Abstract
The molecular scale magnetic proximity effect is proposed in single-molecule magnetic junctions (SMMJs) consisting of a dissociated amine-ended 1,4-benzenediamine (BDA) molecule coupled to two ferromagnetic Co electrodes. Our self-developed JunPy + Landau-Lifshitz-Gilbert simulation combined with first-principles calculation is employed to investigate the role of contact geometry in the magnetotransport properties of SMMJs with the choice of top, bridge, and hollow contact sites. The strong spinterface effect gives rise to distinct angular dependence of equilibrium field-like spin torque (FLST), asymmetric magnetic hysteresis loop and tunable exchange bias. From the analytical derivation of nonequilibrium Keldysh formalism, we believe that a promising way forward is to activate the multi-reflection process via the so-called molecular spinterface that will allow us to conquer as-yet unexplored magnetotransport properties of organic-based spintronics.
1 Introduction
Multi-toggling of magnetism in nanoscale magnetic heterostructures is significant for both fundamental and application in energy-efficient magnetic data storage, such as computer hard disks and magnetic random access memories (MRAMs). Among the hottest topics in contemporary spintronics, the spinterface [–] effect plays a crucial role to modulate the magnetic proximity effects via magnetic field, electric field, mechanical strain, and so on. Much effort has been devoted to solid-state magnetic devices, since the spin polarization and spin-orbit coupling (SOC) are decisive factors in spin transport and magnetic proximity [–], such as magnetic anisotropy, exchange bias (EB), and magnetic coercivity.
Electrical and spin switches across a single organic molecule connecting ferromagnetic electrodes are also burgeoning fields for possible applications in nano-spintronics devices [–], since chemical design offers various ways to incorporate spin degrees of freedom into a molecule to form the so-called molecular spintronics. Currently, most theoretical works [–] focus on the magnetoresistance and the spin-polarized transport in collinear magnetic configurations. The ability to calculate the noncollinear spin torque effect and spin dynamics of magnetic heterostructures remains difficult but important to include the complex structural, electronic, and magnetic properties at spinterfaces for nanoscale spintronics devices.
We introduce in Figure 1 the four steps of DFT + JunPy + LLG calculation procedure, including the density functional theory (DFT) calculation with our self-developed JunPy + LLG simulation, to investigate the magnetoelectric and magnetotransport properties of complex magnetic heterostructures, such as magnetic tunnel junctions (MTJs) [] and single-molecule magnetic junctions (SMMJs) [,]. In this study, we propose the prototypical Co/1,4-benzenediamine (BDA)/Co SMMJs with three kinds of contact geometries for top (BDA-T), bridge (BDA-B), and hollow (BDA-H) contact sites of the N ion bonding to one, two, and three Co apex atoms, respectively. Since the hybridization between Co-d, N-py, and π-orbital of the phenyl ring preserve the spin-up pronounced resonance channel [], the DFT + JunPy + LLG calculation reveals exchange bias toggling via the interplay between spinterface enhanced equilibrium field-like spin torque (FLST) and coercive field of Co electrode. We further use the nonequilibrium Keldysh formalism to clarify the crucial role of multi-reflection processes at interfaces in the non-sinusoidal angular dependence of equilibrium FLST, which may pave the way for unexplored magnetotransport properties of organic-based spintronics.
