Monolayer graphene^[1] has become a promising material for applications in nanoelectronics such as in high speed transistors^[2,3] and spintronics.^[4] This is due to its intriguing electronic properties.^[5] Monolayer graphene can be considered as a gapless semiconductor; its carriers mimic two-dimensional massless Dirac fermions with the speed of light replaced by the Fermi velocity v_F ≈ 10⁶ m/s. The mass of the Dirac fermion in graphene acts as the energy gap. The wave functions of electrons in A- and B-sublattices play the role of pseudo-spin ↑ and pseudo-spin ↓ states, respectively. Electrons in graphene also have a valley degree of freedom k and k′ different from those of 3-dimensional Dirac fermions in a vacuum. Because of its carrier behaving like a relativistic particle, graphene could be considered as a high energy system in condensed matter.^[6] Massless Dirac fermion in graphene can tunnel through the electric-gated barrier without back reflection at the normal incident, which is called “Klein tunneling”.^[7] This effect arises from the conservation of the chirality. It is therefore difficult for a gapless graphene-based field effect transistor to suppress the current for off-state by an electric gate control. In general, graphene is non-magnetic but it can be induced into a ferromagnetic state by means of the proximity effect.^[8–12] This property is very important for application of graphene in spintronic technology.^[4] A lot of attention^[13–17] has been given to magnetotransport in monolayer graphene-ferromagnetic junctions. Spin polarization and magnetoresistance in monolayer graphene nanoribbon^[17] was investigated. It has also been found that the interplay of ferromagnetism and strain leads to controllable spin-valley currents.^[18–20]

Bilayer graphene has some properties that are different from those of monolayer graphene. One of them is the opening of the gap by an applied electric field.^[21,22] When the potential energies V₁₍₂₎ of the electrons in layer 1 (2) are different, the gap opening in AB-stacked bilayer graphene is given by E_gap = (V₂ − V₁)/2.^[21] Monolayer graphene lacks this property since its atomic structure is planar. Due to the differences, transport in bilayer graphene leads to different behavior from those of the monolayer graphene such as anti-Klein tunneling (or perfect back reflection) at the normal incidence.^[23] The current in the bilayer graphene junction can be suppressed easily by an electric gate. Similar to monolayer graphene, bilayer graphene can be induced into the ferromagnetic state by means of the proximity effect.^[24–31] Magnetoresistance and spin polarization in bilayer graphene have been investigated.^[24–26,31] The magnetism in bilayer under the influence of pressure has also been studied.^[27] Experimental study of quantum Hall ferromagnetism in bilayer graphene was recently completed.^[30]

In this paper, we will investigate the effect of layer-dependent ferromagnetism on the magnetotransport property in an AB-stacked bilayer graphene-based N/F/N junction, where N and F are normal and ferromagnetic regions, respectively. We would take the exchange energy in the top layer to be different from that in the bottom layer in the barrier region in the junction. Through this means, the energy gap of the electron in the barrier junction will become spin-dependent when an electric field is applied. This property will be described in the theoretical model. The effect of the interplay of electric and exchange fields on the spin transport property is the main focus of our work, which had not been clarified in previous studies.^{[25,26,29,31]} This effect does not appear in a monolayer graphene ferromagnetic junction.^[8,32] Our work aims to show the potential of layer-dependent ferromagnetism in bilayer graphene, which can generate the carriers with spin-dependent gap controlled by electric field for application in spintronics.

The model of our proposed NFN junction, where N represents a normal bilayer graphene region and F represents a ferromagnetic bilayer graphene region, is depicted in Fig. 1(a). To achieve the latter region, two magnetic insulators are placed above and below a narrow strip of bilayer graphene so that each insulator is in contact with either the top or bottom layer of the bilayer graphene. The proximity effect will induce an exchange field in the graphene sandwiched between the two insulators and will induce the graphene into a ferromagnetic state. This would be the ferromagnetic barrier region (of thickness L) in our NFN bilayer graphene junction. Depending on the direction of the magnetism in the two insulators, two configurations are possible leading to parallel (P) and antiparallel (AP) junctions. The P-(AP-) junction corresponds to the top and bottom exchange fields being parallel (anti-parallel). The gate potential −U/e and perpendicular electric field E_z can also be imposed on the junction. The magnitude of exchange energy “h” induced by the two magnetic insulators in both layers is assumed to have the same value. The wave function of the electron in bilayer graphene will be in the form of a four-state component wave function ψ = (φ_A1 φ_B1 φ_A2 φ_B2)^T, where 1(2) and A (B) denote the top (bottom) layer and the sublattice A (B), respectively.

