Molecular rectification is an important concept in molecular electronics, which has been extensively investigated in the past decades.^[1,2] Various designs have been involved in experimental and theoretical researches, like using asymmetric molecule-electrode interfaces^[3] or asymmetric molecules.^[4–6] Among these, conjugated diblock co-oligomers with a donor–acceptor (D–A) structure are particularly attractive as intrinsic molecular diodes owing to their repeatable performance and controllable rectification behaviors.^[4,5] Especially, the rectifying direction in such diodes may be tuned with chemical methods such as protonation^[7] or changing anchor group.^[8,9]

The exploitation of the electron spin degree of freedom in spintronics also triggered a great deal of interest in pursuit of spin rectifiers.^[10,11] Spin rectifiers mean the spin current (SC), defined as the difference between spin-up and spin-down currents, is asymmetric upon the reversal of bias voltage. The charge current (CC) is a scalar as the sum of spin-up and spin-down currents, and the rectification of CC is easy to understand as the asymmetric amplitude of current. However, the SC is a vector which contains two characteristics: the amplitude of the current and its spin-polarized orientation. The concept of spin-current rectification is more complicated than the normal charge-current rectification. In nature, two kinds of spin-current rectification may exist, asymmetric amplitude or different spin-polarized orientations of the spin current upon the reversal of bias, which can be called parallel spin-current rectification or antiparallel spin-current rectification, respectively.^[12,13] The construction of spin rectifiers requires an asymmetric spatial structure in the spin degree of freedom. The first molecular spin rectifier was designed by Dalgleish and Kirczenow, where an organic molecule is coupled with one ferromagnetic metal and one nonmagnetic metal, where the spatial spin asymmetry comes from the external electrodes.^[12] Recently, with the utilization of organic ferromagnets, we proposed an intrinsic organic spin rectifier with a magnetic/nonmagnetic co-oligomer structure,^[13] where the charge-current rectification and spin-current rectification may be realized simultaneously or separately. The effects of proportion of the co-oligomer and the interfacial coupling with electrodes were also discussed.^[14,15]

Upon the design of molecular spin rectifiers, a controllable rectification with a large rectification ratio is another important issue. As one of non-negligible factors, molecular length can be adjusted conveniently in experiments, which has been proved to tune the rectification in normal charge-current molecular diodes by ab-initio and model calculations.^[16,17] Therefore, the length effect in organic spin diodes deserves to be considered, especially on the spin-current rectification. Here we will explore the length effect on the rectification in organic spin co-oligomer diodes by investigating the asymmetric evolution of the spin-dependent eigenstates under biases. The paper is organized as follows. In Section 2 the model and calculation method will be introduced. In Section 3 the calculated results are discussed and a summary is given in Section 4.

The calculation is based on a quasi-one-dimensional model of magnetic co-oligomer spin rectifier. As shown in Fig. 1, the magnetic co-oligomer is sandwiched between two metallic electrodes. The electrodes are noninteracting semi-infinite one-dimensional metallic chains. The central molecule is composed of the left magnetic molecule (such as poly-BIPO (poly(1,4-bis-(2,2,6,6-tetramethyl-4-oxy-4-piperidyl-1-oxyl)-butAdiin))^[18,19] and the right nonmagnetic one (such as polyacetylene). In the magnetic moiety, magnetic side radicals hang on the even-number sites, each of which contains an unpaired electron and has an uncompensated spin S_nR. The magnetic co-oligomer can be described with an extended SSH+Heisenberg (SSH: Su–Schrieffer–Heeger) model^[20,21]

The two metallic electrodes are symmetric and described by a one-dimensional single band tight-binding model with on-site energy ɛ_f and the nearest neighboring transfer integral t_f. To focus on the intrinsic property of the molecule, the interfacial coupling between the electrodes and the molecule are simplified and set equally as t_lm = t_rm = t_fm.

Under a bias V, an external field E will be generated from the bias between the two electrodes. The spatial potential is described by the following Hamiltonian:

The current through the device with spin σ is calculated with the Landauer–Büttiker formula^[22]

The calculation is performed as follows. First, the ground state of an isolated co-oligomer (poly-BIPO + polyacetylene) molecule is obtained by a self-consistent iteration method.^[20,23] It is found that, in the ground state or under a low temperature, the radical spins in poly-BIPO form a ferromagnetic order.^[20,23] Then, the spin-dependent Green function and the transmission probability are calculated. Finally, the spin-dependent current is obtained from Eq. (3).

We carry out numerical calculations by taking the values of the above parameters as those generally adopted.^[13,20,23] For the magnetic co-oligomer, we use the parameters of poly-BIPO and polyacetylene with t₀ = 2.5 eV, α = 4.1 eV/Å, and J = J_f/t₀ = 1.0. The elastic constant of the carbon atom chain is K = 21.0 eV/Ų. The parameters of the electrodes are taken as ɛ_f = 0, t_f = 2.5 eV, and E_F = 0.3 eV. The coupling between the molecule and electrodes is t_fm = 1.0 eV. A low temperature of T = 10 K is chosen to avoid the spin fluctuation in the electron-transport process.

