Co2(Cr0.6Fe0.4)Al MgO Co0.5Fe0.5 240 90 200731) Co2MnSi MgO Co0.5Fe0.5 753 217 200832) Co2MnSi MgO Co2MnSi 700 180 200933) Co2Fe(Al0.5Si0.5) MgO Co2Fe(Al0.5Si0.5) 832 386 200934) Co2FeAl MgO Co2FeAl 308 130 200935) Co2FeAl MgO CoFe 700 330 201036) Co2Mn1.29Si MgO Co2Mn1.29Si 1995 354 201237) Co2(Fe0.5Mn0.5)Si MgO Co2(Fe0.5Mn0.5)Si 2610 429 201538) Co2FeAl MgAl2O4 Co2Fe(Al0.5Si0.5) 616 342 201639) Co2Mn1.3Si0.84 MgO Co2Mn1.3Si0.84 2010 335 201640) Co2Fe(Ga0.5Ge0.5) Cu(In0.8Ga0.2)Se2 Co2Fe(Ga0.5Ge0.5) 100 40 201641) Co2Fe(Ga0.5Ge0.5) CuGaSe2 Co2FeGa(0.5Ge0.5) 250 100 201942) TTaabbllee  11 Summary of experimental results on TMR ratio [%] at room temperature (RT) and low temperature (LT) for MTJs with Co-based full Heusler compounds Co2YZ  (exact temperatures differ between studies). 92 Journal of the Magnetics Society of Japan Vol.47, No.4, 2023INDEXINDEXThe TMR ratio of Heusler-based MTJs at RT is still perpendicular MCA in Heusler-based MTJs will be also smaller than that of CoFe-based MTJs. Recently, a a future task.  breakthrough in the RT TMR of CoFe-based MTJs was Table 2 summarizes experimental studies on GMR achieved by Scheike et al., where a 631% TMR ratio at  ratios and the difference in resistance-area product RT and 1143% TMR ratio at LT were observed for between parallel and antiparallel magnetizations ΔRA CoFe/MgO/CoFe(001) MTJs.43) To demonstrate the at LT and RT for various CPP-GMR systems with potential of half-metallic Heusler compounds, it is Co-based full Heusler compounds.49-68) Half-metallic important to achieve a larger TMR ratio at RT than that Co2YZ have also been applied in CPP-GMR devices, and of CoFe-based MTJs. The other reason for the decrease relatively large GMR ratios have been observed for in studies is requirement of MTJs with the Heusler-based magnetic junctions with a Cr spacer,49,52) perpendicular magnetic anisotropy (PMA). In Cu spacer,50,58) Ag spacer,51,53–57,59–61) Ag3Mg spacer64,66) spin-transfer torque switching of the magnetization and AgSn spacer.65) Furthermore, insertions of an direction in MTJs, PMA is required in order to reduce additional layer were attempted to enhance the the effective anisotropy field of MTJs. Since the matching of conductive channels between Co-based full Co-based full Heusler compounds show basically cubic Heusler compounds and NM spacers, such as NiAl symmetry, the magneto-crystalline anisotropy (MCA) of insertion in the CFGG/Ag(001) interface,62) Ni insertion Co2YZ is zero. Thus, the magnetic anisotropy of in the CFGG/Ag(001) interface,67) CoFe insertion in the Co2YZ(001) thin films tends to have an in-plane easy CMS/Ag(001) interface63) and Co2(Fe0.4Mn0.6)Si insertion axis due to the shape anisotropy. Some Mn-based in the CoFe/Ag(001) interface,68) which led to an Heusler compounds such as Mn3Ga and Mn3Ge with increase in the CPP-GMR ratios.  D022 structure have tetragonal structures, showing In this review, I discuss the spin-dependent transport perpendicular MCA in the bulk.44–47) However, the properties of TMR and CPP-GMR devices with experimental results for the TMR effects of half-metallic Co2YZ based on first-principles electronic Mn3Ga/MgO- and Mn3Ge/MgO-based MTJs44,48) are very structure and ballistic transport calculations. small compared with those of Co2YZ/MgO-based MTJs Particularly, I focus on the spin-dependent transport of because of the large lattice mismatch of Mn3Ga and magnetic junctions with half-metallic Co2YZ. First, I Mn3Ge with MgO47) and the non-half-metallic behavior discuss the origin of the temperature dependence of the of the Δ1 states of Mn3Ga.44) Therefore, achieving TMR ratios in MTJs. The spin-dependent conductance System GMR % @ LT (ΔRA mΩ· μm2 ) GMR % @ RT (ΔRA mΩ· μm2 ) Year [Reference] Co2MnSi/Cr/Co2MnSi(001)  2.4 (19) 200649) Co2MnSi/Cu/Co2MnSi(001)  9 200850) Co2Fe(Al0.5Si0.5)/Ag/Co2Fe(Al0.5Si0.5)(001) 14 (12.4) 6.9 (7.4) 200851) Co2MnSi/Cr/Co2MnSi(001)  5.2 (6.5) 200952) Co2MnSi/Ag/Co2MnSi(001)  28.8 (8.92) 200953) Co2Fe(Al0.5Si0.5)/Ag/Co2Fe(Al0.5Si0.5)(001) 80 (17) 34 (8) 201054) Co2Mn(Ga0.5Sn0.5)/Ag/Co2Mn(Ga0.5Sn0.5)(001) 17.2 (6.5) 8.8 (4) 201055) Co2MnSi/Ag/Co2MnSi(001) 67.2 36.4 (11.5) 201056) Co2Fe(Ga0.5Ge0.5)/Ag/Co2Fe(Ga0.5Ge0.5)(001) 129.1 (26.4) 41.7 (9.5) 201157) Co2MnGe/Cu/Co2MnGe(001)  9 (6) 201158) Co2(Fe0.4Mn0.6)Si/Ag/Co2(Fe0.4Mn0.6)Si(001)  74.8 201159) Co2(Fe0.4Mn0.6)Si/Ag/Co2(Fe0.4Mn0.6)Si(001) 184 58 (11) 201260) Co2Fe(Ga0.5Ge0.5)/Ag/Co2Fe(Ga0.5Ge0.5)(001) 183 (33) 57 (12) 201361) Co2Fe(Ga0.5Ge0.5)/NiAl/Ag/NiAl /Co2Fe(Ga0.5Ge0.5)(001) 285 (78) 82 (31) 201662) Co2Mn1.45Si0.82/CoFe/Ag/CoFe/Co2Mn1.45Si0.82(001)  20.7 (3.1) 201763) Co2(Fe0.4Mn0.6)Si/Ag3Mg/Co2(Fe0.4Mn0.6)Si(001)  63 (25) 201764) Co2(Fe0.4Mn0.6)Ge/AgSn/Co2(Fe0.4Mn0.6)Ge(001)  25 (7.5) 201865) Co2(Fe0.4Mn0.6)Si/Ag3Mg/Co2(Fe0.4Mn0.6)Si(001) 100 30 201966) Co2Fe(Ga0.5Ge0.5)/Ni/Ag/Ni/Co2Fe(Ga0.5Ge0.5)(001)  32.5 202167) CoFe/Co2(Fe0.4Mn0.6)Si/Ag/Co2(Fe0.4Mn0.6)Si/CoFe(001)  50 (19.1) 202168) TTaabbllee  22 Summary of experimental results on GMR ratio % and ΔRA mΩ· μm2 at room temperature (RT) and low temperature (LT) for CPP-GMR devices with Co2YZ (exact temperatures differ between studies). 93Journal of the Magnetics Society of Japan Vol.47, No.4, 2023INDEXINDEXof MTJs with CMS (or Fe) and MgO was investigated based on first-principles calculations in a noncollinear spin system. Noncollinear magnetic structures are formed as a result of spin fluctuation at finite temperatures. I focus on the spin-flip scattering of conducting electrons caused by the noncollinearity of local spin moments in various interfacial layers, and I discuss the electronic states at the interfaces that contribute to the spin-flip conductance. Our theoretical analysis confirmed the crucial contribution of the spin-flip process at the interface, which leads to a significant reduction in the MR ratio at RT. Then, I discuss the interface spin-asymmetry in CPP-GMR devices. I argue that the matching of the Fermi surface projected to the two-dimensional Brillouin zone in in-plane wave vector kk║ between the CMS and NM spacers is important to enhance the interface spin asymmetry. Furthermore, I clarify the correlation between experimental CPP-GMR ratios and the theoretical interface resistance in parallel magnetization in CPP-GMR.   22..  