High performance bipolar spin filtering and switching functions of poly-(terphenylene-butadiynylene) between zigzag graphene nanoribbon electrodes

Abstract

Using the nonequilibrium Green’s function method combined with spin-polarized density functional theory, we investigate the spin-resolved electronic transport properties of devices made of poly-(terphenylene-butadiynylene) (PTB) between two symmetric ferromagnetic zigzag graphene nanoribbon (ZGNR) electrodes. The bipolar spin filtering effect, rectifying behavior, and negative differential resistance have been found. More interestingly, an on/off ratio in the order of 10⁷ is also predicted by changing the angle between the PTB and ZGNR electrode planes. Further analyses show that the matching of the electronic wave functions among both electrodes and PTB plays a key role in the multi-functional PTB based device. And the coupling between the alkyne triple bond and the phenyl rings of PTB is critical to the value of the spin-resolved current and the on/off ratio. These phenomena suggest that the proposed PTB based devices have potential utilization in molecular spin diodes and molecular switches.

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Introduction

Traditional semi-conductive electronic devices have been confronted with an unprecedented challenge,^1,2 since molecular electronic devices attract many people’s attention, and tremendous efforts have been devoted to them both experimentally^3,4 and theoretically.^5–7 Lots of interesting phenomena have been found, including spin-valve,⁸ spin filtering,⁹ spin crossover,¹⁰ negative differential resistance (NDR),¹¹ switching effect,^12,13 rectification,¹⁴ and so on. The electronic transport properties of molecular devices not only depend on the structure of the molecules, but also have a huge relationship with the properties of both electrodes. Thus, the choice of the electrodes is very important in device designing. Generally, metallic electrodes are some of the most popular ones, and the molecules connect with the metallic electrodes through thiol (–SH) groups. Nevertheless, it is not an ideal choice in view of electrical transparency, mechanical stability and atomic level control over the bonding geometry.^15–17

The discovery and successful preparation of graphene¹⁸ and graphene nanoribbons (GNRs)^19,20 are revolutionary for molecular devices due to their remarkable electronic properties,²¹ high electronic mobility,²² long spin relaxation times and lengths,²³ weak spin–orbital coupling effect,²⁴ gate tunability²⁵ and so on. In particular, much research is concentrating on zigzag graphene nanoribbons (ZGNRs) for their edge magnetism and unique transport properties.^26–28 Jin et al.²⁹ and Rong et al.³⁰ have realized C chains between GNRs in experiments. And then Dong et al.³¹ reported their theoretical results for a C chain between transversely symmetric GNR electrodes, and a perfect spin-filtering effect coupled with a spin rectification ratio in the order of 10⁶ has been found. Li et al.³² found that the oligo(p-phenylene-vinylene) (OPV) molecule without or with different side groups between two ZGNR electrodes can realize the NDR and rectifying effects. What’s more, the changing of OPV conformations with respect to ZGNRs can bring great changes to the spin transport properties.³³ All of this research shows that the spin-filtering, switching, and NDR effects could coexist in a device, which suggests that multi-functional and high performance molecular devices will be a trend in future.

In 2014, Cirera et al.³⁴ successfully synthesized the extended poly-(terphenylene-butadiynylene) (PTB) [–C≡C–Ph₃–C≡C–]_n using vicinal surface templating. The PTB is rich in triple and double bonds of C atoms, which can form π bonds and be beneficial for electron conduction. Theoretical calculations show that it is a semi-conductor with a direct band gap of 1.6 eV,³⁴ which indicates that it has potential utilization in organic molecular devices. However, the electronic transport properties of PTB devices have not been reported. In this paper, we are mainly probing the spin transport properties of PTB based devices. Considering the contact between PTB and the electrodes, while realizing a multi-functional device, we choose symmetric ferromagnetic ZGNRs for both the left and right electrodes. Our calculations show that the bipolar spin-filtering effect, NDR, the rectifying effect, and switching functions can be realized. More importantly, the spin-filtering efficiency is 100%, the rectifying effect can reach 6.8 × 10³ and the on/off ratio is as high as 10⁷.