To clarify the length effect on the rectification, we start our calculations from several specific cases by taking N = 12, 16, 24, and 30, respectively. The CC and SC through each device are calculated and displayed in Fig. 2. Obviously, in each case both the CC and SC are asymmetric upon the reversal of bias, which means that the charge-current rectification and spin-current rectification are realized simultaneously. The spin-current rectification takes the form of parallel spin-current rectification, where only the amplitude of the SC is asymmetric.^[13] Meanwhile, an obvious length-dependent rectification can be observed. For example, in the two shorter cases, N = 12 and 16, the current in the negative bias region is larger than the positive bias region. However, in the longer cases, N = 24, 30, the rectifying direction is opposite with a larger current in the positive bias region. Besides the rectifying direction, the rectification ratio is also length dependent. Here we define the rectification ratio as RR = − I_c (+V)/I_c (−V) for the charge-current rectification and as SRR = − I_s (+V)/I_s (−V) for the spin-current rectification. For N = 12 and 16, the inversion of SRR achieves about 10 at +1.0 V, but the inversion of RR is much smaller with the value below 3.5. The magnitude of RR is close to that of some single-molecule co-oligomer diodes measured in experiments.^[5,24] For the two longer molecules, the SRR follows the RR to increase with the bias, where the maximum value of RR (SRR) achieves 4.5 (3.0) for N = 24 and 7.3 (5.7) for N = 30.

Fig. 2.

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	Fig. 2. Calculated charge current and spin current as a function of bias voltage with (a) N = 12, (b) N = 16, (c) N = 24, (d) N = 30. Solid (red) lines correspond to charge current in units of μA, and dashed (blue) lines indicate spin current in units of ℏ/2e. The inserts are bias-dependent rectification ratios.

To get a clear view of the dependence of the rectification ratio on the molecular length, we calculate the rectification ratio at |V| = 1.0 V as a function of the molecular length. The result is shown in Fig. 3. The length effect on the rectification can be concluded into the following three terms: firstly, the rectification ratio remains fractional until N = 20 and then exceeds 1.0, which means that the rectifying direction is reversed from along the negative bias to along the positive bias; secondly, for a shorter length less than 20, the inverse RR in the inserted plot increases little with the length, i.e., from about 1.0 to about 3.0, but the inverse SRR drops sharply from 70 to about 5.0. Thirdly, when the length is over 20, both the RR and SRR are enhanced obviously by the length, which achieves 8.5 and 7.3 at N = 32. The above results indicate that the molecular length can be used as a valid method not only to control the rectifying direction but also to adjust the rectification ratio. For example, a shorter molecule is advantageous to obtain a large spin-current rectification, but a longer molecule is beneficial for a large charge-current rectification.

Fig. 3.

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	Fig. 3. Dependence of charge-current rectification ratio and spin-current rectification ratio on the molecular length at |V| = 1.0 V. The insert is the inverse ratio at short length.

Next, let us try to understand the mechanism of the length effect on the rectification. Firstly, the length-induced inversion of the rectification can be explored by comparing the spin-dependent transmission of N = 16 and N = 30. Figure 4 shows the spin-dependent transmission at 0 V and ± 1.0 V. At zero bias, one can find that the transmission is spin split for the two cases, and the spin-up transmission spectrum is closer to the Fermi energy than the spin-down one. By examining the molecular eigenstates, we find that the nearest transmission peak to the Fermi level is mainly contributed by the spin-up lowest unoccupied molecular orbital (LUMO). Applying a bias, one can find that for the short molecule the transmission peak of the spin-up LUMO enters the bias window at a negative bias of −1.0 V. However, it is excluded outside the bias window under a positive bias of +1.0 V. Thus, the device rectifies the (spin) current along the direction of negative bias. For the long molecule of N = 30, the spin-up LUMO lies in the bias window no matter whether applying a positive bias of +1.0 V or a negative bias of −1.0 V. However, the transmission peak at the positive bias is much higher than that at the negative bias. As a result, the rectifying direction is reversed with respect to the case of N = 16.

Fig. 4.

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	Fig. 4. Spin-dependent transmission near the Fermi energy at V = 0, ±1.0 V for the different lengths N = 16 (a)–(c) and N = 30 (d)–(f). The vertical dash-dot lines indicate the chemical potentials of electrodes, and the window between them is the conducting bias window.

The mechanism of the length-induced inversion of rectification in magnetic co-oligomer diodes can be deeply understood in reference to that of D/A type co-oligomer charge-current diodes.^[17] In D/A type co-oligomer diodes, there exist two kinds of mechanisms for the rectification: bias-induced asymmetric shift of molecular eigenlevels and bias-induced asymmetric spatial localization of wave functions of the conducting orbitals. The two kinds of mechanisms are very common in molecular rectifiers or even in graphene nanoribbon rectifiers, where the asymmetric evolution of the electronic states may originate from the asymmetric built-in structure of the central molecules or from the asymmetric interfacial contacts with the electrodes.^[25,26] In D/A type diodes, the two mechanisms lead to different rectifying directions, and the dominant mechanism of rectification may exchange with molecular length. The magnetic co-oligomer is not a D/A type diode since there is not obvious structural asymmetry of charge density. However, due to the spin splitting of eigenlevels in the left magnetic moiety, the magnetic co-oligmer can be seen as a combination of an A/D type diode for spin-up channels and a D/A type diode for spin-down channels. In the present case, the spin-up channel dominates the transport. Hence, a similar length-induced inversion of rectification is observed for the spin diode. This can be verified by checking the evolution of the spin-dependent eigenstates under bias, which is shown in Fig. 5. Here the electronic localization of the orbital is defined as^[23] Fig. 5.