CCoommppuuttaattiioonnaall  MMeetthhoodd      I performed first-principles calculations for supercells consisting of Co-based Heusler compounds Co2YZ (Y = Mn or Fe, Z  =  Si, Ga or Ge) and various non- magnetic (NM) layers X such as X = MgO, Ag, Cr, V, Al and Au using the DFT within the generalized-gradient approximation for the exchange-correlation energy.69) In order to facilitate the optimization of atomic positions, which is important for determining the interface structure, I adopted plane-wave basis sets along with the ultra-soft pseudopotential method by using the quantum code ESPRESSO.70) The number of k points is taken to be 15×15×1 for all cases, and Methfessel- Paxton smearing with a broadening parameter of 0.01 Ry is used. The cutoff energy for the wavefunction and charge density is set to 30 Ry and 300 Ry, respectively. These values are large enough to deal with all the elements considered here within the ultra-soft pseudopotential method. Co2YZ/X/Co2YZ(001) magnetic junctions with Co (YZ ) termination were constructed in a multilayer structure containing seventeen (fifteen) atomic layers of Co2YZ and seven or nine atomic layers of NM. The in-plane lattice parameter of the supercell is fixed to that of Co2YZ. These values correspond to a0/√2, where a0 is the lattice constant of the bulk Co2MnSi (5.65 Å). Since the lattice constant of the NM layer X is different from that of Co2YZ, the lattice mismatches between Co2YZ and X on the 45◦ in-plane rotation at the (001) face are considered, leading to tetragonal distortion in the NM layer (X ).   For transport calculations, I calculated the ballistic conductance of the magnetic junctions based on the Landauer formula.71) I considered an open quantum system consisting of a scattering region corresponding to NM spacer X and a junction with Co2YZ attached to left and right semi-infinite electrodes corresponding to bulk Co2YZ. The transmittance can be obtained by solving the scattering equation with infinite boundary conditions, in which the wavefunction of the scattering region and its derivative are connected to the Bloch states of each electrode.72) The potential in the scattering equation can be obtained from self-consistent filed electronic structure calculations for the supercell containing a left electrode and a scattering region. It was confirmed that five atomic layers of Co2YZ between the right edge of the electrode region and the left-hand side of the Co2YZ/X interface are sufficient to represent the shape of the local potential of bulk Co2YZ in the electrode region. Since our system is repeated periodically in the xy plane and propagating states can be assigned by an in-plane wave vector kk║ = (kx, ky) index, different kk║ do not mix and can be treated separately. Furthermore, our approach neglects the spin-orbit interaction and noncollinear spin configuration. Thus, I solved scattering equations for some fixed kk║ and spin index using Choi and Ihm’s method.72, 73)    33..  SSppiinn--ddeeppeennddeenntt  ttrraannssppoorrtt  iinn  TTMMRR  ddeevviicceess      33..11  IInntteerrffaaccee  eelleeccttrroonniicc  ssttrruuccttuurreess    Figure 1(a) shows the majority- and minority-spin local density of states (LDOS) at the Fermi level projected onto each atomic sphere as a function of the distance from the Co2MnSi(CMS)/MgO(001) junction.74) It is found that in the case of MnSi termination, interface states appear in the minority-spin state at the Fermi level, reducing the half-metallicity of CMS. Similar results have been obtained in the Co termination of the CMS/MgO(001) junction.75) Examining the electronic structure at the interface in FFiigg..  11  Local density of states (LDOS) at Fermi level (EF) projected onto  each atomic sphere as a function of the distance from interface for  Co2MnSi(CMS) /MgO(001) with MnSi termination. Positive (negative)  sign of y-axis indicates majority-spin (minority-spin) LDOS. Schematic  figures of supercell of CMS/MgO/CMS(001) MTJ are shown above the  LDOS. (b) The solid lines show LDOS of Mn d at MnSi-terminated  CMS/MgO(001) interface as a function of energy relative to EF. LDOS in bulk (electrode) region is also shown by dashed line as a reference. ref.[74] 94 Journal of the Magnetics Society of Japan Vol.47, No.4, 2023INDEXINDEXmore detail, Fig. 1(b) presents the majority- and minority-spin LDOS of each atom and atomic orbital in the MnSi-terminated CMS/MgO(001). The LDOS in the bulk (electrode) region is also shown as a reference. In Fig. 1(b), it is found that the spin moment of the interfacial Mn (4.06 μB) is larger than those of the bulk CMS (3.31 μB) due to the charge transfer from the minority-spin to the majority-spin states. I considered that the relaxation of the atomic positions and the reduced symmetry at the interface cause charge transfer from the minority-spin to the majority-spin 3d states owing to the localization effect of 3d electrons. Furthermore, in Fig. 1(b), minority-spin states are observed around the Fermi level in the LDOS of the interfacial Mn and Si atoms. The spin polarizations of interface Mn and Si are -0.4 and 0.01, respectively. At the interface, the Mn atom loses half of its first nearest neighbor Co atoms, and non-bonding