Theory of Tunnel Magnetoresistance

J. Magn. Soc. Jpn., 30, 1-37 (2006) Review

Hiroyoshi Itoh and Jun-ichiro Inoue

Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan

We present a review of theories of tunnel magnetoresistance (TMR) putting an emphasis on the role of electron scattering due to randomness. We adopt the linear response theory and generalize the conductance formula to calculate the electrical conductance in layered structures. The effect of randomness is treated in the coherent potential approximation and/or direct numerical simulation. Because the momentum conservation parallel to the junction planes does not hold in the presence of randomness, number of tunneling path through the barrier increases. Using a simple tight-binding model, we demonstrate first how the tunnel conductance and TMR are influenced by the randomness and by the shape of the Fermi surface of the metallic electrodes. The theory is applied to epitaxial tunnel junctions of Fe/MgO/Fe. It is shown that the basic concept of tunneling in the presence of randomness is also hold in realistic junctions. Magnetoresistance of tunnel junctions with semimetals and those with manganites is studied to clarify the effect of their electronic structures and interaction on TMR.

Key words: magnetic tunnel junctions, tunnel conductance, tunnel magnetoresistance, randomness, electron scattering, Kubo- Landauer formula, recursive Green’s function method

1. Introduction ing. In the latter case, the current runs through the insulator Coupling between electrical current and magnetism is a via tunneling, and the junctions is called magnetic tunnel key concept of spin-electronics or magnetoelectronics de- junction (MTJ). veloped in the decade.1) The coupling originates from the The technological relevance of magnetoresistance has fact that the electron possesses both charge and spin de- become evident ever since read heads of high-density data- grees of freedom. The electronics so far has utilized only storage systems are based on GMR or spin-valves. Non- charge degree of freedom of electrons. In the last decade, volatile magnetic random access memories (MRAM) make intensive studies have been performed to understand the use of TMR and are close to commercial application. To fundamental physics of the coupling between charge and this end, high quality junctions of nano-scale size have spin degrees of freedom, and to develop novel devices uti- to be fabricated to produce sufficient functionalities. From lizing them. theoretical point of view, on the other hand, fundamental There are basically two approaches to incorporate the physics of MR should be clarified. Because TMR is highly spin degree of freedom into the charge electronics, one relevant to both MRAM and magnetic sensors, we deal is to introduce spins in semiconductors, and the other is with the theory of TMR in this article. to utilize magnetic multilayers in which the ferromagnetic In the theory of the transport properties of MTJs, one layers are separated by non-magnetic materials. Semicon- has to take several characteristics into consideration; com- ductor spintronics2) attracts much interest after a discov- plex structures of the junctions, electronic structures of ery of high Curie temperature magnetic semiconductors different constituent materials, and effects of magnetism (GaMn)As3,4) and (InMn)As.5) It is expected that the char- or spins. Structural randomness or roughness near the in- acteristic features of magnetic, transport, and optical prop- terface should also be incorporated because randomness erties of the new materials may generate novel functions might be unavoidable even in high quality samples. Quan- for device applications. tum effects become important for nano-scale junctions as The well-known phenomenon in the magnetic multilay- well. In this article, we review theory of TMR and compu- ers is the magnetoresistance (MR). In the multilayers, the tational results performed in our group putting an emphasis alignment of the magnetization of ferromagnetic layers can on the effect of randomness on the tunnel conductance and be controlled by the external magnetic field. When the TMR. Because of no translational invariance in junctions alignment of the magnetizations is parallel (P), the electri- and in systems with randomness, we calculate the conduc- cal resistivity or resistance is low, while it is high when the tivity or conductance in a real space method, that is, using magnetization aligns in antiparallel (AP). Such MR that the recursive Green’s function method for Kubo-Landauer 11–13) depends on the relative magnetization alignment but not formula. The randomness may be treated directly for on the angle between magnetization and current direction finite size systems, or by using the coherent potential ap- 14,15) is called giant magnetoresistance (GMR)6,7) for multilay- proximation. We will show that the effect of random- ers in which the non-magnetic layers are metallic, and it is ness is quite large especially for tunnel junctions. called tunnel magnetoresistance (TMR)8–10) for multilay- The outline of the paper is as follows. In the next section, ers or junctions in which non-magnetic layers are insulat- we present general aspects of TMR, structure of MTJs, definition of MR ratio, a phenomenological theory which is

Journal of the Magnetics Society of Japan Vol. 30, No. 1, 2006 1

related to spin polarization of ferromagnets, and a theory

¢¡ ¤ £¡ of TMR in the free electron model. An emphasis will be ¢¡ put on the momentum ﬁltering eﬀect which is character-

istic in tunneling conductance, and is related to the shape

¥§¦©¨ ¥ ¦©¨ of the Fermi surface. In order to take into account the important characteristics of electronic structures and randomness/roughness which have not treated in the free electron Fig. 1 Schematic figure of ferromagnetic tunnel model, we present in section 3 a formalism of tunnel con- junction for parallel (P) and antiparallel (AP) align- ductance based on the linear response theory.16,17) Some ments of magnetizations. FM and I denote ferromagnet details will be presented to deal with the layered structure and insulator, respectively. and randomness in MTJs. In section 4, we present numerical results of tunnel conductance and TMR in a single band tight-binding (TB) model, and compare the results where R (R ) is the junction resistance for parallel (an- with those obtained in the free electron model. We also P AP tiparallel) alignment of the magnetizations. Instead of the show an importance of electron scattering due to random- pessimistic definition in Eq. (1), MR ratio defined by fol- ness on both tunnel conductance and TMR. As a demon- lowing optimistic definition is also used. stration, we present an analysis of the quantum oscillation of TMR in MTJs with a nonmagnetic spacer.18) Section R − R MR MR∗ ≡ AP P = . (2) 5 is devoted to TMR in realistic MTJs, Fe/MgO/Fe junc- RP 1 − MR tions, for which realistic TB model is adopted in the nu- We use MR ratio MR defined by Eq. (1) throughout this merical calculation. In sections 6 and 7, specific exam- 19) article unless specified. ples of MTJs, semimetal junctions and manganite junc- 27) 20, 21) Although TMR was observed more than decades ago, tions, respectively, will be dealt with. For the former the magnitude of the effect was only a few %.28–31) In junctions, role of the particular shape of the Fermi sur- 1995, high MR ratio as much as 30% at 4.2 K and 10% face of the semimetal in the MR is stressed. For the lat- 8) at room temperature was observed in Fe/Al2O3/Fe and ter case, role of the interaction between itinerant electrons 9) CoFe/Al2O3/Co systems. and localized spins in the manganites is examined. Spin- flip scattering may affect tunnel conductance in manganite 2.2 TMR and spin polarization ff 22) MTJs. E ects of spin-flip scattering on TMR will be ex- TMR originates from the spin dependent tunneling amined in section 8. We present numerical results of TMR 23) across the insulating spacer. A simple explanation of TMR for a limit of strong randomness in section 9. Through- is given by the tunneling Hamiltonian theory.32) In the the- out these sections, we present the results calculated in the ory, the tunnel conductance (Γ = 1/R) is proportional to method based on the linear response theory. Several fac- the product of the densities of states (DOSs) of metallic tors in TMR which have not discussed in these sections electrodes separated by the spacer. By applying the tunnel- will be presented in section 10. Section 11 is devoted to a ing Hamiltonian theory to the magnetic tunnel junctions, summary of this article. the conductance of the junction in P and AP alignment of the magnetization are given as 2. General Aspects of TMR −1 ΓP = RP ∝ DL+DR+ + DL−DR− , (3) 2.1 Magnetic tunnel junctions Γ = R−1 ∝ D D + D D , (4) MTJs are comprised of two ferromagnetic metals sep- AP AP L+ R− L− R+ arated by thin insulating spacer. A schematic figure of an respectively, where DL(R)+ and DL(R)− are the density of MTJ is shown in Fig. 1. The ferromagnetic electrodes of states of majority (+) and minority (−) spin states at the left and right hand sides of the junction are usually made Fermi level for left (right) electrode. MR ratio is expressed of Fe, Co or their alloys. The barrier material used most in terms of the spin polarization (P) of ferromagnetic met- 27, 28) frequently is Al2O3, which presents amorphous like struc- als as 8, 9) tures. Recently, single crystalline MgO barrier has been 2PLPR fabricated to give high MR ratio.24–26) The thickness of the MR = , (5) 1 + PLPR barrier layer is 0.7 ∼ 2 nm. The resistivity (ρ), or resis- with tance (R) of the junction is changed when the magnetiza- − tion alignment of the ferromagnetic electrodes is changed. DL(R)+ DL(R)− PL(R) = . (6) This is the phenomenon call as TMR. The MR ratio is de- DL(R)+ + DL(R)− fined by Although the transmission coefficient governs the tunnel R − R resistance, it does not affect the MR ratio. The spin polar- MR ≡ AP P , (1) RAP ization has been measured for several ferromagnets, and

2 Journal of the Magnetics Society of Japan Vol. 30, No. 1, 2006 with the electron mass m. Here, we have assumed that Table I Spin polarization of ferromagnets (FM) κd 1. measured in superconducting junctions33) and point When the electrodes are ferromagnetic metals in which contacts.34, 35) the density of states are spin split, the transmission coefficient is spin (σ) dependent and is given by replacing E Junction Point contact L(R) with E . We see that the spin dependence appears in the FM 33) 34) 35) σ prefactor of e−2κd. We use in this article the same symbol Fe +0.41 0.42 σ for both electrical conductivity and spin suffix, but we Co +0.45 0.37 0.42 Ni +0.27 0.32 0.43 believe there is no confusion in the presentation. Ni0.8Fe0.2 0.37 In order to obtain the MR ratio, one should calculate the Cu 0.00 transmission coefficient for both in P and AP alignments NiMnSb 0.58 of the magnetization. Slonczewski37) obtained the conduc- La Sr MnO 0.78 0.7 0.3 3 tance in P and AP alignment as CrO2 0.90 2 ΓP ∝ 1 + P , (9) 2

ΓAP ∝ 1 − P , (10)

¦ ¦ with

k − k κ2 − k k P = F+ F− F+ F− , (11) 2

kF+ + kF− κ + kF+kF−

¢ £

¢ √

¡ ¡ where kF± = 2mE± /~ is the Fermi wave number of the

electrodes. Here we have assumed that the left and right

¥ electrodes are equivalent, and that κd 1. MR ratio is

2 2 § § given by ΓP − ΓAP /ΓP = 2P /(1 + P ) which is the same with the expression (5). When we estimate the values of P from the electronic structure calculated in the first princi- Fig. 2 Potential profile of tunnel junction in 1- ples, the value of MR ratio is usually very small as com- dimensional free electron model. Dashed horizontal line pared with the experimental values because of a factor denote the Fermi level. EL(R) and Φ are the Fermi energy 2 38) κ − kF+kF−. The dependences of MR ratio on the bar- measured from the band bottom of the left (right) elec- rier thickness and the applied bias voltage have also been trode and the barrier potential measured from the Fermi discussed.39–43) level, respectively, and d denotes the barrier thickness. It is important to note that the difference between two approaches, the tunneling Hamiltonian theory and the free electron model. In the tunneling Hamiltonian theory, the phase memory of the wave function is assumed to be lost 33–35) the measured values are shown in Table I. When we during tunnel process due to the inelastic scatterings. In use the observed values of P, MR ratio obtained in experi- Slonczewski’s approach, on the other hand, the wave func- ments can be explained semi-quantitatively. tion of the tunneling electron is treated explicitly.

2.3 Free electron model 2.4 Filtering effect TMR can be explained also by using the free electron Equation (7) shows that the spin dependence of the model. Let us first discuss the spin dependent tunneling transmission coefficient is included only in its prefactor in an elementary method described in textbooks. In the of the exponential function. Is there no spin-dependent ef- paramagnetic state (see Fig. 2), the transmission coefficient fect resulted from the exponential function itself, e−2κd? In through the tunnel barrier, thickness of which is d, is given three dimension, as36) q p 16 E E Φ κ = 2m Φ + E(kk) /~ , (12) T = L R e−2κd , (7) (EL + Φ)(ER + Φ) and where E and Φ are the Fermi energy of incident elec- 2 2 L(R) E(kk) = ~ |kk| /(2m) (13) trons measured from the bottom of the band of left (right) where k is the component of the wave vector parallel to electrode and barrier potential measured from the Fermi k ffi energy, respectively, and the exponential decay factor κ is barrier plane. Therefore, for su ciently thick d, the transmission coefficient is the largest at k which makes E(k ) given by k k √ smallest. When the tunneling electrons can take the state κ = 2mΦ/~ , (8) with kk = (0, 0), its contribution to the conductance be-

Journal of the Magnetics Society of Japan Vol. 30, No. 1, 2006 3

comes the largest. For less thicker barrier, wave vectors

¦ § ¤

= , ¤ ¦

near kk (0 0) contribute to the conductance. This kind ¦ ¥

of eﬀect may be called as “ﬁltering” of wave vectors, or ¥

¢¡ £ ¤

“momentum selection” of the tunneling electrons. For complex electronic structure of ferromagnetic electrodes may aﬀect strongly the momentum selection. The momentum selection is also aﬀected by randomness, because the momentum kk parallel to planes need not be con-

served in the presence of randomness.

2.5 Scattering theory in the Green’s function formal-

ism From the argument above, in order to study tunnel conductance and TMR, one should take into account the re- Fig. 3 Schematic figure of layered structure system. alistic electronic structure44–46) and the effects of random- The atomic layer is indicated by `. ness such as interfacial roughness,16,17) disorder in the barrier,47) impurities,48,49) and so on. The most convenient way to deal with these issues is to utilize Green’s function formalism of electrical conductivity based on the Kubo for- In order to treat the layered structure more easily, we mula. In the following sections, we present the formalism use |rk, ì-basis instead of site-index i, where rk and ` are of tunnel conductance and TMR, and numerical results for a positional vector in x-y planes and the layer index in z- junctions with randomness. direction, respectively. The Hamiltonian Hˆ is rewritten as X Wˆ = Wˆ `,` + Wˆ `+1,` + Wˆ `,`+1 , (17) 3. Formalism ` X X 0 rk,rk † The formalism we adopt is based on the linear response Wˆ `,` = w (µ, ν)c c 0 , (18) `,` rk,`,µ rk,`,ν Kubo formula in TB model. In this section, we derive 0 µ,ν rk,rk a general expression of the conductance for the layered X X 0 rk,rk † structures at zero temperature and zero bias limit by us- Wˆ `+1,` = w (µ, ν)c c 0 , (19) `+1,` rk,`+1,µ rk,`,ν 16,17) 0 µ,ν ing Green’s functions. A recursive Green’s function rk,rk method50) is applied to calculate the Green’s function in the Wˆ = Wˆ † , (20) layered structures. The randomness may be treated in the `,`+1 `+1,` X coherent potential approximation (CPA),14, 15) the method Vˆ = Vˆ ` , (21) of which will be explained in detail. The other method ` to treat the randomness is direct simulations for a given X X rk † Vˆ ` = v (µ)c c , (22) arrangement of the random potentials. In this case one ` rk,`,µ rk,`,µ r µ should take an average over random sampling to get well- k converged results. Comparison between the results in CPA where Wˆ `,`, Wˆ `+1,`, and Wˆ `,`+1 denote the transfer integral and direct simulation will also be discussed. within `-th layer, that from ` to `+1-th layer, and that from ` + 1 to `-th layer, respectively. 3.1 Model Hamiltonian We consider a system depicted in Fig. 3 where l indi- 3.2 Kubo formula for layered structures cates the layer along the z-direction. The TB Hamiltonian In this subsection, we rewrite the expression of the Hˆ is given as electrical conductance given by the linear response theory, Kubo formalism,11) to be convenient for the layered Hˆ = Wˆ + Vˆ , (14) X X structures.50) The conductance is expressed by the current- ˆ † 51) W = wi, j(µ, ν)ci,µc j,ν , (15) current correlation function. The conductance Γ for the i, j µ,ν currents perpendicular to the layers (z-direction) per spin X X † channel is Vˆ = vi(µ)c c , (16) i,µ i,µ π~ D h iE i µ Γ = ˆ δ − ˆ ˆ δ − ˆ , 2 2 Tr Jz EF H Jz EF H (23) where i and j denote the sites and µ and ν are orbitals in- N a ˆ cluding spin degree of freedom. Here, wi, j(µ, ν) is the trans- where Jz is the current operator in the z-direction, EF de- fer integral between µ-orbital at i-th site and ν-orbital at j- notes the Fermi energy and Na is the length of the system th site, and vi(µ) is the energy level of µ orbital at i-th site. in the z-direction. The bracket h· · · i denotes the configu- rational average of the quantity ‘··· ’ over the randomness.

