Impact of lattice strain on the tunnel magneto-resistance in Fe/Insulator/Fe and Fe/Insulator/La 0.67 Sr 0.33 MnO 3 magnetic tunnel junctions
aa r X i v : . [ c ond - m a t . m t r l - s c i ] N ov Impact of lattice strain on the tunnel magneto-resistance inFe/Insulator/Fe and Fe/Insulator/La . Sr . MnO magnetic tunnel junctions A. Useinov ∗ , Y. Saeed , N. Singh , N. Useinov , U. Schwingenschl¨ogl † Department of Physics, California State University,Northridge, California 91330, USA PSE Division, King Abdullah University of Scienceand Technology, Thuwal 23955, Saudi Arabia and Department of Solid State Physics,Kazan Federal University, Kazan, Russia (Dated: January 15, 2018)
Abstract
The objective of this work is to describe the tunnel electron current in single barrier magnetictunnel junctions within a new approach that goes beyond the single-band transport model. Wepropose a ballistic multi-channel electron transport model that can explain the influence of in-plane lattice strain on the tunnel magnetoresistance as well as the asymmetric voltage behavior.We consider as an example single crystal magnetic Fe(110) electrodes for Fe/Insulator/Fe andFe/Insulator/La . Sr . MnO tunnel junctions, where the electronic band structures of Fe andLa . Sr . MnO are derived by ab-initio calculations. PACS numbers: 72.10.Fk, 73.40.Gk, 75.45.+j, 75.47.De ∗ [email protected], +1-818-677-2782 † [email protected], +966(0)544700080 . INTRODUCTION One of the fast growing directions in modern magnetic electronics (spintronics) is the fieldof magnetic tunnel junctions (MTJs) and their applications, for example, as basic elementsin magnetic random access memories, read-heads of hard drives, and magnetic field sen-sors. Potential to realize memristors and vortex oscillators creates additional incentive forfuture investments in this area . MTJs such as FM/Insulator/FM and FM/Insulator/HMheterostructures, where FM is a ferromagnet (like Co, Fe, CoFeB), the insulator is ferro-electric (like BaTiO , PbTiO ), and HM is a half-metal (like La . Sr . MnO , Co MnSn),are very promising, because they combine magnetic, ferroelectric, and spin filtering proper-ties. Tunnel electroresistance and tunnel magnetoresistance (TMR) effects may coexist inthese systems. The TMR arises from states of different resistance for parallel and antiparal-lel magnetic alignments, while the tunnel electroresistance relies on the polarization of theferroelectric insulator. The insulating layer has to be thick enough to yield strong ferroelec-tricity, which usually rapidly disappears for decreasing thickness, and has to be thin enoughfor electron tunneling. Moreover, the ferroelectric polarization in thin ferroelectric filmsis conjugated with the magnitude of the lattice strain . A high ferroelectric polarizationis achieved by epitaxial film growth with an initially high difference between the in-planelattice parameters of the substrate and the deposited layers. Obviously, the electronic bandstructures and transport properties of the strained FM and HM layers can be fundamentallydifferent from those without strain.The objective of this work is to establish the interplay between the lattice strain and themagnitude of the TMR using a multi-band approach for the electron transport. We predictthat for strained symmetric MTJs the TMR is reduced, because of changes in the electronicband structure under strain. In general, the tunnel electroresistance in ferroelectric TJsshould logarithmically increase with strain (the ferroelectric polarization increases), as itwas shown, for instance, in the works of Zhuravlev and coworkers . This means there isa balanced configuration of the insulator thickness (potential barrier thickness) and strainthat provides the highest TMR and tunnel electroresistance. To calculate the tunnel currentand TMR we have to go beyond the assumption of two conduction channels (single-bandmodel) similar to Refs. .Investigation of MTJs has a long history . In Ref. 15 Valet and Fert have introduced2 e y x z d kkk kkk FM (HM) Emitter Fe FM CollectorInsulator
FIG. 1: Simplified schema of the multi-channel model of a single crystal MTJ for positive bias(electrons tunnel from left to right). The model assumes independent propagation channels, eachbeing associated with a given spin and symmetry. basic principles for the qualitative and quantitative interpretation of the spin polarizedelectron transport in magnetic multilayer structures, based on Boltzmann-like equations.An alternative theoretical approach of electronic transport through nanocontacts with andwithout domain walls between two FM electrodes has been developed in Ref. 19. Thistheory utilizes quasiclassical as well as quantum mechanical ideas and is based on extendedBoltzmann-like equations. Boundary conditions on the interfaces of the junction are takeninto account as a key part of the solution. The theory can be adapted to the case of ballistictransport through single barrier and double barrier planar junctions.Using the universality of the above technique, we formulate