Mn d states (possibly dyz, dzx and d3z2−r2) can appear around the Fermi level. Since the Mn d3z2−r2 orbital hybridizes with the O pz orbital at the interface, the LDOS of the interfacial Mn dγ states maintain a half-metallic character. Contrary to this, the interfacial Mn dε orbitals (dyz and dzx) become nonbonding in the O-top configuration, reducing the spin polarization at the MnSi termination. Mavropoulos, et al. suggested that the interface states that appear in the minority-spin gap at the junctions of MTJs can also contribute to the tunneling conductance in anti-parallel magnetization, through spin mixing processes such as magnon excitations and inelastic scattering at RT.76-78)   33..22  SSppiinn--fflliipp  ssccaatttteerriinngg  aatt  iinntteerrffaacceess    Since the CMS/MgO(001) interface is not half-metallic due to the appearance of non-bonding states in the minority-spin states around the Fermi level, the spin-flip scattering can be enhanced at finite temperature through the interface states as a result of the thermal fluctuation of magnetic spin moments.79) Thus, the effects of thermal fluctuation and the noncollinear magnetic structure at the interface of MTJ on the spin-dependent transport are considered. The potential for a noncollinear-spin system was obtained by rotating the density matrix of a collinear-spin system with a spin-1/2 rotation matrix. According to Kübler’s formulation,79,80) off-diagonal elements of the effective potential that provide spin-mixing in the ballistic conductance are given by (δExc/δn↑ – δExc/δn↓)(sinθ cosφ ± i sinθ sinφ)/2,  (1) where δExc/δns (s =↑, ↓) is the exchange and correlation potential in a collinear-spin system and θ and φ are the polar and azimuthal angles of a local spin moment with respect to the global spin-quantum axis, respectively. Since our calculations neglected spin-orbit interaction, the azimuthal angle φ did not affect the calculation results and was set to zero. The off-diagonal part of the local potential expressed by equation (1) is given at real-space positions of the supercell within each atomic sphere of interfacial atoms having a noncollinear spin moment.  Fig. 2 (a) shows the model system used for CMS/MgO/CMS MTJs with MnSi termination in the anti-parallel magnetization, where the interface Mn and sub-interface Co spin moments on both sides of the junction were canted by an angle θ in order to provide interfacial spin-flip scattering.81) Figure 2(b) shows the conductance of a CMS/MgO/CMS MTJ with MnSi termination in parallel or anti-parallel magnetization as a function of the angle of the interface spin moments. zero conductance was obtained at θ = 0◦ for anti-parallel magnetization because of the half-metallic character of CMS. On the other hand, the anti-parallel conductance increased with increasing θ, while the parallel conductance decreased. This behavior can be attributed to spin-flip scattering because of the noncollinear interfacial magnetic structures. Fig. 2(c) shows the in-plane wave-vector (kk║) dependence of the tunneling conductance of the MTJ in anti-parallel magnetization with the interfacial Mn spin moment θ = 5◦. It was already shown in ref. [74,75] that bulk CMS has a Δ1 band at the Fermi level in the majority-spin states, and FFiigg..  22  (a) Schematic figure of CMS/MgO/CMS MTJs with MnSi termination and local spin moments for each layer, with a noncollinear magnetic structure on both sides of junction. (b) Parallel and antiparallel conductance of CMS/MgO/CMS MTJ as a function of the angle of interfacial local spin moments θ. (c) In-plane wave-vector (kk║) dependence of antiparallel conductance of CMS/MgO/CMS MTJ with θ = 15◦ for MgO thickness ~ 2.0 nm (nine atomic layers of MgO). ref. [81] FFiigg..  33  Schematic images of LDOS for interface Co or Mn atoms of CMS in (a) collinear spin system (θ = 0) and (b) noncollinear spin system (θ ≠ 0), respectively. Gray areas and cross-hatched black areas indicate the interface d-states and interface s-states, respectively. ref. [81] 95Journal of the Magnetics Society of Japan Vol.47, No.4, 2023INDEXINDEXthe kk║-dependence of the conductance of CMS/MgO/CMS MTJs in parallel magnetization has a broad peak at kk║  = (0,0) because of the slow decay of Δ1 states in the MgO barrier.9) As can be seen in Fig. 2(c), the kk║-dependence of the conductance in anti-parallel magnetization also shows a broad peak at kk║  = (0,0), indicating that the spin-flip conductance is dominated by the Δ1 channel of the conducting electrons. Fig. 3 shows schematic images of the LDOS of interface Mn and sub-interface Co atoms in collinear spin (θ = 0◦) and noncollinear spin (θ ≠ 0◦) cases.81) In the case of a collinear spin system, the CMS/MgO interface is not half-metallic and has interface states in the half-metallic gap of minority-spin states originating from the non-bonding Mn d orbital (see also Fig. 3 in ref. [75]). Then, the noncollinear spin moments of interface Mn and Co generate spin-mixing in the LDOS of CMS/MgO/CMS(001). At the Fermi level, as shown in Fig. 3(b), both bulk s-states and interfacial Mn d-states are projected on the majority-spin and minority-spin states of the global spin-quantum axis, providing spin-flip scattering of the Ballistic conductance. The broad peak at kk║  = (0,0) in Fig. 3(c) indicates that the projected minority-spin Δ1 states mainly contribute to the conductance in anti-parallel magnetization. This means that the contribution of the interfacial d-states that appeared at the MnSi termination of CMS/MgO junctions in the collinear-spin system (cross-hatched black areas in Fig. 3(a) and (b)) was rather small compared to that of the s-states caused by the noncollinearity of interfacial Mn and Co spin moments (light-gray areas in Fig. 3(b)). Since the interface states were mainly composed of dxy and dx2−y2 orbitals of interfacial Mn and Co atoms, they show fast decay in the MgO barrier.  