Models and methods

The structure of PTB is shown in Fig. 1(a), and the structure in the red dashed rectangle is the unit cell. We build the two probe system of M1 by connecting the PTB unit cell with two symmetric ZGNRs of 6 ribbon width (6ZGNRs) within the same plane (i.e. the y–z plane), as shown in Fig. 1(b). The blue shaded areas are the electrode regions. Former research has pointed out that the rotation of phenyl rings can be realized via many methods, such as scanning tunneling microscopy manipulation,³⁵ laser pulses,³⁶ mechanical operation,³⁷ adding an electric field,³⁸ and so on. Therefore, we get the three other devices by rotating the PTB plane around the C chain (z axis) and make the angle between the PTB plane and both 6ZGNR electrode planes changes from 0° to 30°, 60° and 90°, which are marked as M2, M3 and M4, respectively, for short. In the two-probe system, the spin orientation of the left (right) electrode can be controlled using an external magnetic field,^39,40 and plays an important role in spin devices.⁴¹ Here, the edges of the electrode 6ZGNR are assigned to the ferromagnetic state, and show metallic properties.^42,43 And two kinds of spin orientation between the left and right electrodes have been considered: parallel configuration (PC) and antiparallel configuration (APC).

Fig. 1 (a) The supercell of PTB, and (b) and (c) the two-probe systems of M1 and M4. The structure in the red dashed rectangle is the unit cell of PTB. The red curved arrow gives the rotation direction of the center scattering region. The gray and white balls represent the C and H atoms, respectively. The blue shaded areas are the electrode regions.

The geometric structures of the ZGNRs and PTB are firstly optimized using the Vienna Ab initio Simulation Package.^44,45 We construct the two-probe system by composing the optimized structures to a whole system, and the optimization is performed again using the Atomistix ToolKit package until the forces on each atom are smaller than 0.01 eV Å⁻¹. The spin-dependent transport properties of the devices are studied using the Atomistix ToolKit package, which is based on ground state spin density functional theory and the nonequilibrium Green’s function method,^46,47 although, according to C. Zhang et al.,⁴⁸ the method based on steady-state DFT would be more accurate under a bias voltage. Since the difference between both methods is very small under a low bias (<0.5 V in our calculations), methods based on ground state density functional theory are used in the calculations though a bias voltage is added. The Perdew–Zunger parametrization of the local spin density approximation (LSDA) form of the functions is employed for the exchange–correlation potential. In the calculations, the real space grid technique is used with the energy cutoff of 150 Ry as the required cutoff energy in numerical integrations. The k-point grids 1, 1, and 100 are used in the x, y, z directions, respectively, where z is the electron transport direction. Open boundary conditions are used to describe the electronic and transport properties of the devices. A vacuum layer of 12 Å is added to avoid the interaction between adjacent ribbons. The wave functions of all atoms are expanded using the double-zeta polarized basis set. The temperature of the electrodes is set to be 300 K. The spin-polarized current through the system is calculated using the Landauer–Büttiker formula,⁴⁹

Here e is the electron charge, h is the Planck constant, and T_σ is the transmission of an electron with spin σ. f_L(R)(E,V_b) is the Fermi–Dirac distribution function of the left (right) electrode, and μ_L(R)(V_b) is the chemical potential of the left (right) electrode, where V_b denotes the external bias voltage. T_σ can be obtained from the equation,Here G^r(G^a) is the retarded (advanced) Green’s function matrix, and is the retarded self-energy matrix for the left (right) electrode.

