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8-27-2019 Tunneling Anisotropic Magnetoresistance in Ferroelectric Tunnel Junctions Artem Alexandrov Moscow Institute of Physics and Technology

M. Ye. Zhuravlev St. Petersburg State University, [email protected]

Evgeny Y. Tsymbal University of Nebraska at Lincoln, [email protected]

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Alexandrov, Artem; Zhuravlev, M. Ye.; and Tsymbal, Evgeny Y., "Tunneling Anisotropic Magnetoresistance in Ferroelectric Tunnel Junctions" (2019). Evgeny Tsymbal Publications. 98. https://digitalcommons.unl.edu/physicstsymbal/98

This Article is brought to you for free and open access by the Research Papers in Physics and Astronomy at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in Evgeny Tsymbal Publications by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. PHYSICAL REVIEW APPLIED 12, 024056 (2019)

Tunneling Anisotropic Magnetoresistance in Ferroelectric Tunnel Junctions

Artem Alexandrov,1 M. Ye. Zhuravlev,1,2,† and Evgeny Y. Tsymbal 1,3,* 1 Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Region 141700, Russia 2 St. Petersburg State University, St. Petersburg 190000, Russia 3 Department of Physics and Astronomy, University of Nebraska, Lincoln, Nebraska 68588, USA

(Received 30 May 2019; revised manuscript received 3 July 2019; published 27 August 2019)

Using a simple quantum-mechanical model, we explore a tunneling anisotropic magnetoresistance (TAMR) effect in ferroelectric tunnel junctions (FTJs) with a ferromagnetic electrode and a ferroelectric barrier layer, where spontaneous polarization gives rise to the Rashba and Dresselhaus -orbit coupling (SOC). For realistic parameters of the model, we predict sizable TAMR measurable experimentally. For asymmetric FTJs, whose electrodes have different work functions, the built-in electric field affects the SOC parameters and leads to TAMR being dependent on the ferroelectric polarization direction. The SOC change with polarization switching affects tunneling conductance, revealing an alternative mechanism of tunneling electroresistance. These results demonstrate alternative functionalities of FTJs, which can be explored experimentally and used in electronic devices.

DOI: 10.1103/PhysRevApplied.12.024056

I. INTRODUCTION MFTJ constitutes a four-state resistance device where the resistance is controlled both by the FE polarization direc- Ferroelectric tunnel junctions (FTJs) have aroused con- tion of the barrier and the alignment of the siderable interest due to the interesting physics and poten- ferromagnetic electrodes. MFTJs are interesting from the tial applications as nanoscale resistive switching devices point of view of their multifunctional properties, as has [1–3]. A FTJ consists of two metal electrodes separated by been demonstrated in a number of experimental studies a nm-thick ferroelectric (FE) barrier, which allows elec- [25–28]. tron tunneling through it. The key property is the tunneling In parallel with these developments, it has been found electroresistance (TER) effect that is a change in resistance that resistance of MTJs can also depend on the magne- of a FTJ with reversal of the FE polarization. Follow- tization orientation with respect to the crystallographic ing the theoretical predictions [4,5], there have been a axes [29–34]. This phenomenon is known as tunneling number of experimental demonstrations of the TER effect anisotropic magneto-resistance (TAMR). TAMR is the in trilayer junctions [6–10]. It was shown that a sizable manifestation of one of the oldest known effects that cou- TER effect can be achieved by using dissimilar electrodes ple magnetism and electronic transport, that is, anisotropic [11–13], interface engineering [14–16], applied bias [17, magnetoresistance (AMR) [35], in the tunneling regime. 18], or defect control [19]. Contrary to FE capacitors where Both TAMR and AMR are driven by spin-orbit coupling leakage currents are detrimental to the device performance, (SOC), entangling the spin and orbital degrees of freedom. the conductance of a FTJ is the functional characteristic of Contrary to the conventional TMR effect, TAMR may the device [20]. This makes FTJs promising for nonvolatile occur in MTJs with only one ferromagnetic electrode. This memory applications [21,22]. functionality opens alternative possibilities for spintronic The functionality of FTJs can be extended by using devices. ferromagnetic electrodes to create a multiferroic tunnel Exploring a TAMR effect in a FTJ, whose transport junction (MFTJ) [23]. A MFTJ combines the properties properties are strongly dependent on FE polarization, is of a FTJ and a magnetic tunnel junction (MTJ), which interesting due to the interplay between ferroelectricity, exhibits a tunneling magnetoresistance (TMR) effect that magnetism, spin-dependent transport, and SOC. Recent is the dependence of resistance on the relative magnetiza- studies have demonstrated the strong SOC effects in a tion directions in the two ferromagnetic electrodes [24]. number of bulk FE materials. Very large SOC (approxi- Due to the coexistence of the TER and TMR effects, a mately 102–103 meV) has been predicted, resulting from a polarization-induced potential gradient [36–43]. In addi- †[email protected] tion to the sizable SOC favorable for the experimental *[email protected] demonstration of the TAMR effect, these materials have