33..33  EEffffeeccttss  ooff  nnoonnccoolllliinneeaarr  mmaaggnneettiissmm  oonn  TTMMRR    To examine the interfacial structure dependence of the spin-flip conductance, Fig. 4 shows logarithmic plots of TMR ratios as a function of the angle of the local spin moment θ in various interfacial layers of CMS/MgO/ CMS. First, I can confirm that the TMR ratios of the MTJs decreased with increasing θ due to the spin-flip conductance in the interfacial region. In particular, the TMR ratios for CMS/MgO/CMS MTJs with noncollinear first and second interface Co spin moments had values similar to that of Fe/MgO/Fe MTJs for θ > 5◦. This indicates that the advantage of half-metallic electrodes is valid only for small θ if the noncollinearity arises at interfacial Co spin moments. On the other hand, the decrease in the TMR ratio with increasing θ was rather gradual for MTJs with noncollinearity in the first interface Mn spin moment at the MnSi termination and the third interface Co spin moment, compared to that in the first and second Co layers, indicating that the spin-flip conductance was suppressed at the interfacial MnSi-layers and that spin-flip scattering mainly occurred in the first and second Co layers. Since the Si atom is nonmagnetic, the off-diagonal part of the effective potential expressed by (δExc/δn↑ – δExc/δn↓) was rather small, which led to a suppression of the spin-flip probability at the MnSi-layer. In the calculations, the angle dependence of the size of the Co and Mn spin moments was neglected. This is a reasonable approximation for Mn spin moments in view of their localized character in bulk CMS82,83) and MnSi-terminated interfaces.84) For Co-terminated interfaces, the strong hybridization between Co and O atoms can cause angular dependence for the interfacial Co spin moments. As discussed for Ni spin moments in NiMnSb,85) longitudinal fluctuations will increase the length of the local spin moments with increasing angle. This will enhance spin-flip scattering in interfacial regions, resulting in further decrease in the TMR ratio.  33..44  EExxcchhaannggee  ssttiiffffnneessss  aatt  iinntteerrffaacceess   To estimate the exchange stiffness of the interfacial spin moments, The increase in the one-electron band energy E(θ) relative to the collinear-spin system (θ = 0◦) was calculated. Fig. 5(a) shows E(θ) as a function of the angle of local moments θ for various interfacial layers in CMS/MgO(001) and Fe/MgO(001) junctions.81) Then, the results were fitted using E(θ) = B (1–cosθ). Here, B is the inter-atomic-layer exchange stiffness constant; the value in Table 3 for each layer in the interfacial and bulk regions is shown in Table 3. This estimation corresponds to the magnetic force theorem method.86,87)    It was confirmed that the increase in the band energy as a function of the canting angle shown in Fig. 5(a) can be fitted by ~ θ2 up to θ = 30◦. This means that the error in the estimation of the exchange stiffness constant by the magnetic force theorem is very small in the present system.88) First, it can be found that the exchange stiffness of the interfacial Co layer at the Co termination was much smaller than that of the interfacial MnSi-layer at the MnSi termination and that of bulk CMS. These results are consistent with the results of recent first-principles calculations of the exchange constant of CMS/MgO(001).89)  Furthermore, the exchange stiffness of the FFiigg..  44  Logarithmic plots of TMR ratios as a function of angle of local spin moment θ for noncollinear magnetic structures of various interfacial layers in CMS/MgO/CMS MTJs and Fe/MgO/Fe MTJs. ref. [81] 96 Journal of the Magnetics Society of Japan Vol.47, No.4, 2023INDEXINDEXsub-interfacial Co-layer at the MnSi termination was smaller than that of the interfacial MnSi-layer. This means that at the MnSi termination, the sub-interfacial Co spin moments fluctuate more easily than interfacial Mn spin moments. On the other hand, a very high exchange stiffness was found for the interfacial Fe-layer at Fe/MgO junctions compared to those of CMS/MgO junctions, indicating the robustness of interfacial Fe spin moments against thermal fluctuation. One possible reason for this behavior of the exchange stiffness in interfacial regions compared to that of bulk CMS is related to the change in the interfacial spin moments.  As is discussed in ref. [75,84] the majority-spin charge of interfacial Co atoms transfers to the MgO side because of the strong bonding between Co and O atoms. On the other hand, the interfacial Mn spin moments of the CMS/MgO junction74,84) and interfacial Fe spin moments of the Fe/MgO junction9,81) increase in comparison with those of the bulk because of the localization effect (and band-narrowing effect) of the non-bonding d-orbitals in the interfacial regions. Since E(θ) can be characterized by the off-diagonal elements of the local potential expressed by equation (1), the decrease (increase) of local spin moments reduced (enhanced) the off-diagonal terms, leading to the low (high) exchange stiffness.   33..55  TTeemmppeerraattuurree  ddeeppeennddeennccee  ooff  TTMMRR   Finally, I discuss the effect of interfacial spin-flip scattering on the TMR ratios at finite temperatures. To estimate the TMR ratio at a finite temperature, the Boltzmann average of the tunneling conductance was calculated using the following equation, GP(AP)(θ) = ∫ gP(AP)(θ) exp(–2E(θ) / kBT ) sinθ dθ / ∫ exp(–2E(θ) / kBT ) sinθ dθ,          (2) where gP(AP)(θ) is the θ-dependence of the tunneling conductance in parallel (P) and anti-parallel (AP) magnetization, as shown in Fig. 2(b). E(θ) is the increase in the band energy caused by the noncollinearity of the local spin moments at each interfacial layer, as discussed in the previous paragraph. The factor of 2 for E(θ) indicates contribution from both sides of the MTJs.  