Efficient domain wall motion in asymmetric magnetic tunnel junctions with vertical current flow
EEfficient domain wall motion in asymmetric magnetic tunnel junctions with verticalcurrent flow
S. Liu, ∗ D. J. P. de Sousa, † M. Sammon, ‡ J. P. Wang, § and T. Low ¶ Department of Electrical and Computer Engineering,University of Minnesota, Minneapolis, Minnesota 55455, USA (Dated: January 6, 2021)In this paper, we study the domain wall motion induced by vertical current flow in asymmetricmagnetic tunnel junctions. The domain wall motion in the free layer is mainly dictated by thecurrent-induced field-like torque acting on it. We show that as we increase the MTJ asymmetry,by considering dissimilar ferromagnetic contacts, a linear-in-voltage field-like torque behavior is ac-companied by an enhancement in the domain wall displacement efficiency and a higher degree ofbidirectional propagation. Our analysis is based on a combination of a quantum transport modeland magnetization dynamics as described by the Landau-Lifshitz-Gilbert equation, along with com-parison to the intrinsic characteristics of a benchmark in-plane current injection domain wall device.
PACS numbers: 71.10.Pm, 73.22.-f, 73.63.-b
INTRODUCTION
Control of magnetic domain wall (DW) motion innanowire-like structures using current injection has beenof interest for memory applications due to promisingqualities of low power consumption and non-volatility[1]. The typical mode of operation for DW devices iscurrent injection in-plane (CIP) along the propagationdirection of the DW [2–5]. However, current injectionperpendicular to the plane (CPP) has the potential tobe more space-efficient since it allows vertical integrationof device structures. More importantly, CPP DW prop-agation has shown DW velocities up to 100 times fasterthan that of a CIP configuration at similar current den-sities [6–8]. This CPP configuration has the form of aperpendicular magnetic tunnel junction (MTJ), consist-ing of fixed and free magnetization layers that sandwicha MgO insulating layer shown in Fig. 1(a).Current-induced propagation of the DW in the freelayer of the MTJ relies on spin transfer torque (STT),where spin-polarized electrons interact with the mis-aligned magnetization of the DW to induce movement.The contribution of STT to magnetization dynamics inin-plane magnetized MTJs can be decomposed into twoterms: a torque acting in-plane that will be referred to asthe damping-like (DL) torque and an out-of-plane torquewhich is also known as the field-like (FL) torque. Insuch systems, it was demonstrated that the FL torqueis the main contributor to DW motion [7, 8]. This isdue to the fact that in in-plane magnetized thin films,magnetic charges at top and bottom surfaces act to sup-press the DL torque contribution as it tends to rotate themagnetization out-of-plane, leaving only the FL torque-induced DW displacements [7, 8]. In addition, the volt-age characteristics of these current-induced torques andtheir relative magnitudes were shown to depend on thedegree of asymmetry of MTJ devices having dissimi-lar ferromagnetic contacts [9–11]. Most notably, the C u rr e n t f l o w Domain wall displacement xyz
Insulating spacerDomain wall (a) kE Topferromagnet Bottomferromagnet (b) E F δ FIG. 1: (Color online)
Domain wall (DW) device withvertical current flow . (a) We consider a magnetic tunneljunction where an in-plane DW displacement in the free layeris induced by a vertical current flow. The DW on the toplayer is represented by shaded red region. In this picture,tunneling electrons transfer spin angular momentum to themisaligned moments in the free layer via spin transfer torque,giving rise to a DW displacement. Top (free) and bottom(fixed) layer are assumed to be dissimilar ferromagnets, asrepresented by blue and green layers. (b) sketch of the top andbottom layer bands. Red and blue bands represent minorityand majority spin bands, respectively, while the blue shadedregions