2331-7019/19/12(2)/024056(8) 024056-1 © 2019 American Physical Society ALEXANDROV, ZHURAVLEV, and TSYMBAL PHYS. REV. APPLIED 12, 024056 (2019) the advantage of the reversible FE polarization, which can The corresponding Hamiltonian in each region is given be switched by an applied electric field. Since FE materi- by als are noncentrosymmetric, the spin-momentum coupling ⎧ ⎪ 2 J linear in wavevector k is allowed by symmetry, giving ⎪H =− ∇2 − (σ cos φ + σ sin φ), z < 0, ⎪ L x y rise to the linear Rashba and Dresselhaus SOC in the ⎨⎪ 2m 2 bulk of these compounds [44]. As a result, reversal of FE 2 H =− ∇2 + U + H ,0< z < a, polarization changes the SOC parameters, and thus can ⎪ B SOC ⎪ 2m affect the TAMR, similar to the recently predicted effect ⎪ 2 ⎩⎪  of FE polarization of the tunneling anomalous Hall effect H =− ∇2 + W, a > z. R 2m (TAHE) [45]. (1) In this work, we explore the appearance of the TAMR effect in FTJs with a single ferromagnetic electrode and Here, J is the exchange splitting in the FM electrode,σx a FE barrier layer, which spontaneous polarization gives and σy are the Pauli matrices, m is the effective rise to the Rashba and Dresselhaus SOC. For realistic mass, U is the barrier height, and W is the potential in the SOC parameters, we predict the appearance of a sizable NM electrode, as shown schematically in Fig. 1(b).SOC TAMR effect that will be measurable experimentally. For in Eq. (1) is given by FTJs with electrodes of different work functions, TAMR depends on the FE polarization direction due its effect on HSOC = λR(kxσy − ky σx) + λD(kxσy + ky σx),(2) SOC parameters. The latter also affects tunneling resis- λ λ tance, which reveals an alternative mechanism of TER. where R and D are, respectively, the Rashba and Dres- selhaus SOC constants. This form of SOC is typical for FE materials of the C2v point group where the polarization II. THEORETICAL APPROACH is pointing along the z direction. The SOC Hamiltonian We consider a FTJ, which consists of a semi-infinite can be rewritten in terms of the effective k-dependent left (L) ferromagnetic (FM) electrode (z < 0) and a right spin-orbit field (k)coupled to the spin as follows (R) nonmagnetic (NM) electrode (z > a) separated by a FE = ( ) · σ barrier layer of thickness a [Fig. 1(a)]. Magnetization M HSOC k .(3) of the ferromagnet is assumed to lie in the plane of the Compared to Eq. (2), it is seen from that (k) = layer at angle φ with respect to the x axis. FE polariza- (βk , αk ,0), where α = λ + λ and β = λ − λ . tion of the barrier is assumed to be pointing normal to y x R D D R To calculate the conductance, we use the standard the interface. We consider, for simplicity, ideal electrodes, approach based on transmission of the incoming elec- which fully screen polarization charges at the two inter- tron wave across the junction. We employ a free-electron faces. This implies no depolarizing field in the barrier layer model, where electron transport is described by s-like elec- and a constant potential barrier height [Fig. 1(b)][46]. trons. The propagating state of energy E incoming from the left FM electrode and normalized to the unit current (a) density is given by σ m ( + ) σ σ ψ = i kxx iky y ikz zχ L  σ e e φ ,(4) kz σ =↑ ↓ ↑,↓ = where , is the spin index, kz (b) ( ± / )/2 − 2 2m E J 2 k|| is the z component of the wave vector in the FM electrode, k|| = (kx, ky ) is the trans- verse wavevector, which is conserved in the process of − φ/ ↑ ↓ i 2 , √1 e tunneling, and χφ = φ/ is the spinor eigen- 2 ±ei 2 function. The scattering state in the right electrode due to ψσ transmission of L across the junction is given by FIG. 1. (a) Schematic structure of a FTJ, which consists of semi-infinite left (L) ferromagnetic (FM) and right (R) nonmag- m σ σσ iqzz σ σσ¯ iqzz σ¯ ψ ← = (t e χφ + t e χφ ),(5) netic (NM) electrodes separated by a ferroelectric (FE) barrier R L q RL RL of thickness a. Magnetization M lies in the x-y plane at angle φ z with respect to the x axis. (b) Potential profile across the junc- σ¯ =−σ = ( − )/2 − 2 σσ¯ tion. EF is the Fermi energy, U is the barrier height, and J ex is where , qz 2m E W k,andtRL and the exchange splitting. σσ tRL are the transmission amplitudes with and without spin