FIGURE 1
2 Calculation methods
The first step of DFT + JunPy + LLG calculation is to carry out the first-principles calculation including the complex charge transfer and spinterfacial effect via self-consistent process. In the top of Figures 2A–C, these amine-ended SMMJs are composed of a dissociated 1,4-benzenediamine (BDA) sandwiched by two Co hcp [0001] oriented semi-infinite nanowires. To prevent coupling between SMMJs, we set the lateral separation between two neighboring junctions as 7 Å in both x- and y-directions. The junction geometry is optimized by the Vienna Ab initio simulation package (VASP) [–] with DFT based generalized gradient approximation (GGA) in the Perdew, Burke, and Ernzerhof (PBE) [] form. The lattice constant of Co nanowire is fixed at 2.5 Å, and the Co apex atoms and the central BDA molecule are fully relaxed by using the force criteria of 0.02 eV/Å, the cut-off energy of plane wave basis set as 700 eV, a force criterion of 0.02 eV/Å, the total energy difference for electronic steps as 10–5 eV, and the k-point sampling of Γ-point. Based on the Pearson’s principle of hard and soft acids and bases (HSAB) [], the strong coupling between hard metal (Co) and hard base (N) in amine-ended SMMJs favors the covalent bonding between H-dissociated amine linker and Co adatom. This may provide variability in linker–electrode contact geometry, which is crucial but usually not easily controlled during the fabrication of real SMMJs especially for breaking junction techniques []. Here we adopt three possible contact geometries, i.e., top (T-III′), bridge (B-II), and hollow (H-III) contact sites proposed in Figure 1 of Ref. [], as long as the hydrogen ion is dissociated to form a covalent bond between Co-d and N-px,y orbitals. On the one hand the N ion bonds to the central phenyl ring, and on the other hand the amine linker tends to dissociate one (two) H ion to form one (two/three) bonding with Co adatom in T (B/H) case to fulfill the octet rule. This thus gives rise to the shorter optimized Co-N bond length of 1.84 Å in both BDA-T and BDA-B cases but the longer one of 1.95 Å in BDA-H case. By using the optimized junction geometries, we next use the two-probe structure with DFT and non-equilibrium Green’s function (NEGF) formalism implemented in Nanodcal transport package [–] to investigate the spin-polarized transmission spectum and spinterface effect of SMMJs []. The double-ζ double-polarized basis set of local numerical orbitals are applied to all ions, the cut-off energy of real space grid density is 150 Hartree, and the k-point samplings are 1 × 1 × 100 and Γ point for semi-infinite Co electrode and central device, respectively.
FIGURE 2
In the second and third steps, our self-developed JunPy package [, , ] has successfully combined the NEGF and spin torque theory to calculate the angular dependence of equilibrium FLST, , in the noncollinear magnetic configurations, which can be formed by fixing along the z direction but rotating around the y axis with respect to the z axis by an angle θ. Since the weak spin-orbit coupling can be ignored in SMMJs, there are two components of noncollinear spin torque, i.e., the spin-transfer torque (STT, T‖) and the field-like spin torque (FLST, T⊥), originated from the spin accumulation of spin current density [], and only the non-zero equilibrium exists in the absence of an external current. For each angle θ, we repeat Nanodcal + JunPy + NEGF calculation to acquire the self-consistent junction Hamiltonian matrix , the overlap integral , and the reduced Hamiltonian matrix to resolve the lesser Green’s function matrix in the central device region. Here with the subscript n = 0 denotes the spin current passing through the interface between the right NH-linker and the right Co apex atoms, where the spin current density between the two atomic sites i and j can be calculated byand σy is the y-component of Pauli matrices. Thus, the net FLST acting on the right (free) Co electrode is defined as in the directions of .
The last step of macrospin dynamics simulation is to apply the generalized Landau-Lifshitz-Gilbert (LLG) equation with equilibrium FLST component of spin torque [, ] as expressed in the form ofwhere γ is the gyromagnetic ratio, α is the Gilbert damping constant, and are the effective fields along damping and precession directions of right Co electrode, respectively. At equilibrium, and represents the effective field induced by the equilibrium FLST, where Ms, tF and A are the volume magnetization saturation parameter, the thickness and the lateral area of right Co electrode, respectively. For Co electrode [], α = 0.01, μ0Hk = 76 mT, Ms = 1.27 × 106 A/m, the thickness tF = 100 nm, and the lateral area A = 1.63 × 10–19 m2 are chosen in this work.
3 Results and discussion
3.1 Effect of contact geometry on EB effect: DFT + JunPy + LLG
In the bottom of Figures 2A–C, we demonstrate the angular dependence of DFT + JunPy calculated equilibrium FLST fields, , for top (BDA-T), bridge (BDA-B), and hollow (BDA-H) contact geometries. Because of the covalent bonding between H-dissociated amine linker and Co apex atoms as shown in Figure 2 of Ref. [], all three cases exhibit pronounced π-resonant spin-up transmission near Fermi energy. Such spinterface effect assists strong enhancement of which is about one to two orders of magnitude larger than μ0Hk of Co electrode. To further investigate the effect of equilibrium FLST fields on the magnetization switching, our self-developed JunPy + LLG package is employed to solve the generalized LLG equation in Eq. 2. The DFT + JunPy + LLG calculated magnetic hysteresis curves (mz-H) for BDA-T, BDA-B, and BDA-H cases at zero temperature are presented in Figures 3A–C, respectively.
