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[PhysRevB.96.054428.pdf](https://mdr.nims.go.jp/filesets/1ec72444-ad36-4f03-bb11-4fac400ee197/download)

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[Keisuke Masuda](https://orcid.org/0000-0002-6884-6390), [Yoshio Miura](https://orcid.org/0000-0002-5605-5452)

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[Bias voltage effects on tunneling magnetoresistance in Fe/MgAl2O4/Fe(001) junctions: Comparative study with Fe/MgO/Fe(001) junctions](https://mdr.nims.go.jp/datasets/bb83acfd-01f0-42be-a18a-5043a4f3f647)

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PHYSICAL REVIEW B 96, 054428 (2017)Bias voltage effects on tunneling magnetoresistance in Fe/MgAl2O4/Fe(001) junctions:Comparative study with Fe/MgO/Fe(001) junctionsKeisuke MasudaResearch Center for Magnetic and Spintronic Materials, National Institute for Materials Science (NIMS),1-2-1 Sengen, Tsukuba 305-0047, JapanYoshio MiuraResearch Center for Magnetic and Spintronic Materials, National Institute for Materials Science (NIMS),1-2-1 Sengen, Tsukuba 305-0047, Japan;Kyoto Institute of Technology, Electrical Engineering and Electronics, Kyoto 606-8585, Japan;Center for Materials Research by Information Integration, National Institute for Materials Science (NIMS),1-2-1 Sengen, Tsukuba 305-0047, Japan;and Center for Spintronics Research Network (CSRN), Graduate School of Engineering Science, Osaka University,Machikaneyama 1-3, Toyonaka, Osaka 560-8531, Japan(Received 5 April 2017; published 21 August 2017)We investigate bias voltage effects on the spin-dependent transport properties of Fe/MgAl2O4/Fe(001)magnetic tunneling junctions (MTJs) by comparing them with those of Fe/MgO/Fe(001) MTJs. By meansof the nonequilibrium Green’s function method and the density functional theory, we calculate bias voltagedependencies of magnetoresistance (MR) ratios in both the MTJs. We find that in both the MTJs, the MR ratiodecreases as the bias voltage increases and finally vanishes at a critical bias voltage Vc. We also find that thecritical bias voltage Vc of the MgAl2O4-based MTJ is clearly larger than that of the MgO-based MTJ. Sincethe in-plane lattice constant of the Fe/MgAl2O4/Fe(001) supercell is twice that of the Fe/MgO/Fe(001) one,the Fe electrodes in the MgAl2O4-based MTJs have an identical band structure to that obtained by folding theFe band structure of the MgO-based MTJs in the Brillouin zone of the in-plane wave vector. We show that sucha difference in the Fe band structure is the origin of the difference in the critical bias voltage Vc between theMgAl2O4- and MgO-based MTJs.DOI: 10.1103/PhysRevB.96.054428I. INTRODUCTIONMagnetic tunneling junctions (MTJs), in which ferro-magnetic electrodes are separated by an insulating barrier,are key systems for realizing high-performance spintronicdevices such as nonvolatile magnetic random access memories(MRAMs) and read heads of hard disk drives (HDDs). Variouscombinations of ferromagnets and insulating barriers havebeen tested in order to achieve high magnetoresistance (MR)ratios in MTJs. Although amorphous alumina barriers gaveonly low MR ratios [1,2], the use of crystalline MgO barriersincreased MR ratios to 300% at low temperature and 200% atroom temperature [3,4]. Such effectiveness of MgO barrierswas originally proposed in the pioneering theoretical worksby Butler et al. [5] and Mathon et al. [6]. In MgO barriers,the wave functions in the �1 states (s, pz, and d3z2−r2 states)have the slowest decay as evanescent waves [5]. Furthermore,since typical ferromagnetic electrodes (Fe, Co, and someHeusler compounds) have half-metallicity in the �1 states,the majority-spin electrons in the �1 states are selectivelytransmitted from one electrode to another through the MgObarrier, which is the origin of the high MR ratios observed inMgO-based MTJs [5,6]. Namely, the MgO barrier plays therole of a spin filter in MTJs.The spinel