Fig. 5(b) shows the calculated temperature dependence of the TMR ratio for the MTJs of CMS/MgO/CMS and Fe/MgO/Fe MTJs with 2 nm MgO thickness. As can be seen here, the reduction in the TMR ratios of CMS/MgO/CMS MTJs with the 1st layer Co spin fluctuation at Co termination and the 2nd layer Co spin fluctuation at MnSi termination are much faster than that of Fe/MgO/Fe MTJs with 1st layer Fe spin fluctuation. This means that the interfacial spin-flip scattering significantly reduces the TMR ratio at RT.  This result qualitatively agrees with the results of the temperature dependence of the TMR ratio. The temperature dependence of the TMR ratios of CMS/MgO/CMS MTJs with the 1st layer Mn spin fluctuation at the MnSi termination is also significant, but the TMR ratio maintains a large value at RT, indicating that the large exchange stiffness constant at the interface improves the TMR ratio at RT. The tunneling conductance evaluated at finite temperature using equation (2) is 700% at RT for CMS/MgO/CMS MTJs with the thermal fluctuation of the first layer Co spin moment at the Co termination, and 1500% at RT with the thermal fluctuation of the 2nd Co spin moment at the MnSi termination. These TMR ratios are smaller than the TMR ratio of 2500% at 300 K for Fe/MgO/Fe MTJs with thermal fluctuation of the 1st layer Fe spin moment.  From these results, I can conclude that the experimentally observed low TMR ratio of the MTJs at RT can be attributed to the spin-flip conductance caused by thermal fluctuation of the interfacial spin moments. Therefore, to suppress the reduction of the TMR ratios at finite temperature in CMS/MgO/CMS MTJs, the exchange stiffness of Co spin moments at the CMS/MgO(001) interface must be enhanced. In estimating the thermal average of the tunneling conductance, E(θ) was treated as the thermal excitation energy of the local spin moments per in-plane unit cell area, which corresponds to the replacement of various B meV/u.c.a. Co1st Co2nd Mn1st Fe1st Bulk (CMS or Fe) 414 414 565 600 MgO interface 145 347 529 753 FFiigg..  55  (a) Increase in band energy relative to that of collinear-spin system E(θ) as a function of angle of local spin moments θ for first-layer Co spin moment at Co termination, first-layer Mn spin moment and second-layer Co spin moment of CMS/MgO(001), and first-layer Fe spin moment of Fe/MgO(001) junctions. ref. [81] (b) Temperature dependence of TMR ratios of CMS/MgO/CMS(001) and Fe/MgO/Fe(001) calculated by Eq. (2) including noncollinear spin structure of various spin moments. TTaabbllee  33 Inter-layer exchange stiffness constant B meV/u.c.a. fitted to increase in band energies E(θ)=B(1–cosθ) due to noncollinearity of local spin moments for CMS/MgO(001) and Fe/MgO(001) junctions. 1st and 2nd indicate first and second layer at the interface with MgO, respectively. u.c.a. is unit-cell area ~32 Å2. Ref.[81] 97Journal of the Magnetics Society of Japan Vol.47, No.4, 2023INDEXINDEXmagnetic excitation modes at interfacial regions by a single excitation mode of interfacial spin moments given by the in-plane unit cell E(θ). If other magnon excitation modes are considered, the temperature dependence of TMR shown in Fig. 5(b) will change. However, the inclusion of the other excitation modes in equation (2) will lead to a further decrease in the TMR ratio at RT, and the chemical trend of the exchange stiffness at the interface has little dependence on the excitation modes. Therefore, our conclusion that the significant reduction of the TMR ratio of CMS/MgO/CMS MTJs at RT can be attributed to the interfacial spin-flip conductance does not change with a more accurate estimation of the thermally averaged tunneling conductance.  33..66  EEnnhhaanncceemmeenntt  ooff  iinntteerrffaaccee  eexxcchhaannggee  ssttiiffffnneessss   It was concluded that the reduction in the TMR ratio in CMS/MgO/CMS MTJs at RT can be attributed to spin-flip scattering in the interfacial region caused by thermal fluctuation of interfacial and sub-interfacial Co layers, and to suppress reduction in the TMR ratio at RT, it is important to enhance the exchange stiffness of Co layers at CMS/MgO junctions. Thus, I calculated the exchange stiffness constant at the interface of MgO for the other Heusler compounds and B2-CoFe. Fig. 6 shows a bar graph of the inter-atomic layer exchange stiffness constant at each interface. As shown in Fig. 6, very large exchange stiffness constants were obtained for the Co-terminated B2-CoFe/MgO interface, indicating that the B2-CoFe/MgO interface is very robust against thermal fluctuation.  Furthermore, the exchange stiffness of the Co-terminated Co2FeSi(CFS)/MgO and Co2FeAl(CFA)/ MgO interfaces is larger than that of CMS/MgO and Co2MnAl(CMA)/MgO interfaces. Thus, in order to suppress the thermal fluctuation at the interface with MgO, CFS and CFA are better than CMS and CMA. It was considered that the behavior of the exchange stiffness in the interfacial region is related to the behavior of the interfacial spin moments, because the exchange stiffness constant is proportional to the square of the local spin moments. In fact, as can be seen here, the behavior of the interfacial Co spin moments is consistent with the behavior of the interfacial exchange stiffness constant. This means that the enhancement of the interfacial Co spin moment is effective to obtain a large exchange stiffness constant and suppress the thermal fluctuation of Co spin moments at the MgO junction.  