Results and discussion

We calculate the molecular energy spectrum of PTB, as shown in Fig. 2(a). The Fermi level is at zero energy. The highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) are near the Fermi level. The wave functions of both the HOMO and LUMO are also plotted, which can be found in Fig. 2(b) and (c). It is clear that both wave functions are symmetric with respect to the x–z plane. For the 6ZGNR unit cell, the lattice constant is 2.461 Å, and the band structure calculation indicates that it is metallic in the ferromagnetic state. There are two up-spin (down-spin) subbands which are twist with each other and almost degenerated below (above) the Fermi level in the region of 2π/3 ≤ k ≤ π, as can be seen in Fig. 2(d). The wave functions of the π and π* subbands are presented in Fig. 2(e) and (f), respectively. From them, we can find that the wave function of the π subband is antisymmetric and that of the π* subband is symmetric with respect to the x–z plane. We also use the spin generalized gradient approximation (SGGA) of the Perdew–Burke–Ernzerhof functional to have a test about the molecular energy spectrum of PTB and the band structure of the 6ZGNRs. For Fig. 2(a) and (d), we can see that the molecular energy spectrum of PTB and the band structure of the 6ZGNRs calculated using the SGGA (the red line) and LSDA (the black line) are nearly the same. The only difference appears in the band of the 6ZGNRs; the crossed energy point of the up-spin band at the X point is changed from −0.30 to −0.53 eV, and that of the down-spin band is changed from 0.14 to 0.37 eV. And our results about the band structure of the 6ZGNRs calculated using the LSDA method are in good agreement with previous studies.^31,50,51 When PTB is connected to the 6ZGNR electrodes within the same plane (y–z plane) along its central line, as shown in Fig. 1(b), for passing through the device, the electrons are required to have a wave function symmetric with respect to the x–z plane according to the quantum transmission theory.

Fig. 2 (a) The molecular energy spectrum of PTB, (b) and (c) the wave functions of the HOMO and LUMO, (d) the band structure of the 6ZGNRs, (e) and (f) the wave functions (for the π and π* subbands at the Γ point) of the 6ZGNRs, and (g) [(h)] the band structures for the left electrode (left panels), transmission spectra, DOS and PDOS (middle panels), and the band structures for the right electrode (right panels) at zero bias of M1 in PC [APC]. The blue dashed line is the Fermi level. The black dashed line indicates the x–z plane. The blue arrows and crossed arrows represent the conductive and prohibited electron transport channels.

The transmission spectra, density of states (DOS) for M1 and projected density of states (PDOS) for PTB at zero bias with PC and APC have been calculated, as shown in Fig. 2(g) and (h). For the PC situation, the band structures of the left and right electrodes are the same, but for the up-spin and down-spin bands, there is an energy difference of about 0.44 eV between the crossed point of the π and π* subbands. Based on the matching principle, the π subbands of the left and right electrodes and the center scattering region are mismatching, but the π* subbands are matching. Therefore, only the electrons from the π* subbands can pass through the device, so the up-spin transmission peak energy is about 0.44 eV lower than that of the down-spin one (Fig. 2(g)), leading to a perfect spin-filtering effect around the Fermi level. When the magnetic orientation of the right electrode is changed from +x to −x, M1 will be in APC. We can find that the band of the left electrode does not change. Meanwhile, for the right electrode, the up-spin and down-spin bands are changed with each other, thus similar transmission spectra of the up-spin and down-spin states can be observed, as presented in Fig. 2(h).

Interestingly, one can find that there is no transmission spectrum between 0.7 and 1.0 eV for both spins in PC and APC although the wave functions of the left and right electrodes and the center scattering region are matching. We plot the DOS for M1 and the PDOS for PTB, which are the purple and green dashed lines shown in Fig. 2(g) and (h). From them we can see that the DOS for M1 is distributed between 0.7 and 1.0 eV, but there is no PDOS for PTB appearing from 0.7 to 1.0 eV, so there is no contribution of PTB to electron transport through the device. As a result, there is no transmission spectrum appearing between 0.7 and 1.0 eV though the wave functions are matching.