oxides, XY2O4, including more than 200 com-pounds, can also be insulating barriers of MTJs. In particular,MgAl2O4 has been studied from both theoretical and experi-mental points of view. Like in MgO, the wave functions in the�1 states decay most slowly in MgAl2O4 [7,8]. On the otherhand, the first-principles-based transport calculations by Miuraet al. predicted that Fe/MgAl2O4/Fe(001) MTJs have muchsmaller MR ratios (160%) than the Fe/MgO/Fe(001) MTJs(1600%) owing to the “band-folding” effect [8]. Since the in-plane lattice constant of the Fe/MgAl2O4/Fe(001) supercell istwice as long as that of bcc Fe, the original bands of bcc Fe arefolded and not only majority-spin bands but also minority-spinbands cross the Fermi level in the �1 states [8]. This implies theabsence of the half-metallicity in the �1 states, leading to smallMR ratios. Such a theoretical prediction is consistent withearly-stage experiments on Fe/MgAl2O4/Fe(001) MTJs [9],where an MR ratio of 165% was observed at low temperaturefor MgAl2O4 made from Mg/Al bilayers. However, in sub-sequent experiments, Sukegawa et al. clarified that MgAl2O4made from Mg-Al alloys gives much higher MR ratios over300% at low temperature [10], which exceeds the abovetheoretical limit. They also showed that the MgAl2O4 madefrom Mg-Al alloys has disordered cation sites (referred toas the cation-disorder MgAl2O4) by transmission electron mi-croscopy (TEM), giving a lattice constant close to that of bcc Fe[see Figs. 1(a) and 1(b) of Ref. [10] for details]. Therefore, thefolding of the Fe bands does not occur in the cation disorderedMgAl2O4 and hence the MR ratios can be large. Most recently,an even higher MR ratio of 436% at 3 K (245% at 297 K)was achieved for an MTJ with the cation-disorder MgAl2O4prepared by direct sputtering from a sintered target [11].One of the significant features of the MgAl2O4-basedMTJs is the good lattice matching between MgAl2O4 andferromagnetic electrodes. The lattice mismatches of MgAl2O42469-9950/2017/96(5)/054428(8) 054428-1 ©2017 American Physical Societyhttps://doi.org/10.1103/PhysRevB.96.054428KEISUKE MASUDA AND YOSHIO MIURA PHYSICAL REVIEW B 96, 054428 (2017)with typical ferromagnets (e.g., Fe, Co, and the Heusler alloyCo2FeAl) are around 0.1%, which are approximately one ordersmaller than those of the MgO-based MTJs. Furthermore,the lattice constant of MgAl2O4 can be tuned by changingthe Mg/Al composition rate, enabling good lattice matchingwith other ferromagnets with large perpendicular magneticanisotropy, e.g., D022 Mn3Ga [12,13], D022 Mn3Ge [14,15],L10 MnGa [13], and L10 FePt [16], which are advantageous forMRAM applications from the viewpoint of thermal stability.Another important feature of the MgAl2O4-based MTJs isthe robustness of the MR ratios to the applied bias voltage.To estimate the robustness quantitatively, we often use Vhalf ,defined as the bias voltage where an MR ratio becomes halfof the zero-bias value. In the MgAl2O4-based MTJs, quitehigh values of Vhalf , over 1 V, which is about twice that for theMgO-based MTJs [4,17,18], have been observed [9,11,19,20].Such a high Vhalf , leading to a high value of the output voltage,is preferable for device applications. Therefore, the MgAl2O4-based MTJ is one of the most suitable systems for investigatingbias voltage effects on spin-dependent transport properties.However, to the best of our knowledge, no theoretical studyhas been done on this issue.In this work, we theoretically study bias voltage effectson the electronic structures and spin-dependent transportproperties of Fe/MgAl2O4/Fe(001) MTJs. In particular, wefocus on the bias voltage dependence of the MR ratio,which is compared with that of the Fe/MgO/Fe(001) MTJs.First, we investigate zero-bias transport properties of both theMgAl2O4- and MgO-based MTJs, which help in understandingthe bias voltage effects. We clarify which conductive channelis dominant at each energy level for both the MTJs. By