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flip, respectively. The scattering state in the left electrode due to reflection of ψσ from the barrier can be written as L 2 2 ↑,↓ k U = U − EF + ± λRk.(9) eff 2m σ m σσ − σ σ m σσ¯ − σ¯ σ¯ ψ = ( iqzz + ikz z)χ + ikz χ L←L  σ e rLL e φ  σ¯ rLL e φ , kz kz The lowest barrier height occurs in a circular region around 2 (6) the origin at k|| = mλ/ , which is about 0.13 Å for λ = 1 eV Å. The spin follows the effective SOC field σσ¯ σσ  = λR(−ky , kx,0), pointing perpendicular to the wave respectively, where rLL and rLL are the reflection ampli- tudes with and without spin flip, respectively. vector k|| and creating a circular spin texture as schemat- The scattering state in the barrier is given by ically shown by arrows in Fig. 2(a). For thosek|| where the spin points parallel to the magnetization [indicated by red arrows in Fig. 2(a)], the transmission is largest. ψ = i(kxx+ky y) B e This is due to the majority-spin electrons incoming from a+eQ+z + a+e−Q+z + a−eQ−z + a−e−Q−z the FM layer, encountering the lowest barrier height. For × 1 2 1 2 + + − − − − , otherk|| where the spin is misaligned to the magnetization, γ(a eQ+z + a e Q+z − a eQ−z − a e Q−z) 1 2 1 2 the transmission is less efficient due to the admixture of (7) the minority-spin states. When the magnetization angle φ changes, the highest transmission region rotates with the 2 magnetization due to the circular spin texture induced by where Q± = 2m(U − E ± w)/2 + k , γ = (iαk +  x the Rashba SOC. A qualitatively similar behavior occurs β )/ = α2 2 + β2 2 ky w,andw kx ky . The total conductance at for a purely Dresselhaus-type SOC (not shown), when λ = λ = zero temperature is given by D 1 eV Å and R 0. The difference is in the conduc- tance maxima locations and their sense of rotation with the changing angle φ. 2 e In the opposite limit, when λ and λ are equal, for G = G(k||)dk|| = R D (2π)3 example, λ = λ = 1 eV Å, the spin-orbit field is unidi- D R rectional, that is,  = α(0, kx,0), forming a momentum- σσ 2 σσ¯ 2 × (|t | +|t | ) dk||.(8) LR LR E=EF independent spin configuration known as the persistent σ spin texture [43]. In this case, the effective tunneling barrier height is given by The respective transmission amplitudes are obtained by 2 2 matching the wave functions given by Eqs. (4)–(7) at the ↑ ↓  k|| U , = U − E + ± (λ + λ )k , (10) FTJ interfaces. eff F 2m R D x

and the lowest barrier height occurs at ky = 0andkx = III. RESULTS AND DISCUSSION 2 ±m(λR + λD)/ , which is about ±0.26 Å for λR = λD = Next, we perform numerical calculations of the conduc- 1 eV Å. The conductance maxima occur at these k|| points tance for different magnetization angles. In the calcula- with the relative weight being dependent on the magneti- tions, we assume a = 3nm,EF = 3eV,U = 4 eV (which zation angle φ [Fig. 2(b)]. For φ = 0, the effective SOC makes the potential barrier height U − EF = 1eV),W = field is pointing along the y axis and thus the magneti- −10 eV, and J = 3 eV as representative parameters. We zation is normal to the spin direction [shown by arrows start from analyzing the k||-resolved conductance G(k||) in Fig. 2(b)], resulting in the equal conductance at the given by Eq. (8) to demonstrate how the specific form of two maxima. However, for φ = 90◦, the magnetization is 2 SOC affects the conductance and its dependence on the parallel to the spin at kx = m(λR + λD)/ [indicated by magnetization angle. the red arrow in Fig. 2(b)] and antiparallel to the spin at 2 First, we assume a purely Rashba-type SOC in Eq. (3), kx =−m(λR + λD)/ , leading to the higher conductance that is, λR = 1eVÅandλD = 0. Figure 2(a) shows the for the former. calculated k||-resolved conductance for different magneti- The observed changes in the k||conductance with mag- zation angles φ. It is seen that at certain k||, the conduc- netization angle are expected to produce a TAMR effect. tance exhibits a maximum, which follows the changing For λD = 0(orλR = 0), however, the rotational symmetry magnetization direction. This behavior can be understood of the band structure and the spin texture [Fig. 2(a)] lead as follows. The Rashba SOC in the FE layer leads to to the vanishing of TAMR, where the total conductance the splitting of the free electron band, which determines does not depend on angle φ (Fig. 3, gray line). Only when the effective k||-dependent barrier height for tunneling both λR and λD are nonzero, the TAMR effect is finite as

