Synchronously voltage-manipulable spin reversing and selecting assisted by exchange coupling in a monomeric dimer with magnetic interface

Yong-Chen Xiong*ab, Wang-Huai Zhoua, Nan Nanac, Ya-Nan Maa and Wei Lia
aSchool of Science, and Advanced Functional Material and Photoelectric Technology Research Institution, Hubei University of Automotive Technology, Shiyan 442002, People's Republic of China. E-mail: xiongyc_lx@huat.edu.cn
bKey Laboratory of Materials Physics, Institute of Solid State Physics, Chinese Academy of Science, Hefei 230031, People's Republic of China
cState Key Laboratory of Advanced Technology for Materials Synthesis and Processing, Wuhan University of Technology, Wuhan 430070, People's Republic of China

Received 27th September 2019 , Accepted 20th November 2019

First published on 22nd November 2019


The use of the molecular spin state as a quantum of next-generation information technology is receiving impressive research attention, within which the fundamental issues include manipulating the phase transition between the spin-up and -down states and generating spin polarized current. The spinterface between ferromagnetic electrodes and a molecular bridge represents one of the most intriguing elements in this context. Herein, by means of the celebrated numerical renormalization group technique, we present an original way to realize spin reversal in a monomeric dimer. Our scheme is based on the exchange interactions between electronic spins on one monomer and those on the other one or on the electrodes, which could be easily controlled through purely electronic technology. Through a careful engineering of the interfacial parameters, one of the monomers is devoted to the spin reversing, whereas the other one contributes to the spin selecting. The charge numbers of spin-up and -down electrons swap their respective occupancies at some particular points, indicating charge sensing between different spins. The competition between the spinterface and the molecular energy level results in charge oscillating in a single spin channel, which is unfavorable to the spin selecting. The observation may provide a prospective example for a multifunctional magnetoelectronics molecular device, which works without any external magnetic field.


1 Introduction

Molecular spintronics devices (MSDs), which aim at combining the advantages of two effervescent fields: molecular electronics and spintronics, are considered as highly promising candidates for the next generation electronic technologies.1–3 A foundational block of MSDs is the molecular junction (MJ), which basically contains three components: the electrode, the contact interface and the functional molecular centre.4–7 Any potential applications of spin-based quantum information processing, as well as molecular spintronics, rely on the ability to generate transitions between the spin-up and -down states within a localized electron spin on the molecular centre.8,9 Traditionally, this is accomplished by magnetic resonance, with the usage of a resonant oscillating magnetic field. However, developing strong oscillating magnetic fields remains challenging.9–11 An alternative method is the electrical manipulation. Due to the ease of generating locally on-chip and a combination of advantages, such as low power dissipation, high speed, reversibility, and nonvolatility, this technique instead of the magnetic one is particularly attractive.8,12 It has become a highly desirable property of fundamental importance for spintronic devices during the past decade.10,13,14 Many works have been done for such purpose based on molecular magnets.14–17 However, for nonmagnetic molecules, especially those with multilevels and in the strong correlation limit, different generating techniques are mandatory. Here, we report a novel scheme to generate spin reversal within a nonmagnetic molecular architecture in virtue of the exchange coupling. Assisted by the ferromagnetic interface between the local molecular object and the electrodes (also known as spinterfaces18), our scheme could be manipulated by a purely electronic technique, and the spin in one orbital switches abruptly from 1/2 to −1/2, or vice versa, suggesting a very high up–down ratio. Hence our suggested model may have appealing applications in molecular/atomic-scale devices encoding quantum bits.

To implement such a scheme, we invoke a monomeric dimer with double transport-active molecular orbitals connected parallel to two ferromagnetic electrodes, and describe it as a two impurities Anderson-type model with polarized hybridization function (see Fig. 1). We treat the interfacial spin-dependent hybridization as an effective magnetic field, which is resulted from the electron–electron interaction in the molecular orbitals, and could be illustrated as a function of the sweeping orbital (orbital 2). With the help of the state-of-the-art numerical renormalization group (NRG) method, we disclose the competition between the effective field and the sweeping orbital energy, which leads to charge oscillating in a single spin channel. The exchange coupling (or the Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction) between electrons on the fixed orbital (orbital 1) and those on the ferromagnetic electrodes (or orbital 2) results in multiple spin reversals at some particular points. The superposition of the uni-spin-directional charge oscillating and the spin reversing, at the end, gives rise to a charge sensing between opposite spins in the two orbitals. Furthermore, by systematically studying large parameter scales, we also find the linear conductance reveals electronic-controllable spin selecting, which does not favor the charge oscillation, and depends closely on the hybridization strength and the polarization of the electrodes.