When the bias voltage is added on the left and right electrodes, the currents, as a function of the applied bias voltage (I–V curves), can be obtained, as shown in Fig. 3(a) and (d). For M1 in PC, as shown in Fig. 3(a), the up-spin current has a linear increase behavior when |V_b| is lower than 0.2 V, and it reaches the biggest value at |V_b| = 0.3 V, then it begins to decrease with increasing |V_b|, which indicates that the up-spin current has an NDR effect. Meanwhile, for the down-spin current, one can find that it is almost zero over the whole bias region ([−0.5, 0.5 V]). As a result, M1 presents a perfect spin-filtering effect. To have a quantitative understanding about the spin-filtering effect, the spin-filtering efficiency (SFE) is then calculated based on the formula, SFE = [(I_↑ − I_↓)/(I_↑ + I_↓)] × 100%. As shown with the blue line in Fig. 3(a), one can find that the SFE is nearly 100% in the [−0.5, 0.5 V] bias region, which indicates that M1 can act as a perfect spin-filter. For a further understanding of the spin transport behavior, we plot the up-spin and down-spin transmission spectra as a function of the electron energy and bias voltage, as presented in Fig. 3(b) and (c). It is clear that there are up-spin transmission spectra coming into the bias window, but there is no down-spin transmission spectrum appearing in the whole bias window. As the current is determined using the integral of T(E,V_b) within the bias window (eqn 1), the up-spin current is much larger than the down-spin one, leading to the perfect spin-filtering effect.

Fig. 3 (a) [(d)] The spin-dependent I–V curves and SFEs for M1 in PC [APC]; and (b) and (c) [(e) and (f)] the spin-resolved transmission spectra as a function of the electron energy and bias voltage for M1 in PC [APC]. The regions I and II between the black solid lines denote the negative and positive bias windows, respectively.

For M1 in APC, as shown in Fig. 3(d), the I–V curve presents semi-conductive properties with a threshold voltage of 0.1 V. It can be clearly seen that the down-spin current only appears in the positive bias regions, while the up-spin current only appears in the negative bias regions, which indicates that M1 has a perfect bipolar spin-filtering effect and can be used as a spin-diode. The SFE is shown in Fig. 3(d) with a blue line, where one can find that it is nearly 100% in the bias region of [−0.5, −0.1 V], and then undergoes a spin-flip phenomenon, and the SFE becomes −100% in [0.1, 0.5 V]. And our calculation shows that the rectifying ratio can reach 6.8 × 10³. All of these results show that M1 in APC is a perfect multi-functional spin electronic device. From the transmission spectra as a function of the electron energy and bias voltage, as presented in Fig. 3(e) and (f), we can have a further understanding about the spin transport properties of M1 in APC. When |V_b| is smaller than 0.1 V, there is no transmission spectrum appearing in the bias window, so the current is almost zero. But when |V_b| is larger than 0.1 V, one can find that the transmission spectrum begins to come into the bias window, and increases with increasing |V_b|. What’s more, the up-spin transmission spectrum appears in the negative bias region, while the down-spin one appears in the positive bias region, which is in perfect accordance with the I–V curves.

To explore the mechanism of the spin-filtering effect, we present the band structures of the left (right) electrode and the transport spectra of M1 at 0.3 V in PC and APC, as shown in Fig. 4(a) and (d). For M1 in PC, when a 0.3 V bias is applied, the bands of the left electrode shift downward by 0.15 eV, meanwhile the bands of the right electrode shift upward by 0.15 eV. As a result, the π and π* subbands for both spin states of the left and right electrodes are overlapping each other in the bias window (Fig. 4(a)). But because only the up-spin π* of the left electrode can match that of PTB in the scattering region and the right electrode, there is only the up-spin transmission spectrum that appears. Furthermore, we also plot the transmission pathways of the up-spin and down-spin in 0.1 eV for M1 at 0.3 V in PC, presented in Fig. 4(b) and (c), respectively. The transmission pathways T_ij can show us the local bond contributions to the transmission coefficient, for example, the total transmission coefficient between two parts A and B can be expressed as . It can be seen that the up-spin electrons can transfer from the left electrode to the right electrode, and more interestingly, the transport of electrons is mainly along the C bond in the phenyl rings instead of being the hopping transport in pristine ZGNR devices.⁵¹ Since the down-spin transmission pathway is localized, this indicates that the down-spin current channels are blocked.