represent the filled states for each ferromagnetic layer.The band filling of the bottom layer differs from the top oneby a quantity δ , which is the asymmetry parameter. asymmetry-induced tunability of the voltage dependenceof the FL torque has important energy efficiency conse-quences when considering DW motion in the free layerin the CPP configuration, which has not been exploredto-date.In this work, we demonstrate that by increasing theasymmetry between the fixed and free layers of a MTJ,the voltage dependence of the FL torque is modulated,which leads to significantly improved current density andenergy efficiency of DW propagation in the free layer. We a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n also demonstrate that unlike symmetric MTJs, bidirec-tional DW motion can be achieved in asymmetric MTJs,allowing for new switching modality in these devices.Our analysis suggests that the faster DW velocities, inconjunction with improvements in energy efficiency withasymmetric MTJs can allow a CPP device to outperformcounterpart CIP DW devices. THEORETICAL MODEL
The system is sketch in Fig. 1(a). A top ferromagneticlayer with a DW, represented by the red shaded region,is separated from the bottom magnetic layer (fixed layer)with a single domain along the x direction by a thin insu-lating spacer. We assume that the top and bottom ferro-magnets have in-plane magnetization with different bandfillings as sketched by the blue shaded regions in the banddiagrams of Fig. 1(b). The MTJ asymmetry is controlledby the band filling difference via the parameter δ . In thissection, we present the theoretical model employed in thedescription of DW motion due to current-induced torquemediated by the spin polarized tunneling electrons. Adescription of the quantum transport model used for in-vestigating the evolution of spin transfer torques withthe MTJ asymmetry is followed by the analysis of theLandau-Lifshitz-Gilbert (LLG) equation applied to theDW motion in the free layer. Current-induced torque
Following Refs. [9, 10, 12–14], we describe the non-equilibrium properties of the MTJ with a single-orbitaltight-binding (TB) model in combination with quan-tum transport simulations. The system is composedof two semi-infinite magnetic leads coupled to an in-sulating spacer via spin-independent hopping matrices H I,T ( B ) = tσ , where σ = I × is the 2 × t is the associated hopping pa-rameter. The Hamiltonian of the coupled system has theblock tridiagonal form H = H T H T,I H † T,I H I H I,B H † I,B H B , (1)Where H α , with α = T , I , B, refers to the Hamiltonianblock describing the top ferromagnet (T), the insulatingspacer (I) and the bottom ferromagnet (B), respectively.Each region is assumed to be composed of several atomicplanes and is described by the TB Hamiltonian H α = (cid:88) ij,σσ (cid:48) H σσ (cid:48) ij,α c † iσ c jσ (cid:48) , (2)where the sum over atomic plane indexes i, j extends overnearest-neighbors only and c † iσ ( c iσ ) creates (annihilates) an electron with spin σ in the i -th atomic plane. Eachatomic monolayer is assumed to be translationally invari-ant along the in-plane directions. For simplicity, we as-sume that all hopping parameters are spin-independentwith the same value t = − (cid:15) σ ( k || ), where k || refers tothe in-plane momenta, and nearest-neighbor hopping pa-rameter t as H σσ (cid:48) ij = δ σσ (cid:48) δ ij (cid:15) σ ( k || ) + δ σσ (cid:48) ( δ i,j +1 + δ i,j − ) t where σ = ↑ , ↓ is the spin index and δ ij ( σσ (cid:48) ) is the Kro-necker delta for real (spin) space labels. Assuming theinterface is oriented along the xy plane, the translationinvariance along the in-plane directions allows us to ex-press (cid:15) σ ( k || ) = (cid:15) σ + 2 t (cos( k x ) + cos( k y )).For the calculations presented in this paper, we as-sume (cid:15) ↑ − E F = 3 . (cid:15) ↓ − E F = 5 . x (cid:48) direction and (cid:15) ↑ − E F = 3 . δ eV, (cid:15) ↓ − E F = 5 . δ eV for the B ferromagnetic