Introduction

In recent years, the development of nano-electronic devices has attracted a lot of attention. Compared with traditional electronic devices, nano-electronic devices are smaller in size and more integrated, which greatly improves the speed and efficiency of calculations. Especially the spintronic nano-devices, which aim to exploit the spin degree of freedom in electrons, have aroused great research interest because of their characteristics of low energy consumption, high data processing speed and high-density information storage capabilities1,2. In recent years, various types of spintronic devices have been proposed, such as spin filters, spin valves, spin diodes, spin switches, spin transistors, etc3,4,5,6,7,8. Along with the development of nano-electronic devices, multifunctional spin molecular devices with high performance are more expected.

Finding suitable magnetic materials is one of the keys to designing high-performance spintronic devices. The rapid development of two-dimensional (2D) film materials has opened up new ways to explore geometrically limited nano-electronic phenomena. 2D materials, characterized by their outstanding properties including surface flatness, flexibility, ease of cutting and chemical modifications, tunability through electromagnetic field or stress, provide an ideal platform for designing and investigating multifunctional spintronic nano-devices9,10,11,12. So far, different kinds of nanodevices have been designed based on a variety of 2D materials13,14,15,16. As the nearest neighbor element on the left side of carbon, boron is expected to construct borophene, a high-performance single-element 2D material like graphene17,18,19. Because boron has one less electron in its outermost shell than carbon and the main valence state of boron is trivalent, it is difficult to exhibit a uniform honeycomb structure. In 2007, the Ismail-Beigi team at Yale University discovered a two-dimensional borophene structure based on a mixture of triangular close-packed and hexagonal holes, called the α-sheet20. They found that a relatively stable borophene structure can be obtained by mixing the triangular close arrangement with hexagonal holes in a certain proportion, which is the basic starting point for further exploration of new borophene structures. Since then, a large number of borophene allotropes have been predicted based on different hexagonal holes densities20,21,22. In terms of experiments, in 2015, Argonne National Laboratory in the United States cooperated with scholars from three universities used molecular beam epitaxy technology to synthesize borophene on Ag substrate for the first time23. Subsequently, various phases of borophene structures were experimentally synthesized on different metal substrates (Ag, Al, Cu, Au)24,25,26.

The existing experimental and theoretical studies provide a variety of borophene structures, and the calculations show that all of them are metallic22. The abundant electronic states and diverse structures make borophene a potential material for constructing nanodevices. Previous researches have shown that 2D nanosheets can be tailored into quasi one-dimensional nanoribbons, which are eminently suitable for constructing nanojunctions27,28. And recent experiments find super-narrow BNRs can also be synthesized by quantum confinement of the substrate29,30. Many factors, such as transport direction, ribbon width, and edge pattern of nanoribbons affect their spintronic transport properties significantly, which may realize the functionalization of nanostructure31. Inspired by this, we cut out a series of BNRs from a 2D χ-type borophene nanosheet, which was obtained through an extensive search using the first-principles particle swarm optimization global algorithm22. We then systematically study the effects of ribbon width and edge pattern on the spin-resolved electronic properties of the BNRs. Notably, the super-narrow BNR exhibits half-metallic ferromagnetism, paving the way for the design of borophene-based spintronic nanodevices. Based on this, we further construct a super-narrow nanojunction configuration and theoretically calculate its spin-dependent transport properties using the first-principles method. Obvious spin filtering behavior can be realized in the device. Furthermore, regulation of magnetic configurations and stresses during spintronic transport is taken into account to improve the performance of device. In addition, more intriguing spin-dependent transport properties, such as spin rectification, magnetoresistance and piezoresistance, are observed under the control of an external field. Thus, the goal of constructing a multifunctional spin nanodevice with high performance is achieved. This study provides theoretical references for experimental researchers.

Methods

The first-principles calculations in this paper are performed using the Quantum ATK software package32,33. The numerical linear combination of atomic orbitals (LCAO) engine is used for all calculations34, and the basis set is double ζ polarization (DZP). The mode conservation Troullier-Martins pseudopotential is used to describe the internal electron contribution of atoms35, and the exchange correlation functional used the Perdewe-Burke-Ernzerhof (PBE) functional in the spin-dependent generalized gradient approximation (SGGA)36. The structure optimization is based on density functional theory (DFT), and the convergence criterion is less than 0.02 eV/ Å for each atom. After the convergence test, the k-point is set as 1 × 1 × 100, and the real-space grid sampling is chosen with a mesh cutoff of 75 Hartree.