Fig. 4 (a) [(d)] The band structures for the left electrode (left panels), transmission spectra (middle panels), and the band structures for the right electrode (right panels) at V_b = 0.3 V of M1 in PC [APC]; and (b) and (c) [(e) and (f)] the up-spin and down-spin electronic transmission pathways of M1 in PC [APC] in 0.1 eV at V_b = 0.3 V. The region between the blue dashed lines is the energy region contributing to the current, i.e. the bias window.

Similarly to M1 in PC, for M1 in APC, the addition of a 0.3 V bias makes the bands of the left (right) electrode shift downward (upward) by 0.15 eV, as one can see in Fig. 4(d), the π and π* subbands of both spin states appear in the bias window. Due to the mismatching of the π subbands among the left (right) electrode and PTB in the scattering region, only the down-spin transmission spectrum appears in the bias window and the down-spin electrons can pass through. This can also be understood from the transmission pathway, as shown in Fig. 4(e) and (f). The transmission of the up-spin electrons is completely localized in the left part, so there is no up-spin electron passing through the device, and the corresponding transmission coefficient is zero. Meanwhile for the down-spin state, we can see that the transmission pathway is delocalized, and the electrons can easily transfer from the left electrode to the right electrode. As a result, the down-spin electrons are conductive, and a perfect spin-filtering effect can be found in M1.

In the following part, we show research about the effects of the rotation of the PTB plane around the C chain on the electronic transport properties of the device. Fig. 5 shows the spin-dependent I–V curves for M2, M3 and M4 in PC and APC. For the convenience of comparison, the spin-dependent I–V curves of M1, which are the black dashed lines in Fig. 5, are also given. By comparing the up-spin I–V curves of M1–M3 in PC (Fig. 5(a)), one can find a similar transmission behavior, but the value of the current gradually decreases. When it comes to M4, the current is nearly zero (10⁻⁷ μA). So, we can get the conclusion that the currents decrease with an increase in the angle between the PTB plane and the 6ZGNR electrode plane. And a similar phenomenon can also be found for the down-spin I–V curves of M1–M4 in PC (Fig. 5(b)) and the spin-dependent I–V curves of M1–M4 in APC (Fig. 5(c) and (d)). From above, it is clear that the devices can realize the ON and OFF states by changing the angle between the PTB plane and the 6ZGNR electrode plane. M1–M3 can be regarded as the ON state, while M4 can be seen as the OFF state, so the proposed device can also be a perfect molecular switch. And the biggest on/off ratio is between the M1 and M4 devices.

Fig. 5 (a) and (b) [(c) and (d)] The spin-dependent I–V curves for M2-M4 in PC [APC]; and (e) [(f)] the on/off ratio of M1/M4 in PC [APC]. The black dashed lines are the I–V curves of M1.

Therefore, we then calculate the on/off ratio of M1/M4 in PC and APC, which can be found in Fig. 5(e) and (f), respectively. From Fig. 5(e) we can see that the up-spin on/off ratios are all higher than 10⁶ in the bias range of [−0.5, 0.5 V], and tend to increase with decreasing |V_b|. For the on/off ratio of M1/M4 in APC, as shown in Fig. 5(f), the on/off ratio can reach up to 10⁷. More importantly, the on/off ratio in our device is about 10⁵ orders of magnitude higher than that from previous researches.^52–54

To understand the mechanism of the on/off effect in PTB based devices, then we present the molecularly projected self-consistent Hamiltonian (MPSH) of the HOMO and LUMO for the up-spin states of M1–M4 at zero bias in PC, as shown in Fig. 6. For M1, M2, and M3, we can find that the HOMO and LUMO are delocalized, which indicates that the electrons can easily transfer from the left electrode to the right electrode. Meanwhile for M4, one can see that both the HOMO and LUMO are localized, so the transmission of electrons between the left and right electrodes is suppressed. Additionally, comparing the HOMO and LUMO of M2, M3 and M4 with those of M1, it can be found that the coupling between the alkyne triple bond and the phenyl rings of PTB is gradually weakened. So we can get the information that, with the increase in the rotation angle of PTB, the coupling strength between the alkyne triple bond and the phenyl rings of PTB decreases, and nearly disappears at 90°. So the electrons cannot pass through, and the on/off function can be realized.