4 Journal of the Magnetics Society of Japan Vol. 30, No. 1, 2006 Here, the operator δ E − Hˆ is related to the Green’s func- −Wm,m+1 Gm+1,n+1(zF−) − Gm+1,n+1(zF+) tion operator Gˆ(z) as iE ·Wn+1,n Gn,m(zF−) − Gn,m(zF+) , (30) 1 n o δ E − Hˆ = Gˆ(z ) − Gˆ(z ) , (z = E ± i0) , (24) 2πi − + ± where Gm,n is the interlayer Green’s function, i.e., an m-n element of the Green’s function operator Gˆ(zF±) ≡ (EF ± −1 ˆ −1 Gˆ(z) ≡ z · 1ˆ − Hˆ , (25) i0 − H) , which is (Nsite × Norb × 2)-dimensional matrix. Here, Nsite and Norb are the number of sites in a layer and where 1ˆ is the identity operator. The current operator Jˆz is the number of orbitals per site, and the factor 2 comes from 52) described by the local current operator Jˆz(`) as spin. e h i X By using Eq. (30), we can calculate the conductance Jˆ ≡ ez˙ˆ = zˆ, Hˆ = Jˆ (`) , (26) z i~ z for layered system without and with randomness. In the ` absence of the randomness, |kk, ì-representation is use- iea ful instead of |rk, ì because of the translational invariance Jˆ (`) = Wˆ − Wˆ . (27) z ~ `+1,` `,`+1 along directions parallel to the layers. (Of course, there is no translational invariance along the direction perpendicu- It is easily shown that the expectation value of Jˆ (`) is in- z lar to the layers in the layered structures.) Here, k is the dependent of ` as follows. We define the charge density op- k P P † wave vector parallel to the layers. In primitive lattice, kk is erator at layer ` asρ ˆ(`) ≡ e r µ c c . The Heisen- k rk,`,µ rk,`,µ defined by the following Fourier transformation, berg equation gives 1 X ∂ 1 h i 1 n o c† = c† exp−ikk·rk . (31) kk,`,µ rk,`,µ ρˆ(`) = ρˆ(`), Hˆ = Jˆz(` − 1) − Jˆz(`) , (28) Nsite ∂t i~ a kk which is nothing but the continuum equation in the lattice Equation (30) is rewritten as ∂ρ ` /∂ = system. Since ˆ( ) t 0 in the steady state, the expecta- e2 X h n o ˆ kk kk kk tion value of Jz(`) becomes independent of layer `, i.e., the Γ = Tr Wm+1,m Gm,n+1(zF−) − Gm,n+1(zF+) 53,54) 2h current conservation law. By putting Eqs.(24)–(26) kk n o into Eq. (23), we get kk kk kk ·Wn+1,n Gn,m+1(zF−) − Gn,m+1(zF+) π~ n o Γ = − kk kk kk 2 2 (2π) (Na) +Wm,m+1 Gm+1,n(zF−) − Gm+1,n(zF+) X D h n o n o ˆ ˆ ˆ kk kk kk × Tr Jz(m) G(zF−) − G(zF+) ·Wn,n+1 Gn+1,m(zF−) − Gn+1,m(zF+) m,n n o n oiE −Wkk Gkk (z ) − Gkk (z ) ˆ ˆ ˆ m+1,m m,n F− m,n F+ Jz(n) G(zF−) − G(zF+) n o · kk kk − kk π~ D h n o Wn,n+1 Gn+1,m+1(zF−) Gn+1,m+1(zF+) = − ˆ ˆ − ˆ 2 2 Tr Jz(m) G(zF−) G(zF+) n o (2π) a −Wkk Gkk (z ) − Gkk (z ) n oiE m,m+1 m+1,n+1 F− m+1,n+1 F+ ˆ ˆ ˆ n o i Jz(n) G(zF−) − G(zF+) , (29) kk kk kk ·Wn+1,n Gn,m(zF−) − Gn,m(zF+) , (32) where m and n are arbitrary in the second line. In the trans- 0 kk kk rk,r k formation from the first line to the second line of Eq. (29), where Wm,n and Gm,n are Fourier transformations of Wm,n 0 2 rk,r k sums over m and n give rise to factor N since the expec- and Gm,n , respectively, and are (Norb ×2)-dimensional ma- ˆ 55) kk kk tation value of Jz(`) is independent of ` as shown above. trices. In the case of non-primitive lattice, Wm,n and Gm,n Equation (29) is convenient to calculate the conductance are (Natom × Norb × 2)-dimensional matrices where Natom is in layered structures because we use | rk, ` i-representation the number of atoms in the unit cell and kk is defined by for the wave function and the Green’s function. By substi- the periodicity of the unit cell. This expression of the con- tuting Eq. (27) into Eq. (29), we get ductance is only available for layered system without ran- e2 D h domness. On the other hand, the formulation which will be Γ = Tr W G (z ) − G (z ) 2h m+1,m m,n+1 F− m,n+1 F+ presented in subsection 3.4 allow us to calculate the conductance in kk-representation even in disordered systems. ·Wn+1,n Gn,m+1(zF−) − Gn,m+1(zF+) +Wm,m+1 Gm+1,n(zF−) − Gm+1,n(zF+) 3.3 Recursive Green’s function method The interlayer Green’s functions are calculated numer- ·W G (z ) − G (z ) n,n+1 n+1,m F− n+1,m F+ ically using recursive Green’s function technique.50) We − − define GL (z) as the ` -` element of Gˆ(z) for the semi- Wm+1,m Gm,n(zF−) Gm,n(zF+) `0,`0 0 0 L infinite system with l ≤ `0. The Green’s functions G` ,` (z) ·Wn,n+1 Gn+1,m+1(zF−) − Gn+1,m+1(zF+) 0 0 and GL (z) are related with each other by the Dyson `0−1,`0−1

Journal of the Magnetics Society of Japan Vol. 30, No. 1, 2006 5 equation terms of the single site T-matrix as n L X X 0 G (z) = z 11 − V` − W` ,` rk rk `0,`0 0 0 0 ˆ ˆ ˆ ˆ T = t` 1 + G t`0 o 0 0 −1 rk,` (r ,` ) −W GL (z)W , (33) k `0,`0−1 `0−1,`0−1 `0−1,`0 ,(rk,`) ! where 11 is the unit matrix. In order to calculate the con- X r0 X r00 ˆ ˆ k ˆ ˆ k ductance of the sample, two semi-infinite ideal electrodes + G t`0 G t`00 + ··· , (41) (r0 ,`0) (r00,`00) are attached both side of the sample. In the electrodes, k k ,(r ,`) ,(r0 ,`0) the potential is uniform or periodic so that electrons are k k P 0 0 where 0 0 denotes the summation over all r and ` not scattered. The Green’s function for the total system is (rk,` ),(rk,`) k derived by the iteration of Eq. (33) starting from surface except for the site (rk, `). When the random average of the Green’s function of the ideal electrode. For simple models full T-matrix is expressed by the average of the single site such as single band TB model on a simple cubic lattice, T-matrix as, the surface Green’s function is obtained analytically. How- X X 0 D E D r E D r E 56–58) ˆ ˆ k ˆ ˆ k ever, in general, several numerical methods are used T ' t` 1 + G t`0 0 0 rk,` (r ,` ) to calculate the surface Green’s function. k ,(rk,`) ! 3.4 Application of CPA X X D r0 E D r00 E + Gˆ tˆ k Gˆ tˆ k + ··· , (42) 3.4.1 Single site CPA `0 `00 0 0 00 00 (rk,` ) (rk ,` ) Before we evaluate the random average of the local 0 0 ,(rk,`) ,(rk,` ) current-current correlation function, we briefly summarize the coherent potential approximation (CPA)14,15, 59) for sin- the error appears in the terms higherD E than the fourth order of tˆrk . Therefore, the condition tˆrk = 0 gives hTi ' 0. gle band TB model in this subsection. In the CPA, the ran- ` ` dom potential is replaced by a mean field, that is, the co- Now, we get the following equation to determine the coherent potential Σˆ, and the effective Hamiltonian is given herent potential Σ`, D n o−1E as, rk rkrk rk V` − Σ` 1 − G`` V` − Σ` = 0 . (43) ˆ ˆ ˆ H = W + Σ . (34) Here, since G has the translational invariance in the x-y rr Within the single site approximation, the coherent potential directions, Gll is rewritten as is site-diagonal. Since there is no translational symmetry in 1 X rkrk ˆ ˆ ˆ −1 the z-direction in the present system, the coherent potential G`` (z) = hkk, ` | (z · 1 − W − Σ) | kk, ì , (44) Nsite may be dependent on the layer ` as kk X † where kk = (kx, ky) is the element of the wave vector par- Σˆ = Σ`c c . (35) rk,` rk,` allel to the layer, i.e. x-y plane. Since the disorder is in- rk,` cluded only in Vˆ , Wˆ is kk-diagonal. For a single band TB The corresponding effective Green’s function is kk ˆ model on a simple cubic lattice, W`,` ≡ hkk, ` | W | kk, ì = −1 Gˆ(z) ≡ z − Hˆ . (36) −2t(cos(kxa) + cos(kya)). A self-consistent solution of the Eqs. (43) and (44) gives Σ`. This coherent potential is com- ˆ rkrk The Green’s function G(z) is rewritten by using the full T- plex number and depends on the energy z and G`` . matrix Tˆ as When the random potential and the coherent potential are diagonal in orbital and spin, equations derived above Gˆ = Gˆ + GˆTˆGˆ , (37) are applicable to the multi-orbital (including spin) model n o−1 by replacing (rk, `) with (rk, `, µ, σ). Tˆ ≡ Vˆ − Σˆ 1 − Gˆ Vˆ − Σˆ . (38) The general condition to determine the coherent potential 3.4.2 Vertex corrections to the electrical conductance is given by hTî = 0. However, instead of the condition, the In order to calculate the conductance, we need to evalu- rk rk ate the random average of the product of two Green’s func- single site CPA imposes the condition ht` i = 0 where t` is D E the single site T-matrix defined as tions such as Jˆz(m) Gˆ(z1)Jˆz(n)Gˆ(z2) shown in Eq. (29). This can be rewritten as tˆrk ≡ trk c† c , (39) ` ` rk,` rk,` D E Kˆ z1, Jˆz(n), z2 ≡ Gˆ(z1)Jˆz(n)Gˆ(z2) n o−1 D E D E trk ≡ Vrk − Σ 1 − Grkrk Vrk − Σ . (40) ` l ` `` ` ` = Gˆ(z1) Jˆz(n) Gˆ(z2)

rk Dn D Eo n D EoE This approximation is correct within the third order of t ` + Gˆ(z1) − Gˆ(z1) Jˆz(n) Gˆ(z2) − Gˆ(z2) . (45) as shown in below. The full T-matrix can be expanded in

6 Journal of the Magnetics Society of Japan Vol. 30, No. 1, 2006 * * X ! The first term is the products of the two averaged Green’s rk r2k = tˆ 1 + Gˆtˆ + ··· Kˆ functions. The second term is the vertex corrections and ` `2 (r2k,`2) denotes the correlation between scatterings due to the ran- ,(rk,`) ! + + domness. The first term can be calculated within the single X 0 r2k rk site CPA explained in previous subsection. In this subsec- 1 + tˆ 0 Gˆ + ··· tˆ `2 ` 0 0 tion, we evaluate the second term of the right hand side in (r2k,`2) Eq. (45).17) When we define a vertex function Λˆ as15) ,(rk,`) D E D E ˆrk ˆ ˆrk Λˆ z1, Jˆz(n), z2 ≡ Tˆ(z1)Kˆ z1, Jˆz(n), z2 Tˆ(z2) , (46) = t` Kt` * * X X !+ + with r r r r ˆ k ˆ ˆ 2k ˆ ˆ 3k ˆ ˆ k + t` G t` 1 + Gt` + ··· Kt` ˆ ˆ ˆ 2 3 K z1, Jˆz(n), z2 ≡ G(z1)Jˆz(n)G(z2) , (47) (r2k,`2) (r3k,`3) ,(rk,`) ,(r2k,`2) the second term of the right hand side in Eq. (45) can be * * ! + + X X r 0 r 0 rewritten as ˆrk ˆ ˆ 3k ˆ ˆ 2k ˆ ˆrk + t` K 1 + t`0 G + ··· t`0 Gt` Dn D Eo n D EoE 3 2 0 ,`0 0 ,`0 ˆ ˆ ˆ ˆ ˆ (r2k 2) (r3k 3) G(z1) − G(z1) Jz(n) G(z2) − G(z2) 0 0 ,(rk,`) ,(r2k,`2) * * ! 'Gˆ(z1)Λˆ z1, Jˆz(n), z2 Gˆ(z2) , (48) X X + tˆrk Gˆ tˆr2k 1 + Gˆtˆr3k + ··· Kˆ ˆ ˆ ` `2 `3 Here, hG(z)i is approximated as G(z). Now, the problem is (r2k,`2) (r3k,`3) to get an explicit formula to calculate the vertex function ,(rk,`) ,(r2k,`2) ! + + Λˆ . Substitution of Eq. (41) into Eq. (46) enables to express X X 0 0 r3k r2k rk 1 + tˆ 0 Gˆ + ··· tˆ 0 Gˆtˆ the vertex function in terms of the single site T-matrix, `3 `2 ` 0 0 0 0 * (r2k,`2) (r3k,`3) X X ,(r ,`) ,(r 0 ,`0 ) Λˆ = tˆr1k 1 + Gˆtˆr2k + k 2k 2 `1 `2 D E D E rk rk rk rk rk r1k,`1 (r2k,`2) ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ' t` Kt` + t` G Λ − Λ` Gt` . (52) ,(r1k,`1) X X ! Here, the site correlations in the terms higher than the r r Gˆtˆ 2k Gˆtˆ 3k + ··· Kˆ fourth order are neglected as done in Eq. (50). Here, Λ `2 `3 ` (r2k,`2) (r3k,`3) is given by ,(r1k,`1) ,(r2k,`2) Λl z1, Jˆz(n), z2 X X r 0 ˆ ˆ 2k ( 1 + Gt`0 + 2 D r r E r r r 0 ,`0 0 0 k k k k ˆ 1k 1 (r2k,`2) = t (z1)t (z2) K z1, Jz(n), z2 0 0 ` ` `` ,(r1k,`1) ) ! + X 0 0 X X 0 0 0 r r r r r3 r2 r1 k k k k k k k + G Λ 0 , ˆ , G Gˆtˆ 0 Gˆtˆ 0 + ··· tˆ 0 , (49) ``0 (z1) ` z1 Jz(n) z2 `0` (z2) `3 `2 `1 0 0 0 0 (r0 ,`0) (r2k,`2) (r3k,`3) k 0 0 0 0 ,(rk,`) ,(r1 ,` ) ,(r2 ,` ) k 1 k 2 D E n rk rk rkrk ˆ In order to evaluate the vertex function, we use the approx- = t` (z1)t` (z2) K`` z1, Jz(n), z2 imation that is used in the evaluation of the single parti- o −Grkrk (z )Λ z , Jˆ (n), z Grkrk (z ) . (53) cle Green’s function. In this approximation, the product of `` 1 ` 1 z 2 `` 2 successive single site T-matrices is approximated as rkrk rkrk ˆ where K`` and K`` are the matrix elements of K in 0 0 0 r r r r3 r2 r1 ˆ ˆ 1k ˆ 2k ˆ 3k ˆ k ˆ k ˆ k Eq. (45) and K in Eq. (47), respectively. Since we have not t` · t` · t` ··· t`0 · t`0 · t`0 1 2 3 3 2 1 used specific feature of Jˆ (n) to derive Eq. (53), Eq. (53) z r 0 r 0 r 0 r1k r2k r3k 3k 2k 1k holds if we replace Jˆz(n) with any operator. By replacing ' tˆ · tˆ · tˆ ··· tˆ 0 · tˆ 0 · tˆ 0 · δr ,r 0 δ` ,`0 . (50) `1 `2 `3 ` ` ` 1k 1k 1 1 15) 3 2 1 Jˆz(n) in Eq. (53) with 1ˆ and using the Ward identity Therefore, within this approximation, the vertex function ˆ Σ`(z1) − Σ`(z2) becomes site-diagonal, that is Λ` z1, 1, z2 = − , (54) X X z1 − z2 rk † ˆ ˆ rkrk rkrk Λ ' Λ` = Λ`cr ,`cr ,` . (51) G (z1) − G (z2) k k rkrk ˆ `` `` rk,` rk,` K`` z1, 1, z2 = − , (55) z1 − z2 ˆ rk The site diagonal operator Λ` is rewritten as we obtain r D E Λˆ k rk rk ` t` (z1)t` (z2) n o h rkrk rkrk = Σ`(z1) − Σ`(z2) G`` (z1) − G`` (z2)

Journal of the Magnetics Society of Japan Vol. 30, No. 1, 2006 7 i i rkrk rkrk −1 kk −K (z −, Jˆ (m), z −)Λ`(z −, Jˆ (n), z −) , (61) − G`` (z1)G`` (z2) {Σ`(z1) − Σ`(z2)} . (56) `` F z F F z F

Equation (53) is a simultaneous equation for Λ`, which can The non-vertex part Γ0(m, n) of the conductance describes be solved by noting that Λ` is non-zero only for the layers specular transmission where kk is conserved. In contrast, rkrk ˆ the vertex part Γv(m, n) of the conductance describes diffu- with randomness. From the definition of K, the term K`` can be calculated using the definition of the local current sive transmission where kk is not conserved. operator, Eq. (27), as 1 X 4. Single band model rkrk ˆ kk ˆ K`` z1, Jz(n), z2 = K`` z1, Jz(n), z2 , (57) Nsite In this section we present calculated results of the tunnel kk conductance and TMR using a single band TB model,16,17) iea n K kk z , Jˆ (n), z = Gkk (z )Wkk Gkk (z ) the Hamiltonian of which is given by `` 1 z 2 ~ `,n+1 1 n+1,n n,` 2 X X o † † kk kk kk H = −t ciσc jσ + Viσciσciσ , (62) − G (z1)W G (z2) , (58) `,n n,n+1 n+1,` (i, j),σ i,σ

kk † The interlayer Green’s function Gmn is derived by the re- where ciσ(ciσ) is the annihilation (creation) operator of an cursive Green’s function method in the last subsection. electron with spin σ at site i, t the hopping integral between kkkk Once we get the interlayer Green’s function, K`` in nearest neighbor sites, and Viσ the on-site potential for an Eq. (57) is calculated and then the vertex function Λ` is electron with spin σ at site i. obtained from Eq. (53). The vertex corrections to the con- In the calculation of the tunnel conductance, we first ductance are then obtained from Eq. (48). show the difference between the results in the free electron The expression Eq. (29) of the conductance per spin and TB models, and show that in the presence of random- channel is rewritten as ness the conductance may increase due to electron scattering. This suggests that MR ratio is also strongly affected Γ(m, n) = Γ0(m, n) + Γv(m, n) , (59) by randomness. It will be shown that the influence of the where Γ0(m, n) and Γv(m, n) comes from non-vertex and randomness on TMR is related to the shape of the Fermi vertex correction terms, that is, the first and second terms in surface. In the last subsection, we will present an expla- Eq. (45), respectively. The first term in Eq. (59) is written nation of the quantum oscillation of MR ratio observed in as epitaxial MTJs with a nonmagnetic spacer. This is a specific example which shows the importance of the random- Γ0(m, n) ness on the TMR. e2 Xh n o = Wkk Gkk (z ) − Gkk (z ) 2h m+1,m m,n+1 F− m,n+1 F+ 4.1 Tunnel conductance in single band model kk n o In this subsection, we consider tunnel junctions which kk kk kk ·Wn+1,n Gn,m+1(zF−) − Gn,m+1(zF+) consist of two paramagnetic metals separated by a thin in- n o sulator of M atomic planes. The schematic figure of the +Wkk Gkk (z ) − Gkk (z ) m,m+1 m+1,n F− m+1,n F+ potential and the density of states is shown in Fig. 4. The n o kk kk kk potentials of the electrodes VL and VR are both chosen to ·Wn,n+1 Gn+1,m(zF−) − Gn+1,m(zF+) n o be zero. Here, we define the barrier height Φ as the energy kk kk kk difference between the Fermi energy EF and the bottom −Wm+1,m Gm,n(zF−) − Gm,n(zF+) n o of the band in the insulator. We introduce the randomness kk kk kk ·Wn,n+1 Gn+1,m+1(zF−) − Gn+1,m+1(zF+) into each atomic plane at the interfaces between the metal- n o lic electrodes and the insulating barrier. Then, the conduc- −Wkk Gkk (z ) − Gkk (z ) m,m+1 m+1,n+1 F− m+1,n+1 F+ tance Γ depends on the four parameters, i.e., the thickness n o i kk kk kk of the insulator d(≡ Ma), Φ, EF and the concentration c ·W Gn,m(zF−) − Gn,m(zF+) , (60) n+1,n which is the measure of the randomness. This expression can be obtained by replacing true Green’s First, we examine the tunnel conductance without ran- kk kk function Gm,n(z) in Eq. (32) with effective one Gm,n(z) The domness. We have studied the dependences of Γ on Φ and second term in Eq. (59) is written as d. The relations ln Γ vs Φ and ln Γ vs d depend on EF in general. Because the EF dependence can be seen more Γv(m, n) clearly in the relation between ln Γ and d than in that be- ~ Xh = K kk (z , Jˆ (m), z )Λ (z , Jˆ (n), z ) tween ln Γ and Φ, the calculated results of ln Γ are plot- 4πa2 `` F+ z F− ` F− z F+ kk,` ted as functions of d for various values of EF in Fig. 5. In these calculations, Φ is kept constant to be t by shifting the kk ˆ ˆ +K`` (zF−, Jz(m), zF+)Λ`(zF+, Jz(n), zF−) band in the insulator when the Fermi energies is varied. Al- kk ˆ ˆ though ln Γ depends almost linearly on d, the slope of ln Γ −K`` (zF+, Jz(m), zF+)Λ`(zF+, Jz(n), zF+) with respect to d depends on EF. This is in contrast to the