a multi-channel (or multi-band) approach following the ideas of Ref. 21. The tunneling conductance in MTJs can bewritten in terms of the averaged spin-dependent tunneling probabilities of the conductionchannels for parallel (P) and antiparallel (AP) magnetizations. According to our ab-initio calculations for Fe, several minority and majority spin bands cross the Fermi level, rep-resenting different electron wave functions. We extract the dispersion relations along thetunneling direction (perpendicular to the Fe(110) interface) from the bulk band structure.For simplicity, the insulator is considered to be homogeneous. Our approach does not in-corporate filtering effects inside the barrier, which are important in the case of MgO or forthe splitting of the valence band in SrTiO and BaTiO , for instance .3 I. THE MULTI-CHANNEL APPROACH
The ideas of the multi-channel approach are demonstrated in Fig. 1. In this model eachpropagating channel is associated with a given spin and symmetry of the wave function. Theemitter provides electrons with different Fermi vectors, which tunnel across the barrier intothe states of the collector. We employ a formula for the current density originally derivedfor transport through a magnetic planar junction . For the single-band model the currentdensity is proportional to the integral of the product of the transmission coefficient, D P(AP) ,and the cosine of the incidence angle of the electron trajectory, cos (cid:16) θ ↑ , ↓ L (cid:17) . The angle θ ↑ , ↓ L ismeasured from the normal (transport direction) to the interface plane ( L : left, R : right).The integral is taken over d Ω L = sin ( θ L ) dθ L dφ : J P(AP) ↑ , ↓ = e V (cid:16) k ↑ , ↓ L (cid:17) π ~ D cos (cid:16) θ ↑ , ↓ L (cid:17) D P(AP) ↑ , ↓ E Ω L . (1)Here k ↑ , ↓ L is the absolute value of the Fermi vector of the left-hand electrode and ↑ , ↓ isthe spin index. The transmission coefficient is a function of the applied bias voltage V , of θ ↑ , ↓ L = 0 ... arccos s(cid:12)(cid:12)(cid:12)(cid:12) − (cid:16) k ↑ , ↓ R /k ↑ , ↓ L (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)! , and of k ↑ , ↓ L ( R ) . With x ↑ , ↓ = cos (cid:16) θ ↑ , ↓ L (cid:17) we can write D x ↑ , ↓ D P(AP) ↑ , ↓ E Ω L = Z X ↑ , ↓ x ↑ , ↓ D P(AP) ↑ , ↓ dx ↑ , ↓ , where the lower limit X ↑ , ↓ for the integration arises from the conservation of the projectionof the Fermi vector in the xy-plane: k ↑ , ↓k = k ↑ , ↓ L sin (cid:16) θ ↑ , ↓ L (cid:17) = k ↑ , ↓ R sin (cid:16) θ ↑ , ↓ R (cid:17) . It equals zerowhen the electrons tunnel from the left minority into the right majority conduction band and X ↑ , ↓ = s(cid:12)(cid:12)(cid:12)(cid:12) − (cid:16) k ↑ , ↓ R /k ↑ , ↓ L (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) when they tunnel from the left majority into the right minorityconduction band. For the multi-band approach the majority and minority bands can beboth spin up and down for any magnetic configuration.To achieve a multi-channel model (or model with multi-band tunnel relations) for singlecrystal junctions we redefine the current density in Eq. (1): J P(AP) ↑ , ↓ = e V π ~ N X η =1 M X µ =1 (cid:0) k ↑ , ↓ η (cid:1) (cid:10) cos ( θ η ) D P(AP) η,µ (cid:0) k ↑ , ↓ η , k ↑ , ↓ ( ↓ , ↑ ) µ (cid:1)(cid:11) Ω L . (2)Here η and µ are the indices of the left-hand and right-hand bands, respectively, and N and M are the numbers of bands. The combinations { η, µ } , see Fig. 1, identify the conduction4elations between the bands through the barrier. Equation (2) is valid for positive bias. Thesolution for negative bias is derived using symmetric relations of the system, i.e., the collectorand emitter are exchanged ( k η → k µ , k µ → k η ). We assume that there is no spin flip leakageand that a conduction channel is available between any left-hand and right-hand bands withthe same spin. Otherwise the electrons are reflected back, giving rise to a resistance. Notethat the lowest conductance corresponds to the largest difference in the density of states atthe Fermi level between the left and right electrodes. Regarding the transmission coefficientfor the single barrier system, the basic mathematical expressions can be found in Ref. 12,where an exact quantum mechanical solution has been derived employing Airy functions forthe tunnel barrier.The band structures obtained from ab-initio calculations for bulk Fe (space goup Cmmm ) and La . Sr . MnO are shown in Figs. 2 and 3, as derived using the WIEN2kpackage . The exchange-correlation potential is parametrized in the generalized gradi-ent approximation . For the wave function expansion inside the atomic spheres a maxi-mum value of the angular momentum of ℓ max = 12 is employed and a plane-wave cutoff of R mt K max = 9 with G max = 24 is used. Self-consistency is assumed when the total energyvariation reaches less than 10 − Ry. We use a mesh of 10 × × k -points for calculatingthe electronic structure in order to describe the ground states of the compounds with highaccuracy.Figure 2 shows the band structure of Fe for different in-plane lattice parameters a =3 .