usingthe nonequilibrium Green’s function method in combinationwith the density functional theory, we calculate the bias voltagedependencies of the currents, MR ratios, and transmittancesin both the MTJs. We found that in both the MTJs, theMR ratio decreases with increasing the bias voltage V andeventually vanishes at a critical value Vc. We also found thatthe MgAl2O4-based MTJ has a larger Vc than the MgO-basedone. As mentioned above, in Fe/MgAl2O4/Fe(001) MTJs, theoriginal bands of Fe electrodes are folded owing to the largesupercell. We clarify that this band folding effect is the originof the larger Vc in the MgAl2O4-based MTJs.II. STRUCTURE OPTIMIZATIONTo properly estimate transport properties of MTJs, weneed to optimize atomic configurations in the supercells usedin our transport calculations. We prepared the supercellsFe(5)/MgO(5)/Fe(5) and Fe(5)/MgAl2O4(9)/Fe(5) shown inFig. 1, where each number represents the layer number of eachcompound. In a previous study [8], one of us confirmed thata layer consisting of Al and O atoms is energetically favoredas the termination layer of MgAl2O4 in Fe/MgAl2O4(001).Moreover, it was also confirmed that the interfacial atomicconfiguration in which O atoms are on top of Fe atoms has thelowest energy. Considering these facts, we adopted the atomicconfiguration shown in Fig. 1(b) for the Fe/MgAl2O4/Fe(001)supercell. The in-plane lattice constants of the MgO- andMgAl2O4-based supercells were fixed to one and two timesthat of bcc Fe (2.866 Å), respectively. As initial atomic(a)(b) FeFeFeFe FeFeFeFeFe FeFe FeFeFeFeFe FeFeFeFeFe FeFe FeMgMgMg MgMg MgAlAlAlAlAlAl AlAl AlFe Fe FeFeFeFeFe FeFe Fe FeFeFeFeFe FeMgMgMgMgMgMgMgddyzyzFIG. 1. Supercells of (a) Fe(5)/MgO(5)/Fe(5) and (b)Fe(5)/MgAl2O4(9)/Fe(5), optimized by the procedures of Sec. II.configurations, we utilized the bulk atomic positions of Fe,MgO, and MgAl2O4. Note that the distance d between theelectrode and the barrier (see Fig. 1) also needs to be optimizedin addition to the atomic configurations. First, for some fixedvalues of d, we calculated the ground-state energy E0 of thesupercell relaxing the atomic positions. We next obtained a dversus E0 curve with the help of the spline interpolation. Fromthe curve, the distance d1 with a minimum value of E0 wasdetermined. Finally, we calculated the ground-state energy ofthe supercell at d1, from which we obtained the optimizedatomic positions in the supercell. All these calculations weredone using the density functional theory in combination withthe generalized gradient approximation, implemented in theVienna ab initio simulation program (VASP) [21,22]. We used20 × 20 × 3 and 10 × 10 × 3 k-point meshes in the MgO andMgAl2O4 cases, respectively. We also assumed that the spinsof all Fe atoms in the supercell align parallel to each other inboth cases. As a result of the calculations, the distance betweenthe electrode and the barrier was determined to be 2.2 Å and2.0 Å in the MgO and MgAl2O4 cases, respectively.III. ZERO-BIAS CASEFor each of the MgO- and MgAl2O4-based MTJs withzero-bias voltage, we considered the quantum open systemcomposed of the supercell (obtained in Sec. II) attached tothe left and right semi-infinite electrodes of Fe atoms. Wecalculated the transmittance of each quantum open systemusing the quantum code ESPRESSO [23]. This code can analyzethe band-resolved transmittance at zero-bias voltage, whichgives valuable information to understand bias voltage effectson the transport properties of MTJs. First, we obtained theself-consistent potential of the quantum open system by meansof the density functional theory and the generalized gradi-ent approximation, where a 10 × 10 × 1 k-point mesh andMethfessel-Paxton smearing with the broadening parameter0.01 Ry were used. We also set the cutoff energies for thewave functions and the charge density to 30 and 300 Ry,respectively. Owing to the two-dimensional periodicity of oursystems, the