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¢ ¢ e 16E Φ ¢ Γ ' F −2κd 2 e (63) h (EF + Φ) √ for κd 1, where κ is deﬁned as κ = 2mΦ/~ with mass Fig. 6 Schematic ﬁgures of the Fermi surfaces projected on the k -k plane of the Brillouin zone for (a) m of the free electron. Therefore, the ln Γ weakly depends x y E < −2t and (b) E > −2t. The shaded regions are the on E in the free electron model. F F F occupied states. The strong dependence of ln Γ on EF in the present case is explained as follows: Because we are dealing with three dimensional system, we should take a sum over kk in order makes κ and W minima. For E smaller than −2t, the to calculate the conductance. In TB model, the expression kk F of the conductance for κd 1 is given as Fermi surface projected to kx-ky plane is as shown in

Fig. 6(a). In this case, the minimum of Wkk is realized at e2 X sinh2(κa) sin2(qa) Γ ' e−2κd , (64) kk = (0, 0). When EF is larger than −2t, the Fermi surface h (cosh(κa) − cos(qa))2 kk is as shown in Fig. 6(b). A hole pocket appears around kk. Then, the minimum of W is realized by the k vectors de- where κ and q is deﬁned by kk k noted by the thick line in the ﬁgure, that is, kk points which

cosh(κa) = (Φ + 6t + Wkk )/2t , (65) satisfy Wkk = EF − 2t. Therefore, when EF increases more than −2t, W and κ increase. Then, Γ decreases rapidly, cos(qa) = (W − E )/2t . (66) kk kk F in distinction to the results in the free electron model. The pEquation (65), which determines κ, corresponds to κ = weak EF dependence of Γ for EF below −2t is due to the Φ + Wkk in the free electron model. Because of the expo- EF dependence of the prefactor of the exponential function nential decay factor exp(−2κd), the wave vector kk which in Γ. This is the same as the weak EF dependence in the makes κ minimum contributes predominantly in the sum- free electron model. mation over kk. Physically this means that the electrons Next, we consider the tunnel junction with interfacial whose incident energy (the kinetic energy devoted to the randomness. The atoms of the insulator and the electrodes motion in the z-direction) is the highest contribute dom- are assumed to be distributed randomly with the concen- inantly to the conductance. Now, let us obtain which kk tration c and 1 − c, respectively, at interface layers ` = 0

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atoms at the interface in Figs. 7(a) and 7(b). Here, c = 0 ¢¡¤£ ; and c = 1 denote the perfect interface, and the width of the insulator layer are thicker by two atomic layer for c = 1 than that for c = 0. Therefore, the conductance at c = 1 Fig. 9 Comparison between the calculated results of is much smaller than that at c = 0. For c & 0.3, the diffu- the conductance Γ in the CPA theory (solid curve) and sive part (vertex correction Γv) is larger than the specular the numerical simulations (circles) as functions of the one (non-vertex part Γ0) and the conductance is governed concentration c. The cluster size, i.e. the number of by the former. The increase in the diffusive transmission atoms on each atomic layer is 20 ×20 (•) and 40 ×40 causes a maximum in the concentration dependence of Γ (◦) in the numerical simulations. The error bars are the as shown in Fig. 7(b). This result indicates that the vertex standard deviations of the Γ due to the statistical average corrections are important for the tunnel conductance when over 40 samples. the randomness exists. The important mechanism for the increase of the conductance due to the diffusive scattering is summarized in numerical simulations. We find that the agreement is suffi- Fig. 8. In the absence of the randomness, tunneling through ciently good. The discrepancy between CPA and numerical the state with kk = (0, 0) is not allowed because of the simulation becomes smaller when the system size Nsite in- kk-conservation. Therefore, the state with a kk away form creases in the numerical simulation. (0, 0) contribute to the conductance. However, the state de- The conclusion of this subsection is that the diffusive cays faster in the barrier than kk-state. In the presence of transmission caused by the interfacial randomness can in- the randomness, kk is not necessarily conserved, anymore. crease the conductance drastically. This is the effect of the Therefore, the electron injected from the electrode with shape of the Fermi surface in TB model. kk , (0, 0) can be scattered by the interfacial randomness into kk = (0, 0)-state in the barrier. Due to this diffusive 4.2 TMR in single band model scattering, the conductance increases. In this subsection, we consider magnetic tunnel junc- In Fig. 9, we compare the calculated results of the con- tions in which the electrodes are made of ferromagnetic centration dependence of Γ with those calculated by the metals. Because the potentials in the ferromagnetic elec-

10 Journal of the Magnetics Society of Japan Vol. 30, No. 1, 2006

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! #" of the left electrode is the majority (minority) spin state ! #" in the left electrode but is the minority (majority) spin state in the right electrode. Fig. 11 Calculated results of conductance without (filled symbols) and with (open symbols) randomness. Conductances Γ++, Γ−−, and Γ+−(= Γ−+) are plotted in trodes are uniform, they are denoted as V+(−) for majority (a), (b), and (c), respectively. Results for EF = −5t and (minority) spin state. The conductance is calculated in the EF = 0 are shown by circles and squares, respectively. two current model for two different magnetization alignments depicted in Figs. 10(a) and 10(b). We use Γσσ0 to denote the conductance for an electron having σ-spin in the left electrode and σ0-spin in the right electrode, where σ randomness due to the mechanism mentioned previously and σ0 are defined by local spin axises (see Fig. 10). Since while the other conductances are not affected much by the the spin-flip scattering is not included, the electron spin randomness. is conserved. When the magnetizations of the electrodes Now, we mention the dependence of the TMR on the align in parallel (Fig. 10(a)), Γ++ and Γ−− are calculated thickness and the height of the barrier with and without for the majority and minority spin states. The total conduc- randomness. Figure 12 shows the calculated results of MR tance, ΓP, is given as ΓP = Γ++ + Γ−−. When the magneti- as functions of the barrier thickness d for EF = −5t and 0. zations of the electrodes align in anti-parallel (Fig. 10(b)), We can see that the MR either decreases or increases with the electron spin parallel (antiparallel) to the magnetization d depending on the position of EF. The results for lower of the left electrode is the majority (minority) spin state in EF are qualitatively the same with that in the free electron the left electrode but is the minority (majority) spin state model. This is because the electronic structure of the TB in the right electrode. The conductance Γ+− for an electron model is similar to that of the free electron model when EF having + spin in the left electrode is calculated by using is close to the band bottom. Those for higher EF is related VL = V+ and VR = V−. Since the conductance Γ+− for to the characteristic feature of the Fermi surface in the TB an electron having − spin in the left electrode is the same model. As can be seen in Fig. 12, the MR ratio tends to as that for + spin electrons, the total conductance, ΓAP, is be constant with increasing the barrier width. The conduc- ΓAP = 2Γ+−. The tunnel magnetoresistance (TMR) ratio is tance itself strongly depends on the barrier width, the MR defined as before ratio, in contrast, is rather stable against the change in the

ΓP − ΓAP barrier width, which is in agreement with the experimental MR = . (67) 61) ΓP observation. The importance of the combined eﬀect of the position of Calculated results of the spin-dependent conductance the EF and the randomness can also be seen in the depen- are shown in Fig. 11 as a function of the barrier thick- dence of MR on the barrier height Φ as shown in Fig. 13. ness. Conductances Γ++, Γ−−, and Γ+− for EF = 0 are much When EF = 0, MR increase with Φ except for small val- smaller than those for EF = −5t. This is because there ex- ues of Φ irrespective to the randomness. The qualitative ists a hole pocket around kk = (0, 0) on the projected Fermi future in this case is similar to the results in the free elec- surface of +spin states when EF = 0. Conductances Γ++ tron model. When EF = 0, MR without randomness de- and Γ−− for EF = 0 are much increased by the interfacial creases with Φ. In contrast, MR with randomness increases

Journal of the Magnetics Society of Japan Vol. 30, No. 1, 2006 11

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Fig. 12 Calculated results of the MR ratio as func- Fig. 13 Calculated results of the MR ratio as functions of barrier thickness d for EF = −5t (• and ◦) and tions of the barrier height for EF = −5t (• and ◦) and EF = 0 ( and ). Filled and open symbols denote the EF = 0 ( and ). Filled and open symbols denote the results without (c = 0) and with (c = 0.5) random- results without (c = 0) and with (c = 0.5) randomness, ness, respectively. Results with randomness are plotted respectively. as functions of the average thickness of the barrier.

ing, since in addition to the Fermi wave vector,64) another with Φ. The latter result is related also to the shape of the wave vector, i.e., the cut-off k-point kcp, given by the depth Fermi surfaces which are spin dependent. This is a clear of the quantum well, is also known to contribute to the 65) indication of the combined effect of the position of EF and conductance oscillations. This wave vector dominates randomness. the predicted oscillations of CPP-GMR in a Co/Cu/Co tri- Summarizing the results, the diffusive conductance orig- layer.66) In fact, the calculated oscillations of TMR for a inated from the interfacial randomness crucially depends clean junction67) cannot be explained by a single period on the shape of the Fermi surface, and dominates tunnel determined by kF only. Furthermore, the asymptotic value conductance for EF when the hole pocket opens around of the MR ratio calculated for a thick spacer layer is finite, kk = (0, 0) in the two dimensional Fermi surface. The MR which disagrees with the observed results. The purpose of ratio are also dependent on the EF in such a way that it the present subsection is to reconcile the theoretical results can be increased by the randomness when EF is close to with the observed ones and thus deepen our understanding the band bottom, while it is decreased when EF is higher of the TMR effect. than −2t + V+(= −3.5t in the present case). The variation of the MR ratio with the barrier height and thickness also 4.3.2 Present approach depends on the position of EF. These results are closely re- In this subsection we will show that the combined ef- lated to the dependence of the shape of the Fermi surface fects of barrier thickness and randomness can explain the in the TB model. The MR ratio is found to be rather stable experimental results.18,68–70) In particular, we will demon- against the change in the barrier thickness and height for strate that i) increasing barrier thickness increases the am- relatively thick and high barriers. This is sharp contrast to plitude of the kF oscillation period relative to the kcp os- the thickness and height dependence of the conductance. cillation period, ii) randomness introduced in the barrier also weakens the amplitude of the kcp oscillation period, 4.3 Quantum oscillation of TMR in single band model and iii) the randomness decreases the asymptotic value of 4.3.1 Contradiction between Experimental results and the MR ratio. These results are interpreted in terms of the previous theories momentum selection of electrons incident on the barrier Recent experiments by Yuasa et al.62) show clear oscil- interface and in terms of the diffusive scattering due to ran- lations of the MR ratio as a function of Cu layer thickness domness which opens additional conduction channels via for high quality NiFe/Al2O3/Cu/Co junctions in which the quantum well states. These effects will be demonstrated Co/Cu electrode is a single crystal. In their experiments, first by numerical calculation for a single band TB model. two characteristic features of the oscillations have been ob- We will then demonstrate that the calculated results can be served: (i) the average MR ratio decays to zero with in- reproduced by the stationary phase approximation.66) This creasing nonmagnetic layer thickness; (ii) the period of the implies that this technique is applicable to a realistic multi- oscillations is determined solely by the belly or long period orbital tunneling junction, where a purely numerical calcu- Fermi wave vector kF of Cu. The observed period agrees lation would be unfeasible. The results indicate that kF of quite well with that of the oscillations of photoemission Cu is really responsible for the oscillation period observed. spectra caused by quantum well states in Co/Cu multilay- Let us consider a FM/I/NM/FM junction on a simple cu- ers.63) From the theoretical point of view this is confus- bic lattice with lattice spacing a, where FM, I, and NM de-

12 Journal of the Magnetics Society of Japan Vol. 30, No. 1, 2006

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~~Journal of the Magnetics Society of Japan Vol. 30, No. 1, 2006 13~~

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f ness on the magnitude of Γ+−, and on the period of oscillations, we have calculated the dependence of the transmission coefficient on kk. Figures 17(a) and 17(b) show Fig. 18 Schematic diagram of diffusive scattering in the transmission coefficients T+−(kk), where Tσσ0 (kk) ≡ AP alignment. + spin electron injected from left FM P 0 0 0 tσσ0 (kk → k ) is the transmission coefficient for an electrode with k are scattered into k state of right kk k k k electron incident from the left FM with momentum kk electrode. on the barrier without and with randomness, respectively. When there is no randomness, the contribution to T+− in the momentum space is concentrated near kk = (0, 0). ness, the oscillatory part of the conductance is derived en- However, inclusion of randomness gives rise to additional tirely from states in the region of kk = (0, 0). In order to contributions to T+− of momenta outside this area. This is check this hypothesis, we use the stationary phase method 66) due to the fact that kk need not be conserved in diffusive (see Ref. and references therein), which is able to de- scattering. In the absence of randomness, only kk points on termine the oscillatory contributions from isolated regions the Fermi surfaces, that satisfy the kk conservation, may of the Brillouin zone, for thick spacers. The results, de- contribute to the conductance. For diffusive scattering, on picted by solid curves in Figs. 15(c) and 15(d) and Fig. 16, the other hand, the entire set of kk points on the Fermi sur- are in excellent agreement with the numerical calculations face contributes to the conductance (see Fig. 18). More for LNM & 5. This fact indicates that the oscillation pe- precisely, kk points corresponding to these quantum well riod observed in the experiments may be determined in the states contribute to the conductance. These kk points ap- stationary phase approximation for realistic systems. The pear as spiky peaks in Fig. 17(b) and they fall on concentric experimental finding62) that the oscillation period is deter- rings in the kk space. mined by kF of Cu spacer is thus naturally explained. It is clear from Fig. 17(b) that the number of open kk channels contributing to T+−, is the same as that contribut- 4.3.5 Summary of the role of randomness ing to T++. This explains the increase in the constant part We have shown that the observed period of TMR oscil- of the conductance Γ+− to a value of approximately same as lations and the oscillation of the MR ratio around an av- Γ++. In addition, the introduction of diffusive scattering has eraged value close to zero are explained by the combined almost eliminated the sharp momentum cut-off observed in effects of tunneling barrier and randomness. Randomness Fig. 17(a), which explains why the kcp oscillation period of introduces new conductance channels via quantum well Γ+− is weakened by randomness. The other transmission states, which are confined at the nonmagnetic spacer in coefficients are not greatly affected by the introduction of clean junctions. The new conduction channels increase the randomness, as scattering cannot open new kk channels for conductance of AP alignment to approximately that of P these cases. This explains why the introduction of random- alignment, and average MR ratio decreases to almost zero. ness has little affect on the conductances Γ++, Γ−−, and Γ−+. Selection of perpendicular transmission due to the barrier We therefore expect that in the presence of random- and elimination of the sharp Fermi surface cut off due to