875 ˚A, 3.937 ˚A, 3.999 ˚A, and 4.030 ˚A (bulk value), where red and green color representthe two spins. As an example, we consider the symmetric Fe/Insulator/Fe junction anddemonstrate how to collect the conducting spin channels via the applied bias V . The bandsof the left electrode (emitter) are the same as those of the right electrode (collector) and theFermi energies E LF = E RF = E F are equal at zero bias. Horizontal dashed lines represent E F ,which intersects with the bands at the Fermi vectors k ↑ , ↓ η ( µ ) . In particular, in Figs. 2(a), 2(b)and Figs. 2(e), 2(f) the system has two k ↓ and two k ↑ vectors at zero bias, while in the caseof Figs. 2(c), 2(d), 2(g), 2(h) E F is intersected by three spin up and three spin down bands.We thus have the Fermi vector set { k ↑ , ↓ L ( R ) , k ↑ , ↓ L ( R ) , k ↑ , ↓ L ( R ) } . In the case of positive (negative)bias, by definition, E F of the left electrode shifts up (down) in energy, while for the rightelectrode it shifts down (up) by the same amount. The voltage drop is (cid:12)(cid:12) E LF − E RF (cid:12)(cid:12) = | eV | .As a result the Fermi vector set is changed. As an example, let us set V = +0 . (d) (b)(a) Spin down bands (c) (f)(e)(g)(h)
Spin up bands -0.40.00.4 ( E - E F ) ( e V ) -0.40.00.4 kk (¯ -1 ) z k z k (¯ -1 ) FIG. 2: (Color online) Electronic bands for bulk Fe along the Γ-Z direction. E − E F = 0 correspondsto zero bias. Data are derived for different lattice parameters, which correspond to different latticestrains. The four band structures refer to (a),(e) a = 3 .
875 ˚A, c = 3 .
083 ˚A; (b),(f) a = 3 .
937 ˚A, c = 2 .
986 ˚A; (c),(g) a = 3 .
999 ˚A, c = 2 .
894 ˚A; (d),(h) a = 4 .
030 ˚A, c = 2 .
850 ˚A. The Γ point islocated at k ↑ , ↓ z = 0 and the Z point is shown by vertical dotted lines. E LF = 0 . E RF = − . {
0, 0, k ↑ L } and { k ↓ L , k ↓ L , k ↓ L } and for the right electrodein the sets { k ↑ R , k ↑ R , k ↑ R and k ↓ R , k ↓ R } , which generates 1 × × × × × N × M ). When the Fermi vectors vanish we have,of course, a non-conducting channel with vanishing current density.Figure 3 shows the band structure of La . Sr . MnO along the Γ-Z direction for twosets of lattice parameters: a = 3 .
875 ˚A, c = 23 .
250 ˚A and a = 4 .
030 ˚A, c = 21 .
496 ˚A.The Fermi vector, transmission coefficient, and current density for each band are derivedas demonstrated before. However, some of the spin down bands are very flat with energy6 ( E - E F ) ( e V ) gaps (a)(b) Spin down bands Spin up bands (c)(d) (¯ -1 ) z k z k (¯ -1 ) FIG. 3: (Color online) Electronic bands for bulk La . Sr . MnO along the Γ-Z direction. E − E F = 0 corresponds to zero bias. Data are derived for different lattice parameters, whichcorrespond to different lattice strains. The two band structures refer to (a),(c) a = 3 .
875 ˚A, c = 23 .
250 ˚A and (b),(d) a = 4 .