Fig. 1.

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Fig. 1. Schematic illustration of bilayer graphene-based normal/ferromagnetic/normal junction (a) where h1 and h2 are the exchange energies in the top and bottom layers, respectively. The spin-dependent energy levels in each region of the P-junction and spin-dependent energy levels in each region of the AP-junction are illustrated in panels (b) and (c), respectively. In the case of |h1| = |h2| = h = ΔM, the spin-dependent gap is generated only when the junction is an AP-junction and there is no spin splitting.

The Hamiltonian for electrons in the barrier region of the AB-stacked bilayer graphene ferromagnetic junction^[21,33] in the P-(AP-) configuration is

In the barrier, the spin-dependent Eigen energies for the Hamiltonian for P- and AP-junctions in Eqs. (2) and (3) are, respectively,

For the N-regions,

Here, p and q represent momentums in F and N regions, respectively and E is the energy of the electron.

In the normal region, the bilayer graphene becomes a gapless semiconductor. The energy levels for spin up and down electrons in each region are described in Fig. 1(b). The exchange fields in the P-junction generate the spin-splitting. The energy gap ε_P,σ = 2|Δ_E| of electron in the P-junction is generated solely by the electric field. The junction leads to magnetic-induced gap Δ_M = h only in the case of the AP-junction. This is due to the mismatch of exchange fields in the top and bottom layers in the barrier region. The spin-dependent energy gap in the AP-junction has the form

As we have seen in Eq. (8), for Δ_E = Δ_M, the electron with spin ↑ acquires an energy gap of 2Δ_M, while the electron with spin ↓ will be gapless. The strong differences in energy gaps of spin-up and spin-down electrons in the AP-junction would give rise to an intriguing magnetotransport property.

In this section, we study the scattering process in an AB-stacked bilayer graphene-based N/F/N junction. The current is taken to flow in the x direction. The wave functions of the carriers in the system are combinations of the eigen solutions of the two-component Hamiltonian defined by Eqs. (2)–(4). If the electrons are being injected at an angle θ to the plane of the interface between the different regions of the junction, the momentum in the y direction will be conserved and is given by with E being the excitation energy. The wave functions of the electrons in the left-N, and right-N for the P and AP cases are

The wave function in the ferromagnetic barrier of a P-junction is given by

The wave function in the ferromagnetic barrier of an AP-junction is obtained by changing the subscripted notation P → AP, where the coefficients are now

All amplitudes of the wave function or coefficients r, r′, t, t′, a_1,2, and b_1,2 in Eqs. (9) and (10) can be determined by matching the wave functions at the interfaces located at x = 0 and x = L, i.e., by requiring that at the two interfaces

The transmitted coefficient in the case of the P (AP) junction is given by t → t_P(AP) in Eq. (9) corresponding to the P (AP) type.

The current density in the x direction is given by

The total conductance is defined as

The spin polarization in the junction is defined by the ratio of the difference between the conductance of spin-up and spin-down currents, as given by