~~14 Journal of the Magnetics Society of Japan Vol. 30, No. 1, 2006~~

~~ the randomness remove the cut oﬀ oscillation period, and, ~~

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the observed oscillation period in experiments can be explained in terms of the spanning vector kF of the Cu Fermi surface. Fig. 19 Dispersion curves of (a) MgO, (b) majority spin states of Fe, and (c) minority spin state of Fe along 5. TMR of Fe/MgO/Fe kz direction. Recently, a huge MR ratio up to 200%, in the optimistic definition, has been observed at room temperature √ in single-crystalline Fe/MgO/Fe junctions24,26) and highly for simplicity that the lattice constant of MgO is 2 times oriented CoFe/MgO/CoFe junctions.25) The MR ratio ob- as long as that of Fe. In (001)-stacking, Fe lattice is ro- served in magnetic tunnel junction using MgO24–26,71–73) tated by π/4 relative to MgO lattice. We further assume is much larger than that obtained in conventional junction that Fe atoms lie above O atoms at the interface and Fe-O using amorphous Al2O3 as the insulating barrier. Since the distance is the same with Mg-O distance. This assumption MR ratio is much larger than that expected in the DOS is supported by experimental and theoretical works.74,78–81) model, the experimental results indicate ballistic or coher- Transfer integrals at interface between Fe and MgO are cal- ent transport of conduction electrons through the junction. culated by Harrison’s method.82) Ballistic theories based on the first-principle calculation74) As for the randomness in MgO, we consider distortion and the realistic TB model75) have also expected that the of the perfect MgO lattice. We deal with the distortion phe- MR ratio is over 1,000%. In the theories, the large MR nomenologically as on-site random potential and assume ratio is attributed to the momentum (kk) conservation and that s-level vMg(s) of Mg and p-level vO(p) of O take ran- also the symmetry of the Bloch wave function. However, dom values uniformly distributed in an energy width Vrand, in the experiments, there exists randomness such as defects i.e., in MgO and lattice mismatch at the interfaces. Low barrier v − / ≤ v ≤ v + / , height of 0.37 ∼ 0.40 eV analyzed from the I-V relation ¯Mg(s) Vrand 2 Mg(s) ¯Mg(s) Vrand 2 (69) and from the dependence of the tunnel conductance on the v¯O(p) − Vrand/2 ≤ vO(p) ≤ v¯O(p) + Vrand/2 , (70) MgO thickness suggests the existence of the randomness. Here, we clarify the effects of the randomness on TMR wherev ¯Mg(s) andv ¯O(p) are average energy levels, and Vrand of Fe/MgO/Fe junction by performing numerical simula- describes the degree of the randomness in MgO. We calcu- tion based on a realistic TB model. We also study how the late the conductance of Fe/MgO/Fe junction using Eq. (68) MR ratio depends on the magnetic moment of Fe at inter- in the absence of randomness (Vrand = 0). In the presence face. of the randomness, we use finite size cluster and calculate the conductance using Eq. (68). 5.1 Model and method In order to treat realistic band structures of Fe and MgO, 5.2 Results without randomness we use multi-orbital TB model (see Eq. (16)) used by First, we show calculated results of the conductance Γ in Mathon and Umerski:75) the absence of randomness as a function of MgO thickness X X X X dMgO in Fig. 20. Conductances decay in exponential with = w µ, ν † + v µ † . H i, j( )ci,µc j,ν i( )ci,µci,µ (68) increasing dMgO. The slope of ln Γ depends of the spin and i, j µ,ν i µ magnetization alignment. These results can be explained TB parameters for bulk Fe and MgO are taken from lit- in terms of the electronic structure of Fe. Since the wave 76, 77) eratures. Produced dispersion curves of bulk Fe and vector kk is conserved in a clean junction and the barrier MgO are shown in Fig. 19. In our model, the band gap of height takes minimum at Γ-point as shown in Fig. 19(a), MgO is 7.7 eV and the Fermi level locates at the middle we concentrate on our attention on the energy-dispersion of the gap. The barrier height is almost consistent with the along kz-axis. 78) experimental value. The energy dispersion along kz-axis, i.e., Γ − H line, of Since the bond length of O-O in NaCl type MgO is about Fe for + and −spin states are shown in Figs. 19 (b) and the same with the lattice constant of bcc Fe, we assume 19(c), respectively. We can see that ∆1 band crosses the

~~Journal of the Magnetics Society of Japan Vol. 30, No. 1, 2006 15~~

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Fig. 21 kk-dependence of transmission coeﬃcients Fig. 23 Spin dependent conductance in P alignment (a) T++, (b) T−−, and (c) T+− for junctions without disorder. calculated for Fe/MgO/Fe junction with disorder as functions of MgO thickness.

Fermi level in +spin state. Since the ∆1 Bloch states consists of s, pz, and d3z2−r2 -orbitals, it hybridizes with MgO All of these results are consistent with those calculated in orbitals. Therefore, transmission coefficient T++(kk) takes the ballistic theories previously. maximum at kk = (0, 0) in the surface Brillouin zone (see Fig. 21(a)). 5.3 Results with randomness In the −spin state, instead of ∆1 band, ∆2 band crosses Calculated results of the conductance for the disordered the Fermi level. However, the ∆2 band consists of dx2−y2 - junction are shown in Fig. 23 as a function of dMgO. Here, orbital, it does not hybridize with MgO orbitals. Therefore, the cross section of the cluster used is 24aFe ×24aFe, where −spin state with kk = (0, 0) does not contribute to the con- aFe is the lattice constant of Fe. The degree of the ran- ductance. This results in a hole pocket around kk = (0, 0) domness that Vrand = 4 eV adopted in the model may in T−−(kk) in P alignment and T+−(kk) in AP alignment as correspond to a substitution of Mg ions with O ions by shown in Fig. 21(b) and 21(c), respectively. Furthermore, 1.0 atomic % if we evaluate the life time of electrons us- for kk states away from (0, 0), the effective barrier is higher ing the Born approximation. With increasing randomness than the real barrier height, and the slope of the ln Γ−− is Vrand, Γ−− increases while Γ++ does not change so much. larger than that of ln Γ++ as shown in Fig. 20. This behavior can be understood from the diffusive scatter- MR ratio evaluated from the conductances is shown in ing explained in section 4. The diffusive scattering enables Fig. 22. The MR ratio is about 0.9∼0.94. Although the ap- − spin electron injected from the electrode with kk , (0, 0) parent increase in MR ratio with increasing dMgO seems to to tunnel through kk = (0, 0) state in MgO, therefore Γ−− be small in this definition of the MR ratio, it increases from increases with increasing randomness. 900% to 1,600% in the optimistic definition of the MR ra- We show MR ratio calculated for various value of Vrand ∗ tio (MR ) given by Eq. (2). The reason of the large MR in Fig. 24. The MR ratio decreases with increasing Vrand. ratio is that Γ++ Γ+− Γ−−. Change in the the slope of For example, it decreases from 0.94 to 0.88 at dMgO = the spin dependent conductance shown in Fig. 20 yields an 10, which corresponds to a change from 1,600% to 800% ∗ increase in the MR ratio with increasing barrier thickness. in the optimistic MR ratio MR . Since the Γ−− (and Γ+−)

~~16 Journal of the Magnetics Society of Japan Vol. 30, No. 1, 2006~~

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5.5 Discussion and summary We have obtained large MR ratio, more than 1,000% served, and the diffusive scattering increases Γ−− and Γ+−, in clean Fe/MgO/Fe junctions. The large MR ratio is ex- and, then, decreases the MR ratio. We have also shown the plained in term of the kk conservation and the electronic relation between the MR ratio and the interface Fe mag- structure of Fe, especially half-metallic nature of ∆1 band. netic moment. Once we take into account the symmetry Because of the Bloch wave function symmetry, transmis- of the Bloch wave function (which is neglected in single sion coefficients T−− and T+− have a hole pocket around band model), all of the results mentioned above can be un- kk = (0, 0). This yields large MR ratios. Calculated re- derstood in term of the basic concepts presented in section sults are consistent with those in the previous ballistic the- 3, i.e., kk-conservation, shape of the Fermi surface, and the ories.74, 75) The effect of randomness in MgO on MR ratio diffusive scattering. is also clarified. As the degree of the randomness increases, Although we have assumed perfect interface between Fe 83) MR ratio decreases. In the presence of randomness, kk and and MgO, the existence of FeO layer has been reported. symmetry of the wave function are not necessarily con- First principle calculation has shown that FeOx layer re-

Journal of the Magnetics Society of Japan Vol. 30, No. 1, 2006 17 duces the MR ratio due to the bonding of Fe with O (al- rier density. Typical examples are Bi and Sb, in which the though the vertex correction is not included in the conduc- overlap of the electron- and the hole-pockets is about 10 tance calculation).84) We have also neglected lattice misfit meV; carrier density is 1017 ∼ 1018/cm3; and the effec- between Fe and MgO. The lattice misfit might cause distor- tive mass is about 1/100 of the electron mass. Another tion of the lattice at the interface. The distortion increases type of semimetals, rare-earth pnictides such as GdAs and the contribution from bands other than ∆1 symmetry and ErP, have been predicted by the first principles band cal- reduces the MR ratio. Asymmetry in the bias dependence culations to be semimetallic.89, 90) These materials have of the MR ratio might be attributed to the asymmetric in- the electron- and hole-pockets at X and Γ points, respec- terface structure due to the distortion. tively, on the Fermi surface with an effective mass simi- The lattice misfit also causes strain. It is known that lar to the bare electron mass. Thus far a few studies have MgO layers deposited on Fe are contracted in plane and been performed on the electronic state and the transport the interlayer distances are expanded.83) The strain might properties of these materials; spin-dependent resonant tun- give rise to the dislocation and defect near the interface. neling through ErAs quantum wells,91) GdAs/GaAs su- The dislocation and defect might also exist in MgO al- perlattices,92) and STM measurement of a crossover from though only lattice distortion in MgO is considered in our semimetallic to semiconductive in quantum dots of ErP on model. Our results is not applicable when in-gap states InP.93) The crossover from semimetallic to semiconductive are formed. Gibson et al. have shown that vacancy defects in the quantum dots is attributed to the quantum size effect such as F and V centers in MgO form in-gap states and on the electronic structure. An increase in the energy band raise the Fermi level.85) When the Fermi level is raised rel- gap in semiconductors has also been observed in nano- ative to the MgO band, the conductance and the MR ra- scale semiconductors.94–96) tio might be affected since the effective barrier height of In spite of these works on semimetals, no study has been MgO is reduced. Furthermore, if the in-gap states locate reported thus far on the spin-dependent transport properties near the Fermi level, the resonant tunneling occurs, which of junctions made of semimetals. Given that the techni- might give significant effects on the I-V characteristics of cal applications require optimization of resistance and MR the conductance and MR ratio. of ferromagnetic junctions, detailed study of the transport In the previous ballistic theories,74, 75, 86) an importance properties of FM/semimetal/FM junctions is desirable. In of the interface resonance so called hot-spot is pointed out. such study, it should be noted that the electronic structure However, it is not well understood how the hot-spot con- of the semimetal spacer could be changed by the quantum tribute to the conductance in disordered junctions. Ran- confinement because of the nm size of the junction, and domness might harm the formation of the resonant states that the transport properties of these junctions are affected but might increase conductance by hybridizing localized due to the change in the electronic states. The purpose of resonant states with propagating states. In order to treat this section is to study the transport properties of FM/NM these issues, we should perform numerical simulations spacer/FM junctions, emphasizing on the role of the elec- with much larger cluster size or by using the supercell tronic states of the NM spacer in the MR effect of the junc- method. Alternatively, development of the CPA code for tion.19,97) realistic TB model might be necessary. 6.1 Model and method We adopt a TB model on a simple cubic lattice for 6. Semimetal junction the FM/NM spacer/FM junctions. The FM electrodes are In order to apply the TMR and CPP-GMR to MRAM treated in a single band model with exchange splitting, and/or to magnetic sensors, optimization of the MR ra- and the NM spacer is dealt within a two-band model. The tios and resistance is required. Optimization of the MR Hamiltonian of the model is given as ratio may be achieved by selecting suitable FM materi- H = H + H + H0 , (71) als. Optimization of the resistance, however, is difficult FM NM due to the fact that the materials of the NM spacer are where HFM and HNM are the Hamiltonians for FM elec- limited to insulators or metals. Some attempts have been trodes and the NM spacer, respectively, and H0 repre- made to remove the limitation; to utilize single crystal tun- sents electron transfer between FM electrodes and the NM nel junctions,24–26, 71–73, 87) for example, and to fabricate spacer. They are given explicitly as tunnel junctions of ferromagnetic semiconductors such as X X H = −t c† c + v c† c , (72) (GaMn)As with a semiconducting spacer, in which the bar- FM iσ jσ iσ iσ iσ 88) i, j,σ i,σ rier thickness and height might be controlled. Another X X † † way to remove this limitation, though it has attracted much HNM = tµciµσc jµσ + viµciµσciµσ, (73) less attention, is to utilize disparate materials – that is, i, j,µ,σ i,µ,σ semimetals – for the NM spacer. X H0 = t0 c† c + h.c. (74) The electronic structures of the semimetal at the Fermi µ iσ jµσ i∈FM, j∈NM,µ,σ level consist of electron- and hole-pockets with a low car-

~~18 Journal of the Magnetics Society of Japan Vol. 30, No. 1, 2006~~

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Fig. 27 Schematic ﬁgures of electronic structures: (a) Energy dispersion of a bulk semimetal, (b) Den- Fig. 28 Calculated results of the conductance as sities of states of paramagnetic electrode (PM) and a functions of band overlap ∆E/t in the semimetal with semimetal (SM) in a paramagnetic junction. ∆ex = 0.0 and Eb/t = 2.0.

ing |∆E|. The decrease is stronger for thicker spacers, as Here, t and viσ are the hopping integral and the spin (σ = +, −)-dependent potential of the FM electrodes, respec- expected. In the metallic region, Γ oscillates with varying the thickness L of the spacer. The period of oscillation is tively. The latter is given as vi± = v0 ∓ ∆ex/2, with an ex- determined by the Fermi wave length of the semimetals. change potential ∆ex. In the second equation, viµ is a spin- independent, but orbital (µ = 1, 2)-dependent potential of The momentum states of the propagating waves con- 0 tribute differently to the conductance depending on the the spacer, and tµ, and tµ are the hopping integrals within the spacer and between the FM electrodes and the spacer, spacer thickness L. Figures 29(a) and 29(b) show the cal- 0 culated results of the momentum-resolved conductance respectively. The values of t, tµ, and tµare assumed to be the same, and the energy is measured in units of t through- T (kk), defined in such a way that out this study. A schematic figure of the energy-momentum e2 X Γ = T (k ) (75) dispersion relation of the semimetal is shown in Fig. 27(a), h k and a schematic density of states (DOS) of the paramag- kk netic junctions is shown in Fig. 27(b). for paramagnetic junctions where L = 1 and L = 5 with In the present model, the electronic state of semimetals ∆E/t = 1.0 and Eb/t = 4.0, respectively. When L = 1, is given as an electron- and hole-pocket at R and Γ points, the conductance shows a broad peak near kx = ky ∼ π/4, respectively, in the Brillouin zone of the simple cubic lat- whereas when L = 5, the conductance shows a high peak tice. The overlap of the valence and conduction bands is near kx = ky ∼ 0. It should be noted that the state of characterized by ∆E (= energy at the top of the valence kk = (0, 0) is excluded in the latter case. Because we are − energy at the bottom of the conduction band). When dealing with the ballistic conductance, the momentum kk ∆E > 0, there is an overlap of the valence and conduc- of the propagating wave is conserved, and the wave is re- tion bands and the spacer is semimetallic; when ∆E < 0, flected when there is no available state in the spacer. Fig- the system is semiconductive. The important parameters ures 29(c) and 29(d) shows the Fermi surfaces of the elec- are ∆E, position of the Fermi level (Eb) measured from trode and the spacer projected onto kk space, respectively. the top of the FM band in the paramagnetic state, and the It can be seen that there are no states with common kk near exchange splitting ∆ex. kx = ky ∼ π/4. Nevertheless, the total conductance Γ shows The conductance is calculated by using the Kubo- a broad peak near kx = ky ∼ π/4 for L = 1. This is due to Landauer formula within the ballistic limit for parallel tunneling from the left electrode to the right one. Because (P) and anti-parallel (AP) alignment of FM magnetiza- the spacer is very thin, the tunneling probability is recog- 16, 17) tions. The number of kk = (kx, ky)-points, used in the nizably large. With increasing spacer thickness, the con- mixed representation of (kk, `) where ` stands for the layer tribution from the tunneling decreases exponentially, and, index, is 200 × 200 unless specified otherwise. consequently, Γ shows a peak in the region where kk of the propagating wave is available in both the electrode and the 6.2 Calculated results spacer. These results indicate the importance of matching 6.2.1 Conductance of paramagnetic junctions the electronic states of the electrodes and the spacer at the Figure 28 shows the calculated results of conductance Fermi level in the ballistic transport. Γ of junctions with paramagnetic electrodes with ∆ex = 0 and Eb/t = 1.0 as functions of the band overlap ∆E. When 6.2.2 Magnetoresistance in ferromagnetic junctions ∆E > 0, the spacer is semimetallic, and the conductance is Now we move to ferromagnetic junctions. The + and − large, while when ∆E < 0, the spacer is semiconductive, spin bands of the ferromagnetic electrodes are exchange and the conductance decreases exponentially with increas- split by ∆ex. The MR ratios are calculated by taking

~~Journal of the Magnetics Society of Japan Vol. 30, No. 1, 2006 19~~

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¨ © ¨ © ¨ © 0) but for the semimetallic one, the analysis may be reduced to one-dimensional (1-D) propagation of a wave with k = (0, 0). Fig. 29 k -dependence of transmission coefficients k k The MR ratios calculated for the 1-D model (see for (a) L = 1 and (b) L = 5, with ∆E/t = 1.0 and Fig. 31(b)) as functions of ∆E are compared with those E /t = 4.0. Here, a is the lattice spacing of the sim- b for the 3-D model. We find that the 1-D results are in good ple cubic lattice used. (c) and (d) are schematic figures of 2-dimensional Fermi surface of the electrode and the agreement with the 3-D ones except for the oscillation in semimetal, respectively, in the surface Brillouin zone. MR for ∆E > 0. The oscillation in 3-D results is rather Shaded regions are occupied states. obscure due to contributions from many wave vectors to the conductance. It should be noted that the resonant states in the spacer do not necessarily form peaks in MR ratios in the 1-D case, but, rather, give rise to dips because the ∆ex/t = 2.0 and Eb/t = 2.0, and the results are shown in conductances in both P and AP alignments of magnetiza- Fig. 30 as functions of ∆E for several values of L. Several tion are high due to the resonance. Sharp peaks of narrow characteristics can be seen in the results. width often appear due to the subtle balance between P and i) For positive values of ∆E, MR ratio is about 0.2 and AP conductance. These results indicate that the peak near oscillates with increasing ∆E. The oscillation is attributed ∆E = 0 cannot be attributed to an effect of the resonance. to a change in the Fermi wavelength of semimetals. In order to gain a more insight into the dependence of ii) There exists a peak of MR ratio near ∆E ∼ 0, where the MR ratio on ∆E, we calculated the partial density of a crossover from semimetal to semiconductor occurs. states (PDOS) of the FM electrodes at the interface with iii) The position of the peaks, however, depends on the kk = (0, 0) as a function of ∆E. The results are shown in thickness of the spacer. This is simply due to the quantum Fig. 31(a). We found there was a big difference between the confinement of the electrons within the spacer. When the PDOS of the + and − spin states near ∆E = 0. By defining spacer is thin, the effective band-width shrinks due to the the spin polarization of the left and right electrodes as PL quantum size effect, and the crossover from semimetals to and PR using the spin-dependent PDOS, the MR ratio can semiconductors occurs for positive values of ∆E. For L & be given by Eq. (5). The calculated result of the MR ratios 15, the effect of the quantum confinement is weak, and the given by the expression are compared with those for the crossover occurs at ∆E ∼ 0. 1-D model in Fig. 31(b). The results of the 1-D and the The pronounced feature is that the MR ratio shows a 3-D model agree quite well for ∆E ≤ 0. Thus, we may maximum at the crossover from semimetal to semiconduc- conclude that the PDOS at the interface governs the MR of tor. The analysis of the results will be given in the next these junctions at least for ∆E ≤ 0. subsection. The reason that the MR ratio is well described by the PDOS of the electrodes at the interface can be under- 6.2.3 Analysis of the calculated results of MR stood in terms of a conductance formula, as follows. In the As mentioned earlier, the conductance is caused pre- tunneling regime, the transmission coefficient in the 1-D dominantly by propagating wave vectors common to both model is given by16) the FM electrodes and the spacer. Figure 29 showed that 2 2 the conductance for thicker spacers is caused predomi- T = 4t ImgLImgR sinh (κa)