layer (fixedlayer) with spin quantization axis along the x direction,where the Fermi level is fixed at E F = 0 eV and the pa-rameter δ controls the asymmetry between the magneticlayers [9–11]. Additionally, the tunnel barrier is assumedto be composed of N = 3 atomic planes with onsite ener-gies (cid:15) ↑ = (cid:15) ↓ = 9 . t ↑ = t ↓ = − . i -th atomic plane of the top lead is T i = − [ ∇ · Q ] i , where [ ∇ · Q ] i = Q i − ,i − Q i,i +1 is thedivergence of the spin current where Q ij = 14 π (cid:90) BZ d k || (2 π ) (cid:90) dE Tr σ [( H ji G 2, in order to obtain their dependenceover all important parameters for our analysis. When -0.4 -0.2 0.0 0.2 0.4-0.20.00.20.4 Voltage (V) T D L ( µ e V / Å ) δ = 0.0 eV δ = 0.6 eV δ = 1.2 eV δ = 1.8 eV -0.4 -0.2 0.0 0.2 0.4-0.6-0.4-0.20.00.2 T F L ( µ e V / Å ) Voltage (V) -0.4 -0.2 0.0 0.2 0.4-2.0-1.5-1.0-0.50.00.51.01.52.0 Voltage (V) J ( x A / c m ) (a) (b) (c) FIG. 2: (Color online) Voltage dependence of current-induced torques and current density . Voltage dependence ofthe (a) DL, (b) FL current-induced torques and (c) current density for asymmetric MTJs for several asymmetry parameters δ .The horizontal dashed lines highlight the equilibrium value of these quantities. the system is driven out of equilibrium under an appliedvoltage µ T − µ B = eV we assume that the potential dropslinearly inside the oxide layer.Figure 2 displays the voltage dependence of DL and FLtorques, as well as the charge current density for severaldifferent values of the asymmetry parameter δ . The hor-izontal dashed lines highlight the equilibrium values (at µ T = µ B ) of these quantities. As is apparent from theblue symbols in Fig. 2(a), the current-induced DL torqueexerted on the free layer of a symmetric MTJ ( δ = 0eV) switches sign as one reverses the voltage polarity,being more efficient when spin polarized electrons flowfrom the fixed to the free layer, i.e., at V > δ (cid:54) = 0 eV cases), but nowwith a smaller voltage modulation, i.e., smaller torquestrength at a given voltage. It is worth emphasizing thatthe voltage modulation of the current density also dropswith increasing δ such that the torque efficiency, definedas T DL /J , might actually increase within some asymme-try window where the current density drops faster [11].The voltage dependence of the current-induced FLtorque acting on the free layer is displayed in Fig. 2(b) forseveral asymmetry parameters δ . First, one notices thatthe interlayer exchange coupling (IEC), being the mag-nitude of this torque at equilibrium, i.e., at V = 0 V,changes as one tune the asymmetry of the MTJ. Thisfeature, that is apparent by the shift in the horizon-tal dashed lines, can be understood in terms of Bruno’smodel for IEC [20], where its strength depends on therelative value of the reflection coefficients of majorityand minority spins at both oxide/ferromagnet interfaces.Hence, as δ increases, electrons scatter differently as theyencounter a larger energy barrier at the B interface, mod- ifying the strength of the IEC between fixed and freelayer mediated by the insulating spacer. Secondly, whilethe DL torque always presents qualitatively similar volt-age dependencies with varying δ , the FL torque mightcompletely change its voltage dependence from purelyquadratic ( δ = 0 eV) to linear (see, for example, the δ = 1 . Landau-Lifshitz-Gilbert equation The magnetization dynamics are described by the LLGequation with an added spin transfer torque term d m dt = − γ m × H eff + α m × d m dt + γµ M S t free T ST T , (4)where m is the magnetization of the free layer normalizedwith saturation magnetization M S , γ is the gyromagneticratio, α is the Gilbert damping parameter, µ is the vac-uum permeability, and t free is the thickness of the freelayer. H eff is the effective magnetic field due to magne-tostatic, exchange, and anisotropy energies added to anyexternal magnetic field.The total torque density due to STT is described as T ST T = T