scattering states can be classified by an in-planewave vector k‖ = (kx,ky). For each k‖ and spin index, wesolved the scattering equations derived under the conditionthat the wave function and its derivative of the supercell are054428-2BIAS VOLTAGE EFFECTS ON TUNNELING . . . PHYSICAL REVIEW B 96, 054428 (2017)-3-2-1 0 1 2 3 -16 -14  -12  -10 -8 -6 -4 -2  0  0  0.1  0.2  0.3  0.4  0.5-20-15-10-5 0(a) (b)-3-2-1 0 1 2 3 -16 -14  -12  -10 -8 -6 -4 -2  0-20-15-10-5 0 0  0.1  0.2  0.3  0.4  0.5(c) (d)-3-2-1 0 1 2 3 -20  -15 -10  -5  0(e)parallel, majority-spin parallel, minority-spin antiparallelFIG. 2. Majority-spin transmittance at k‖ = (0,0) in Fe/MgO/Fe(001) MTJs with parallel magnetization of electrodes. (a) The energydependence of the total transmittance in the [001] direction. (b) The band-resolved transmittance in the [001] direction. At each point of the Femajority-spin bands in the [001] direction, the logarithmic magnitude of the resolved transmittance is shown as a color in the color palette. Ineach band, constituent orbitals are listed from largest to smallest contribution. (c) and (d) The same as (a) and (b) for minority-spin channel,respectively. (e) The same as (a) for antiparallel magnetization of electrodes.connected to those of the Fe electrodes [24,25]. In this section,we focus only on the transmittance at k‖ = (0,0) since thiscomponent provides the dominant contribution to the totaltransmittance in coherent tunneling processes.Figure 2(a) shows the energy dependence of the majority-spin transmittance in the MgO-based MTJ with parallel mag-netization of Fe electrodes. We can see a clear three-step struc-ture: T (E) � 10−2 for E − EF � −1.2 eV, 10−6 � T (E) �10−8 for −1.2 eV � E − EF � −2.4 eV, and 10−15 � T (E)for −2.4 eV � E − EF. To understand this behavior, we showthe band-resolved transmittance in Fig. 2(b). By comparingFigs. 2(a) and 2(b), we find that the drops in the transmittancein Fig. 2(a) occur at the band edges of the majority-spinFe bands. In the highest energy region (E − EF � −1.2 eV),the majority-spin Fe band mainly from the d3z2−r2 stategives quite large transmittances. In the middle-energy region(−1.2 eV � E − EF � −2.4 eV), the band from the dxz anddyz states provides the dominant contribution to the totaltransmittance although the presence of another band from thedx2−y2 state. The lowest energy region (−2.4 eV � E − EF)has only the band from the dx2−y2 state, which gives quite smalltransmittances. A previous theoretical study [5] showed that�1 (d3z2−r2 , pz, s), �5 (dxz, dyz, px, py), and �2 (dx2−y2 , dxy)states of Fe have the first, second, and third slowest decaysas evanescent waves in the MgO barrier, respectively, whichis consistent with our present results. Figure 2(c) shows theenergy dependence of the minority-spin transmittance. Asseen from the band-resolved transmittance in Fig. 2(d), thed3z2−r2 state gives quite large transmittances in the high- andlow-energy regions away from the Fermi level. The secondlargest transmittances are given by the dxz and dyz statesin the region of 2 eV � E − EF � 0 eV. Below the Fermilevel (0 eV � E − EF � −2.4 eV), the dx2−y2 state providesthe smallest transmittances. These are also consistent with theprevious theoretical results [5]. As is clear from Figs. 2(b) and2(d), the Fe electrodes have half-metallicity in the �1 states(d3z2−r2 -based states), which gives a large spin-dependenttransmittance at the Fermi level, namely, a relatively highMR ratio [5,6]. In Fig. 2(e), we show the transmittance in theantiparallel magnetization state of Fe electrodes, where the leftand right electrodes have opposite directions of magnetization.In this case, tunneling of electrons occurs between majority-and minority-spin bands. We see that the transmittance haslarge values T (E) � 10−2 for E − EF � 2 eV, in which boththe majority- and minority-spin states have d3z2−r2 -based bandgiving large transmittance as shown in Figs. 2(b) and 