~~20 Journal of the Magnetics Society of Japan Vol. 30, No. 1, 2006~~

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Fig. 31 Calculated results of (a) the PDOS at the FM interface, and (b) the MR ratio as a function of ∆E/t for that the PDOSs at the interface are much different from ∆ex/t = 2.0 and Eb/t = 2.0. those at the surfaces. This strong modification of PDOS at the interface is due to mixing between the electronic states of FM metal and semimetal. As a consequence, a large difference appears between the + and − spin PDOSs at the × sinh κ(L + 1)a + t(gL + gR) sinh(κLa) Fermi level and the PDOS at the surface. The occurrence −2 of the peak in MR ratio as a function of ∆E may now be + t2g g sinh κ(L − 1)a , (76) L R understood as the result of PDOS at the FM interface. The where gL(R) is the surface Green’s function of the isolated result suggests that the approximate expression of the spin left (right) electrode, that is, the semi-infinite ideal elec- polarization P used in equation (5) should be given by the trode, and κ denotes the exponential decay factor in the PDOS at the interface rather than the DOS of the surface. spacer defined by the height of the band gap. In the case of The modification of PDOS shown in Fig. 32 may be ex- the large semimetal thickness, i.e., κLa 1, equation (76) plained in the following way. Because ∆E/t = −1.0, there can be reduced to is no state for E & −1.0t in the semimetal spacer. There- Img fore, the PDOS of the FM interface for E & −1.0t is half- T ≈ 2 −2κLa L 4t e 2 |1 + tgL exp(−κa)| elliptic, similar to that at the surface. Below E ∼ −1.5t, the confined states in the spacer mix with the + spin states ImgR × . (77) in the FM electrodes, and the form of PDOS is broadened. |1 + tg exp(−κa)|2 R In contrast, the − spin states confined in the spacer cannot On the other hand, the Green’s functiong ˜ L(R) at the left find available states in the − spin states in the FM elec- (right) electrode interface of the whole system is given by trodes, and form sharp peaks. (Only a portion of these is shown in the figure.) g˜L(R) =gL(R){sinh κ(L + 1)a + tgR(L)} × sinh κ(L + 1)a + t(gL + gR) sinh(κLa) 6.2.4 Increase in MR ratio due to Fermi surface mis- 2 −1 match + t gLgR sinh κ(L − 1)a Increase of MR ratio may be expected for cases in which g L(R) the matching of Fermi surfaces between the FM electrodes ≈ −κa . (78) 1 + tgL(R)e and the semimetal is spin-dependent. In the present model, As a result, the transmission coefficient is written as T = a hole pocket appears around Γ point of the Brillouin zone 2 4t exp(−2κLa)Im˜gLIm˜gR, and is proportional to the local when the FM bands are shifted upwards; that is, with in- DOS of the electrodes at the interface. creasing Eb. Because of the exchange splitting between the Calculated results of the details of the PDOS at the in- + and − spin bands, a situation occurs in which a hole- terface between FM metal and semimetal in P alignment pocket at Γ point exists in the − spin band, while no hole- are shown in Fig. 32 as functions of energy E/t. Solid and pocket exists in the + spin band. In this case, there is no short broken curves are the + and − spin PDOSs at the in- contribution from the − spin electrons to the conductance, terface, and the half-elliptic curves shown by the chained and the MR ratio can therefore be large. Numerical re- curves are those at the surfaces for the 1-D model. We find sults actually show that the MR ratio increases with in-

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Fig. 34 MR ratio calculated for (a) Eb/t = 2.0 and creasing Eb/t from 2.0 to 3.0 as shown in Fig. 33. With in- (b) Eb/t = 3.3 in clean (filled circles) and disordered creasing Eb further, the MR ratio reaches almost 1.0. Thus, (open symbols) junctions. the matching and/or ill-matching of the Fermi surfaces between FM electrodes and semimetal is crucial to the MR ratio. are important for the optimization: the matching of the 6.2.5 Effect of randomness on TMR Fermi surfaces of the FM electrodes with the spacer, and So far, no randomness has been introduced in the junc- the PDOS at the FM interface. In the previous section, we tions. Even in epitaxial junctions, however, randomness, showed that when the wave vectors near kk = (0, 0) are especially at interfaces, may not be avoidable. The previ- available for propagation, a suitable position of the Fermi ous sections showed that the contribution to the MR ra- level gives rise to a high MR ratio. We also showed that tio from diffusive conductance, which does not conserve MR ratio is not affected much by randomness when the momentum in the tunneling event, is large in tunnel junc- state with kk = (0, 0) exists on the Fermi surface of the tions with randomness. This is because the randomness electrodes. This result is consistent with that shown in sec- makes the transmission of electrons via momentum states tion 3. 98, 99) near kk = (0, 0), prohibited in the ballistic tunneling, pos- Figure 35 is our preliminary results calculated by sible. When wave vectors near kk = (0, 0) are available using realistic TB model for n-type semiconductor/ GdAs for electron propagation in the ballistic case, the effects semimetal/ n-type semiconductor junction. Obtained trans- of randomness may be small. In Fig. 34, we show the mission coefficient has a peak at small region around kk = calculated results of MR with randomness. We assume (0, 0), which is consistent with results shown in Fig. 29(b). † random potentials viciσciσ at the interfaces of the elec- Therefore, our results obtained by simple model does not trodes. The randomness is introduced in the region of five change qualitatively even if we take into account realistic atomic layers from the interfaces, and vi takes +Vrand/2 or electronic structures. 75) −Vrand/2 randomly in the region. We find that the MR ra- Tunnel conductance calculated for Fe/MgO/Fe and 100) tio for Eb/t = 2.0 is not affected much by the random- Fe/ZnSe/Fe tunnel junctions in P alignment shows that ness while the MR ratio for Eb/t = 3.3 decreases with the Γ point of the + spin bands contributes to the tun- increasing random potential. This is because the wave vec- nel conductance, while that of the down − bands does tors near kk = (0, 0) are available on the Fermi surface for not. This may indicate that the MR ratio can be large for Eb/t = 2.0 whereas a hole pocket exists around kk = (0, 0) Fe/semimetal/Fe junctions in far as the Fermi surface of for Eb/t = 3.3 in the present model of semimetals. De- the semimetal is sufficiently small near the Γ point. The tailed calculations that take into account realistic electronic conductance itself can be high because there is no energy structures and randomness is desired in the near future. gap in the spacer. The lattice matching between Fe and a semimetal such as ErP, however, may not be good. From 6.3 Discussion and Summary the viewpoint of lattice matching, MnAs might be more ap- We studied the conductance and MR ratio in FM/NM propriate ferromagnetic materials. The energy dispersion spacer/FM junctions using a simple model, and found that relation of NiAs-type MnAs,101) however, is rather compli- the MR shows a peak near the region of crossover from cated, and more studies may be necessary to acquire quan- a semimetallic to a semiconductive NM spacer. The result titative estimate of the MR ratios for MnAs junctions. indicates that both the MR and the resistance may be op- In summary, we studied in detail the conductance and timized; that is, high MR and low resistance are optional magnetoresistance of FM/NM spacer/FM junctions within conditions. Two characteristics of the electronic structure the ballistic limit for the current perpendicular to the junc-

~~22 Journal of the Magnetics Society of Japan Vol. 30, No. 1, 2006 intergrain MR109) is due to a much stronger T-dependence of P at interfaces than that of the bulk.~~

~~Furthermore, the value of P itself at low tempera- § tures is an issue to be resolved. Wei et al.110) and Park~~

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~~toemission experiments, respectively. In contrast, measure- ¡ ment of P using point contact Andreev reﬂection technique has shown that P ∼ 0.8 for manganites,35) being consis-~~

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¨ © also shown that P depends on the contact resistance, the residual resistivity of the ferromagnet, and the superconducting gap.112, 113) larger spin polarization P & 0.9 has Fig. 35 kk-dependence of transmission coefficients been obtained for samples with high residual resistivity and for n-type semiconductor/ GdAs semimetal/ n-type small superconducting gap. Therefore, in order to clarify semiconductor junctions. The thickness of GdAs is 10 the characteristics of the TMR of MTJs, it is of interest to atomic layers. calculate the T-dependence of P at interfaces of MTJs and to study their interrelation. It is well known that the basic physics of manganites re- tion planes. Emphasis was placed on the role of the elec- sides in the so-called double exchange (DE) model,114–116) tronic structures of the junctions in the conductance and though the orbital degree of freedom and the Jahn-Teller MR ratio. It was shown that the MR ratio can be large and effect in manganites are responsible for the variety of mag- 117,118) shows a maximum when the character of the nonmagnetic netic states and the colossal MR near TC. The DE spacer changes from semimetallic to semiconductive. This model, however, contains many-body interaction and its result has well been reproduced in the one-dimensional ground state properties are not fully understood yet as model, and be attributed to a change in the partial density shown in a recent numerical study on the DE model.119) In of states at the Γ point of the Brillouin zone. It has been view of these circumstances, here we focus our attention on pointed out that the characteristic electronic structure im- how Hund’s rule (exchange) coupling between the itinerant portant to govern the MR ratio is the spin-dependence of electrons and localized spins in the DE model influences the Fermi surface matching between ferromagnetic elec- the T-dependence of P and the MR ratio, and treat it in trodes and semimetals. The MR ratio can be large when an approximate way where the exchange coupling is dealt the matching of Fermi surfaces between the ferromagnetic with as spin fluctuation acting on the tunneling electrons at electrodes and semimetals is good in one spin component, the interfaces.20) In the following, we calculate the values while that in the other component is ill-matched. of P and the tunnel conductance Γ in the linear response theory (Kubo formula)11) and study the interrelation between the MR ratio and P. We will show that the exchange 7. Manganite MTJs coupling has an intrinsic effect on the T-dependence of the In the expression of MR ratio given by Eq. (5) the largest MR ratio and P in these MTJs, and that the MR ratio de- MR ratio is given for the so-called half-metallic ferro- pends more strongly on temperature than P does. Never- magnets where P = 1, that is, only one spin component theless, the MR ratio may be large enough even at high has non-zero density of states (DOS) at the Fermi level. temperatures. The ab initio band calculations have predicted so far that Heusler compounds102) and several metallic oxides such as 7.1 Hamiltonian 103) 39, 104) CrO2 and (La-Sr)MnO3 show the half-metallicity. We consider a junction of manganese oxide/ insulator/ Therefore, it is natural to utilize these materials for elec- manganese oxide. In manganese oxide, conduction electrodes of MTJs to obtain a high MR ratio. Lu et al.105) and tron (eg electron) couples ferromagnetically to localized Viret et al.106) fabricated MTJs using (La-Sr)MnO to ob- 3 Mn core spin (t2g electron) as shown in Fig. 36. The dou- tain MR ratios as high as 0.44 and 0.8, respectively. Re- ble exchange mechanism yields the ferromagnetic phase. cently, a huge MR ratio as much as 0.95 has been observed We consider the double exchange model at 4K.107) However, the MR ratio has been found to de- X X X † † crease rapidly with increasing temperature (T) and to van- H = −t ciσc jσ + viciσciσ − K Si · si, (79) (i, j),σ iσ i ish far below the Curie temperature (TC). The rapid decrease in the MR ratio with increasing T has been also where t and vi are the nearest neighbor hopping and the observed in the intergrain MR of perovskite manganites; on-site potential, respectively, and the last term denotes the 108) Park et al. have suggested on the basis of spin resolved Hund’s rule coupling between the localized spin Si and the P † photoemission experiments that the rapid decrease in the itinerant electron spin si ≡ σ,σ0 ciσσciσ0 with a positive

~~Journal of the Magnetics Society of Japan Vol. 30, No. 1, 2006 23 £ ¡ ~~

~~ £ ¡ diﬃculty may be avoided by including the eﬀect of spin~~

~~ ﬂuctuation at the interfacial layers only. The approximation may be justiﬁed when we note that the tunnel MR is~~

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~~results; the experiments shows that the MR ratio is strongly~~

¢¡¤£¦¥§ ©¨ ¦ © ¨ influenced by magnon excitations,41, 122) and theories show that the tunnel conductance is correlated with the interfa- Fig. 36 Schematic figure of double exchange model. cial electronic state123) and is affected by the interfacial 17) Localized spin formed by t2g electron couples with spin roughness. As the spin fluctuation may be considered of conduction (eg) electron. to be thermally induced randomness, its effect on the electronic state of the tunneling electrons may be the largest at the interfaces. Although we use the bulk values of hS zi value of K. The localized spins are taken to be S = 3/2. and spin fluctuation, the contribution of the spin fluctua- z The other notations are standard. When we take into action relative to that of hS i can be properly included, at least, qualitatively, in our treatment. count the degeneracy and the anisotropy of the eg orbitals, we use the following two band DE model instead of the The expression of Γ given by the Kubo formula is rewrit- single band one (Eq. (79)). ten in terms of intra- and inter-layer Green’s functions, X X which are calculated by the recursive Green’s function † † H = ti j(µ, ν)ciµσc jνσ + vi(µ)ciµσciµσ method as describe in section 3. The vertex correction (i, j),µν,σ i,µ,σ X is taken into account in a way which is consistent with that used for the self-energy. The asymmetry of the trans- − K Si · si, (80) i verse spin fluctuation terms, however, breaks the current conservation in the AP alignment of the magnetization of where µ(ν) denotes d3z2−r2 and dx2−y2 -orbitals and si ≡ P † MTJs even though the vertex function is determined self- 0 c σc . µ,σ,σ iµσ iµσ0 consistently. This fact has not been noticed in previous 120,121) 7.2 SCBA for spin fluctuation work. Therefore, we apply semi-classical approxi- + − − + 2 z 2 In order to treat the exchange coupling, we adopt the mation such that hS S i = hS S i ∼ S − h(S ) i. This local approximation developed by Kubo120) and followed approximation may not be unreasonable except for low later by Takahashi and Mitsui.121) For simplicity, we con- temperatures as the spin values S are large in our case. sider the single band DE model. The exchange interaction The current conservation has been confirmed numerically is rewritten as as well. X X z † σσ0 † Si · σi = hS i iciσciσ + Vi ciσciσ0 , (81) 7.3 Calculated results in SCBA σ σσ0 In the calculations, we chose two values of Fermi level, σσ z z +− − −+ + with Vi = σ(S i − hS i i), Vi = S i and Vi = S i , EF = −5t and −4t (see inset of Fig. 37), K = 2t, the bar- σσ0 where h· · · i denotes the thermal average. We treat Vi rier height Φ = 2t, and barrier thickness d = 5a with a perturbatively in the self-consistent Born approximation lattice constant a for a simple cubic lattice. The number of σσ0 σ0σ (SCBA) where terms such as hVi Vi i are retained as itinerant electrons is thus determined to be 0.08 and 0.16 spin fluctuation acting on the itinerant electrons subject per atom for EF = −5t and −4t, respectively. We have ne- σσ0 to that hVi i = 0. In this approximation, the self-energy glected the T-dependence of the Fermi distribution func- σ included in the effective Green’s function Gii of the itin- tion. Since the electronic structure of the electrode (man- P σσ0 σ0 σ0σ erant electrons is given as σ0 hVi Gii Vi i. Therefore, ganese oxide) varies with temperature, we shift the energy σ0σ higher order terms in the perturbative expansion of Vi bands of the electrode so that the charge neutrality condi- are taken into account similarly to the CPA. Since the self- tion is satisfied. ± ± energy includes the effective Green’s function, the self- In this subsection, we use ΓP and ΓAP instead of Γ±± energy and the Green’s function must be determined self- and Γ±∓ since the spin flip scattering is included and the consistently for given magnetization hS zi and spin fluctua- electron spin is not conserved during the transport process. σσ0 σ0σ σ σ tion hVi Vi i. Because we do not include the Green’s ΓP (ΓAP) denotes the conductance in P (AP) alignment for function of the localized spins, the self-consistency be- an electron having σ spin injected from the left electrode, + − + tween the localized spin and the itinerant electrons is not and the total conductance is ΓP = ΓP + ΓP and ΓAP = ΓAP + − complete, that is, the Curie temperature is taken as a given ΓAP. parameter. Figure 37 shows the T-dependence of the spin polariza- Despite the approximation to the exchange interaction, tion P at the interface for the itinerant electrons. The solid the treatment of this term still meets a difficulty due to non- and broken curves are the calculated values of P with and existence of the translational invariance of the MTJs. The without spin fluctuation, respectively. When EF = −5t, the

~~24 Journal of the Magnetics Society of Japan Vol. 30, No. 1, 2006~~

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Fig. 37 Temperature dependence of P with (solid Fig. 38 Temperature dependence of the MR ratio curves) and without (broken curves) spin ﬂuctuations with (solid curves) and without (broken curves) spin ﬂuctuation for EF = −5t (thick curves) and −4t (thin for EF = −5t (thick curves) and −4t (thin curves). Inset shows the density of states of the bulk electrodes. curves). Inset is the MR ratio as functions of the square of the magnetization M of the bulk electrodes.