F L m × m p + T DL m × ( m × m p ) , (5)where m ( m p ) is the magnetization direction of the free(pinned) layer, T DL is the magnitude of DL torque, and T F L is the magnitude of FL torque. The FL torque is alsounderstood as the non-equilibrium interlayer exchangecoupling.With the results of T DL and T F L from the quan-tum transport calculation in the previous section, themagnetization dynamics are solved using MuMax3, aGPU-accelerated micromagnetics simulator [21]. Thesystem simulated in MuMax3 is a Fe/MgO/Fe rectan-gular nanowire where the dimensions of the free layerare 15 nm wide and 3 nm thick. To evaluate the ve-locity of the DW, the length of the system is treated asinfinite. The magnetic parameters used are of Fe, with M S = 1 . × A/m, exchange stiffness A ex = 1 . × − J/m, and Gilbert damping constant α = 0 . . × . × . . Current isinjected in the +z direction and is uniformly distributedacross the wire cross-section. The calculation assumeszero temperature. The non-equilibrium torque densitycontributions as a function of voltage from the quantumtransport calculation are interpolated to be used in themicromagnetics simulation. RESULTS In this section, we present the DW propagation resultsalong with analysis using a 1-dimensional model. We alsopresent intrinsic delay and energy efficiency benchmarksof a proposed vertical current injection logic device anddiscuss the advantages of a MTJ structure. Domain wall velocity To produce the micromagnetics results of DW velocitydependence on current density, a constant voltage wasapplied perpendicularly across the MTJ where a positivevoltage leads to electron flow from the fixed layer to thefree layer. The position of the DW as a function of timewas tracked using the average +x-direction magnetiza-tion of the nanowire and the steady-state velocity of theDW was recorded. This was repeated multiple times,changing the voltage for each trial. Using the averagecurrent density per voltage shown in Fig. 2, the depen-dence of DW velocity on current density can be obtained.It can be seen in Fig. 3(a) that for the symmetric case,DW velocity is characteristically quadratic, leading tounidirectional motion. With increasing asymmetry, thedependence of DW velocity on current density becomescharacteristically linear which leads to bidirectional DWmotion in the small current density regime. The change -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5-300-200-1000100200 δ = 0.0 eV δ = 0.6 eV δ = 1.2 eV δ = 1.8 eV v D W ( m / s ) J (x10 A/cm ) (a) (b) v D W / J ( x c m / A ⋅ s ) δ (eV) J = 0.5 x 10 A/cm2J = 1.0 x 10 A/cm2J = 1.5 x 10 A/cm2J = 2.0 x 10 A/cm2 UnidirectionalBidirectional (c) -0.100.10 1 2 3 4 5 6-40-2002040 V Pulse D W po s i t i on ( n m ) time (ns) δ = 0.0 eV δ = 1.8 eV V o l t age ( V ) FIG. 3: (Color online) DW velocity characteristics forasymmetric MTJs . (a) DW velocity as a function of the av-erage current density for different asymmetries δ s. The sym-bols were obtained numerically by solving the LLG equationfor the rectangular nanowire with current-induced torques ob-tained from the quantum transport simulations. The solidlines are the analytical toy model results. (b) Asymmetry-dependent DW velocity per applied current density. Differentsymbols/curves were obtained by considering different appliedcurrent densities. Unidirectional and bidirectional highlightedregions for negative and positive V DW /J , respectively, refersto the directionality of DW motion for each asymmetry case.