2(d). Wealso see that other energy regions have quite small transmit-tances T (E) � 10−12 owing to the absence of the d3z2−r2 -basedband in either the majority- or minority-spin state.Next, we show in Fig. 3(a) the energy dependence ofthe majority-spin transmittance in the MgAl2O4-based MTJwith parallel magnetization of Fe electrodes. We also showthe band-resolved transmittance in Fig. 3(b), where a largernumber of majority-spin Fe bands are seen than for theMgO case [Fig. 2(b)]. This is due to the band-folding effectpointed out in previous studies [8,10]. Since the in-plane latticeconstant of the Fe/MgAl2O4/Fe(001) supercell is twice aslong as that of bcc Fe [see Fig. 1(b)], the original Fe bandsare folded in the k‖ space, resulting in the large numberof bands. In Fig. 3(b), we find that the band mainly fromthe d3z2−r2 state provides the dominant contribution to thetotal transmittance in the energy region above the Fermilevel. On the other hand, below the Fermi level, the totaltransmittance has contributions from various bands, but thed3z2−r2 -based bands play the dominant role [26]. Previoustheoretical studies on Fe/MgAl2O4/Fe(001) MTJs [7,8] haveshown that the d3z2−r2 -based Fe state has the slowest decay inthe MgAl2O4 barrier, which is consistent with our results.Note here again that in Fig. 3(b), the d3z2−r2 -based bandsexist in the whole energy region around the Fermi levelbecause of the band-folding effect. This is clearly differentfrom the MgO case [Fig. 2(b)] and is the origin of the smoothenergy dependence of the majority-spin transmittance shownin Fig. 3(a). Figure 3(c) shows the energy dependence of054428-3KEISUKE MASUDA AND YOSHIO MIURA PHYSICAL REVIEW B 96, 054428 (2017)-3-2-1 0 1 2 3 -16 -14  -12  -10 -8 -6 -4 -2  0-12-10-8-6-4-2 0 0  0.1  0.2  0.3  0.4  0.5(a) (b)-3-2-1 0 1 2 3 -16 -14  -12  -10 -8 -6 -4 -2  0-12-10-8-6-4-2 0 0  0.1  0.2  0.3  0.4  0.5(c) (d)-3-2-1 0 1 2 3-16 -14 -12 -10 -8 -6 -4  -2  0(e)parallel, majority-spin parallel, minority-spin antiparallelFIG. 3. The same as Fig. 2 for Fe/MgAl2O4/Fe(001) MTJs.the minority-spin transmittance. From Fig. 3(d), we see thatthe d3z2−r2 -based bands provide the dominant contribution tothe total transmittance. We also find that the half-metallicityin the d3z2−r2 -based bands is absent, unlike the MgO-basedMTJs, because the folded minority-spin band mainly fromthe d3z2−r2 state crosses the Fermi level. Here, it is worthyto note that the minority-spin d3z2−r2 -based folded bandcrossing the Fermi level of bcc Fe additionally includesthe px and py states without the total symmetry about thez-axis rotation [see Fig. 3(d)]. This is in contrast to themajority-spin case [Fig. 3(b)], where the d3z2−r2 -based bandcrossing the Fermi level includes only the d3z2−r2 , pz, ands states. Since the mixing of the px and py states affectsthe spatial distribution of the wave function of the foldedminority-spin band, the majority- and minority-spin states inFe electrodes are expected to have different overlaps with thewave functions of the �1 evanescent state of the MgAl2O4barrier. In the case of parallel magnetization, the scatteringwave function of the �1 evanescent minority-spin state showsa strong reduction at both sides of the Fe/MgAl2O4 interface,due to the partial overlaps between the folded minority-spinstate of Fe (including d3z2−r2 , px , and py) and the evanescent�1 state of MgAl2O4 [see Fig. 6(b) of Ref. [8]]. This leads tothe small transmittance compared with the majority-spin state,in spite of the same decay rate of the �1 evanescent state for themajority- and minority-spin channels in the barrier region. Onthe other hand, in the case of the antiparallel magnetization,significant reduction of the scattering wave function at theFe/MgAl2O4 interface occurs only at one side of the junction,because of the complete matching between the majority-spin�1 state of Fe and the evanescent �1 state of MgAl2O4 atthe other side of