The tunneling magnetoresistance (TMR) of the junction is defined as

In the numerical calculation, the ferromagnetism in bilayer graphene is assumed to be proximity-induced by magnetic insulators of exchange energy h = 5 meV.^[8] The thickness of the barrier L is taken to be 100 nm. We first study the spin-dependent transmissions for P and AP configurations, which are plotted in Fig. 2. In the case of no electric field-induced gap Δ_E = 0, the transmissions for two configurations are seen in Figs. 2(a) and 2(b). In this case, the carriers are gapless particles. The transmission of the AP-junction is not spin-dependent because there is no spin-dependent gap and exchange-induced-spin-splitting (see Fig. 2(b)). The spin-dependent transmission in the P-junction arisen explicitly from the exchange-induced-spin-splitting (see Fig. 2(a)). When an electric field is applied Δ_E = h, a strong spin-dependent gap E_gap↑ = 2h and E_gap↓ = 0 occurs in the case of the AP-junction. This gives rise to a strong insulating barrier for the spin-↑ electron but to a good conducting barrier for the spin-↓ electron (see Fig. 2(d)). Electrons in the P-junction acquire the same energy gap E_{gap ↑,↓} = 2Δ_E, but at a different Fermi level. This would cause a different transmission of the electrons (see Fig. 2(c)). It can be concluded that the spin-dependent transmission in P- and AP-junctions is generated in different ways. In the P-junction, the spin-dependent transmission is generated by an exchange-field induced shift of the Fermi level μ_↑ ≠ μ_↓ with μ_σ = σh − U, just like a Zeeman splitting. In the case of the AP-junction, the spin-dependent transmission is due to the exchange-field effect on the spin-dependent-gap, which causes E_gap↑ ≠ E_gap↓, since E_gap,σ = Δ_E + σh.

Fig. 2.

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Fig. 2. Comparison of transmissions in the P and AP configurations at zero gate potential. P-and AP-junctions. Panels (a) and (b) show the transmissions for the two configurations when the electric field is absent (i.e., when ΔE = 0). Panels (c) and (d) show the transmission for the P and AP configurations in the presence of an electric field (i.e., ΔE = h ≠ 0) respectively. Spin-dependent transmissions occur when the electric field and exchange field are applied in the magnetic AP-junction, ΔE ≠ 0 and h ≠ 0.

We next study the dependence of the spin-conductance G_↑,↓ on the electric field Δ_E = eE_zd for both the P and AP configurations with E = U in Fig. 3. As seen in Figs. 3(b) and 3(d), the perfect spin filter, (i.e. the spin conductance splits into two peaks perfectly) occurs when the junction is anti-parallel. This behavior is stable under a varying E ≫ h. The P-junction does not exhibit such behavior. The junction exhibits a strong conducting junction when E is inside the interval − h ≤ Δ_E ≤ + h, as plotted in Figs. 3(a) and 3(c). In Fig. 4, we show that there is no spin-conductance splitting in the P-junction, even when varying U ≠ E. For Δ_E is in the interval − |E + h − U| ≤ Δ_E ≤ |E + h − U|, a spin-up electron will flow through the junction, while a spin-down electron will flow in the junction when Δ_E is in the interval − |E − h − U| ≤ Δ_E ≤ |E − h − U|. In Fig. 5, we see that an AP- junction with a perfect spin filter having a smaller spin-current band width is possible when E = U (see Fig. 5(c)). This result points to the condition needed for an AP junction to exhibit the perfect spin filtering effect, which can be controlled by an electric field.

Fig. 3.

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	Fig. 3. Plot of spin conductance versus electric field-induced gap ΔE in the case of (a) a P-junction with E = U = 5 meV, (b) an AP-junction with E = U = 5 meV, (c) a P-junction with E = U = 30 meV, and (d) an AP-junction with E = U = 30 meV, respectively. As is seen, the perfect (or complete) spin filter by varying electric field occurs only for the AP configuration.

Fig. 4.

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	Fig. 4. Plot of spin conductance versus electric field-induced gap ΔE in the case of E = h = 5 meV, for the P-junction with different values of U: (a) 0 meV, (b) 2.5 meV, (c) 5 meV, and (d) 7.5 meV.

Fig. 5.

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	Fig. 5. Plot of spin conductance versus electric field-induced gap ΔE in the case of E = h = 5 meV, for the AP-junction with different values of U: (a) 0 meV, (b) 2.5 meV, (c) 5 meV, and (d) 7.5 meV.