FIGURE 3
For the BDA-T case in Figure 2A, its exhibits the non-sinusoidal angular dependence with a positive value and a maximum below π/2. Note that the positive and negative magnitudes refer to the field-like and anti-field-like equilibrium fields, respectively. When θ⩽π/2, the large and positive magnitude implies that the free tends to move toward the parallel (P) magnetic configuration with θ = 0. In other words, such strong and positive equilibrium FLST field, i.e., where HK is cohesive field of right Co electrode, significantly postpones the P-to-AP magnetic switching (red line) at a much more negative external magnetic field (μ0Hext) and hence in turn causes large EB effect as shown in Figure 3A. Instead, for θ > π/2 the positive but smaller magnitude pushes the free away from the anti-parallel (AP) magnetic configuration with θ = π, that is to say, assists the AP-to-P magnetic switching (blue line) at less positive field and thus in turn leads to a highly asymmetric magnetic hysteresis loop as shown in Figure 3A.
Unlike BDA-T case, the smaller magnitude with nearly sin 2θ angular dependence of for BDA-B case is presented in Figure 2B. Its negative value for θ < π/2 and positive value for θ > π/2 both result in the fact of . Interestingly, their comparable magnitudes but opposite signs assist both P-to-AP (red line) and AP-to-P (blue-line) magnetic switching to form a symmetric but narrower magnetic hysteresis loop as shown in Figure 3B. On the other hand, the BDA-H case retains the sinusoidal angular dependence of similar to conventional MTJs [], due to its relatively larger Co-N bond length (1.95 Å) compared to those of BDA-T (1.84 Å) and BDA-B (1.84 Å) cases. However, the existence of spinterface effect of BDA-H case still gives notable and positive value of |H(0)|∼ HK for both θ < π/2 and θ > π/2 and an asymmetric and slightly shifted magnetic hysteresis loop as shown in Figure 3C.
To further demonstrate the validity of our self-developed LLG simulation, we carry out the well-known OOMMF software [] with the extension of SpinXferEvolve, but this extension simply uses default sinusoidal angular dependence of FLST in MTJs. Therefore, we revise it to include the user-defined angular dependence of FLST by fitting to our DFT + JunPy calculation in Co/BDA/Co SMMJs with three kinds of contact geometries as presented in Figures 2A–C. It is clear to find the excellent agreement between OOMMF simulation in Figures 4A–C and our own LLG calculation in Figures 4A–C.
FIGURE 4
3.2 Angular dependence of equilibrium FLST : Analytical derivation
Finally, we turn to investigate the underlying mechanism of non-sinusoidal angular dependence of equilibrium FLST field by using the NEGF method to derive analytical formalism of equilibrium FLST in Co/Barrier/Co MTJ with noncollinear magnetic configuration. In Figures 5A,B, the central barrier is considered as 1) the resonant tunneling barrier for BDA-based MTJs with strong spinterface effect and 2) the direct tunneling barrier for Co/BDMA/Co SMMJs where additional methylene (CH2) units are inserted between N-atom and the phenyl ring to form the 1,4-benzenedimethanamine (BDMA) molecule and then eventually destroy the spinterface effect. This is because CH2 unit well separates N-px,y orbital and π orbital of central phenyl ring near Fermi energy as shown in Figure 3 of Ref. [].
FIGURE 5
The net equilibrium FLST acting on the right (free) Co electrode can be defined by the one-dimensional tight-binding model with non-equilibrium Keldysh formalism [],where b and α′ denote the last site of barrier and first site of right Co electrode, respectively, and t is the coupling between barrier and the two Co electrodes. In consideration of multi-reflection process between two Co leads, the , , and presented in the bottom of Figure 5A denote the equilibrium FLST contributed by multi-reflection at one, two, and three left (L) or right (R) interfaces, respectively. The details of analytical derivation are arranged in the following. For Co/Barrier/Co MTJ, are the retarded, advanced, and lesser Green’s functions for the isolated left (L) Co, central barrier (B), and right (R) Co; t is the coupling between the two neighboring regions; and the coupled Green’s function matrix of the MTJ can be written as
Note that is a 4 × 4 matrix with matrix elements of , , , and , and all Green’s functions are expanded as 2 × 2 matrices in spin space. Following by the Dyson equation, we can use the equations of , , and to recastwhere
By substituting Eqs 5, 6 into Eq. 3, we can derive the corresponding , , and contributions of net in consideration of multi-reflection processes. It is worth to mention that Eq. 6 is similar to Eqs 7–9 of Xiao et al. [] for metallic spin valves due to its diffusive component to the spin-dependent reflection at interfaces.