the junction. Thus, the transmittance in theantiparallel magnetization state of Fe/MgAl2O4/Fe(001) MTJsis larger than that in the minority-spin state in the case ofthe parallel magnetization [see Figs. 3(c) and 3(e)], which isclearly different from the case of Fe/MgO/Fe(001) MTJs.IV. FINITE-BIAS CASETransport calculations under finite-bias voltages were car-ried out using the nonequilibrium Green’s function method incombination with the density functional theory, implementedin the Atomistix ToolKit package (ATK) [27–29]. First, weconstructed a Hamiltonian of the supercell attached to theleft and right semi-infinite Fe electrodes, between which afinite voltage V was applied. As the supercells, we usedFe(7)/MgO(5)/Fe(7) and Fe(7)/MgAl2O4(9)/Fe(7), optimizedby the procedure of Sec. II [30]. In addition to the pseudopoten-tial and exchange-correlation potential, the Hamiltonian alsoincludes the Hartree potential determined from the electrondensity by solving the Poisson equation [28].From the Hamiltonian, we can obtain the following retardedGreen’s function matrix Gσ (ε) with spin σ and energy ε:Gσ (ε) = 1(ε + iδ)Sσ − Hσ − �L,σ (ε) − �R,σ (ε), (1)where Sσ and Hσ are the overlap and Hamiltonian matrices,respectively. Here, �L,σ (ε) and �R,σ (ε) represent the self-energy matrices from the left and right electrodes, respectively.Using the Green’s function matrix Gσ (ε), the density matrixDσ ≡ DLσ + DRσ can be calculated, where DLσ and DRσ aregiven byDLσ = 12π∫Gσ (ε)f (ε − μL)�Lσ (ε)G†σ (ε) dε, (2)DRσ = 12π∫Gσ (ε)f (ε − μR)�Rσ (ε)G†σ (ε) dε. (3)Here, μL (μR) is the chemical potential in the left (right)electrode and f (ε) = 1/(eβε + 1) with β = 1/kBT is theFermi distribution function. In this work, we assume thatelectrons flow from the left to right electrode by a positivebias V , namely, μL − μR = eV . The matrices �Lσ (ε) and�Rσ (ε) are defined as�Lσ (ε) = i(�L,σ (ε) − �†L,σ (ε)), (4)�Rσ (ε) = i(�R,σ (ε) − �†R,σ (ε)). (5)054428-4BIAS VOLTAGE EFFECTS ON TUNNELING . . . PHYSICAL REVIEW B 96, 054428 (2017)From the density matrix Dσ , we obtain the electron densitynσ (r) for spin σ through the following expression:nσ (r) =∑μ,νφμ(r) Re[(Dσ )μ,ν]φν(r), (6)where φμ(r) is a real basis function. All these procedureswere repeated iteratively until the electron density converged.In these self-consistent calculations, we set the electrontemperature (thermal smearing of the Fermi distribution) to1200 K and the density mesh cutoff to 75 Hartree. For thecalculations of Fe electrodes, we used a 7 × 7 × 50 k-pointmesh. Such a large number of kz points is required to minimizethe mismatch of the Fermi energy between the Fe electrodeand scattering region including the barrier, where an openboundary condition is adopted to treat finite-bias voltages. Wealso utilized the double-ζ and single-ζ polarized basis sets forthe MgO- and MgAl2O4-based MTJs, respectively. By usingthe Green’s function matrix Gσ (ε) with the converged electrondensity nσ (r), the current Iσ (V ) at zero temperature can becalculated as follows:Iσ (V ) = eh∫ μLμRTσ (ε) dε, (7)where Tσ (ε) is the transmittance given byTσ (ε) = Tr[�Rσ (ε)G†σ (ε) �Lσ (ε)Gσ (ε)]. (8)In such transport calculations, k‖-point samplings were per-formed using 101 × 101 and 151 × 151 points in the k‖Brillouin zones for the MgO and MgAl2O4 cases, respectively.Figures 4(a) and 4(b) show the bias voltage dependenciesof the currents in the MgO- and MgAl2O4-based MTJs,respectively. Here, IP and IAP represent the total (majority-spinplus minority-spin) currents in the case of the parallel andantiparallel magnetization of electrodes. Using these IP andIAP, the optimistic MR ratio is defined byMR ratio [%] = 100 × IP − IAPIAP. (9)We show in Fig. 4(c) the bias voltage dependencies of theMR ratios in the MgO- and MgAl2O4-based MTJs. We seethat in both the MTJs, the MR ratio decreases as the biasvoltage V increases and eventually vanishes at the criticalvalue Vc. We can understand