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(a)

(b)

FIG. 2. k||-resolved conductance G(kx, ky ) for different magnetization angles φ for λR = 1eVÅandλD = 0 (a) and λR = λD = 1 eV Å (b). Arrows indicate spin-orbit field orientation. Red color specifies the spin-orbit field being parallel to magnetization. revealed in Fig. 3 by the oscillatory variation of the nor- −P is equivalent to the space inversion operation, which malized conductance g(φ) = [(G(φ) − G(0))/G(0)] with changes the wavevector from k to −k but preserves the the magnetization angle. This variation is well described spin σ . Applying the time-reversal symmetry operation to by g(φ) ∝ sin2φ typical for AMR in general. For a given this state with reversed polarization, we transform −k back value of λR (e.g., λR = 1 eV Å in Fig. 3), the amplitude of to k, but flip the spin, changing it from σ to −σ . Thus, the ◦ TAMR, which is defined as η = g(90 ), increases with λD reversed-polarization state is identical to the original state in a linear fashion (Fig. 3). with the same k, but reversed σ . For a FTJ, this property Figure 4(a) shows the calculated amplitude of TAMR as has been predicted to be responsible for the change of sign a function of λR and λD. We find that this variation quan- of the TAHE [45]. titatively follows the relationship η ∝ λRλD. Overall, both In case of TAMR, however, the situation is different. the amplitude of TAMR as a function of the SOC constants Due to the TAMR effect being a second order in SOC (the and its angular dependence are well described by a simple first-order term in the expansion is an odd function of k formula and thus vanishes after integration over k||)[47], reversal

2 g(φ) ∝ λRλDsin φ. (11)

This formula follows from the expression for the conduc- tance derived using second-order perturbation theory for SOC [47]. Figure 4(b) shows the TAMR amplitude η as a func- tion of the exchange constant J in the FM layer for λR = λD = 1 eV Å. As expected, η increases with J from zero to finite values due to the increasing disbalance between the number of majority and minority spin carriers in the FM electrode. As follows from the discussion above, the TAMR effect in FTJs is driven by the SOC in the FE barrier layer due to broken inversion symmetry, producing linearity in the wavevector k Rashba and Dresselhaus SOC effects. Switching FE polarization P in bulk ferroelectric leads to a FIG. 3. Normalized conductanceg(φ) = [G(φ) − G(0)]/G(0) full reversal of the spin texture as was originally proposed as a function of magnetization angle φ for λR = 1eVÅand for GeTe [36]. This is due to the fact that a change of P to different values of λD.