image file: c9cp05316f-f1.tif
Fig. 1 Schematic view of the molecular junction consisting of a double non-magnetic monomeric molecule sandwiched between two ferromagnetic electrodes. U and εi (i = 1, 2) are the electron–electron interaction and the single-electron energy within level i, respectively. Gate-i (i = 1, 2) is the external gate voltage, which controls εi. Γσ is the spin-dependent electrode–molecule hybridization.

It is noticed that generating spin currents in organic molecules and their functionalities is also an emerging field with a wide range of applications, from molecular spintronics to quantum computing. Many methods have been suggested for such purpose, for instance, the usage of an external magnetic field,19,20 the electrical spin injection,21,22 the spin–orbit interaction in molecular semiconductors23,24 and the chirality induced spin selectivity.25,26 Apart from these techniques, the spinterfaces has become another key ingredient.18,27 The importance of the spinterface in the quantum transport was officially born in 2010, when C. Barraud et al. reported a systematic study of the giant tunnel magnetoresistance in Alq3-based nanojunctions.18,28 During the last few years, the spinterface-related behaviors have attracted considerable research interest both in theoretical and experimental aspects, including the spin-dependent trapping,29 the strong magnetoresistance effects,30–33 the spin filtering,32,34,35 and the spin signal reversing,36,37 and so on. However, on the other side, the microscopic mechanisms behind these phenomena and their relationship to the interfacial spin-dependent properties is far from saturated, especially the inter-correlation between the spinterface and the strong local interactions in multilevel molecular architectures. Additionally, in the context of molecular spintronics, the implementation of a multifunctional spintronic device is one of the most momentous challenges.38,39 Our designed device may meet this purpose, which reveals spin reversing and spin selecting synchronously, and even charge sensing between different spins. Meaningfully, for appropriate parameters, one of the monomers contributes to the spin reversing, whereas the other one devotes to the spin selecting.

2 Model and method

The second quantized Hamiltonian of the molecular structure presented in Fig. 1 is written as a two impurities Anderson-type model,40–42 which contains three constituents:
 
H = Hmol + Hele + Htul. (1)
Here, Hmol illustrates the contribution of the monomeric dimer, each contains a single transport-active molecular orbital: image file: c9cp05316f-t1.tif. εi and U are the single-electron energy and the electron–electron interaction within orbital i respectively. d (i = 1, 2) is the fermion creation operator, n = dd (σ = ↑, ↓) is the number operator for spin-σ. Experimentally, εi is controllable through chemical technologies or external gate voltage,43,44 U is determined by the molecule-type. Due to the discrete energy spectrum, the molecular dimer could be treated physically as quantum dots.41 Hele describes the ferromagnetic electrodes with Fermi energy εF = 0. For typical ferromagnetic materials, such as Fe, Co, Ni, one could adopt non-interacting s electrons to model them, due to the strong spatial confinement of electrons on the 3d sub-bands, which mainly relate to the electron correlation and contribute to the magnetic order.45 Thus Hele could be written as image file: c9cp05316f-t2.tif, with ν = L, R corresponding to the left and right electrodes respectively. cνkσ creates an electron with energy ε, wave vector k, and spin σ. Htul is the coupling of the electrodes to the molecular orbitals: image file: c9cp05316f-t3.tif. τνki, which could be controlled by the distance between the molecule and electrodes, labels the tunneling matrix element between the k, σ electrons belonging to electrode ν and those on molecular orbital i. Without losing generality, we assume each molecular level is connected to two electrodes with identical amplitude, which is chosen to be real and k independent τνki = τ. The ferromagnetic electrode/molecule interface is illustrated by a spin-dependent hybridization Γσ ≡ πρστ2, which contains all the information about spin asymmetry in the electrodes with spin-dependent density of states (DOS) ρσ.28,45,46 Γσ could be parameterized through polarization parameter p with Γ↑(↓) = Γ(1 ± p)/2. Here, +, − corresponds to up- (↑) and down-spin (↓) respectively, and Γ = Γ + Γ. Experimentally, p could be tuned through a process of placing it in close proximity to some magnets.60