(a) Bias-dependent eigenvalues near the Fermi energy. (b) Evolution of the electronic localization of the spin-up LUMO and spin-down LUMO under biases. Here the molecular length is N = 12.

where φ_μ,σ,n(V) is the wave function of the eigenstate μ with spin σ in the Wannier representation. A larger electronic localization means a more localized orbital. It can be found that at negative biases the spin-up LUMO turns closer to the Fermi level accompanied with enhanced electronic localization, and goes far away from the Fermi level with weakened electronic localization. Such evolution of eigenstates corresponds to the change of transmission shown in Fig. 4, which is similar to that in co-oligomer charge-current diodes. The evolution of the spin-down LUMO is opposite to the spin-up LUMO.

Now we are in the position to clarify the reason of the different length dependence of the rectification ratio in each rectifying direction. Before that, one should make it clear about the length-induced change of molecular eigenstates. In Fig. 6, we plot the length-dependent eigenvalues and electronic localization of the spin-up LUMO at ± 1.0 V. Obviously, with the increase of length, the spin-up LUMO becomes closer to the Fermi energy, while the electronic localization is enhanced at the same time. The change of this eigenstate looks similar at ±1.0 V. However, one can still get some qualitative understanding about the length effect on the rectification ratio by comparing the evolution at +1.0 V and −1.0 V. From Fig. 6(b), we notice that the difference between the electronic localization at +1.0 V and −1.0 V turns larger and larger with length. The electronic localization is usually related to the height of transmission peak. Thus, a larger difference of the transmission height will be generated with length. For a longer chain (N > 20), the spin-up LUMO has been included in the bias window at both +1.0 V and −1.0 V, and the magnitude of transmission dominates both the values of CC and SC. This is why the RR and SRR increase with length in this region.

Fig. 6.

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	Fig. 6. Length dependence of (a) the eigenvalue and (b) the electronic localization of the spin-up LUMO. The results at both +1.0 V and −1.0 V are given. The rectangle in panel (a) indicates the bias window.

In the short-length region (N < 20), we notice from Fig. 6(a) that the spin-up LUMO at −1.0 V is closer to the Fermi level than that at +1.0 V. With the increase of length (from 12 to 20), at −1.0 V this orbital enters the bias window gradually, while it is still excluded out the bias window at +1.0 V. Thus, more efficient transmission is contributed at −1.0 V and the charge current is increased. Hence, the RR will be increased, whose value relies on the portion of the transmission in the bias window at ±1.0 V. However, the drop of the SRR cannot be understood easily from this picture. Then we plotted and compared the spin-dependent transmission spectrum at N = 10 and 16 (see Fig. 7). It is found that at a very short length of N = 10, the conducting orbitals are far away from the Fermi energy at +1.0 V, and the current is only contributed by the tails of the transmission peaks of spin-up LUMO and spin-down LUMO. The spin splitting is not obvious in the tails, and thus the SC is extremely small at +1.0 V. This leads to an extremely high inverse SRR at very short length. When the length is increased to N = 16, the spin splitting of the transmission involved in the bias window at +1.0 V becomes very obvious. Thus, the SC at +1.0 V of N = 16 is much more enhanced than that of N = 10, as shown in Fig. 7(b). This is the reason for the sharp drop of the inverse SRR. The spin splitting of the transmission peaks from spin-up and spin-down LUMO at +1.0 V may be even larger with the increase of length (see Fig. 4(f) for N = 30).

Fig. 7.

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	Fig. 7. Spin-dependent transmission near the Fermi energy at V = ± 1.0 V with (a), (b) N = 10 and (c), (d) N = 16. The window between the vertical dash-dot lines is the conducting bias window.

In summary, we have investigated the length effect on organic co-oligomer spin rectifiers. The results show that both the rectifying direction and the rectification ratio may be strongly affected by the molecular length. There exists a critical length where the rectifying direction is reversed. In each rectifying direction, the length dependence of the rectification ratio is quite different. For short co-oligomers below the critical length, the inverse RR increases little, while the inverse SRR drops sharply with the length. For long co-oligomers, both the RR and SRR are increased quickly with the length. By analyzing the spin-dependent transmission and the asymmetric evolution of molecular eigenstates, we demonstrate that the origin for the different length dependence of the rectification ratio comes from two different rectification mechanisms, that is, asymmetric shift of molecular eigenlevels and asymmetric spatial electronics localization of the conducting orbitals upon the reversal of bias. This work indicates a possible way to control the rectification in organic spin diodes by molecular length and deserves to be tested by future experiments.