33..77  NNeeww  pphhyyssiiccaall  pprrooppeerrttyy  ffoorr  tteemmppeerraattuurree  ddeeppeennddeennccee  ooff  TTMMRR   Recently, a new physical property characterizing the temperature dependence of TMR in MTJs was proposed in ref. [90] It was found that the intra-atomic s−d exchange coupling Jsd at the interface was also a key parameter in the reduction of TMR at RT. Masuda described the temperature dependent spin-flip Hamiltonian including Jsd at the interface Fe of Fe/MgO/Fe(001) MTJs by the coherent potential approximation of various magnetization directions according to the temperature dependence of the magnetization up to the Curie temperature. A smaller Jsd can disconnect the coupling between conductive s electrons and localized d electrons, leading to suppression of the spin-flip scattering at the interface. Therefore, ferromagnetic materials with smaller Jsd will be preferable to obtain larger TMR at RT. This new physical parameter Jsd will also be important for improving the large temperature dependence of the TMR of Heusler based-MTJs.  44..  SSppiinn--ddeeppeennddeenntt  ttrraannssppoorrtt  iinn  CCPPPP--GGMMRR  ddeevviicceess   44..11  VVaalleett--FFeerrtt  mmooddeell      According to Valet and Fert’s two-current model,91) the resistance change-area product between parallel and antiparallel magnetization congurations (∆RA = RAPA – RPA) can be expressed by two intrinsic factors in spin-dependent electron scattering. They are the bulk spin-asymmetry coefficient (β) in ferromagnetic (FM) layers and the interfacial spin-asymmetry coefficient (γ), which are included in the following equation,  ∆RA = (β ρ*F + 2 γ ρ*N r*b)/( ρ*F tF + ρ*N tN + 2 r*b)  (3) where ρ*F and ρ*N are the resistivity of the ferromagnetic electrode and the non-magnetic (NM) spacer layer in the bulk, and the tF and tN are the thickness of the FM and NM layers, respectively. The r*b is the spin-independent interface resistance. The β is enhanced by using the HMF in the FM layer, while enhancement of the γ is accomplished by reduction of the interface resistance RFM/NM through the choice of non-ferromagnetic (NM) spacers. Here, I mainly discuss the γ in HFM/NM interface, because the temperature dependence of is much larger than that of the bulk spin FFiigg..  66  (a) Exchange stiffness constant B of Co atom at Co termination of various MgO interfaces with Co-based Heusler compounds and B2-CoFe. Estimation of exchange stiffness constant B is same as in Fig. 5(a). (b) Interfacial Co spin moments at Co terminations of various MgO interfaces with Co-based Heusler compounds and B2-CoFe. 98 Journal of the Magnetics Society of Japan Vol.47, No.4, 2023INDEXINDEXasymmetry β. The γ is expressed by the majority-spin resistance (RP↑) and minority-spin interface resistance (RP↓) in the parallel magnetization conguration as follows,  γ = (RP↓ – RP↑)/(RP↓ + RP↑)              (4) The RP↓ at the HMF/NM interface is determined by the interface state-mediated spin-ip scattering, as discussed in the previous section, because there are no states around the Fermi level of HMF minority-spin states in the bulk. In HMF/NM junctions, there are many scattering processes, making them complicated compared with HMF/insulator junctions such as Heusler/MgO. This results in a nite value for RP↓ in the HMF/NM junctions, which will be difficult to control experimentally. On the other hand, the RP↑ will be determined by the matching of the conductive channel between the majority-spin states of HMF and NM. Thus, in CPP-GMR devices with HMFs, matching of the majority-spin band dispersions with the NM spacer is important to reduce the RP↑ of HMF/NM interfaces. As shown in the inset of Fig. 7(a), γ will rapidly increase with decreasing RP↑ under xed RP↓. This means that the choice of NM spacer layer is very important to enhance the CPP-GMR with HFM Heusler compounds.   44..22  CCoommppaarriissoonn  ooff  tthhee  ttrraannssmmiittttaannccee  ffoorr  CCMMSS//XX//CCMMSS    As discussed in ref., [56] the transport properties obtained from the Landauer formula do not give the real conductance and resistance for a three-dimensional metallic multilayer. However, our aim in this paper is not to obtain the quantitatively correct conductance and resistance of HFM/NM/HFM junctions, but to clarify the differences in the conductance and resistance thorough HFM/NM junctions depending on the NM spacer and the interfacial termination. If the electrode region (ferromagnetic layer) consists of Co2MnSi (CMS) for all cases, the difference in resistance between HFM/NM/ HMF and HFM/NM’/HFM junctions indicates the difference in the interfacial resistance between the HFM/NM and HFM/NM’ junctions, which can originate from electron scattering due to changes in the local potential and band structures at the interfacial region. Furthermore, a completely epitaxial multi-layer was assumed, where the crystal momentum parallel to the layer (or kk║) is conserved because of the two-dimensional periodicity of the system, and the number of conductive channels perpendicular to the plane is less than about ve or six per kk║. Therefore, the ballistic transport calculations from the Landauer formula can be applied to evaluate the difference in the interfacial resistance depending on the band structures of the materials on both sides of the interface. This point has already been confirmed in our previous work in ref. [56] where the difference in the resistance-area products in the ballistic transport calculation between CMS/Cr/CMS(001) and CMS/Ag/CMS(001) is roughly comparable to that obtained experimentally. It is considered that this justies the use of the Landauer formula for the investigation of the interfacial resistance of all metallic multilayers depending on the non-ferromagnetic spacer