The spin polarizations for the P- and AP-configurations are shown in Fig. 6. Interestingly, the behavior of the spin polarizations in the P-junction is quite different from that of the AP junction. It is seen that, the spin polarization in the P-junction, S_P, oscillates strongly above the zero point for U ≪ E. For U ≫ E, S_P oscillates below the zero point. At U = E, S_P oscillates about the zero point. In contrast to that in the P-junction, S_AP in an AP junction exhibits perfect spin-polarization switching, instead of oscillating. Switching of spin polarization from +100% to −100% occurs when the electric field is reversed (remembering that Δ_E = eE_zd).

Fig. 6.

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	Fig. 6. Plot of spin polarization versus electric field-induced gap ΔE in the case of E = h for different values of U: (a) 0 meV, (b) 5 meV, and (c) 10 meV, respectively. The SP(AP) represents spin polarization of the P-(AP-) junction. Perfect switching of spin polarization from +100% to −100% controlled by electric field is predicted only in the AP-junction.

Figure 7 shows that the TMR of the junction will oscillate as a function of Δ_E when Δ_E is in the interval of − h ≥ Δ_E ≥ h (with U = E = h), but a 100% TMR occurs when Δ_E is in the range of −h ≤ Δ_E ≤ h (see Figs. 7(a) and 7(d)). The TMR = 100% for arbitrary U ≤ h is predicted to occur in the range of −U ≤ Δ_E ≤ U (see Figs. 7(b)–7(d). The result will be clarified by considering the approximated momentums of electrons in P- and AP- barriers

Fig. 7.

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	Fig. 7. Tunneling magnetoresistance (TMR) as a function of electric field-induced gap ΔE in the case of E = h = 5 meV for (a) U = 5 meV, −20 meV < ΔE < + 20 meV, (b) U = 0, −54 meV < ΔE < + 5 meV, (c) U = 2.5 meV, −5 meV < ΔE < + 5 meV, and (d) U = 5 meV, −5 meV < ΔE < + 5 meV. The condition with the most stability for TMR=100% under a varying electric field is found at E = h = U.

For E = U = h (see Figs. 7(a) and 7(d)), we have . This causes the P-junction to become a good conductor having G_P ≫ 0 when − h ≤ Δ_E ≤ h. This condition also gives rise to the AP-junction becoming a strong insulator since will be imaginary for all values of Δ_E. This leads to G_AP → 0 and to TMR = 100% for − h ≤ Δ_E ≤ h. The predicted result indicates that a junction with a 100% TMR could easily be realized by setting E = h = U in the regime of |Δ_E| ≤ h.

Finally, we show that when the Fermi energy E is very large, spin polarization and magnetoresistance of the junction may go to zero (see Figs. 8(a) and 8(b)). This is due to the fact that when the energy of the electron is very large, the difference in wave vector of electrons with spin-up and spin-down is very small. The large difference in behavior of S_P in the P-junction and AP-junction is found for E being small (see Fig. 8(a)). In addition, large S_P and TMR occur when the magnitude of E is comparable with h. The very large oscillation of TMR is predicted in the case of Δ_E ≠ 0. It may be said that strong S_P and TMR may be achieved by using the condition of E being comparable with the net energy gap of the electron in the barrier.

Fig. 8.

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	Fig. 8. Plot of spin-magnetic effect as a function of Fermi energy E. (a) Spin polarization SP and (b) tunneling magnetoresistance TMR. There is no magnetic effect for large Fermi energy E.

We have investigated the spin and magnetotransport properties of an AB-stacked bilayer graphene junction, which has a ferromagnetic-gated control barrier. The ferromagnetism in bilayer graphene can be induced by the proximity effect. The exchange fields in top and bottom layers can be either parallel (P) or antiparallel (AP). In this work, we have focused on the effect of an exchange-field-induced spin-dependent gap, which occurs only in the AP-junctions. It was found that, the AP-junction leads to several interesting results, which are important for spintronic applications such as the perfect spin filter where the perfect switching of spin polarization can be achieved by reversing the electric field. The junction under study also exhibits a giant tunneling magneto-resistance (a TMR of 100%). It was shown that the interesting behaviors occur because the energy dispersion relationship for the carriers in bilayer graphene acquires a spin-dependent gap. The gap is induced by a mismatch of exchange energy in the top and bottom layers. This work reveals the potential of bilayer graphene to be useful for applications in the area of spintronics.