~~bulk electrodes are half-metallic up to ∼ 0.4TC, and there- ¥ fore the thick broken curve shows P = 1 below ∼ 0.4TC. ¥~~

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F half-metallicity and the existence of the spin fluctuation are responsible for the strong T-dependence of P. A linear T-dependence at low temperatures is due to the classical Fig. 39 Spin dependent conductances as functions of approximation. the temperature. Conductances in P and AP alignments Figure 38 shows the calculated results of the MR ratio as are shown by filled and open symbols, respectively. σ σ functions of T/TC. When EF = −5t, the MR ratio without ΓP (ΓAP) denotes the conductance in P (AP) alignment spin fluctuation (thick broken curve) takes the maximum for an electron having σ spin injected injected from the left electrode. Because of the spin-flip scattering, value 1 below ∼ 0.4TC and decreases rapidly above this temperature where the half-metallicity is lost. Inclusion of Γ+− , Γ−+ in AP alignment (see the text). spin fluctuation makes the MR ratio smaller than that without spin fluctuation for T . 0.7TC as shown by the thick solid curve. The origin of the strong T-dependence of the MR ratio ∝ M2 approximately. Noting that the intergrain 109) MR ratio for half-metallic electrodes is the same with that TMR in pyrochlore Tl2Mn2O7 and double perovskite 125) of the T-dependence of P. In contrast, when EF = −4t, Sr2FeMoMnO6 shows a power law behavior, indicating the MR ratio decreases gradually with increasing temper- that the interfacial effects on magnetism may be smaller, ature. A small difference between the solid and broken MTJs with these materials might show a large MR ratio at curves in this case may be due to a cancellation of two temperatures close to TC. effects; a decrease in the MR ratio due to spin flip tun- Deeper insight into the T-dependence of the MR ratio neling and an increase in the MR ratio due to diffusive can be obtained by plotting the spin dependent conduc- tunneling,17) both of which are caused by the spin fluctu- tance as a function of T, as shown in Fig. 39. The increase ation. It is worthwhile to note that the T-dependence of in Γ above T ' 0.4TC is due to the disappearance of the the MR ratio for half-metallic electrodes (EF = −5t) at half-metallicity and the decrease in Γ above T & 0.7TC is high temperatures is similar to that for normal ferromag- due to the spin fluctuation. One should note that the spin netic electrodes (EF = −4t). This indicates that the MR fluctuation makes Γ++ smaller with T at low temperatures, ratio for MTJs with half-metallic electrodes can be large but Γ+− increases with T in this temperature range. even at high temperatures near TC, which disagrees with experimental results105–107) although there are some excep- 7.4 Numerical simulation of conductance and TMR tions.124) The disagreement is discussed later. 7.4.1 Model of alloy analogy The inset of Fig. 38 shows the MR ratio vs M2 with In previous subsection, the effect of spin fluctuation was M being the magnetization of the electrodes. We see the introduced only in the interfacial layers of manganites. In

~~Journal of the Magnetics Society of Japan Vol. 30, No. 1, 2006 25~~

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' ¢()*¨ +,)- average value of the conductance by averaging over the ¢()*¨ +,)- snapshots (samples).21,126) Fig. 41 Schematic figure of a ferromagnetic tunnel 7.4.2 Systems for numerical calculation junction comprised of (La-Sr)MnO3 (LSMO), SrTiO3 We consider a trilayer consisting of two manganites (STO), and semi-infinite electrodes. Thicknesses of LSMO separated by an insulator STO and assume that LSMO and STO are L and d u.c., respectively. Here, ideal electrodes are connected to both sides of the junc- 1 u.c. is about 4Å. The cross section of the system is tion (see Fig. 41). We use the two-band DE model given Nx × Ny. by Eq. (80) for the numerical calculation. Cluster size is as follows. The thickness of LSMO is L = 50, the thickness of STO is d = 6, the cross section of the cluster is samples. Nx × Ny = 30 × 30 in the unit of u.c., where 1 u.c. is the length of a side of perovskite unit cell and is about 4 Å. 7.4.3 Calculated results We assume that doping rate x of Sr is 0.3, i.e., the number Calculated results of the conductance are shown in of conduction electron per site is n = 0.7. We also assume Fig. 42 as a function of temperature. The electronic struc- that the barrier height is Φ = 4t where t is the transfer in- ture of LSMO for K = 1.0t is of weak ferromagnet whereas tegral between dx2−y2 orbitals at nearest neighbor sites in it is half-metallic for K = 2.0t at T = 0 as shown in insets x-y plane. We use ferromagnetic metal LSMO treated by of Figs. 42(a) and 42(b), respectively. With increasing tem- the mean field approximation and also paramagnetic metal perature, the conductances ΓP and ΓAP increase. However, LaNiO3 for the electrodes, although we only show the for- in the case of K = 2.0t, ΓAP = 0 at T . 0.6TC. This is mer results in this article. In order to obtain the average because the electronic structure of LSMO for K = 2.0t value of conductance in the numerical simulation, we use is half-metallic at T . 0.6TC. As temperature increases 10 snapshots (samples). We have confirmed that the aver- further, the electronic structure changes to weakly ferro- age of the conductance converges sufficiently even in 10 magnetic and, then, ΓAP increases. By comparing the re-

~~26 Journal of the Magnetics Society of Japan Vol. 30, No. 1, 2006~~

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VWcd lmuv of K = 2.0t, MR ratio takes the maximum value 1.0 at T ≤ 0.6TC because ΓAP = 0 due to half-metallicity of the electronic structure. As the temperature increases and the Fig. 44 (a) Density of states and projected Fermi sur- electronic structure changes from half-metallic to weakly faces of LSMO at T = 0 (E = −1.3t, + spin) and ferromagnetic, MR ratio decreases. In both Figs. 43(a) and T > TC (E = −2.6t, + and − spins) for K = 2.0t. In 43(b), temperature dependence of the MR ratio is weak and (b) and (c), Fermi surface is projected onto plane and the value remains finite up to TC of bulk LSMO. Figure 43 shown by shaded area in the first surface Brillouin zone. also indicates that the spin fluctuation does not give much effects on the MR ratio. Here, we only showed numerical results obtained by using ferromagnetic electrodes. We also calculated MR ratio These terms bring about spin mixing and spin-flip scatter- by using nonmagnetic metal such as LaNiO3 for the elec- ing. When the alignment of the localized spins are non- trodes. The obtained results do not differ form those shown collinear, the + and − spin states of the conduction elec- here. The reason that the spin fluctuation does not affect trons are no more good quantum states and mix together. the conductance and the MR ratio very much can be un- This is the spin mixing. In order to study the effects of derstood when we look at the Fermi surface of the LSMO. spin√ mixing√ on TMR, we assume that S = (0, 0, ±3/2), Fermi surfaces of the LSMO projected onto kx-ky plane are ( 2 cos φi, 2 sin φi, ±1/2), where φi is the azimuthal an- shown in Fig. 44. Shaded area in Figs. 44(b) and (c) cor- gle and takes random values. The randomness is replaced respond to the projected Fermi surface. We can see that with uniform random digits in the computation. kk = (0, 0)-state exists on the Fermi surface. As explained The calculated results of the conductance and MR ratios in section 3, the effect of the randomness is weak when are shown in Fig. 45. We find that the conductance in the kk = (0, 0)-state exists on the Fermi surface. P alignment is not changed so much even if we introduce So far, we have neglected the transverse components of the spin mixing effect. On the other hand, the conductance x y the spin fluctuation, that is, terms of S and S in Eq. (79). in the AP alignment increases near T = 0.6TC. This is be-

~~Journal of the Magnetics Society of Japan Vol. 30, No. 1, 2006 27~~

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. /0.21 mixing and spin-flip scattering correctly into account, we presume that these effects are not too strong to make the Fig. 45 Temperature dependence of (a) the conduc- MR ratio vanish far below TC. The origin of the vanish- tance and (b) the MR ratio calculated by including ing TMR far below TC in experiments might be a change spin-mixing effect. In (a), conductances ΓP and ΓAP for in the electronic states and magnetism at the interface. We K = 2.0t are shown by ◦ and , respectively. In (b), MR believe that the MR ratio remains up to TC when the carrier ratio obtained for K = 1.0t, 1.5t, and 2.0t are shown by concentration is kept constant in LSMO by the interface ◦, , and 4, respectively. engineering. cause the current can flow even in the half-metallic state 8. Spin-flip tunneling due to the spin-mixing. The result that ΓAP is unchanged at The interaction between the tunneling electrons with low temperatures, 0 < T ≤ 0.2TC, is due to the fact that localized spins gives rise to spin-flip tunneling, which z 22,128) probability for S i to be ±1/2 is very small. At high tem- is known to reduce the magnitude of TMR. Spin- peratures, 0.8TC ≤ T < TC, ΓAP is also unchanged because flip tunneling has so far been studied for systems in LSMO is not half-metallic anymore and both + and − spin which the localized spins are located at or near the in- electrons contribute the conductance. The effect of spin- terfaces,20,129–132) or paramagnetic impurities.133) There is mixing is also small for K = 1.0t and 1.5t because of the another situation in which the tunneling electron inter- same reason. Figure 45(b) show the MR ratio calculated acts with many localized spins. An example may be grain from the results of the conductance shown in Fig. 45(a). boundary tunneling in Mn oxides. In this case, the extent When K = 2.0t, the MR ratio decreases strongly at tem- to which the spin-flip tunneling influences the TMR may peratures T ∼ 0.6TC because of the spin-mixing. This is still be nontrivial. because the half-metallicity is lost near this temperature We deal herein with the interaction between a tunneling region and the ΓAP increases. When K = 1.0t and 1.5t, the electron and quantum spins of finite number in the insulat- change in the MR ratio is small because both + and − spin ing barrier, and study the effects of the spin-configurations electrons contribute to the conductance. and spin-flip tunneling on TMR. The linear-response theory is adopted to calculate the tunnel conductance at a 7.5 Discussion and summary zero-bias limit in a one-dimensional model. Because the We have shown that the MR ratio have finite values near localized spins themselves interact with each other, one TC even if we introduce the effects of scattering by spin should basically treat a many-body problem to calculate fluctuation, spin mixing and spin flip scattering. Our re- the tunnel conductance. We will discuss the relevance of sults disagree with the experimental fact that the MR ra- the results to the experimental results of grain boundary tio becomes zero at lower temperature than TC of the bulk TMR as well as the TMR of manganite MTJ. sample. The disagreement might be attributed to charge transfer at the interface of the junctions127) which causes 8.1 Model a change in the electronic states and/or magnetism at the We adopt herein a one-dimensional TB model for MTJs interface. Although we have taken into account the effect consisting of two electrodes of ferromagnetic metal sepa- of spin-flip scattering at only one layer by the interface, the rated by an insulating barrier. Localized spins exist within effect might be small as discussed in the next section. the barrier and interact with the tunneling electrons. A Summarizing the results, it has been shown that the ef- schematic figure of the system is presented in Fig. 46. fects of the electron scattering due to the spin fluctuation The Hamiltonian is given by on TMR are small, MR ratio survives near T , and the tem- C H = H + H + H + H . (83) perature dependence of the MR ratio is small at low tem- t v K J peratures. Although we have not taken the effects of spin- Ht represents electron hopping between nearest-neighbor

~~28 Journal of the Magnetics Society of Japan Vol. 30, No. 1, 2006 sites and is given as, X † Ht = ti jciσc jσ, (84)~~

~~(i, j),σ~~

© ¥ ./' -)0%- / ©!+,'!- "¢#¤$¦%¨& ' (¨)*+,'- where σ = + or −. The second term, ¢¡¤£¦¥¨§ © ¨ © X † Hv = viσciσciσ, (85) i,σ Fig. 47 An example of basis sets of spin configuration. Small solid arrow and large open arrow indi- gives site- and spin-dependent energy levels viσ, which also represent the exchange-splitting of the ferromagnetic elec- cate the itinerant electron spin and localized spin, respectively. Each box denotes the spin configuration of trodes. HK and HJ indicate the exchange interaction be- z tween tunneling electrons and localized spins, and between S tot = +1/2 generated from two localized spins and one itinerant spin. localized spins within the insulating barrier, respectively. They can be written as X HK = −K Si · si, (86) i spins is shown in Fig. 47, where there are six states gen- and erated from an antiferromagnetic alignment of two local- X X ized spins. Because the number of basis functions thus z ex HJ = J Si · S j − µB S i h . (87) constructed increases rapidly with increases in the num- hi, ji i ber of localized spins, we deal with only eight localized Here, S and s are the localized spin and the spin of the spins with S = 1/2 each in the present work. The tunnel tunneling electron, respectively, K and J indicate the cou- conductance at the zero-bias limit is calculated by taking pling between localized spins and the tunneling electron the intermediate states thus constructed into consideration. and between localized spins, respectively, and hex is the The following parameter values have been used in units external magnetic field acting on the localized spins. The of t; K = 2 (ferromagnetic interaction), J = 0.1 (antiferro- g-factor is taken to be 2.0. The external magnetic field is magnetic interaction), and u = 2.0 for the barrier potential. introduced only to control the spin configurations of the The specific choice of the sign of K and J has been made localized spins. The Zeeman effect on the itinerant elec- by considering the grain boundary tunneling in mangan- trons is neglected. We also assume that AP alignment of ites. The exchange field acting on the nearest-neighbor lo- the magnetization of the ferromagnetic electrodes can be calized spins from the ferromagnetic electrodes is assumed altered to a P alignment by an infinitesimally small mag- to be negligibly small in the present case. The spin config- netic field. uration of the localized spins in the ground state is determined by diagonalizing HJ for a given external magnetic 8.2 Method of calculation field hex. In order to clarify the contribution of the spin- We now deal with the full Hamiltonian; a uniform hop- flip process to the tunnel conductance, we have separately ping t between the nearest-neighbor sites is assumed for treated the constant K for the longitudinal (Kz) and trans- Ht, and Hv includes uniform exchange-splitting ∓∆ex for ± verse (K+−) components of the exchange interaction. spin electrons in the ferromagnetic electrodes and a spin- independent potential u in the barrier region. 8.3 Calculated results In order to treat HK and HJ, we prepare the following First, we treat systems with paramagnetic electrodes many-body basis function for the TB basis set; where the Fermi level EF is taken to be −1.0. Figure 48 † shows the calculated results of the spin-asymmetry ratio Ψiσµ = ciσ|µi, (88) (Γ+ − Γ−)/(Γ+ + Γ−) of the tunnel conductance, where Γσ where i and |µi indicate the electron position and spin con- is the σ component of the tunnel conductance Γ. The in- figurations of the localized spins in the barrier, respec- set shows the dependence of Γσ on K+− for a fixed value tively. When site i is on the electrodes, |µi represents the of Kz = 2.0. With increases in the value of K+−, the tun- ground state of the localized spins determined by HJ, while nel conductance increases, because the lowest energy level there are many intermediate states when the electron is in- for the tunnel electron interacting with the localized spins side the barrier region because the electron creates new decreases due to the transverse terms of the exchange inter- states via the exchange interaction. The new states can action. The ratio (Γ+ − Γ−)/(Γ+ + Γ−) plotted as a function ex z be generated by operating H to the ground state of the of the external magnetic field h (or S tot) increases with z z z localized spins, keeping s + S tot constant where S tot is increasing magnetic field. This result is quite reasonable, the z-component of the total spin angular momentum of as the + spin electrons become more transparent due to the the localized spins.134) All of these new states are taken ferromagnetic exchange interaction with increasing align- into consideration to calculate the tunnel conductance. An ment of the localized spins. We see that the spin-flip tun- example of the basis set for a barrier with two localized neling, i.e., the contribution from the K+− term, decreases

~~Journal of the Magnetics Society of Japan Vol. 30, No. 1, 2006 29~~

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F G Fig. 49 Calculated results of the MR ratio as functions of the Fermi energy for S z = 0 and 1 with Fig. 48 Calculated results of the spin-asymmetry ra- F tot K = 0.0 tio of tunnel conductance as functions of the external +− z

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O +-,P-Q change splitting, the bottom of the minority spin density of * states is −1.0. When hex = 0, the localized spins align anti- z ferromagnetically so that S tot = 0. By changing the magne- Fig. 50 Calculated results of the MR ratio as func- z tization direction of the ferromagnetic electrodes, keeping tions of the external field or S tot for EF = −0.99 (circles) the antiferromagnetic spin alignment of the localized spins and −0.5 (triangles) with K+− = 0 (closed symbols) and +− unchanged, the MR ratio can be calculated. With further K = 2.0 (open symbols). ex z increases in h , S tot increases. The MR ratio at hex = 0 is dependent on the position of the Fermi level EF. Figure 49 shows the dependence of tunneling decreases the ratio by only ∼ 10%. z the MR ratio on EF for S tot = 0.0 and 1.0 with K+− = 0.0. z The solid curve for S tot = 0.0 corresponds to a tunneling 8.4 Discussion and summary through the usual potential barrier because no spin-filtering In Fig. 49, we have shown that large MR ratio is ob- effect exists in this case, and the result is quite similar to tained when the Fermi level is close to the band bot- that obtained previously in the free-electron model.37) For tom. Since the ferromagnetic manganites are nearly half- 35, 112, 113, 135) EF . −0.8, Γ++ > Γ−− is satisfied. Especially EF ∼ −1.0, metallic with a spin polarization of 0.7 ∼ 0.9, Γ++ is much higher than Γ−−, resulting in a quite high MR the position of the Fermi level should be close to the band ratio. As EF is raised, Γ−− becomes higher and the MR ratio bottom. Therefore, large MR ratio is expected in man- becomes zero around EF ∼ −0.8. For EF > −0.8, we get ganite grain boundary junctions. Actually, large MR ratio z Γ++ < Γ−−. When S tot = 1.0, the + spin electrons becomes has been observed in grain boundary junctions by Philipp 136) more transparent, and Γ++ therefore increases. Therefore, et al. z the MR ratio at S tot = 1 is large for EF . −0.7, while it Furthermore, the approximately linear increase in the is small for EF & −0.7 as compared with the MR ratio at MR ratio with increasing magnetic field shown in Fig. 50 is z 109, 137) S tot = 0. in qualitative agreement with experimental results. Figure 50 shows the MR ratio with and without spin- Lee et al.137) have attributed the dependence of grain ex z ex flip tunneling as a function of h (or S tot) for EF = −0.99 boundary TMR on h at high magnetic fields to a second- and −0.5. The MR ratio increases monotonically with in- order process, pointing out that a hcos2θi term appears in ex creasing h when EF = −0.99, while it becomes zero once the expression of the tunnel conductance. z around S tot = 1 for EF = −0.5, as the value Γ++ being Finally, we would like to remark on the method we have smaller than Γ−− is compensated for due to the exchange adopted. As the exchange interaction in the Hamiltonian interaction, as mentioned above. We see that the spin-flip is of a many-body interaction, our method can be applied