this behavior from Figs. 4(a)and 4(b) and Eq. (9) as follows. In a small bias region,the parallel current IP has larger values than the antiparallelcurrent IAP, leading to positive values of MR ratios. Thevalue of the MR ratio decreases as V increases, owing tothe increase in IAP, as seen from Figs. 4(a) and 4(b) andthe denominator of Eq. (9). As V increases further, IAPincreases more rapidly and finally has the same value as IPat Vc, where the MR ratio becomes zero. Note that sucha rapid increase of IAP in V > 1.5 V is a common featurefor both the MTJs, which is attributed to the enhancementof the electron tunneling between the d3z2−r2 -based majority-and minority-spin bands, whose lower band edges are locatedaround E − EF = −1.2 eV and 2 eV at V = 0 V, respectively[see Figs. 2(b) and 2(d) for the MgO-based MTJ, and Figs. 3(b)and 3(d) for the MgAl2O4-based MTJ]. In Fig. 4(d), weshow the bias voltage dependencies of the current polarizationdefined by P [%] = 100 × (IP,maj − IP,min)/(IP,maj + IP,min),IPIPIAPIAP(a) MgO(b) MgAl2O40.02.04.06.08.010.02.52.01.51.00.50.0Current I [10-8A]Current I [10-7A] 101 100 102 103 104MR ratio [%]MgOMgAl2O4Bias Voltage V [V] 50 60 70 80 90 100 0  0.2  0.4  0.6  0.8  1.0  1.2  1.4  1.6  1.8  2.0Current polarization P [%] MgOMgAl2O4(c)(d)FIG. 4. The bias voltage dependencies of currents in (a)Fe/MgO/Fe(001) and (b) Fe/MgAl2O4/Fe(001) MTJs. The biasvoltage dependencies of (c) MR ratios and (d) current polarizationsin both the MTJs (see text for details).where IP,maj and IP,min are the majority- and minority-spincurrents in the case of parallel magnetization, respectively.In both the MTJs, the polarization sharply increases forsmall V and has an almost constant value for intermediateV . For large V , the current smoothly decreases in both theMTJs.Next, let us focus on the clear difference in Vc (∼0.3 V)between the MgO- and MgAl2O4-based MTJs shown inFig. 4(c). By comparing Figs. 4(a) and 4(b), we find that054428-5KEISUKE MASUDA AND YOSHIO MIURA PHYSICAL REVIEW B 96, 054428 (2017)-1.5 -1 -0.5  0  0.5  1  1.5-1-2-3-4-5-6-7-8-1-2-3-4-5-6-7-8-1-2-3-4-5-6-7-8-1-2-3-4-5-6-7-8(a)(b)(c)(d)P, majority-spinP, minority-spinAP, majority-spinAP, minority-spinFIG. 5. Energy dependencies of transmittance integrated over k‖in Fe/MgO/Fe(001) MTJs for (a) V = 0.0 V, (b) V = 0.5 V, (c) V =1.0 V, and (d) V = 1.6 V. In each panel, the red solid (dashed) curvecorresponds to the majority- (minority)-spin transmittance in the caseof parallel (P) magnetization of electrodes. The blue solid (dashed)curve represents the majority- (minority)-spin transmittance in thecase of antiparallel (AP) magnetization of electrodes. The shadedarea represents the integral region for the calculation of the currentfollowing Eq. (7).this difference mainly comes from the different behavior ofIP in a high-V region (V > 1 V): the IP−V curve maintainsits slope beyond V = 1 V in the MgAl2O4 case, while itdoes not in the MgO case. To discuss such a difference inIP, we need to remember that the current is obtained byintegrating the transmittance Tσ (ε) from ε = μR to μL, asshown in Eq. (7). In our calculations, since the origin of theenergy ε was set to the average of the chemical potentials,0 = μL + μR , the integration was carried out from ε =−e V/2 to e V/2. Figures 5(a)–5(d) show the energy depen-dencies of the transmittance integrated over k‖ for variousbias voltages in the MgO-based MTJ [31]. Each shaded arearepresents each integral region. In Fig. 5(a), we can see aclear steplike structure at E − EF ≈ −1.2 eV in the majority-spin transmittance in the parallel magnetization case, whoseintegrated value provides the dominant contribution to IP. Thesteplike structure is identical to that at E − EF ≈ −1.2 eV inFig 2(a), which reflects the band edge of the d3z2−r2 -basedband leading to large values of transmittance, mentioned inthe previous section. As V increases, the steplike structuremoves to higher energies, which is due to the upper shift of thedensity of states by e V/2 in the left Fe electrode [32]. Around-1 -0.5  0  0.5  1-1-2-3-4-5-6-7-8-1-2-3-4-5-6-7-8-1-2-3-4-5-6-7-8-1-2-3-4-5-6-7-8(a)(b)(c)(d)P, majority-spinP, minority-spinAP, majority-spinAP, minority-spinFIG. 