and the interfacial termination. Figure 7 shows the majority-spin resistance-area product of CMS/X/CMS(001) with MnSi termination and Co termination for NM spacer X = fcc-Au, fcc-Ag, fcc-Al, bcc-V, and anti-ferromagnetic bcc-Cr in the parallel magnetization conguration.92) First, Fig. 7 shows that the majority-spin resistance-area product RP↑A of the junctions with X = Au, Ag, Al, and V is about 3~5 mΩµm2, which corresponds to a conductance of 0.6~1.0 G0, where G0 = e2/h is the ballistic conductance without scattering in the Landauer formula. On the other hand, the junctions with anti-ferromagnetic bcc-Cr spacer are about 13~16 mΩµm2, which is 3~5 larger than that with the other spacer. These results qualitatively agree with recent experimental results on CPP-GMR devices incorporating epitaxial CMS/Ag/CMS and CMS/Cr/CMS, where a smaller resistance-area product has been obtained for the CMS/Ag(001) interface than for CMS/Cr(001). The other feature of the transmittance of CMS/X/CMS(001) shown in Fig. 7 is the interface-structure dependence, i.e., the resistance-area product with Co termination is larger than that with MnSi termination except for the anti-ferromagnetic bcc-Cr spacer. Analyzing the kk║-dependence of the majority-spin transmittance at E = EF for CMS/X/CMS with Co and MnSi termination, I found that the difference in transmittance between the two terminations is significant at kx ≠ 0, especially for X = Ag, Au, Al and V. This means that the transmittance in the whole kk║ region contributes to the interface-structure dependence. To understand this, I show in Fig. 7(b) the local density of states (LDOS) of interfacial Co-d at the Co-terminated interface and interfacial Mn-3d at the MnSi-terminated interface in CMS/Ag(001), together with the transmittance averaged FFiigg..  77  (a) Bar graph comparing transmittance of CMS/X/CMS(001) for X = Au, Ag, Al, V and Cr with MnSi and Co termination. (b) LDOS of interfacial Co-d at Co termination and interfacial Mn-d at MnSi termination in CMS/Ag/CMS(001), together with transmittance averaged over the whole kk║ region as function of energy relative to Fermi energy. ref. [92]. 99Journal of the Magnetics Society of Japan Vol.47, No.4, 2023INDEXINDEXover the whole kk║  region as a function of the energy relative to the Fermi energy. It is evident that the transmittance increases with the decrease in the d component of the LDOS of interfacial atoms, indicating that the interfacial d-orbitals act as a scatterer of electrons. Furthermore, I found that the LDOS of Co-d at the Co termination shows large components at the Fermi level compared with those of Mn at the MnSi termination. The large d-components at the interfacial regions cause additional reection of propagating electrons, leading to the large interfacial resistance in metallic multilayer. These features can be observed also in the LDOS of CMS/Au, CMS/Al and CMS/V interfaces.   44..33  FFeerrmmii  ssuurrffaaccee  mmaattcchhiinngg  aatt  HHFFMM//NNMM  iinntteerrffaaccee   Figures 8(a) and (b) show the kk║-dependence of the majority-spin conductance in the parallel magnetization conguration of CMS/Ag/CMS and CMS/Cr/CMS. It is clearly seen that the conducting channels of CMS/Cr/CMS are restricted to a small region around kk║ = (0,0). On the other hand, highly conducting channels spread over almost the entire region in the kk║ plane for CMS/Ag/CMS. The highly conducting region in the kk║ plane can be understood qualitatively by considering the matching of the Fermi surface over the kk║-plane between CMS and Ag.  Figures 8(c)-(e) show the Fermi surfaces in the Brillouin zones of L21-CMS, fcc-Ag, and AFM bcc-Cr with a tetragonal unit cell. A large overlapping area of the Fermi surface can be seen on the kk║-plane between CMS and Ag, resulting in the small resistance-area product (2RCMS/AgA = 3.21 mΩ·µm2) for CMS/Ag/CMS, while the mismatch of the Fermi surface between CMS and Cr causes the large resistance-area product (2RCMS/CrA = 16.1 mΩ·µm2) of CMS/Cr/CMS. Although no conductance (innite resistance) is expected in the minority-spin channel in ideal half-metals, a nite resistance-area product could appear in the minority-spin channel in real half-metals owing to such reasons such as atomic disorder, thermal uctuation of magnetic moments, spin-orbit coupling, and so on. The minority-spin resistance-area product causes a reduction in both β and γ. Thus, one can expect a larger value of γ for CMS/Ag/CMS compared to CMS/Cr/CMS, since the majority-spin resistance-area product of CMS/Ag/CMS is much smaller than that of CMS/Cr/CMS. Experimentally, the CMS/Ag/CMS and CMS/Cr/CMS junctions with the same annealing temperature (350◦C) and exactly the same stacking structure except for the spacer layer showed RA of 51.4 and 66.9 mΩ·µm2, respectively.56) Here, the difference of RA (~ 15.5 mΩ·µm2) between them is mainly caused by the difference in interface resistances, i.e., RCMS/AgA and RCMS/CrA, because the contribution of the bulk resistance in Ag and Cr spacers is negligibly small. Although the experimental RA is much higher than the calculated 2RCMS/NMA (NM = Ag or Cr) since it includes all of resistances in the CPP-pillar, the difference in RA, i.e., 2RCMS/CrA − 2RCMS/AgA of 15.5 mΩ·µm2 is roughly comparable to the calculated one (~ 12.9 mΩ·µm2). This agreement is evidence of a small interface resistance and large γ between CMS and Ag due to good Fermi-surface matching, as predicted in our calculations.   44..44  CCoorrrreellaattiioonn  bbeettwweeeenn  tthheeoorreettiiccaall  RRPP  aanndd  eexxppeerriimmeennttaall  CCPPPP--GGMMRR    Fig. 9 shows the experimental CPP-GMR in various systems as a function of the calculated majority-spin resistance-area product in parallel magnetization for the corresponding systems. As can be seen in Fig. 9, the FFiigg..  