The electron transport calculation is based on the non-equilibrium Green’s function method combined with density functional theory (NEGF + DFT).37 At a certain bias, the spin-dependent current through the central scattering region is calculated by the Landauer-Büttiker formula38, which is shown as follows:

$$\:{I}_{\sigma\:}=\frac{e}{h}\int\:{T}_{\sigma\:}(E,V)\left[{f}_{L}\right(E-{\mu\:}_{L})-{f}_{R}(E-{\mu\:}_{R}\left)\right]dE.$$
(1)

Where the subscript \(\:\sigma\:\) represents the spin-up (α) state or spin-down (β) state, \(\:{f}_{L\left(R\right)}\) is the Fermi-Dirac distribution function, and \(\:{\mu\:}_{L\left(R\right)}\) is the electrochemical potential of the left and right electrodes. \(\:{T}_{\sigma\:}(E,V)\) is the spin-dependent transmission coefficient at bias V, which is defined as:

$$\:{T}_{\sigma\:}=Tr\left[{\varGamma\:}_{L\sigma\:}{G}_{M}^{r\sigma\:}{\varGamma\:}_{R\sigma\:}{G}_{M}^{a\sigma\:}\right].$$
(2)

Where \(\:{\varGamma\:}_{L\sigma\:\left(R\sigma\:\right)}\) is the coupling matrix of the left (right) electrode related to the spin of the central scattering region, and \(\:{G}_{M}^{r\sigma\:\left(a\sigma\:\right)}\) is the retarded (advanced) Green’s function with spin index.

Results & discussion

Electronic properties of χ-type BNRs

Fig. 1
figure 1

(a) Structure of the χ-type borophene nanosheet, “a” and “b” indicate rows of boron atoms with different coordination numbers in the 2D film. (b) Band structure of the χ-type borophene nanosheet. The red solid and blue dashed lines represent the subbands of spin-up (α) and spin-down (β) electronic states, respectively.

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The geometry of the target χ-type borophene nanosheet is shown in Fig. 1a, which was proposed through structural searches in previous studies22. The thermal stability illustrated in the Supplementary Information Fig. S1 supports the possibility of its experimental fabrication. Two types of boron atoms are present in this χ-type borophene film. The rows of boron atoms with coordination numbers (CNs) of 5 and 4 are labeled “a” and “b”, respectively. Using the first-principles method, we explore the band structure of the 2D borophene film and find that it is a conductor and its electronic states are spin-degenerate (see Fig. 1b). Hence, the 2D borophene is not suitable for constructing spin nanodevices.

Fig. 2
figure 2

(a) Band structure and spin-resolved density of states of 4aaZχ BNR. The inset illustrates its geometrical configuration. (b) The density of α, β and α-β (net spin) electronic states of 4aaZχ BNR. (c) Fat band structure of 4aaZχ BNR. The subbands around the Fermi level are labeled 1—5.

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Considering that 2D nanosheets can be tailored into quasi-one-dimensional nanoribbons, and that the electronic properties can be effectively regulated by the ribbon width and edge pattern, we cut the borophene nanosheet into narrow ribbons and calculate their spin-resolved band structures one by one. The results are illustrated in the Supplementary Information. Most intriguingly, by analyzing the band structure and density of states in Fig. 2a, we find that the super-narrow 4aaZχ BNR exhibits remarkable half-metallic properties, where α electrons behave like a semiconductor while β electrons behave like a conductor39. This intrinsic spin selectivity provides assistance in designing spin nanodevices.

The spin-resolved electronic properties of wider BNRs with different edge patterns are presented in the Supplementary Information (Figs. S2-S5), where none of them show half-metallic ferromagnetism. In this study, we primarily focus on the method for constructing functional nanodevices by utilizing the half-metallic properties of materials. Therefore, these BNRs are not recommended for subsequent studies of spintronic nanodevices.

To inspect the source of magnetism in 4aaZχ BNR, we calculate the spin density, as shown in Fig. 2b. It is clear that the spin magnetic moment is mainly contributed by the atoms in the edge hexagonal holes, especially the b-type atom labeled 6. Besides, the spin-up electrons have one more occupying subband than spin-down electrons below the Fermi level. This can be demonstrated by the 0.997 µΒ magnetic moments of 4aaZχBNR unit cell.