(c) DW position for two cases δ = 0 . δ = 1 . from even to odd dependence in voltage is reflective of thequadratic and linear FL torque dependence with voltagein symmetric and asymmetric MTJs, respectively. Addi-tionally, the results indicate that a velocity of over 100m/s can be obtained with current density 1 × A/cm for a more asymmetric MTJ, which is more efficient thanthat of in-plane current injection [1].A 1-dimensional analytical approximation described byKhvalkovskiy et al. can be used to characterize the DWmotion [7]: u ∆ − α ˙Φ − γ T DL M S t free = γM S K x sin(2Φ) , (6a) − α u ∆ − ˙Φ + γ T F L M S t free = 0 . (6b)Here, u is the DW velocity, ∆ is the width of the DW, K x is the anisotropy constant in the x-direction, and Φis the precessional angle of magnetization of the DW. u = γT F L αM S t free ∆ (7)The steady state solution of the analytical approximationis ˙Φ = 0, which yields the expression shown in Eq. (7).This approximation indicates that DW velocity in thisconfiguration is solely dependent on FL torque. UtilizingDW width ∆ as the fitting parameter, ∆ = 11 . . δ = 0 eV) is negative, indicating unidirectionalDW propagation. This is in contrast to the case of anasymmetric MTJ, where bidirecitonal DW propagationallows for greater control of DW dynamics.An important point to emphasize is the bidirection-ality of the DW motion at different asymmetry levels.Figure 3(a) and (b) suggest that DW motion in symmet-ric MTJs, having δ = 0 . δ (cid:54) = 0 eV) are moreappropriate for applications were the bidirectionality ofthe DW movement via electrical means is required. Toillustrate this difference, we plotted the DW position asa function of time for symmetric ( δ = 0 . δ = 1 . . t = 3 ns, as shown in Fig. 3(c). Before t = 3 ns wherethe voltage is positive, DWs for symmetric and asym-metric devices move in opposite directions. After thevoltage polarity reversal at t = 3 ns, one observes thatDW motion of symmetric MTJ is not reversed while thatof the asymmetric device is, i.e., asymmetric devices offera greater flexibility when considering voltage control ofDW motion under vertical current flow. We also observethat the DW in asymmetric MTJs propagates much far-ther than DW in symmetric devices, as is apparent inFig. 3(c). Eint (x10-16 J) d ( e V ) R e t a n g u l a r 0 . 2 5 n s r a m p G a u s s i a n 0 . 5 0 n s r a m p B e n c h 0 . 7 5 n s r a m p 1 . 0 0 n s r a m p FIG. 4: (Color online) CPP vs. CIP . CPP device intrinsicenergy dissipation considering different voltage pulses: Rect-angular (open squares), ramp with different rise times rangingfrom 0.25 ns to 1.00 ns (filled symbols) and Gaussian (opencircle) shaped voltage pulses. The horizontal dashed line isthe CIP benchmark from Ref. [22] Intrinsic device benchmarking In this section, a comparison in terms of energy dissi-pation is made between the presented CPP device and abenchmark CIP device. While most benchmarks requirea device architecture beyond the scope of this paper, in-trinsic benchmark comparisons can be made. The bench-mark device is a 3-terminal MTJ device utilizing in-planecurrent injection [5]. Using the same methodology as inNikonov et al., the intrinsic energy dissipation for a logicoperation E int can be calculated as follows [22]: E int = (cid:90) t int V ( t ) ∗ [ J ( t ) ∗ L ∗ W ] dt (8)where t int is the time delay taken for the DW to propa-gate past the sensing MTJ, V ( t ) and I ( t ) are the voltageand current, and L and W are the length and width ofthe channel.In order to make comparisons to the in-plane currentinjection device, the time delay is set at t int = 1 . W = 15 nm and t = 1 . δ = 0 eV) is not included because it is not possible toobtain a time delay t int = 1 . 77 ns. For the degrees ofasymmetry presented, the energy dissipation is generallymonotonically decreasing for increasing asymmetry, indi-cating increased device efficiency. However, for the 0.75and 1.00 ns ramps, there is a slight increase in energydissipation between δ = 1 . δ = 1 . δ = 1 . δ , it can be concluded that device intrinsic efficiencyincreases as device asymmetry increases. Additionally,the efficiency of the proposed CPP device can outper-form the CIP benchmark device (horizontal dashed line)with appropriate tuning of device asymmetry. It is alsofound that the Gaussian pulse (more adiabatic) results inthe most efficient switching while the rectangular pulse(more abrupt) generally results