88  Majority-spin conductance in parallel magnetization configuration calculated for (a) CMS/Ag/CMS and (b) CMS/Cr/CMS as function of kk║ = (kx, ky). Fermi surfaces in Brillouin zone corresponding to tetragonal unit cell of (c) L21-CMS, (d) fcc-Ag, and (e) anti-ferromagnetic (AFM) bcc-Cr plotted by XCRYSDEN.93) ref.[56] FFiigg..  99  Correlation between experimental CPP-GMR ratios at room temperature vs. theoretical resistance-area product RP for various CPP-GMR devices in experiments. Broken line indicates interface spin asymmetry γ as function of majority-spin interface resistance in parallel magnetization RP↑. 100 Journal of the Magnetics Society of Japan Vol.47, No.4, 2023INDEXINDEXexperimental CPP-GMR ratio increases with the decrease in the theoretical RPA, which is roughly consistent with the RP↑ dependence of the interface spin asymmetry γ as shown in the inset of Fig. 7. This means that the enhancement of γ by the matching of the conductive channel between HFM and NM through Fermi surface matching is very important to obtain large CPP-GMR. In experiments, insertion of B2-type NiAl between Co2FeGa0.5Ge0.5 (CFGG) and the Ag spacer dramatically enhanced the CPP-GMR and ∆RA both at LT and RT62). This modication was motivated by a theoretical calculation showing that the Fermi surface matching between CFGG and B2-type NiAl is much better than that of CFGG and Ag. Furthermore, the experimental results showed that not only NiAl but also Ni insertion between CFGG and the Ag spacer enhanced the CPP-GMR67). This result is also conrmed by theoretical calculations showing that CFGG/Ni/Ag/Ni/CFGG exhibits smaller RP↑ than CFGG/Ag/CFGG. More recently, half-metallic Co2FeMnSi (CFMS) was inserted between A2-CoFe and Ag spacer, and the CPP-GMR was maximized when the thickness of CFMS was around 7 nm68). One possible reason for the large CPP-GMR was attributed to the better Fermi surface matching of CFMS/CoFe as compared with CoFe/Ag. All these results indicate that Fermi surface matching between HFM and the NM spacer layer is a key factor in enhancing the CPP-GMR for future ultra-sensitive magneto-resistive sensors.  55..  SSuummmmaarryy    In this paper, rst-principles studies on electronic structures and ballistic transport properties of HMF Co-based Heusler compound-based magneto-resistive devices were reviewed. The effects of spin-ip scattering by interfacial noncollinear magnetic structures on the TMR ratios of MTJs with Co2MnSi(CMS) and MgO were investigated using rst-principles calculations. It was found that the effects of the noncollinearity of interfacial Co spin moments on the TMR are signicant in determining the spin-flip conductance. Furthermore, the inter-atomic layer exchange stiffness constant of interfacial and sub-interfacial Co spin moments at CMS/MgO junctions is much smaller than that in bulk regions. From these results, I conclude that the low TMR ratio at room temperature in MTJs with half-metallic Co-based full Heusler compounds can be attributed to spin-flip scattering by the noncollinear magnetic structures of interfacial Co as a result of thermal uctuations. These results are validated by recent experimental results, i.e., the signicant reduction in the TMR ratios of CMS/MgO/CMS MTJs at RT compared to those of Fe/MgO/Fe MTJs. This means that the spin-flip scattering at the interface is at least as important as any other effect. I showed that the interfacial exchange stiffness can be strongly enhanced by inserting an ultrathin B2-CoFe layer in CMS/MgO junctions, leading to a larger TMR ratio at room temperature. For CPP-GMR, the ballistic conductance of HFM/NM/ HFM was calculated in order to clarify the origin of interfacial resistance. The majority-spin transmittance of CMS/X/CMS(001) in the parallel magnetization conguration was calculated by the Landauer formula, and it was found that the matching of the Fermi surface projected to the two-dimensional Brillouin zone in in-plane wave vector kk║ between CMS and NM spacers is a main contributing factor to the interfacial resistance among spacers. The experimental CPP-GMR was well reproduced as a function of the theoretical resistance-area product in parallel magnetization through the equation of interface spin asymmetry γ. All these ndings suggest that the TMR ratios of MTJs and CPP-GMR with half-metallic Co-based full Heusler compounds can be designed by controlling their interfaces, which is worth further investigation.  AAcckknnoowwlleeddggeemmeennttss    I am grateful to M. Shirai and K. Abe of Tohoku University, K. Hono, S. Mitani, Y. Sakuraba, H. Sukegawa, S. Kasai, K. Masuda and I. Kurniawan of the Na-tional Institute for Materials Science, and K. Nawa of Mie University for valuable discussions on this work. This work was partly supported by Grants-in-Aid for Scientic Research (Grant Nos. 17H06152, 20H00299, 20H02190, 20H02186, 21H01750 and 22H04966) from the Japan Society for the Pro-motion of Science, Center for Spintronics Research Network (CSRN) of Osaka University, and the Cooperative Research Project Program of the Research Institute of Electrical Communication in Tohoku University.  RReeffeerreenncceess   1) M. N. Baibich, J. M. Broto, A. Fert, F. Nguyen Van Dau, F. Petroff, P. Etienne, G. Creuzet, A. Friederich, and J. Chazelas: Phys. Rev. Lett. 6611, 2472 (1988). 2) G. Binasch, P. Grünberg, F. Saurenbach, and W. Zinn: Phys. Rev. B 3399, 4828 (1989). 3) T. Miyazaki and N. Tezuka: J. Magn. Magn. Mater. 113399, L231 (1995). 4) T. Hanyu, T. Endoh, D. Suzuki, H. Koike, Y. Ma, N. Onizawa, M. Natsui, S. Ikeda, and H. Ohno: Proc. IEEE 110044, 1844 (2016). 5) D. Ielmini and S. 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