Based on the above, 4aaZχ BNR is a promising candidate for creating functional spin nanodevices. In Fig. 2c, we present the fat band structure of 4aaZχ BNR to clarify the source of electronic states around the Fermi level. It can be observed that the 2s and 2p orbitals of the boron atoms mainly contribute to the subbands within the energy range from − 1 eV to 1 eV. Among them, subbands 1, 3 and 4 are primarily contributed by the in-plane orbitals s, py and pz, while subbands 2 and 5 are solely contributed by the out-of-plane orbital px. The different spatial distributions of atomic orbitals cause the Bloch wave functions of the subbands to have distinct parity characteristics under the mirror operation in the nanoribbon plane (yz-plane). Thus, we hope to achieve the design and improvement of nanodevices by manipulating wave functions with different parities.

Spin transport properties of BNR-based nanojunction

Fig. 3
figure 3

Schematic of borophene-based spintronic nanodevices. (a) Device M1 represents the pristine structure, in which two electrodes have the same spin polarization directions. (b) The spin polarization directions of the two electrodes are antiparallel in device M2. (c) The width of right electrode is compressed by 5% in device M3.

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The configuration of the 4aaZχ BNR-based device is shown in Fig. 3. The left and right electrodes of device M1 are 4aaZχ BNRs. Compared with a common boron single chain, the two-atom wide boron ribbon has a more stable structure and is more conducive to electron transport40. Then, the left and right electrodes are connected by a two-atom-wide boron ribbon in the middle. The entire structure can be tailored from the χ-type borophene nanosheet22. In device M1, the two electrodes exhibit a ground state ferromagnetic order, and their magnetic configurations are readily set to be parallel. On the basis of M1, structures M2 and M3 are prepared by manipulating the magnetic configuration of electrodes and compressing the electrode width, respectively. In detail, the magnetic configurations of two electrodes are set to be antiparallel in M2 by flipping the spin polarization direction of right electrode. In M3, the right electrode is compressed 5% of its original width along the Y direction, while maintaining a two-dimensional planar structure. Based on the first-principles calculation, it is found that both these two control methods can change the spin transport properties of nanojunction significantly. The I-V curves of device M1, M2 and M3 are shown in Fig. 4. From the spin-resolved current, it can be clear witnessed that the conduction current is polarized in these three devices. And the remarkable spin polarized I-V curves suggest their high spin filtering efficiency. Additionally, the spin-resolved currents in M2 and M3 exhibit unidirectional conductive properties, indicating their spin rectification characteristics. Moreover, the opening voltage of the current is reduced in M2 and M3. In other words, the device can exhibit giant magnetoresistance or piezoresistance effects within specific bias ranges. The causes of these phenomena will be analyzed in detail in the following sections.

Fig. 4
figure 4

Spin-resolved I-V curves for devices (a) M1, (b) M2, and (c) M3. Iα and Iβ represent the currents of α and β electronic states, respectively.

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Spin filtering effect

As shown in Fig. 4a, under positive bias voltages, the spin-up current (Iα) is filtered out within our calculated bias range, while the spin-down current (Iβ) rises to a considerable value after the threshold opening voltage of 0.3 V, i.e., the conduction current is spin-polarized within the bias range (0.4 V, 0.9 V). Similar results are observed with negative bias voltages. Thus, device M1 demonstrates excellent spin filtering performance. In Fig. 4b, under the positive bias voltage, M2 also has excellent ability to filter out Iα, and the threshold opening voltage for the conduction current Iβ is reduced to 0.1 V. Because of the symmetrical geometric structure of M2, the total current under negative bias has a similar evolutionary trend as that under positive bias. However, the spin properties of conduction current and cutoff current are reversed at positive and negative bias voltages, due to the opposite magnetization directions of two electrodes. The major conduction current becomes Iα under negative biases. Therefore, M2 has bidirectional spin filtering properties under positive and negative bias voltages. In Fig. 4c, under positive bias voltages, a similar filtered spin current to that of M2 can be witnessed in M3, and reversed spin filtration under negative bias voltages can also be achieved. Besides, it is interesting to see that the spin-resolved current of M3 shows oscillating characteristics. As the negative bias increases, the conduction current is dominated by Iα and Iβ successively. Therefore, the filtered spin properties in M3 can be manipulated by adjusting the negative bias voltage.