In the present work, we mainly focus on the quantum behavior when U is strong enough, i.e., UΓ. Therefore, we invoke the NRG technique47–49 to treat the equation practically. The NRG is considered as an ideal method to study the low temperature properties of the Anderson model, and is also identified as a notable approach for multi-level molecular systems in the strong correlation limit.42 The key procedure of the NRG technique includes the logarithmic discretization of the conduction band, which then maps to a one dimensional tight-binding chain. The molecular cluster corresponds to the starting group of this chain, and the whole system could be diagonalized iteratively. In our NRG treatment, we choose the discretization parameter to be 2.0, save about 2000 low-lying states in each iteration, and iterate about 100 steps insuring the eigenvalues converge well enough for physical quantities calculation.

3 Results and discussion

We choose the half bandwidth of the electrode D (typically 100–101 eV in bulk metals) as the energy unit. We first fix Γ = 0.01, U = 0.15, p = 0.05 and ε1 at the particle–hole (p–h) symmetric point ε1 = −U/2. In Fig. 2(a), we give the charge occupation number for spin-up, -down and total spin on both orbitals image file: c9cp05316f-t5.tif at zero temperature as functions of ε2. It is seen the total number of both spins image file: c9cp05316f-t6.tif increases from 1.0 to 3.0 continuously in a step-manner due to strong U when ε2 is adjusted downwards. Interestingly, the numbers of spin-up 〈n〉 and -down 〈n〉 electrons reveal charge sensing,50 where they swap the respective occupancies at some particular points, i.e., ε2 ≈ 0, −0.075 and −0.15. More details could be found out in the charge number of each orbital 〈n〉. As depicted in Fig. 2(b) and (c), the total number of orbital 1 〈n1〉 remains at 1.0 due to fixed ε1 = −U/2, while 〈n1↑〉 alternates with 〈n1↓〉 and its spin z component S1z = (n1↑n1↓)/2 jumps from −1/2 to 1/2 (or the reverse) at the above points, indicating perfect local spin reversal. On the other hand, 〈n2〉 grows from the empty state to the fully occupied state in a step-manner, whereas the evolutions of 〈n2σ〉 are more complex. To be exact, 〈n2↑〉 increases from 0.0 to 1.0 continuously around ε2 ≈ 0 and −0.15, where 〈n2↓〉 oscillates with respect to decreasing ε2, suggesting a nonmonotonic charge filling. Besides, they also alternate with each other at about −U/2.
image file: c9cp05316f-f2.tif
Fig. 2 Charge occupation number for spin-up, -down and total spin on (a) both levels image file: c9cp05316f-t4.tif, (b) level 1 〈n1σ〉, and (c) level 2 〈n2σ〉 respectively at zero temperature as functions of ε2. Here, the legend ‘up’ (‘down’, ‘total’) stands for up-spin σ = ↑ (down-spin σ = ↓, total spin σ = ↑ + ↓). The other parameters are given by Γ = 0.01, U = 0.15, ε1 = −U/2, and p = 0.05 in unit of the half bandwidth of the electrodes D.

The underlying physical picture of the above phenomena could be understood as follows. According to Haldane's scaling method,51 when p ≠ 0, the spin splitting in each level εi is generated by virtual processes due to spin-dependent quantum charge fluctuations. Which then could be considered as an effective field Beff,i = δεiδεi, with46,52

 
image file: c9cp05316f-t7.tif(2)
Here, f(ω) is the Fermi–Dirac distribution function and ω is the energy variable. The first term in the curly brackets illustrates the charge fluctuations between an empty state and a singly occupied one, i.e., the electron-like processes, and the second term is that between a singly occupied state and a fully occupied one, i.e., the hole-like processes. At zero temperature and with a flat band, eqn (2) could be integrated analytically, giving
 