Fig. 6 The MPSH of the HOMO and LUMO for the up-spin state of M1–M4 at zero bias in PC. The red and blue colors indicate the positive and negative signs of the wave functions, respectively. The isovalue is 0.03 Å^−3/2.

Moreover, to further understand the current suppression behavior, we also plot the up-spin transmission spectra and up-spin local density of states (LDOS) at the Fermi level of M1–M4 in PC at 0.2 V, as shown in Fig. 7. We can see the transmission area in the bias window becoming smaller from M1 to M4, so the corresponding current also becomes smaller. In particular, for M4, there is no transmission spectrum appearing in the bias window, so the current is almost zero. Then we present the up-spin LDOS of M1 to M4 at the Fermi level, as shown in Fig. 7(e)–(h); it can be found that the LDOS of M1 and M2 is delocalized, which offers the transmission states for electrons. For M3, though the LDOS is still delocalized, the distribution on the phenyl rings becomes weaker, so the current becomes smaller. Meanwhile, the LDOS of M4 is completely localized in the left and right electrodes and the alkyne triple bonds (see Fig. 7(h)), so the transport channel of the electrons is blocked.

Fig. 7 (a)–(d) The up-spin transmission spectra of M1–M4 at 0.2 V in PC and (e)–(h) the up-spin LDOS of M1–M4 at the Fermi level. The isovalue is 0.05 Å⁻³. The region between the blue dashed lines is the energy region contributing to the current, i.e. the bias window.

To have a further understanding about the high on/off ratio, we also consider the other conditions for the rotation of the three phenyl rings in PTB. Based on the structure of M1, we firstly rotate two of the phenyl rings of PTB from 0 to 90°, as shown in Fig. 8(a), marked as M5. Then we rotate only one of the phenyl rings from 0 to 90°, as shown in Fig. 8(b), labeled as M6. The on/off ratios of the up-spin currents between the states of M1 with M5 and M6 in PC have been calculated, as one can see in Fig. 8(c). For a clear comparison, we also present that of M1/M4, shown as the black line in Fig. 8(c). Obviously, the on/off ratio of M1/M5 is about 10³, which is higher than that of M1/M6 (about 10²), but three orders of magnitude smaller than that of M1/M4. So M4 can effectively improve the on/off ratio. For M4, the length of the perpendicular phenyl rings is 1.14 nm, and that is 0.71 nm and 0.28 nm for M5 and M6, respectively. When the bias is applied, the probability of electrons tunneling through the scattering region is bigger as the length of the perpendicular phenyl rings decreases, which in turn contributes to the current, thus the on/off ratio is decreased.

Fig. 8 (a) and (b) Two-probe systems of M5 and M6 and (c) the up-spin on/off ratio of M1/M4, M1/M5, and M1/M6 in PC.

Conclusions

In conclusion, using DFT and the NEGF method, we have investigated the spin-dependent electronic transport properties of PTB connecting two 6ZGNR electrodes. Our results show that the device has a 100% SFE and NDR effect in PC, while it has a bipolar spin-filtering effect and spin rectifying functions in APC. We also find the transport properties have a significant relationship with the angles between the PTB plane and the 6ZGNR electrode plane. The currents decrease with increasing angle, and are finally close to zero when the angle is 90°. As a result, the on/off function can be realized, and the on/off ratio is as high as 10⁷. The theoretical analyses show that the matching of the electronic wave functions among both the 6ZGNR electrode and PTB scattering region, in addition to the coupling between the alkyne triple bond and the phenyl rings of PTB, is critical to the spin transport properties. These results suggest that the PTB based devices have potential utilization in designing multi-functional and high performance molecular devices.

Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant No. 61306149, 11274260), the Natural Science Foundation of Hunan Province (No. 14JJ3026), Hong Kong Scholars Program (No. XJ2013003), the Research Grants Council of Hong Kong SAR (Project No. CityU 100311/11P), and the Fundamental Research Funds for the Central Universities of Central South University (No. 2015zzts014).

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