We finally discuss the angular dependence of equilibrium FLST for noncollinear Co/BDA/Co(θ) SMMJ in Figure 5A. To simulate its strong spin-up dominated π-resonant tunneling, both and are complex numbers for (i,j)=(a,b) inside the resonant barrier, namely, the equilibrium FLST must consider all multi-reflection processes. We then summarize a general expression asto represent the t2N-th order of equilibrium FLST including the Nth multi-reflection processes of M and (N − M) times at left Co/N and right N/Co interfaces, respectively, and a0 and an’s are real numbers. Notably, this allows us to modulate the spinterface induced non-sinusoidal angular dependence of equilibrium FLST fields via the contact geometry in BDA-based SMMJs as shown in Figures 2A–C.
It is interesting to recall that BDA-H case exhibits pronounced π-resonant spin-up transmission near Fermi energy even though its Co-N bond length (1.95 Å) is relatively larger than those of BDA-T (1.84 Å) and BDA-B (1.84 Å) cases. This gives rise to a weaker spinterface effect of BDA-H case that still can assist the enhancement of equilibrium FLST, i.e., |H(0)|∼ HK, but the larger Co-N bond length significantly weakens those contributions from multi-reflection process and hence in turn preserves the sinusoidal angular dependence of as shown in Figure 2C. In sharp contrast, the central BDMA molecule of noncollinear Co/BDMA/Co(θ) SMMJ can be simplified as the direct tunneling barrier, due to the elimination of spinterface effect by inserting additional CH2 units as shown in Figure 5B. Similar to the insulating barrier in MTJs [], it has been known that exhibits sinusoidal angular dependence and a much smaller magnitude which is about two orders of magnitude smaller than that of BDA-H case. Therefore, the fact of does not cause EB effect and then remains symmetric P-to-AP and AP-to-P magnetic switching at ∓ HK as presented in Figures 5C,D.
4 Conclusion
We summarize the four steps of DFT + JunPy + LLG calculation, which successfully resolve computational difficulties in spin torque, magnetotransport and magnetic proximity for complex magnetic heterojunctions with noncollinear magnetic configurations. Here we propose three types of dissociated amine-ended Co/BDA/Co SMMJs with top, bridge, and hollow contact sites together with strong equilibrium FLST fields, i.e., . Our calculation results illustrate the underlying mechanism in an important aspect, namely, molecular scale exchange bias effect, via the modulation of angular dependence of equilibrium FLST. In consideration of spinterface induced resonant tunneling via central BDA molecule, the nonequilibrium Keldysh formalism is applied to derive the non-sinusoidal angular dependence of equilibrium FLST resulting from the multi-reflection processes at interfaces.
Statements
Data availability statement
The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.
Author contributions
Y-HT conceived the study and carried out the DFT + JunPy + LLG calculation. Y-HT and Y-CC carry out the analytical derivation and the macrospin dynamics simulation. B-HH developed JunPy + LLG code. All authors discussed the results and wrote the manuscript.
Funding
This work is supported by the Ministry of Science and Technology (MOST 107-2633-M-008-004- and 108-2628-M-008-004-MY3) and the National Center for Theoretical Sciences (NCTS).
Acknowledgments
We thank Yu-Sheng Lin to support the analytical derivation of spin torque effect via resonant tunneling. We thank to National Center for High-performance Computing (NCHC) of National Applied Research Laboratories (NARLabs) in Taiwan for providing computational and storage resources.
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher’s note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
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Summary
Keywords
exchange bias, field-like spin torque, contact geometry, first-principles, magnetotransport, spin dynamics, single-molecule magnetic junction
Citation
Tang Y-H, Chuang Y-C and Huang B-H (2022) Exchange bias toggling in amine-ended single-molecule magnetic junctions by contact geometry. Front. Phys. 10:967406. doi: 10.3389/fphy.2022.967406
Received
12 June 2022
Accepted
25 July 2022
Published
09 September 2022
Volume
10 - 2022
Edited by
Dongzhe Li, UPR8011 Centre d’Élaboration de Matériaux et d’Etudes Structurales (CEMES), France
Reviewed by
Alexey Kartsev, Russian Academy of Sciences (RAS), Russia
Shuai Qiu, Shandong Normal University, China
Updates
Copyright
© 2022 Tang, Chuang and Huang .
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Yu-Hui Tang, yhtang@cc.ncu.edu.tw
This article was submitted to Condensed Matter Physics, a section of the journal Frontiers in Physics
Disclaimer
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.