Fig. 5
figure 5

Spin filtering efficiency curves of devices (a) M1, (b) M2 and (c) M3, respectively. (The SFEs at non-conductive bias voltages are treated as 0.)

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In order to quantitatively describe the spin filtration performance of these devices, the spin filtering efficiency (SFE) is introduced and defined as.

$$SFE = [(I_{\alpha} - I_{\beta} )/(I_{\alpha} + I_{\beta} )] \times 100\%,$$
(3)

as shown in Fig. 5. Herein, the maximum SFE of M1 is close to -100%, which occurs in the bias range of 0.4 V to 0.9 V (or -0.4 V to -0.9 V). In devices M2 and M3, significant bidirectional spin filtering effect can be achieved by applying reverse voltages, and the SFE reaches − 100% and 100% respectively under specific positive and negative biases. Besides, bidirectional spin filtration can also be achieved in M3 by adjusting the magnitude of negative bias voltages.

Fig. 6
figure 6

Transmission spectra and band structures of both electrodes for (a) M1 at 0.5 V, (b) M2 at -0.5 V, (c) M3 at -0.2 V, and (d) M3 at -0.4 V. The black dashed lines indicate the bias window. The red and blue triangle symbols point to the eigenvalues of α and β states of the central two-atom-wide boron ribbon. (e) Bloch states corresponding to the noted locations of each subband. (f) The spatial states of the two frontier molecular orbitals. (g) Spin-resolved transmission eigenstates of M3 at an energy of -0.13 eV under − 0.4 V bias voltage.

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In order to figure out the spin filtering mechanism in these devices, we present the spin-polarized transmission spectra and band structures of both electrodes for M1 at 0.5 V, M2 at -0.5 V, M3 at -0.2 V, and M3 at -0.4 V in Fig. 6a, b, c and d, respectively. When a bias voltage V is applied, the energies of subbands in the left electrode shift downwards by eV/2 and the energies of subbands in the right electrode shift upwards by eV/2. In Fig. 6a, the β transmission spectrum contributes to the current because both β subbands of two electrodes and the frontier orbitals of β spin are involved into the bias window. To thoroughly investigate the electron tunneling mechanism, the Bloch wave functions of subbands and the spatial distributions of frontier orbitals are plotted in Fig. 6e and f. Herein, both Bloch wave functions and frontier molecular orbitals have parity characteristics under the yz-plane mirror operation. Due to the parity conservation rule for the wave function of tunneling electrons, only the tunneling between two electronic states with the same symmetry is allowed41. Based on this, β electrons in Fig. 6a can tunnel from subband 1 (1b and 1c) of left electrode to subband 3 (3a and 3b) of right electrode through the frontier molecular orbital HOMO-1 (the one below the highest occupied molecular orbital), contributing to β current. However, no α subband of left electrode aligns with the bias window, resulting in an incomplete tunneling channel for α electrons. Thus, M1 shows remarkable spin filtering performance. Similar cases can be observed in devices M2 and M3 under positive bias voltages.

In M2, because the spin polarization directions are antiparallel, the spin characteristics of subbands in the right electrode are opposite to those in the left electrode, as shown in Fig. 6b. This leads to the absence of β subbands within the bias window at negative bias voltages. Therefore, when the reverse current passes through M2, it will filter out Iβ. Finally, M2 shows bidirectional spin filtering behavior under positive and negative biases.

For M3, after the right electrode is compressed by 5% of the original width, it is found that the band structure of the right electrode changes, as shown in Fig. 6c and d. The original subbands at 1a and 4a shift downward to become 6a and 7a, while the subbands at 1c and 4c shift upward to become 6c and 7c. It can be observed from Fig. 6e that although the positions of these subbands are shifted, their spatial distributions of Bloch functions still maintain a high consistency. In Fig. 6c, α electrons within the bias window can tunnel from subband 4 (4a, 4b and 4c) of the left electrode to subband 7c of the right electrode through the frontier molecular orbital HOMO-1, contributing to a distinct transmission peak. However, the tunneling between electronic states 2a (2b) and 6a, which have different parities under the yz-plane mirror operation, is forbidden. As a result, the transmission peaks cannot be obtained even though β subbands align with each other within the bias window.