6. The same as Fig. 5 for Fe/MgAl2O4/Fe(001) MTJs with(a) V = 0.0 V, (b) V = 0.5 V, (c) V = 1.0 V, and (d) V = 1.9 V.1 V, the structure is involved in the integral region as shownin Fig. 5(d). This causes the decrease in the slope of the IP−Vcurve around 1 V [see Fig. 4(a)], which is the origin of thesmaller Vc. On the other hand, the majority-spin transmittancein the MgAl2O4-based MTJ with parallel magnetization ofthe Fe electrodes exhibits a very smooth energy dependencewith no steplike structure as shown in Fig. 6(a), whichcorresponds to that in Fig. 3(a). As explained in the previoussection, this is because the d3z2−r2 -based band exists in thewhole energy region around the Fermi level because ofthe band-folding effect in the Fe electrodes [8,10]. Owingto such a smooth energy dependence in the majority-spintransmittance, the slope of the IP−V curve in Fig. 4(b) hardlychanges, even when V exceeds 1 V, yielding the larger Vc.From all these results and considerations, we can concludethat the difference in the critical bias Vc between MgO-and MgAl2O4-based MTJs mainly originates from the band-folding effect [8,10], which occurs only in MgAl2O4-basedMTJs.Finally, we comment on the transmittance in the case ofantiparallel magnetization of electrodes. Here, for simplicity,let us focus on the electron transmission from majority-spinbands in the left electrode to minority-spin bands in theright electrode, which is expressed by the blue solid curvesin Figs. 5 and 6. In the MgO-based MTJ at zero bias, themajority-spin bands have the d3z2−r2 -based component in E −EF � −1.2 eV, as shown in Fig. 2(b). On the other hand, theminority-spin bands have that component in E − EF � 2.0 eV054428-6BIAS VOLTAGE EFFECTS ON TUNNELING . . . PHYSICAL REVIEW B 96, 054428 (2017)and E − EF � −2.4 eV, as shown in Fig. 2(d). Therefore,electrons with the d3z2−r2 -based component can tunnel throughthe barrier in the shared region E − EF � 2.0 eV, which is themain origin of the humplike structure of the transmittance atE − EF � 1.5 eV shown in Fig. 5(a). When the bias voltageis increased, the humplike structure moves to lower energies,as shown in Figs. 5(a)–5(d). This is because the shared energyregion moves to lower energies due to the lower shift of thedensity of states in the right Fe electrode. In the case ofMgAl2O4-based MTJ, relatively smooth energy dependencieswith no specific features were found in the transmittances,as shown in Figs. 6(a)–6(d). This reflects the fact that thed3z2−r2 -based component exists in wide energy regions aroundthe Fermi level in both the majority- and minority-spinbands, owing to the band-folding effect [see Figs. 3(b)and 3(d)].V. SUMMARYWe studied bias voltage effects on the spin-dependenttransport properties of Fe/MgAl2O4/Fe(001) MTJs by meansof the nonequilibrium Green’s function method and the densityfunctional theory. In particular, the bias voltage dependence ofthe MR ratio was investigated in detail by comparing it withthat of the Fe/MgO/Fe(001) MTJ. In both MTJs, the MR ratiodecreases as the bias voltage increases and eventually vanishesat a critical bias voltage Vc. We also found that the MgAl2O4-based MTJ has a larger Vc than the MgO-based MTJ. Inorder to understand such a difference in Vc, we analyzedthe energy dependencies of the current and transmittancefor both the MTJs. From these results, it was revealedthat the MgAl2O4-based MTJ with parallel magnetizationof electrodes has a large majority-spin transmittance in awide energy region around the Fermi level, owing to theband-folding effect in the electrodes, which is the origin ofthe larger Vc. 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