~~30 Journal of the Magnetics Society of Japan Vol. 30, No. 1, 2006 only for the low-carrier (tunneling electron) density limit for a ﬁnite number of localized spins. Because we have~~

dealt with a zero-bias limit, we assumed that the ﬁnal states are the same with the initial ones in the calculation of the Trace in the expression of the conductance. Despite these limitations, we believe that the present calculation may

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¨ © from those considered by Appelbaum and Anderson ¨ where the electrodes are paramagnetic. In summary, we have studied TMR in MTJs in which the tunneling electrons interact with localized quantum spins in the barrier by extending the treatment of tunnel con- Fig. 51 Schematic figures of (a) tunnel junction, (b) ductance to take into account the many-body effects of the randomness, (c) and electronic structures. exchange interaction. We found that the MR ratio is decreased by approximately 10% due to the spin-flip process caused by the exchange interaction. A unified theory to treat the exchange interaction between the tunnel- the product of the surface DOSs of metals separated by the ing electrons and localized spins in larger regions remains spacer. to be studied in future. Our theory has been restricted to zero-bias limit and therefore any relevance to tunnel spec- 9.1 Model troscopy for spin excitations etc. may be given in a future We consider a trilayer consisting of two semi-infinite work. metallic electrodes separated by a spacer of L atomic layers. This trilayer is described by the single band TB model 9. Tunneling in strong randomness on a simple cubic lattice and (001) axis is taken for stacking direction. In order to take into account the random- In this section, we concentrate on the conductance in a ness and band gap, we assume following substitutional- model including strong randomness keeping the band gap type randomness in the spacer. The Hamiltonian of the sys- 23,140) open in the spacer. In real systems, however, the band tem is gap does not open by randomness but rather by s-p-d mix- X † Hˆ = − t c c 0 0 + H.c. ing, ordering of spins/atoms, or electron interaction. Here rk,`,σ rk,` ,σ h(r ,`),(r0 ,`0)i,σ we consider a model with strong Coulomb interaction be- k k tween itinerant electrons,141) and study the tunnel conduc- X + u c† c , (89) tance and magnetoresistance in junctions in which the bar- rk,`,σ rk,`,σ rk,`,σ rk,`,σ rier is made of a material with strong electron interaction. where c(†) is annihilation (creation) operator of electron It has been known that the electronic state of the Hubbard rk,`,σ model, which includes strong Coulomb repulsion, may ap- at site rk in `-th plane with spin σ. Here, ` labels the proximately be described in an alloy analogy model. In this layer plane normal to the (001) axis and rk denotes the model, the strong Coulomb interaction between electrons site within the layer plane. In Eq. (89), t is transfer inte- 0 0 is replaced with random potentials. gral and the summation h(rk, `), (rk, ` )i runs over nearest- neighbor sites. The on-site potential u at site r in `-th We use scattering theory to treat the wave function cor- rk,`,σ k rectly and clarify the important factors which govern the plane takes constant value u0 in both left (` ≤ 0) and right tunnel conductance through the disordered spacer. Linear (` ≥ L + 1) electrodes while it takes u or −u randomly with response theory, i.e., Kubo formula and CPA are used to equal probability in the spacer (1 ≤ ` ≤ L). We take the calculate the conductance at zero temperature and at zero random potential u large to open the gap in the band of the bias limit. One of the advantages of the CPA is that the spacer. The model of the junctions and electronic structure band gap of the spacer can be reproduced for large random are shown in Fig. 51. potentials. Because there is no translational invariance in the current direction, the vertex correction to the conduc- 9.2 Results tance does not disappear. The vertex correction is calcu- First, we show the calculated results of the DOSs of lated by the approximation consistent with the CPA in or- spacer and electrode in Figs. 52(a) and (b), respectively. der to satisfy the current conservation. Here, the chemical potential (Fermi energy) is chosen at It will be shown that the conductance is proportional to EF = 0. In Fig. 51(c), the gap appears around the chemical

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~~32 Journal of the Magnetics Society of Japan Vol. 30, No. 1, 2006 &)(~~

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, - . might be finite bias drop at the insulating barrier and in- coherent tunneling described in terms of DOS. The latter, however, may not be the characteristics of the tunneling Fig. 55 Calculated results of the MR ratio for several any more because the recent measurement of TMR in epi- values of random potential u. Chained, dotted and solid taxial MTJs has evidenced the existence of coherent tun- curves are MR ratio for u/t = 2, 8 and 10. Broken curve neling. Although the finite bias drop at the barrier is im- show the evaluated value of MR from Eq. (6) with the portant for technical applications, basic feature of the TMR spin polarization of surface DOS instead of that of the may be captured for the zero bias limit. Under a finite bias, DOS of ferromagnetic electrodes. the tunneling electrons should dissipate their energy after tunneling. The time scale of the energy dissipation, however, is not necessarily the same with that of transmission by ΓP = Γ++ + Γ−− by using the two current model. Simi- through the barrier. The tunneling conductance itself might larly, the conductance ΓAP in the anti-parallel alignment is be governed by the energy conserved transmission. This is given by ΓAP = Γ+− + Γ−+. the reason why the linear response theory has been adopted Calculated results of the MR ratio as functions of E are in our work. shown in Fig. 55 for several values of u. As u increases, The present method, however, is easily extended to tun- 2 2 MR ratio becomes close to 2PSF/(1 + PSF) where PSF is neling under finite bias. In this case, the shape of the en- the spin polarization of the surface DOS of the metallic ergy dispersion relations and DOS near the Fermi level + − + − electrodes defined as PSF ≡ (DSF − DSF)/(DSF + DSF). of the insulator and electrodes materials might be decisive Results obtained in strong disordered limit are similar factors for the TMR. This does not necessarily mean that to those obtained by using the tunnel Hamiltonian theory, the coupling of the tunneling electrons with e.g. spin wave but the conductance is proportional to the product of the excitations can be ignored. Actually, bias dependence of surface DOSs of metallic electrodes. As for the TMR, P the MR ratio has been well explained by introducing spin- in Eq. (5) should be interpreted as the spin polarization of wave excitations by tunneling electrons.41) The coupling the surface DOS of the ferromagnetic metals for strongly of tunneling electrons with phonon excitations may be im- disordered junctions. portant as well.60) Observed results of the bias dependence of the TMR is usually dependent on the current direction. 9.3 Discussion and summary This may be due to the difference between the crystalline In the linear response theory, the conductance at zero structures at the left and right interfaces and/or electrodes. temperature is expressed as17, 142) The barrier height estimated from the I-V relation and

2 X 0 0 from the dependence of the tunnel conductance on the e kk,k k 4 k 2 kk k 26) Γ = 4t h|G1,L | i Im fL Im fR (90) barrier thickness is rather low as compared with that h 0 kk,kk predicted in the first principles band calculations. Atom- 0 istic defects such as oxygen defects may be responsible to kk,kk where G1,L is the interlayer Green’s function between 1- the low barrier height. These defects may produce in-gap st and L-th planes and fL(R) is the surface Green’s function states and therefore the effective barrier height may be re- of the left (right) electrode. Here, kk = (kx, ky) is the el- duced. The position of the energy levels of the in-gap states ement of the wave vector parallel to the layer planes and may dependent on barrier materials and therefore might be the bracket h· · · i denotes the statistical average due to the ffi k ,k0 a decisive factor of the e ciency of the tunnel barriers. 0 k k 2 randomness. If the kk, kk dependence of h|G1,L | i is ne- In the present article, we have dealt MTJs with sim- 0 glected, the summations over kk and kk run independently, ple structures. Interesting results of TMR have been re- then, the conductance is proportional to the product of the ported for MTJs with double barriers junctions, both lay- surface DOSs of the electrodes. In our results, Γ is propor- ered structures and quantum dots structures.143–149) In the 2 tional to DSF when u is large. Then, it is considered that latter systems, the spin accumulation under finite bias plays k ,k0 k k 2 0 a dominant role on the TMR.150, 151) It is noted that the h|G1,L | i in Eq. (90) becomes independent of kk and kk in strong disordered limit. TMR is realized even for paramagnetic dots so far as the In conclusion, the conductance through the strong disor- electrodes are ferromagnetic. It is also shown that the MR dered spacer is proportional to the products of the surface ratio oscillate as a function of bias voltage which is related

Journal of the Magnetics Society of Japan Vol. 30, No. 1, 2006 33 to Coulomb blockade and spin accumulation in the quan- One of the important progress in the research of TMR re- tum dots.144, 147) sides in that a quantitative comparison between theory and MTJs made of ferromagnetic semiconductors experiment has become possible after fabrication of epi- (GaMn)As also show high MR ratio, however, the taxial junctions and construction of theories of TMR in- appearance of the TMR is restricted at low temperature at cluding the effect of randomness. present, because of low Curie temperature of (GaMn)As. MR of MTJs with semimetals and of those with man- Because ferromagnetic semiconductors can be grown ganites has been studied to clarify the effect of their elec- epitaxially and is easy for microscopic processing, room tronic structure and interaction on the MR. In the former temperature ferromagnetic semiconductors will open a case, the control of the energy gap of the semimetal may way to realize novel spintronics devices. provide useful features of MR for technical applications. Heusler alloys102, 152, 153) are another example of mate- In the latter case, it has been shown that the so-called dou- rials in which the half metallicity is predicted in the first ble exchange interaction between itinerant electrons and principles calculation. Because of the reason, MTJs with localized spins is not sufficient to explain the temperature Heusler alloys have been made to get higher MR ratios. Re- dependence of the TMR. Effects of spin mixing and spin- cently rather high MR ratios, 30 ∼ 40% have been realized flip tunneling have been studied, however, more study may at room temperature. Improvement of the sample quality be desired to clarify the TMR in manganite junctions. may provide higher MR ratios in these MTJs.155,156) Other Finally, we have presented numerical results of TMR for half metallic oxides materials have been used to MTJs, MTJs with strong randomness. In this case, the MR ratio however, the MR ratio is rather low at room temperature can be expressed in terms of density of states at the sur- at present. faces of the electrodes. The strong randomness can be a High MR ratios as much as 200% observed in epitaxial model of insulators realized by strong Coulomb interac- Fe/MgO/Fe and CoFe/MgO/CoFe MTJs is a breakthrough tion. It should be examined by using more sophisticated to realize magnetic random access memories (MRAM). methods whether the present results hold for MTJs with Progress in the fundamental physics on current induced strong Coulomb interaction. magnetization switching (CIMS)157–161) is a promising way to fabricate smaller and faster MRAMs that consume Acknowledgements Authors acknowledge the financial less power and operate with higher reliability. Because of supports of Grants-in-Aid for Scientific Research (C) and the high sensitivity, that is high MR ratio in MTJs as com- for Scientific Research in Priority Areas “Semiconduc- pared with CPP-GMR junctions, realization of CIMS in tor Nanospintronics” of MEXT, CREST Project “Func- MTJs is highly desired. Although there appeared some re- 162–164) tion Evolution of Materials and Devices based on Electron/ ports on CIMS in MTJs, the current density to pro- Photon Related Phenomena” (Y. Suzuki Team) of JST, the ff duce magnetization switching is still high. Further e orts International Joint Research Program “Nano-scale Magne- to reduce the current density may be desired. toelectronics” of NEDO, “Grid Application Research in Nanoscience” of NAREGI, and Nagoya University 21st 11. Summary Century COE Program “Frontiers of Computational Sci- ence”. Authors also express sincere thanks to collabora- We present a review of theories of TMR putting an em- tors; P. Burno, S. Honda, M. Ikeda, T. Kondo, T. Kumazaki, phasis on the role of electron scattering due to randomness S. Maekawa, J. Mathon, Y. Naito, N. Nishimura, T. Oh- on TMR. In the ballistic limit, the tunneling electrons pre- sawa, J. Ozeki, A. Shibata, A. Umerski, and A. Watanabe. serve the coherency of the wave functions, and momen- Authors wish to thank computer centers of NUCC and IMS tum conservation parallel to the planes holds. In this case, NanoGrid. the important concept is the momentum filtering effect in which electrons with wave vector kk = (0, 0) transmit most frequently through the tunnel barrier so long as the state References kk = (0, 0) exists in the propagating states. Randomness relaxes the restriction of the momentum conservation, and 1) Spin dependent transport in magnetic nanostructures, Eds. therefore number of tunneling paths (channel number) in- S. Maekawa and T. Shinjo, (Taylor and Francis, New York, creases resulting in an increase in the tunnel conductance. 2002). The TMR is thus affected strongly by the presence of ran- 2) S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. domness in MTJs. We have demonstrated how the tunnel Daughton, S. von Molnar, M. L. Roukes, A. Y. Chtchelka- conductance and TMR are influenced by the randomness nova, and D. M. Treger: Science, 294, 1488 (2001). and the shape of the Fermi surface of the metallic elec- 3) H. Ohno, A. Shen, F. Matsukura, A. Oiwa, A. Endo, S. Kat- sumoto, and Y. Iye: Appl. Phys. Lett., 69, 363 (1996). trodes. 4) H. Ohno: Science, 281, 951 (1998). The theory has been applied to epitaxial tunnel junctions 5) H. Munekata, H. Ohno, S. von Molnar, Armin Segmuller, Fe/MgO/Fe, and shown that the concepts studied in the L. L. Chang, and L. Esaki: Phys. Rev. Lett., 63, 1849 simple model are also applicable to the realistic junctions. (1989).