As the negative bias voltage increases, the subband 3 (3a and 3b) of the left electrode is involved into the bias window gradually, as shown in Fig. 6d. Henceforth, β electrons can tunnel from subband 3 (3a and 3b) of the left electrode to subband 6a of the right electrode through HOMO-1, resulting in a β transmission peak emerging. However, the transmission peak is very small. This is closely related to the geometric structure of the nanojunction. In these devices, the left and right electrodes are connected by a double-layer boron strip, and the anchoring positions are two layers of atoms in the lower part of both electrodes. This results in the lower two layers of atoms having a relatively large contribution to electron tunneling. Observing the spatial distribution of Bloch function 6a in Fig. 6e, its wave function is mainly localized in the upper part of the electrode. Thus, the tunneling capability of β electron transmission channel contributed by 6a is not strong. In order to visualize the different tunneling probabilities of α and β transmission peaks in device M3, the transmission eigenstates of different spin states at -0.13 eV energy level under a bias voltage of -0.4 V are illustrated in Fig. 6g. It is clear that the transmission eigenstate of the β electron has large amplitude on the left electrode and the middle double-layer boron strip, but only few states are scattered on the right electrode, corresponding to a small β transmission value. Therefore, under low negative bias voltages, M3 can filter out Iβ.

With further lifting the negative bias voltage, the α overlap range between subband 4 (4b and 4c) and subband 7c will decrease gradually, while the β overlap range between subband 3 (3a and 3b) and subband 6 (6a, 6b and 6c) will increase gradually. Thus, as the negative bias voltage increases, spin filtered current of M3 changes from Iα to Iβ, showing a spin selectable filtering phenomenon.

Spin rectification

Fig. 7
figure 7

Spin-resolved rectification ratio curves of (a, b) M2 and (c, d) M3.

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Focusing on the I-V curve of M2 and M3, one of the most noteworthy features is that every spin-resolved current is unidirectionally conductive. Namely, M2 and M3 have obvious spin rectifying behavior. To quantify the rectifying performance of these devices, the rectification ratio is defined as.

$$RR = [I_{{\alpha(\beta)}} \left( V \right)/I_{{\alpha(\beta)}} \left( { - V} \right)],$$
(4)

where Iα(β)(V) and Iα(β)(-V) are the currents at V and -V bias, respectively. The rectification ratio curves of M2 and M3 are drawn in Fig. 7, and two interesting features can be witnessed from them. Firstly, because the current is completely cut off in a certain range of bias voltage, both M2 and M3 show giant rectifying properties, in which the maximum 1/RR reaches 7.6 × 105 and 8.3 × 106. Secondly, bipolar rectifying effect is evidently observed both in M2 and M3.

Fig. 8
figure 8

Transmission spectra and band structures of both electrodes for (a) M2 at 0.5 V and (b) M3 at 0.4 V. (c) Isosurface plots of Bloch states of the spin-resolved subbands 8 and 9.

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The spin rectifying behavior in M2 and M3 devices can also be analyzed by the spin-polarized transmission spectra and band structures of both electrodes. In M2 (Fig. 8a), β electrons tunnel from subband 1 (1b and 1c) of the left electrode to subband 4 (4a, 4b and 4c) of the right electrode through the frontier orbital HOMO-1, indicating that the β spin state remains conductive under positive bias. However, β spin subbands of right electrode vanish within the bias window under negative bias, resulting in a current cutoff of β spin, as shown in Fig. 6b. Thus, β current of M2 shows forward rectifying behavior. Furthermore, by comparing the electron tunneling channel under positive and negative biases, it is not difficult to find that α current has a reverse rectifying direction compared to β current. In M3, according to the parity characteristics of Bloch states (Figs. 6e and 8c), β current is conductive under positive bias, which benefits from the tunneling channel formed by subband 1 (1b and 1c), HOMO-1 and subband 8 (8a and 8b), as shown in Fig. 8b. However, β current is limited under negative bias, because of the localized spatial distribution of Bloch function of subband 6a (Fig. 6). Therefore, β current of M3 shows forward rectification. For α current of M3, there is a complete electron tunneling channel under negative bias, while the electron tunneling channel is blocked at left electrode under positive bias. Thus, α current of M3 shows reverse rectification.