030 ˚A, c = 21 .
496 ˚A. The Γ point is located at k ↑ , ↓ z = 0 and the Zpoint is shown by vertical dotted lines. gaps between them, in contrast to the spin up bands. As a function of the bias the systemtherefore switches between a HM and FM. However, there are also energies at which neitherspin up nor spin down states exist. III. TUNNEL MAGNETORESISTANCE UNDER STRAIN
Physical parameters that characterize the properties of MTJs are the total tunnel cur-rent density J P(AP) = ( J ↑ + J ↓ ) P(AP) , the TMR = (cid:0) J P − J AP (cid:1) /J AP × n = (cid:0) J P − J AP (cid:1) /J AP × TMR − ( V = 0), and the output voltage V out = V (cid:0) J P − J AP (cid:1) /J AP , which can be obtained from free-electron or tight-binding mod-els. However, unfortunately these models do not reproduce the experimental effect of strainon the charge transport characteristics. A single-band approach is sufficient to model theTMR in amorphous sputtered MTJs and can satisfactorily describe the TMR n and V out of epitaxial single and double barrier FeCoB/MgO junctions . In our case we have to gobeyond parabolic dispersions and the single-band model, however, keeping the simplicity ofthe approach. For the Fermi vectors derived above as well as for typical parameters of anAl O tunnel barrier, TMR results derived by Eq. (2) are shown in Figs. 4 to 6. The barrier7 B B B T M R ( % ) Bias (V) o A o A o A FIG. 4: (Color online) TMR versus applied voltage for Fe/Insulator/Fe MTJs with the latticeparameters: a = 3 .
875 ˚A, 3.937 ˚A, and 4.030 ˚A. The barrier parameters are d = 1 . U B = 2 . thickness is set to d = 1 . E F to U B = 2 . m B = 0 . . In our calculations for metals the effective mass is equal to the freeelectron mass.Figure 4 presents the TMR as function of the bias for different lattice parameters, showingthat the TMR, in general, behaves non-monotonically. For unstrained Fe ( a = 4 .
030 ˚A) adecreasing in-plane lattice parameter (increasing strain) leads to a lower TMR. Figure 5 givesthe TMR as a function of the lattice parameter for 0.1 mV and 0.1 V bias. Interestingly,we observe deviations from a linear behavior: For almost zero bias the TMR increases upto 31.1% for a = 3 .
999 ˚A, 28.6% for a = 4 .
030 ˚A, and 27.3% for a = 3 .
968 ˚A. This behavioris related to modifications in the reflection of the majority states at the Z point, wherethe Fermi vector achieves its maximal magnitude (Fig. 2, dashed rectangles). Note thatthese states give the main contribution to the tunnel current. The observed differences fordifferent in-plane lattice parameters are explained by variations of the band structure. Thedashed rectangles in Figs. 2(e-h) demonstrate the bands near the Z point. For a = 3 .
999 ˚A,see Fig. 2(g), the majority band intersects the Fermi level at the Z point, favoring J P over J AP , in contrast to the other lattice parameters. The maximal TMR value close to zero biasis in good agreement with the results of Yuasa and coworkers for Fe(110)/Al O /Fe Co ,see Fig. 3(b) in Ref. 29, and of Hauch and coworkers for Fe(110)/MgO(111)/Fe(110), 28%8 .90 3.95 4.00202530 (¯) T M R ( % ) a FIG. 5: (Color online) TMR as function of the lattice parameter a for Fe/Insulator/Fe MTJs.Black and red color refer to biases of 0.1 mV and 0.1 V, respectively. at T = 300 K .In the case of the Fe/Insulator/La . Sr . MnO MTJ our model gives a positive TMR for
V > .
11 V as well as a negative TMR below, see Fig. 6. The TMR curves are qualitativelysimilar to those obtained experimentally for Co/SrTiO /La . Sr . MnO and agree with theroom temperature TMR in Fe / MgO / Co MnSn (about −
5% at a bias of 0.1 mV). However,according to these authors the TMR is suppressed in the voltage range | V | ≥ . a = 4 .
030 ˚A and a = 3 .