image file: c9cp05316f-t8.tif(3)
Note Beff,i vanishes when U = 0 and is completely resulting from the electron–electron interaction. It also shows a logarithmic divergence when εi → 0 and εi + U → 0. Furthermore, with finite U, Beff,i also disappears at the p–h symmetric point εi = −U/2. This is an important feature, which will be explained below. In Fig. 3, we describe the ε2-dependent Beff,2. Due to Beff,2, the charge occupancy of orbital 2 is totally spin split in the singly occupied regime 〈n2〉 ∼ 1.0. One notices Beff,2 changes its sign at about ε2 = −U/2, this explains the spin flip process in 〈n2σ〉 at ε2 ∼ −U/2. Besides, we attribute the charge oscillating in 〈n2↓〉 to the competition between ε2 and Beff,2. Take ε2 in the vicinity of 0.0 for example. When ε2 > 0, level 2 is almost empty, 〈n2σ〉 is not sensitive to Beff,2. Thus 〈n2↑〉 and 〈n2↓〉 increases simultaneously as ε2 decreases. When ε2 exceeds 0.0, level 2 begins to be singly occupied significantly and the effect of Beff,2 becomes obvious. In this case, the renormalized energy of spin-up (-down) electrons in level 2 is ε2↑* ≡ ε2 + Beff,2 (ε2↑* ≡ ε2Beff,2). Therefore, 〈n2↑〉 grows to 1.0 continuously, while 〈n2↓〉 weakens back to zero, due to the Zeeman effect, leading to a nonmonotonic filling. When ε2 + U crosses the Fermi surface, i.e., ε2 ≈ −0.15, level 2 tends to be fully occupied. The charge oscillating in the down-spin channel could be understood in a similar way based on hole-like processes. Furthermore, it is worth noting that the spin splitting and reversing in 〈n1σ〉 is not induced by Beff,1, since ε1 = −U/2, whereas it is resulted from the indirect (direct) exchange coupling between electrons on level 1 and those on level 2 (the ferromagnetic leads). For instance, in the regime −U/2 < ε2 < 0, both levels are nearly singly occupied and the z component of local spin on level 2 reads S2z ≈ 1/2. Hence the ground state of the molecule is more likely to be Sz = S1z + S2z ≈ 1 of the triplet, due to the RKKY interaction (∼Γ2/U) mediated by the coupling between electrons on the leads and those on the local orbitals,53 giving S1z ≈ 1/2. When ε2 > 0, the RKKY interaction, which demands singly occupied states in both levels, becomes invalid. However, the spin-dependent Γσ induces a spin imbalance in the leads, i.e., ρ > ρ. The effective anti-ferromagnetic exchange coupling between level 1 and the leads Jkk′1 = −8τ2/U results in a polarized spin singlet ground state within the interface,40,53 hence S1z ≈ −1/2. For ε2 < −U/2, one could make similar analyses.


image file: c9cp05316f-f3.tif
Fig. 3 Effective magnetic field in level 2 evaluated from eqn (3). The other parameters are the same as in Fig. 2.

For the sake of checking the robustness of the above spin reversal and spin-dependent charge oscillation, we give 〈n1σ〉 and 〈n2σ〉 in terms of different Γ and p in Fig. 4. Here, 〈nσ〉 are omitted since they are similar to the case of Fig. 2(a) with only some quantitative differences. Besides, our NRG results show the charge sensing between different spins could also be found in these cases. One may see the spin reversal still exists in 〈n1σ〉 in large regimes of Γ and p. However, when Γ is large enough and ε2 is far from the divergence points, the up–down (or the reverse) ratio decreases. Because in this case, the level width Γσ covers the level difference between the renormalized spin-up and -down levels in orbital 1, and the spin degeneracy is enhanced, see Γ = 0.02 in Fig. 4(c). On the other hand, one could also find oscillations within 〈n2↓〉 when ε2 and ε2 + U cross the Fermi level, but they are strongly affected by Γ and p. Specifically, with increasing Γ and p, the oscillation is weakened due to the following picture. As Γ or p increases, the effective field Beff,i is enhanced, cf. eqn (3), thus the difference between the renormalized energy ε2↑* and ε2↓ becomes more obvious. As a result, the possibilities of 〈n2↓〉 grows simultaneously with 〈n2↑〉 reduced, and the oscillation is suppressed. Furthermore, one may also notice 〈ni〉 ≈ 〈ni〉 around ε2 = −0.075, when Γ and p are small, see Fig. 4(a), (b), (e) and (f). Since in this case, Beff,i is so weak, even reaching to zero theoretically, thus it's difficult to separate εi and εi. However, for larger Γ and p, this behavior does not occur, resulting from the fact that the imbalanced density of states for different spins in the electrodes is enhanced, hence a very small change in ε2 may break the ground-state spin degeneracy in the molecule. We argue there may exist theoretically a spin degeneracy point around ε2 ∼ −U/2, whereas it is extremely difficult to capture it in related experiments. In fact, in our present numerical realization, even the interval in the vicinity of −U/2 reaches to 10−8, the spin degeneracy point does not arise for Γ = 0.01 and p = 0.05, cf. Fig. 2.