Giant magnetoresistance and piezoresistance

Fig. 9
figure 9

(a) I-V curves of the total current for M1, M2 and M3. (b) The magnetoresistance ratio curve calculated from the currents of M1 and M2. (c) The piezoresistance ratio curve calculated from the currents of M1 and M3.

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The total current curves of three devices within a bias range of (-0.4 V, 0.4 V) are illustrated in Fig. 9a. It is clear that the current of M1 is significantly lower than that of M2 and M3. Namely, magnetoresistance and piezoresistance can be obtained. To quantify these two resistance changes, the magnetoresistance ratio (MR) and piezoresistance ratio (PR) are plotted in Fig. 9b and c, where.

$${\text{MR}} = \left| {\left( {I_{{{\text{M2}}}} - I_{{{\text{M1}}}} } \right)/I_{{{\text{M1}}}} } \right|,$$
(5)
$${\text{PR}} = \left| {\left( {I_{{{\text{M3}}}} - I_{{{\text{M1}}}} } \right)/I_{{{\text{M1}}}} } \right|.$$
(6)

Nevertheless, at zero bias, there is no current in these devices. The resistive ratios cannot be obtained from the equations above. Therefore, MR and PR are calculated by.

$${\text{MR}} = \left[ {T_{{{\text{M2}}}} \left( {E_{{\text{F}}} } \right) - T_{{{\text{M1}}}} \left( {E_{{\text{F}}} } \right)} \right]/T_{{{\text{M1}}}} \left( {E_{{\text{F}}} } \right),$$
(7)
$${\text{PR}} = \left[ {T_{{{\text{M3}}}} \left( {E_{{\text{F}}} } \right) - T_{{{\text{M1}}}} \left( {E_{{\text{F}}} } \right)} \right]/T_{{{\text{M1}}}} \left( {E_{{\text{F}}} } \right),$$
(8)

where TM1(EF), TM2(EF) and TM3(EF) are transmission coefficients of M1, M2 and M3 configurations at EF, respectively. According to the above formulas, the maximum MR values occur at ± 0.2 V bias, which exceed 4.0 × 103. The maximum PR reaches 6.9 × 103 under − 0.2 V bias.

Fig. 10
figure 10

Transmission spectra and band structures of both electrodes for (a) M1 at -0.2 V and (b) M2 at -0.2 V. Under a bias voltage of -0.2 V, the transmission pathways for (c) M1, (d) M2 and (e) M3 at 0 eV, -0.04 eV and − 0.10 eV, respectively.

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To clarify the giant magnetoresistance and piezoresistance behaviors, the overlapping subbands of M1, M2 and M3 under − 0.2 V bias are analyzed. In Fig. 10a, the tunneling channel is blocked for M1 under − 0.2 V bias, because the overlapping subbands 1 (1b and 1c) and 2 (2a and 2b) within the bias window have different parities under the yz-plane mirror operation. Thus, there is no current flowing across junction M1 under − 0.2 V bias. Whereas in Fig. 10b, electrons tunnel through M2 within the bias window, where two subbands (the subband 4 (4b and 4c) of the left electrode and the subband 1 (1b and 1c) of the right electrode) with the same parities align with each other. This electron tunneling channel contributes to the obvious current of M2 under − 0.2 V bias. According to the above analysis, the electrical conductance of the junction can be tuned by changing the magnetization configuration of right electrode. Hence, giant magnetoresistance is realized in our device. Similar analysis goes to M1 and M3. By comparing the subband overlap range of M1 and M3 within the − 0.2 V bias window, it is clear that the compressed right electrode provides an electron tunneling channel. Therefore, the target 4aaZχ BNR model exhibits giant piezoresistance behavior.

The different conductivity can also be confirmed by the corresponding transmission pathways, plotted in Fig. 10c-e. Herein, we draw the transmission pathways of M1, M2 and M3 at 0 eV, -0.04 eV and − 0.10 eV under − 0.2 V bias voltage. The transmission pathway of M1 is blocked in the left electrode, while the transmission pathways of M2 and M3 are continuous throughout the whole backbone. Therefore, the conductivity of the device can be significantly enhanced either by applying an antiparallel magnetic configuration to the electrodes or by narrowing the width of either electrode.