875 ˚A, where the lattercorresponds to unstrained La . Sr . MnO . For positive bias the magnitude of the TMRdecreases with the Fe lattice strain, whereas for negative bias the situation is reversed. Forthe circled points in Fig. 6, where the TMR goes to zero, both spin channels are closed,compare the energy gaps in Fig. 3, because of J P = J AP = 0. There are other points wherethe TMR is zero as J P = J AP . Variation of the effective mass in the tunnel barrier leadsto a weak response of the TMR in symmetric (1.5% decrease) and a strong response inasymmetric (14% increase) junctions, for all lattice parameters close to zero bias, m B = 1.9 T M R ( % ) Bias (V) o A o A FIG. 6: (Color online) TMR versus applied bias for the Fe/Insulator/La . Sr . MnO MTJ. Thebarrier parameters are d = 1 . U B = 2 . m B = 0 . IV. CONCLUSION
We have extended an established quasi-classical ballistic transport model to multi-channelconductance, which has enabled us to investigate the role of the electronic band structureand the effect of strain on the transport properties of single crystal Fe/Insulator/Fe andFe/Insulator/La . Sr . MnO MTJs. Our approach takes into account all bands of the FMand HM along the Γ-Z direction (direction of tunneling). We have demonstrated for typicalparameters of an Al O tunnel barrier a maximal TMR of 31.1% for the Fe/Insulator/FeMTJ, which is in good agreement with the experiment. A negative TMR of 5% is foundfor the Fe/Insulator/La . Sr . MnO MTJ close to zero bias, where the dependence onthe bias reproduces experimental findings. The developed technique thus has demonstratedgreat potential for further studies on transport properties (including the spin transfer torque)in simple and magnetic TJs.Strain effects on the TMR have been explored theoretically for the first time by a multi-band approach. For the Fe/Insulator/Fe MTJ it turnes out that for small bias the TMRdecreases linearly with the in-plane strain at the interface, whereas in the case of theFe/Insulator/La . Sr . MnO MTJ the strain effects strongly depend on the sign of theapplied bias. For positive bias it is positive and maximal for unstrained Fe, while for neg-ative bias it is negative and the amplitude increases with strain and bias. The observed10elations between the strain and the TMR are explained by variations of the band structure.We have demonstrated that in-plane strain can increase and decrease the TMR and thereforemakes it possible to obtain optimal regimes for MTJ applications. A. Dussaux, B. Georges, J. Grollier, V. Cros, A. V. Khvalkovskiy, A. Fukushima, M. Konoto,H. Kubota, K. Yakushiji, S. Yuasa, K. A. Zvezdin, K. Ando, and A. Fert, Nat. Commun. , 10(2010). A. Chanthbouala, R. Matsumoto, J. Grollier, V. Cros, A. Anane, A. Fert, A. V. Khvalkovskiy,K. A. Zvezdin, K. Nishimura, Y. Nagamine, H. Maehara, K. Tsunekawa, A. Fukushima, and S.Yuasa, Nat. Phys. , 626 (2011). A. Petraru, N. A. Pertsev, and H. Kohlstedt, J. Appl. Phys. , 114106 (2007). N. A. Pertsev, A. G. Zembilgotov, and A. K. Tagantsev, Phys. Rev. Lett. , 1988 (1998). Y. S. Kim, D. H. Kim, J. D. Kim, and Y. J. Chang, Appl. Phys. Lett. , 102907 (2005). M. Zhuravlev, R. Sabirianov, S. Jaswal, and E. Tsymbal, Phys. Rev. Lett. , 246802 (2005). M. Zhuravlev, S. Maekawa, and E. Tsymbal, Phys. Rev. B , 104419 (2010). N. Mott, Proc. R. Soc. London , 699 (1936). I. Campbell, A. Fert, and A. Pomeroy, Philos. Mag. , 977 (1967). M. Julliere, Phys. Lett. , 225 (1975). A. Manchon, N. Ryzhanova, N. Strelkov, A. Vedyayev, and B. Dieny, J. Phys. Condens. Mat. , 165212 (2007). A. Useinov, R. Deminov, N. Useinov, and L. Tagirov, Phys. Stat. Sol. B , 1797 (2010). A. Kalitsov, M. Chshiev, I. Theodonis, N. Kioussis, and W. H. Butler, Phys. Rev. B , 174416(2009). Y.-H. Tang, N. Kioussis, A. Kalitsov, W. H. Butler, and R. Car, Phys. Rev. B , 054437(2010). T. Valet and A. Fert, Phys. Rev. B , 7099 (1993). W. H. Butler, X.-G. Zhang, T. C. Schulthess, and J. M. MacLaren, Phys. Rev. B , 054416(2001). J. Mathon and A. Umerski, Phys. Rev. B , 220403 (2001). P. Seneor, A. Fert, J.-L. Maurice, F. Montaigne, F. Petro, and A. Vaures, Appl. Phys. Lett. ,
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