image file: c9cp05316f-f4.tif
Fig. 4 n1σ〉 and 〈n2σ〉 in terms of other parameters: (a) and (b) Γ = 0.002, (c) and (d) Γ = 0.02, (e) and (f) p = 0.005, and (g) and (h) p = 0.6 as functions of ε2 respectively. The other parameters are the same as in Fig. 2.

To study the transport property through the molecule, we depict the spin-dependent linear conductance Gσ at zero temperature as a function of ε2 in Fig. 5(a) for Γ = 0.01 and p = 0.05, the same as in Fig. 2. Here, Gσ is obtained from the Landauer formula:54

 
image file: c9cp05316f-t9.tif(4)
with Tσ(ω) being the spin-σ transmission probability. At zero temperature, Gσ is determined by the behavior of Tσ(ω) at the Fermi surface in the limit of zero bias. It is seen as ε2 sweeps downwards, two peaks belonging to both down- and up-spin conductances are found at about ε2 = 0.0, and −U, respectively, where G and G increase from 0.0 to their maximums continuously, then decrease back to zero. At these points, G tends to reach its unitary limit e2/h since in these cases, 〈n2↑〉 ≈ 0.5, thus electrons with up-spin could transmit the molecule due to the spin-up Kondo effect. On the other hand, G could not reach the unitary limit due to 〈n2↓〉 not reaching to 0.5 and the spin-down Kondo effect is incomplete. In other regions, Gσ remains almost at zero, because level 2 is either empty (ε2 > 0), or totally polarized (ε2 ∼ −U/2), or fully occupied (ε2 < −U), while level 1 is totally polarized, hence neither of them contribute to Gσ. It is noted that since both G and G reach to their maximums at about 0.0 and −U, i.e., G↑,m ≈ 1.0 and G↓,m ≈ 0.4, it is not an ideal spin selector which should contain high selecting efficiency image file: c9cp05316f-t10.tif. In panels (b)–(e), we also show Gσ for other parameters, one finds the evolution of Ge is nearly opposite to the charge oscillating with respect to increasing Γ and p. Namely, the charge oscillating in the spin-down channel of orbital 2 is to the disadvantage of spin selecting. Besides, we also find additional peaks for both spins in Gσ locate at about ε2 = −U/2 for smaller Γ and p, e.g., Γ = 0.002 and p = 0.005 in panels (b) and (d) respectively. Distinct from ε2 ≈ 0.0 and −U, the physical reason for this peak is resulting from the fact that at this point both levels are singly occupied with nearly equal spin-up and -down electrons, thus a spin-1 Kondo effect tends to be generated. However, due to Beff,i, the Kondo peak becomes spin split and does not locate at the Fermi surface precisely.45 As a result, the Kondo peak is suppressed, and Gσ could not reach to its unitary limit. Note panels (a)–(c) also describe that the width of the peak in Gσ depends closely on the level broadening (Γ) of the molecular orbital due to hybridization, which determines a finite lifetime ∼ħ/Γ for electrons on the orbital.55 Therefore, only a sharp regime of the molecular orbital could contribute to Gσ for smaller Γ. While for ΓU, Γ broadens the local electronic states2 and covers both levels for both spins, thus all of them may contribute to the conductance, and Gσ is finite in large regimes of ε2, cf. Fig. 5(c). As a brief summary, to generate the perfect spin selector, i.e., with high selecting efficiency Ge and G (or G) tends to reach its unitary limit, large p is recommended, cf. Fig. 5(e). Under such circumstances, only orbital 2 contributes to the spin current, whereas orbital 1 contributes to the spin reversal.


image file: c9cp05316f-f5.tif
Fig. 5 Spin-resolved linear conductance Gσ in unit of e2/h at zero temperature as functions of ε2 for various parameters. Parameters in panels (a)–(e) are the same as in Fig. 2, and panels (a), (c), (e) and (g) in Fig. 4, respectively.