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(a) (b) orientation. This nearly two-fold change in the abso- lute value of the polarization is mirrored in the change λ+ = of the SOC parameters. For example, if R(D) 1eVÅ λ ≈ λ− ≈− (i.e., R(D) 0.78 eV Å) then R(D) 0.56 eV Å. Thus, reversal of the FE polarization leads to a nearly two-fold change in the SOC constants, which according to Eq. (11) is expected to produce about a four-fold change in TAMR. Figure 5 shows the calculated TAMR curves in polar coordinates, which illustrate this behavior [49]. An interesting observation that follows from our calculations is a sizable effect of the SOC parameters on conductance. Since the SOC parameters are affected FIG. 4. Amplitude of TAMR η = [G(90◦) − G(0)]/G(0) as a by the orientation of FE polarization, we argue that this function of λR and λD (a) and exchange constant J for λR = λD = manifests an alternative mechanism of TER. Figures 6(a) 1 eV Å (orange line) and λR = λD = 0.5 eV Å (green line) (b). and 6(b) show the calculated k||-resolved conductance for purely Rashba SOC (λD = 0), when it changes from λD = 1 eV Å [Fig. 6(a)]toλR =−0.56 eV Å [Fig. 6(b)] of FE polarization leading to the reversal of spin at a given with reversal of the FE polarization. We find that the k|| does not change the conductance and hence the TAMR. total conductance of the FTJ changes from about 27.4 to Nevertheless, in realistic FTJs, the effect of polarization 9.9 −1 cm−2. The effect stems entirely from changes in switching on TAMR is expected to occur due to the intrin- the electronic band structure of the FE barrier driven by sic asymmetry of FTJs. This asymmetry is known to appear SOC. As evident from comparison of Figs. 6(a) and 6(b), as a result of different screening lengths and/or work func- the reduction in the Rashba SOC shrinks the k|| space tions of the electrodes or different interface terminations, where the conductance is sizable, and also, according to leading to the TER effect [3]. Eq. (9), enhances the barrier height. The effect is even To illustrate the effect of asymmetry, we assume that a more pronounced for λR = λD [Figs. 6(c) and 6(d)]. When FTJ has FM and NM electrodes with different work func- the SOC parameters change from 1 eV Å [Fig. 6(c)]to tions, resulting in a built-in electric field. This electric −0.56 eV Å [Fig. 6(d)], the conductance minima become field, pointing in a fixed direction, is expected to affect the closer, and the effective barrier height, according to Eq. SOC parameters when the FE polarization is reversed, and (10), is reduced (stronger than in the case of pure Rashba thus to change TAMR. If W is the difference in work SOC) resulting in a conductance change from about 318 functions of the two electrodes, the induced electric field to 19.1 −1 cm−2. Such a large change in the conductance = / in a FE barrier layer is given by Ei W ea, where e corresponds to an ON/OFF ratio of about 17, which is is an elementary charge and a is the barrier thickness. comparable to the TER values predicted earlier [4]. This field induces polarization Pi = Ei( − 0), where  is the background dielectric permittivity of the ferroelectric. The induced polarization has a fixed orientation and hence adds to or subtracts from the spontaneous polarization P. The total polarization is given by P + Pi or −P + Pi, depending on the spontaneous polarization being parallel or antiparallel to the built-in electric field, respectively. For simplicity, we assume that the spin-orbit field and the SOC constants are proportional to the total polarization P [48]. If the spontaneous polarization P of a bulk ferro- electric gives rise to the SOC constants λR(D) in a FTJ with the induced polarization Pi due to the built-in electric field, λ± =±λ ( ± )/ these constants become R(D) R(D) P Pi P, where the sign +(–) designates polarization pointing parallel (antiparallel) to the built-in electric field. For an estimate, we consider representative parame- ters W = 1eV, a = 2 nm, and  = 300, which lead = to the built-in field Ei 0.5 V/nm and induced polar- FIG. 5. Conductance of a FTJ as a function of magnetization ≈ μ 2 ization Pi 14 C/cm . If the spontaneous polarization angle φ (given in degrees) for polarization P parallel (λR = λD = = 2 + = is P 50 μC/cm , the total polarization is P Pi 1 eV Å, orange line) and antiparallel (λR = λD =−0.56 eV Å, 2 2 64 μC/cm or −P + Pi =−36 μC/cm , depending on its cyan line) to the built-in electric field Ei.

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(a) (b) magnetization angles, we analyze the effect of the SOC. For realistic parameters of the model, we find sizable TAMR values, which can be measured experimentally. We argue that a built-in electric field in asymmetric FTJs affects the SOC parameters and leads to a change in TAMR when FE polarization of the barrier layer is reversed. This prediction is elaborated, using an example of a FTJ with electrodes of different work functions. Finally, we find (c) (d) that a change in the SOC parameters with polarization switching leads to a sizable change in conductance, which manifests an alternative mechanism of TER. We hope that our theoretical predictions will stimulate experimental studies of TAMR in FTJs.

ACKNOWLEDGMENTS The authors thank Dr. Andrei Zenkevich and Dr. Lin- gling Tao for helpful discussions. This work was finan- cially supported by the Moscow Institute of Physics and FIG. 6. k||-resolved conductance for λ = 1eVÅandλ = 0 R D Technology (MIPT) and by the Russian Science Founda- (a), λ =−0.56 eV Å and λ = 0(b),λ = λ = 1eVÅ,(c) R D R D tion (Grant No. 18-12-00434). and λR = λD =−0.56 eV Å (d). The magnetization angle is fixed at φ = 0.

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