in the least energy effi-cient switching. CONCLUSIONS To summarize our results, we use a quantum trans-port model in conjunction with a micromagnetics modeldescribed by the LLG equation to demonstrate thatDW propagation efficiency in a vertical current injectionMTJ with in-plane magnetization can be significantly in-creased by increasing the asymmetry of the ferromagneticcontacts. This is due to the fact that DW propagationwith this geometry is dominated by the FL torque; asthe asymmetry of the MTJ is increased, the voltage de-pendence of the FL torque acting on the free layer is al-tered. We present two different methods to quantify DWpropagation efficiency, with current density as well as en-ergy dissipation of a proposed device that utilizes verticalcurrent injection. In both cases, there is a clear generaltrend of increased efficiency with increased asymmetry.In particular, this increased asymmetry can significantlyimprove the performance of the proposed device and po-tentially outperform a CIP benchmark device. Due tothe current efficiency limitations of vertical current in-jection, it may not be perfectly suitable to implementin a logic device. However, these results have promis-ing implications for applications like spintronic intercon-nects, where vertical current injection has already beenshown to be effective [23]. We show that by tuning theasymmetry of a vertical current injection magnetizationin-plane MTJ, increased DW propagation efficiency aswell as bidirectional DW motion can be achieved. Acknowledgments . DS, TL, and JPW were partiallysupported by DARPA ERI FRANC program. We ac-knowledge useful discussions with D. E. Nikonov fromIntel Corporation. ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] ¶ Electronic address: [email protected][1] F. Matsukura, Y. Tokura and H. Ohno, Nat. Nanotech-nol. , 209 (2015).[2] G. S. D. Beach, M. Tsoi and J. L. Erskine, J. Magn.Magn. Mater. , 1272 (2008).[3] M. Yamanouchi, D. Chiba, F. Matsukura and H. Ohno,Nature , 539 (2004).[4] J. Grollier, P. Boulenc, V. Cros, A. Hamzi´c, A. Vaur`es,A. Fert and G. Faini, Appl. Phys. Lett. , 509 (2003).[5] J. A. Currivan-Incorvia, S. Siddiqui, S. Dutta, E. R.Evarts, J. Zhang, D. Bono, C. A. Ross and M. A. Baldo,Nat. Comm. , 3 (2016).[6] 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).[7] A. V. Khvalkovskiy, K. A. Zvezdin, Ya. V. Gorbunov, V.Cros, J. Grollier, A. Fert and A. K. Zvezdin, Phys. Rev.Lett. , 1 (2009).[8] C. Boone, J. A. Katine, J. R. Childress, X. Cheng and I.N. Krivotorov, Phys. Rev. Lett. , 1 (2010).[9] Y.-H. Tang, Nicholas Kioussis, Alan Kalitsov, W. H. But-ler, and Roberto Car, Phys. Rev. B , 054437 (2010).[10] Y.-H. Tang, Nicholas Kioussis, Alan Kalitsov, W. H.Butler, and Roberto Car, Phys. Rev. Lett. , 057206(2009).[11] D. J. P. de Sousa, P. M. Haney, D. L. Zhang, J. P. Wangand T. Low, Phys. Rev. B , 81404 (2020).[12] Ioannis Theodonis, Nicholas Kioussis, Alan Kalitsov,Mairbek Chshiev, and W. H. Butler, Phys. Rev. Lett. , 237205 (2006).[13] Alan Kalitsov, Mairbek Chshiev, Ioannis Theodonis,Nicholas Kioussis, and W. H. Butler, Phys. Rev. B ,174416 (2009).[14] D. Datta, B. Behin-Aein, S. Datta and Sayeef Salahud-din, IEEE. Trans. Nanotech. , 1536 (2011).[15] H. X. Yang, M. Chshiev, A. Kalitsov, A. Schuhl, and W.H. Butler, Appl. Phys. Lett. , 262509 (2010).[16] A. Kalitsov, P.-J. Zermatten, F. Bonell, G. Gaudin, S.Andrieu, C. Tiusan, M. Chshiev, and J. P. Velev, J.Phys.: Condens. Matter , 496005 (2013).[17] A. Kalitsov, W. Silvestre, M. Chshiev, and J. P. Velev,Phys. Rev. B , 104430 (2013).[18] J. Z. Sun, Phys. Rev. B , 570 (2000).[19] J. Xiao, A. Zangwill, and M. D. Stiles, Phys. Rev. B ,014446 (2005).[20] P. Bruno, Phys. Rev. B , 411 (1995).[21] A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F.Garcia-Sanchez and B. Van Waeyenberge, AIP Adv. ,107133 (2014).[22] D. E. Nikonov and I. A. Young, Proc. IEEE , 2498 (2013).[23] D. E. Nikonov, S. Manipatruni and I. A. Young, J. Appl. Phys.115