The above discussions are restricted at U = 0.15 and ε1 = −U/2, and it is interesting to compare them with other cases. In Fig. 6(a), (b) and (g), we present 〈n〉 (i = 1, 2) and Gσ for fixed ε1 far from the p–h symmetric point, e.g., ε1 = −0.2. One finds orbital 1 keeps in a doubly occupied state, hence has no contribution to the linear conductance. Whereas 〈n2σ〉 and Gσ behave similar to the case of ε1 = −U/2 (e.g., Fig. 2(c)), indicating the above spin selection and spin-dependent charge oscillation may also be found in the single orbital molecular junction. For identical εi, the evolution of the electron occupancies for both spins becomes discontinuous at ε2 ≈ 0 and U, indicating a first order quantum phase transition, due to symmetric Γ between the electrode and two molecular orbitals,56 see Fig. 6(c) and (d). Correspondingly, there exist abrupt changes in Gσ, see Fig. 6(h). As U increases, i.e., U = 0.25 in panels (e), (f), and (i), the evolutions of 〈n〉 and Gσ resemble respectively the results in Fig. 2(b), (c) and 5(a), where high ratio spin reversal in 〈n1σ〉 occurs at about 0, −U/2, and U, and spin selecting (spin-down charge oscillating) contributed mainly by 〈n2σ〉 could be found at ε2 ≈ 0 and U. However, the influence of U on the oscillation amplitude is not obvious. Besides, our NRG results also show that the charge sensing between different spins does not occur for those parameter scales as in panels (a) and (c).


image file: c9cp05316f-f6.tif
Fig. 6 n〉 (i = 1, 2) and Gσ under different conditions: (a), (b) and (g) ε1 = −0.2, (c), (d) and (h) ε1 = ε2, (e), (f) and (i) U = 0.25 as functions of ε2 respectively. The other parameters are the same as Fig. 2.

4 Conclusion

In conclusion, we have designed a novel scheme to implement spin reversal in a nonmagnetic monomeric dimer with strong electron–electron interaction. Such a molecular device reveals ideal spin reversing and spin selecting simultaneously. The spinterfaces between the molecule and the magnetic electrode can be considered as an effective magnetic field, which competes with the tuning orbital, resulting in a charge oscillation within the down-spin channel. The exchange coupling (or the RKKY interaction) between electrons on the fixed level (at the p–h symmetric point) and those on the ferromagnetic leads (or the tuned level) leads to spin flip at some particular points. The accumulation of the spin-dependent charge oscillating and the spin reversing, at the end, gives rise to charge sensing between different spins. The linear conductance behaves as spin selecting, which dislikes the charge oscillation, and is strongly affected by the hybridization strength Γ and polarization p. To achieve high efficiency spin selecting and reversing, larger p is of the essence. Interestingly, with appropriate Γ and p, one of the orbitals contributes to the spin reversal, while the other one contributes to the spin selection, both of which could be manipulated through purely electrical methods, indicating a promising multifunctional molecular spintronics device.

To be realized experimentally, we suggest 3,4,9,10-perylenetetracarboxylic-dianhydride (PTCDA) complexes as potential candidates for the molecular framework, within which large electron–electron interactions have been observed.57,58 Typical ferromagnetic materials, such as Fe, Co, Ni, which induce a spinterface, could be adopted for the electrodes. The spinterface could be engineered through the bonding nature of the molecule and the surface,59 or by external excitation, electrical, optical and even magnetic technologies.39

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This work is financially supported by NSFC (No. 11504102), the Natural Science Foundation of Hubei Province (No. 2019CFB788 and 2019CFB259), and the Foundation of Hubei Educational Committee (No. B2019074 and B2016091).

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