34 Journal of the Magnetics Society of Japan Vol. 30, No. 1, 2006 6) M. N. Baibich, J. M. Broto, A. Fert, Nguyen Van Dau, F. ence, 282, 85 (1998). Petroﬀ, P.Etienne, G. Creuzet, A. Friederich, and J. Chaze- 36) W. A. Harrison: Phys. Rev., 123, 85 (1961). las: Phys. Rev. Lett., 61, 2472 (1988). 37) J. C. Slonczewski: Phys. Rev. B, 39, 6995 (1989). 7) For a review on GMR, see P. M. Levy: Solid State Phys., 38) H. Itoh and J. Inoue: J. Magn. Soc. Jpn., 24, 583 (2000), 47, 367, Eds. H. Ehrenreich and D. Turnbull, (Academic (in Japanese). Press, San Diego, 1994); M. A. M. Gijs and G. E. W.Bauer: 39) A. M. Bratkovsky: Phys. Rev. B, 56, 2344 (1997). Adv. Phys., 46, 285 (1997); E. Y. Tsymbal and D. G. Petti- 40) X. Zhang, B.-Z. Li, G. Sun, and F.-C. Pu: Phys. Rev. B, 56, for: Solid State Phys., 56, 113, Eds. H. Ehrenreich and F. 5484 (1997). Spaepen, (Academic Press, San Diego, 2001). 41) S. Zhang, P. M. Levy, A. C. Marley, and S. S. P. Parkin: 8) T. Miyazaki and N. Tezuka: J. Magn. Magn. Mat., 139, Phys. Rev. Lett., 79, 3744 (1997). L231 (1955). 42) T. Kishi and K. Inomata: J. Magn. Soc. Jpn., 22, 1150 9) J. S. Moodera, L. R. Kinder, T. M. Wong, and R. Meservey: (1998), (in Japanese). Phys. Rev. Lett., 74, 3273 (1995). 43) Y.Utsumi, Y.Shimizu, and H. Miyazaki: J. Phys. Soc. Jpn., 10) For a review on TMR, see J. S. Moodera and G. Mathon: 68, 3444 (1999). J. Magn. Magn. Mat., 200, 248 (1999); E. Tsymbal, O. 44) H. Itoh and J. Inoue: J. Magn. Magn. Mat., 226-230, 930 N. Mryasov, and P. R. LeClarir: J. Phys.: Cond. Mat., 15, (2001). R109 (2003). 45) H. Itoh and J. Inoue: Surf. Sci., 493, 748 (2001). 11) R. Kubo: J. Phys. Soc. Jpn., 17, 975 (1957). 46) J. Inoue and H. Itoh: J. Phys. D: Appl. Phys., 35, 2432 12) R. Landauer: IBM J. Res. & Dev., 1, 223 (1957). (2002). 13) S. Datta: Electronic transoirt in mesoscopic systems, 47) H. Itoh, Y. Naito, and J. Inoue: J. Magn. Soc. Jpn., 24, 587 (Cambridge University Press, Cambridge, 1995). (2000), (in Japanese). 14) P. Soven: Phys. Rev., 156, 809 (1967). 48) S. Nonoyama, H. Itoh, A. Oguri, J. Inoue, and P. Bruno: J. 15) B. Velicky: Phys. Rev., 184, 614 (1969). Magn. Soc. Jpn., 23, 61 (1999). 16) H. Itoh, T. Kumazaki, J. Inoue, and S. Maekawa: Jpn. J. 49) S. Nonoyama, H. Itoh, A. Oguri, J. Inoue, and P. Bruno: J. Appl. Phys., 37, 5554 (1998). Magn. Magn. Mat., 198-199, 158 (1999). 17) H. Itoh, A. Shibata, T. Kumazaki, J. Inoue, and S. 50) P. Lee and D. S. Fisher: Phys. Rev. Lett., 47, 882 (1981). Maekawa: J. Phys. Soc. Jpn., 68, 1632 (1999). 51) G. D. Mahan: Many-Particle Physics (Kluwer Aca- 18) H. Itoh, J. Inoue, A. Umerski, and J. Mathon: Phys. Rev. B, demic/Plenum Publishers, New York, 1981). 68, 174421 (2003). 52) H. U. Baranger and A. D. Stone: Phys. Rev. B, 40, 8169 19) A. Watanabe, H. Itoh, and J. Inoue: Jpn. J. Appl. Phys., 43, (1989). 540 (2004). 53) C. L. Kane, A. Serota, and P.A. Lee: Phys. Rev. B, 37, 6701 20) H. Itoh, T. Ohsawa, and J. Inoue: Phys. Rev. Lett., 84, 2501 (1988). (2000). 54) S. Maekawa, J. Inoue, and H. Itoh: J. Appl. Phys., 79, 4730 21) J. Ozeki, T. Ohsawa, H. Itoh, and J. Inoue: J. Magn. Soc. (1996). Jpn., 29, 286 (2005), (in Japanese). 55) H. Itoh, J. Inoue, and S. Maekawa: Phys. Rev. B, 51, 342 22) J. Inoue, N. Nishimura, and H. Itoh: Phys. Rev. B, 65, (1995). 104433 (2002). 56) D. Kalkstein and P. Soven: Surf. Sci., 26, 85 (1971). 23) H. Itoh, J. Inoue, S. Maekawa, and P. Bruno: J. Magn. 57) T. Ando: Phys. Rev. B, 44, 8017 (1991). Magn. Mat., 198-199, 545 (1999). 58) A. Umerski: Phys. Rev. B, 55, 5266 (1997). 24) S. Yuasa, A. Fukushima, T. Nagahama, K. Ando, and Y. 59) K. Levin, B. Velicky, and H. Ehrenreich: Phys. Rev. B, 2, Suzuki: Jpn. J. Appl. Phys., 43, L588 (2004) 1771 (1979). 25) S. S. Parkin, C. Kaiser, A. Panchula, P. R. Rice, B. Hughes, 60) C. B. Duke: Solid State Phys. Suppl., 10, Eds. F. Seits, D. M. Samant, and S. H. Yang: Nature Mat., 3, 862 (2004). Turnbull, and H. Ehrenreich, (Academic Press, New York 26) S. Yuasa, T. Nagahama, A. Fukushima, Y. Suzuki, and K. and London, 1969). Ando: Nature Mat., 3, 868 (2004). 61) W.J. Gallagher, S. S. P.Parkin, Y.Lu, X. P.Bian, A. Marley, 27) M. Julliere: Phys. Lett., 54A, 225 (1975). K. P. Roche, R. A. Altman, S. A. Rishton, C. Jahnes, T. M. 28) S. Maekawa and U. Gafvert:¨ IEEE Trans. Magn., IMAG- Shaw, and G. Xiao: J. Appl. Phys., 81, 3741 (1997). 18, 707 (1982). 62) S. Yuasa, T. Nagahama, and Y. Suzuki: Science, 297, 234 29) Y. Suezawa and Y. Gondo:¯ Proc. Int. Symp. Physics of (2002). Magnetic Materials, Sendai, 303 (1987); Y. Suezawa, F. 63) J. E. Ortega and F. J. Himpsel: Phys. Rev. Lett., 69, 844 Takahashi, and Y. Gondo:¯ Jpn. J. Appl. Phys., 31, L1415 (1992). (1992). 64) A. Vedyayev, N. Ryzhanova, C. Lacroix, L. Giacomoni, 30) T. Miyazaki, T. Yaoi, and S. Ishio: J. Magn. Magn. Mat., and B. Dieny: Europhys. Lett., 39, 219 (1997). 98, L7 (1991). 65) J. Mathon, M. Villeret, and H. Itoh: Phys. Rev. B, 52, R6983 31) J. Nowak and J. Rauluszkiewicz: J. Magn. Magn. Mat., (1995). 109, 79 (1992). 66) J. Mathon, A. Umerski, and M. Villeret: Phys. Rev. B, 55, 32) J. Bardeen: Phys. Rev. Lett., 6, 57 (1961). 14378 (1997). 33) R. Meservy and P.M. Tedrow: Phys. Rep., 283, 173 (1994). 67) J. Mathon and A. Umerski: Phys. Rev. B, 60, 1117 (1999). 34) S. K. Upadhyay, A. Palanisami, R. N. Louie, and R. A. 68) H. Itoh, J. Inoue, A. Umerski，and J. Mathon: J. Magn. Buhrman: Phys. Rev. Lett., 81, 3247 (1998). Soc. Jpn., 27, 133 (2003), (in Japanese). 35) R. J. Soulen Jr., J. M. Byers, M. S. Osofsky, B. Nadgorny, 69) H. Itoh and J. Inoue: J. Magn. Soc. Jpn., 27, 1013 (2003), T. Ambrose, S. F. Cheng, P. R. Broussard, C. T. Tanaka, J. (in Japanese). Nowak, J. S. Moodera, A. Barry, and J. M. D. Coey: Sci- 70) H. Itoh, J. Inoue, A. Umerski, J. Mathon: J. Magn. Magn.

Journal of the Magnetics Society of Japan Vol. 30, No. 1, 2006 35 Mat., 272-276, e1467 (2004). Phys. Rev. B, 59, 5470 (1999). 71) M. Bowen, V.Cros, F. Petroﬀ, A. Fert, C. M. Boubeta, J. L. 101) M. Shirai: unpublished. Costa-Kramer,¨ J. V. Anguita, A. Cebollada, F. Briones, J. 102) R. A. de Groot, F. M. Mueller, P. G. van Engen, and K. H. M. de Teresa, L. Morellon,´ M. R. Ibarra, F. Guell,¨ F. Peiro,´ J. Buschow: Phys. Rev. Lett., 50, 2024 (1983). and A. Cornet: Appl. Phys. Lett., 79, 1655 (2001). 103) K. Schwarz: J. Phys. F: Metal Phys., 16, L211 (1986). 72) S. Mitani, T. Moriyama, and K. Takanashi: J. Appl. Phys., 104) S. Satpathy, Z. S. Popovic, and F. R. Vukajlovic: Phys. Rev. 93, 8041 (2003). Lett., 76, 960 (1996). 73) J. F. Vincent, C. Tiusan, E. Jouguelet, F. Canet, M. Sajied- 105) Yu Lu, X. W.Li, G. Q. Gong, G. Xiao, A. Gupta, P.Lecoeur, dine, C. Bellouard, E. Popova, M. Hehn, F. Montaigne, and J. Z. Sun, Y. Y. Wang, and V. P. Dravid: Phys. Rev. B, 54, A. Schuhl: Appl. Phys. Lett., 82, 4507 (2003). R8357 (1996). 74) W. H. Butler, X.-G. Zhang, T. C. Schulthess, and J. M. Ma- 106) M. Viret, M. Drouet, J. Nassar, J. P. Contour, C. Fermon, cLaren: Phys. Rev. B, 63, 054416 (2001). and A. Fert: Europhys. Lett., 39, 545 (1997). 75) J. Mathon and A. Umerski: Phys. Rev. B, 63, 220403 107) M. Bowen, M. Bibes, A. Barthel´ emy,´ J.-P. Contour, A. (2001). Anane, Y. Lemaˆıre, and A. Fert: Appl. Phys. Lett., 82, 233 76) D. A. Papaconstantopoulos: Handbook of the Band Struc- (2003). ture of Elemental Solids, (Plenum, New York, 1986). 108) J.-H. Park, E. Vescovo, H.-J. Kim, C. Kwon, R. Ramesh, 77) V. -C. Lee and H. -S. Wong: J. Phys. Soc. Jpn., 45, 895 and T. Venkatesan: Phys. Rev. Lett., 81, 1953 (1998). (1978). 109) H. Y. Hwang, S-W. Cheong, N. P. Ong, and B. Batlogg: 78) W. Wulfhekel, M. Klaua, D. Ullmann, F. Zavaliche, J. Phys. Rev. Lett., 77, 2041 (1996). Kirschner, R. Urban, T. Monchesky, and B. Heinrich: Appl. 110) J. Y. T. Wei, N.-C. Yeh, and R. P. Vasquez: Phys. Rev. Lett., Phys. Lett., 78, 509 (2001). 79, 5150 (1997). 79) T. Urano and T. Kanaji: J. Phys. Soc. Jpn., 57, 3403 (1988). 111) J.-H. Park, E. Vescovo, H.-J. Kim, C. Kwon, R. Ramesh, 80) C. Li and A. J. Freeman: Phys. Rev. B, 43, 780 (1991). and T. Venkatesan: Nature, 392, 794 (1998). 81) H. Itoh, J. Ozeki, and J. Inoue: J. Magn. Magn. Mat., in 112) B. Nadgorny, I. I. Mazin, M. Osofsky, R. J. Soulen, Jr., press. P. Broussard, R. M. Stroud, D. J. Singh, V. G. Harris, A. 82) W. Harrison: Electronic structure and the properties of Arsenov, and Ya. Mukovskii: Phys. Rev. B, 63, 184433 solids, (W. H. Freeman and Company, San Francisco, (2001). 1980). 113) G. T. Woods, R. J. Soulen, Jr., I. Mazin, B. Nadgorny, M. 83) H. L. Meyerheim, R. Popescu, J. Kirschner, N. Jedrecy, M. S. Osofsky, J. Sanders, H. Srikanth, W. F. Egelhoﬀ, and R. Sauvage-Simkin, B. Heinrich, and R. Pinchaux: Phys. Rev. Datla: Phys. Rev. B, 70, 054416 (2004). Lett., 87, 076102 (2001). 114) C. Zener: Phys. Rev., 82, 403 (1951). 84) X.-G. Zhang, W. H. Butler, and A. Bandyopadhyay: Phys. 115) P.-G. de Gennes: Phys. Rev., 118, (1960) 141. Rev. B, 68, 092402 (2003). 116) J. Inoue and S. Maekawa: Phys. Rev. Lett., 74, 3407 (1995). 85) A. Gibson, R. Haydock, and J. P. LaFemina: Phys. Rev. B, 117) S. Ishihara, J. Inoue, and S. Maekawa: Phys. Rev. B, 55, 50, 2582 (1994). 8280 (1997). 86) P. H. Dederichs, Ph. Mavropoulos, O. Wunnicke, N. Pa- 118) A. J. Millis, P.B. Littlewood, and B. I. Shraiman: Phys. Rev. panikolaou, V. Bellini, R. Zeller, V. Drchal, and J. Ku- Lett., 74, 5144 (1995). drnovsky:´ J. Magn. Magn. Mat., 240, 108 (2002). 119) A. Moreo, S. Yunoki, and E. Dagotto: Science, 283, 2034 87) S. Yuasa, T. Sato, E. Tamura, Y. Suzuki, H. Yamamori, K. (1999). Ando, and T. Katayama: Europhys. Lett., 52, 344 (2000). 120) K. Kubo: J. Phys. Soc. Jpn., 36, 32 (1974), see also K. Kubo 88) M. Tanaka and Y. Higo: Phys. Rev. Lett., 87, 026602 and N. Ohata: J. Phys. Soc. Jpn., 33, 21 (1972). (2001). 121) M. Takahashi and K. Mitsui: Phys. Rev. B, 54, 11298 89) A. Hasegawa: J. Phys. Soc. Jpn., 54, 677 (1985). (1996). 90) A. G. Petukhov, W. R. L. Lambrecht, and B. Segall: Phys. 122) J. S. Moodera, J. Nowak, and R. J. M. van de Veerdonk: Rev. B, 53, 4324 (1996). Phys. Rev. Lett., 80, 2941 (1998) 91) A. G. Petukhov, W. R. L. Lambrecht, and B. Segall: Phys. 123) E. Yu. Tsymbal and D. G. Pettifor: J. Phys.: Cond. Mat., 9, Rev. B, 53, 3646 (1996). L411 (1997). 92) J.-B. Xia, S.-F. Ren, and Y.-C. Chang: Phys. Rev. B, 43, 124) T. Obata, T. Manako, Y. Shimakawa, and Y. Kubo: Appl. 1692 (1991). Phys. Lett., 74, 290 (1999). 93) L. Bolotov, T. Tsuchiya, and A. Nakamura: Phys. Rev. B, 125) K.-I. Kobayashi, T. Kimura, H. Sawada, K. Terakura, and 59, 12236 (1999). Y. Tokura: Nature, 395, 677 (1998). 94) T. Yamauchi, Y. Matsuba, L. Bolotov, M. Tabuchi, and A. 126) T. Ohsawa, J. Inoue, and H. Itoh: unpublished. Nakamura: Appl. Phys. Lett., 77, 4368 (2000). 127) H. Yamada,Y. Ogawa, Y. Ishii, H. Sato, M. Kawasaki, H. 95) T. Yamauchi, Y. Matsuba, Y. Ohyama, M. Tabuchi, and A. Akoh, and Y. Tokura: Science, 305, 646 (2004). Nakamura: Jpn. J. Appl. Phys., 40, 2069 (2001). 128) H. Itoh, N. Nishimura, and J. Inoue: J. Magn. Magn. Mat., 96) T. Yamauchi, Y. Ohyama, Y. Matsuba, M. Tabuchi, and A. 240, 121 (2002). Nakamura: Appl. Phys. Lett., 79, 2465 (2001). 129) F. Guinea: Phys. Rev. B, 58, 9212 (1998). 97) A. Watanabe, H. Itoh, and J. Inoue: J. Magn. Magn. Mat., 130) J. Inoue and S. Maekawa: J. Magn. Magn. Mat., 198-199, 272-276, 1720 (2004). 158 (1999). 98) J. Inoue, H. Itoh, S. Honda, K. Yamamoto, and T. Ohsawa: 131) P.Lyu, D. Y.Xing, and J. Dong: Phys. Rev.B, 58, 54 (1998). J. Phys.: Cond. Mat., 16, S5563 (2004). 132) R. Jansen and J. S. Moodera: Phys. Rev. B, 61, 9047 (2000). 99) S. Honda, T. Kondo, H. Itoh, J. Inoue: unpublished. 133) A. Vedyayev, D. Bagrets, A. Bagrets, and B. Dieny: Phys. 100) J. M. MacLaren, X.-G. Zhang, W. H. Butler, and X. Wang: Rev. B, 63, 64429 (2001).

36 Journal of the Magnetics Society of Japan Vol. 30, No. 1, 2006 134) S. Akazawa, J. Inoue, and S. Maekawa: J. Phys. Soc. Jpn., Takanashi, Y.K. Takahashi, and K. Hono: Appl. Phys. Lett., 66, 2758 (1997). 84, 3106 (2004). 135) D. C. Worledge and T. H. Geballe: Appl. Phys. Lett., 76, 152) K. Inomata, S. Okamura, R. Goto, and N. Tezuka: Jpn. J. 900 (2000). Appl. Phys., 42, L419 (2003). 136) J. B. Philipp, C.Hofener,¨ S. Thienhaus, J. Klein, L. Alff, 153) S. Kammerer,¨ A. Thomas, A. Hutten,¨ and G. Reiss: Appl. and R. Gross: Phys. Rev. B, 62, R9248 (2000). Phys. Lett., 85, 79 (2004). 137) S. Lee, H. Y. Hwang, B. I. Shraiman, W. D. Ratcliff II, and 154) H. Kubota, J. Nakata, M. Oogane, Y. Ando, A. Sakuma, S-W. Cheong: Phys. Rev. Lett., 82, 4508 (1999). and T. Miyazaki: Jpn. J. Appl. Phys., 43, L984 (2004). 138) J. Appelbaum: Phys. Rev. Lett., 17, 91 (1966). 155) I. Galanakis: J. Phys.: Cond. Mat., 14, 6329 (2002). 139) P. W. Anderson: Phys. Rev. Lett., 17, 95 (1966). 156) P. A. Dowben and R. Skomski: J. Appl. Phys., 95, 7453 140) H. Itoh, J. Inoue, S. Maekawa, and P. Bruno: J. Magn. Soc. (2004). Jpn., 23, 52 (1999). 157) J. C. Slonczewski: J. Magn. Magn. Mat., 159, L1 (1996), 141) J. K. Freericks: Phys. Rev. B, 70, 195342 (2004). 247, 324 (2002). 142) C. Caroli, R. Combescot, P. Nozieres, and D. Saint-James: 158) L. Berger: Phys. Rev. B, 54, 9353 (1996). J. Phys. C: Solid State Phys., 4, 916 (1971). 159) E. B. Myers, D. C. Ralph, J. A. Katine, R. N. Louie, and R. 143) K. Ono, H. Shimada, Y. Ootuka: J. Phys. Soc. Jpn., 66, A. Buhrman: Science, 285, 867 (1999). 1261 (1997). 160) A. Brataas, Y. V. Nazarov, and G. E. W. Bauer: Phys. Rev. 144) J. Barnas and A. Fert: Phys. Rev. Lett., 80, 1058 (1998). Lett., 84, 2481 (2000). 145) K. Majumdar and S. Hershfield: Phys. Rev. B, 57, 11521 161) Y. Jiang, T. Nozaki, S. Abe, T. Ochiai, A. Hirohata, N. (1998). Tezuka, and K. Inomata: Nature Mat., 3, 361 (2004). 146) S. Takahashi and S. Maekawa: Phys. Rev. Lett., 80, 1758 162) Y.Liu, Z. Zhang, P.P.Freitas, and J. L. Martins: Appl. Phys. (1998). Lett., 82, 2871 (2003). 147) A. Brataas, Yu. V. Nazarov, J. Inoue, G. E. W. Bauer: Phys. 163) G. D. Fuchs, N. C. Emley, I. N. Krivorotov, P.M. Braganca, Rev. B, 59, 93 (1999). E. M. Ryan, S. I. Kiselev, J. C. Sankey, J. A. Katine, D. 148) H. Wang and A. Brataas: Phys. Rev. Lett., 83, 5138 (1999). C. Ralph, and R. A. Buhrman: Appl. Phys. Lett., 85, 1205 149) A. Btaraas, M. Hirano, J. Inoue, Yu. V. Nazarov, and G. E. (2004). W. Bauer: Jpn. J. Appl. Phys., 40, 2329 (2001). 164) Y. Huai, F. Albert, P. Nguyen, M. Pakala, and T. Valet: 150) K. Yakushiji, S. Mitani, K. Takanashi, and H. Fujimori: J. Appl. Phys. Lett., 84, 3118 (2004). Appl. Phys., 91, 7038 (2002) 151) F. Ernult, K. Yamane, S. Mitani, K. Yakushiji, K. Received Nov. 05, 2004; Accepted Oct. 19, 2005.

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