Tunneling conductance regulated by the electrostatic barrier in monolayer black phosphorene

We investigate the tunneling conductance of monolayer black phosphorene through double magnetic barriers modulated by the electrostatic barrier. This magnetic barrier is generated by two ferromagnetic strips deposited on monolayer phosphorene and can form parallel (P) and antiparallel (AP) structures. Applying a voltage to a ferromagnetic strip will create an electrostatic barrier. The transmission of P and AP magnetic structures is calculated using the transfer matrix. Tunneling conductance is given by Landauer–Büttiker theory. Our results show that under double electrostatic barriers modulation, the AP conductance of the system is zero except for sharp peaks, while the P conductance has a certain value, which implies a large tunneling magnetoresistance. However, in the case of single electrostatic barrier modulation, the conductance becomes flat. For the Fermi level in the valence band, the movement of the Fermi level leads to the severe mismatching of transmitted wave vectors in the barrier region, resulting a positive and negative oscillation magnetoresistance. This may be useful for designing memory using a two-dimensional electron gas system.


Introduction
The tunneling properties of inhomogeneous magnetic fields in two-dimensional electron gases through the deposition of ferromagnetic strips on heterogeneous structures have been studied for a long time [1].Theoretical and experimental studies have shown that such a magnetic field can be regarded as a δ-function magnetic field when the edge effects are ignored [2,3].If a voltage is applied across the ferromagnetic strip, a rectangular electrostatic barrier is created [1].Tunneling properties of many two-dimensional materials have been studied under such magnetic fields, such as black phosphorene, graphene, and so on [4].Black phosphorus, a layered material, has a band gap that increases with the number of layers, from 0.3 eV for bulk to 2 eV for monolayer black phosphorus [5].Monolayer black phosphorus is often called phosphorene, which is formed by each P atom and its three neighbors through sp3 hybridization to form a stable ring structure [6].The arrangement of atoms in the plane produces two different directions, namely the armchair direction (x) with Dirac-like electron behavior and the zigzag direction (y) with parabolic dispersion.The special structure of black phosphorus material makes it different from other two-dimensional nanomaterials in its specific physical properties and applications, so it has attracted more and more theoretical researchers in condensed matter physics to engage in this field of research.The tunneling properties of black phosphorene are also varied, some of which are briefly listed below.The anisotropic tunneling magnetoresistance in the phosphorene-based magnetic barrier was studied, and the result shows that the magnetoresistance is enhanced by the reduction of the band gap under the same effective mass components.The anisotropy and regulation of the conductance of monolayer black phosphorene in response to an electric field or chemical potential were also studied.The transmission and conductance of a rectangular vector potential modulated by a continuous laser and revealing the strong light-matter interaction under the magnetic vector potential have been theoretically explained [7].The positive and negative magnetoresistance regulation of monolayer black phosphorus armchair nanoribbon using the external gate voltage was realized, which showed periodic positive and negative magnetoresistance for slight voltage changes [8].The double barrier structure was used and the high anisotropy and Klein tunneling characteristics, and the conductance and transmission show oscillatory behavior concerning the barrier width were investigated [9].The tunneling properties of monolayer black phosphorene through double magnetic barriers under electrostatic barrier modulation have not been explored.
In this work, we investigate the tunneling conductance of monolayer black phosphorene through double magnetic barriers under the influence of an electrostatic barrier.When the edge effect is ignored, the δ function double barriers can be generated on the monolayer of black phosphorene by depositing two identical ferromagnetic strips.In addition, the electrostatic barrier is caused by the application of gate voltage on the ferromagnetic strips.Our system considers two cases, one is to apply voltage to two ferromagnetic strips to form a double electrostatic barrier, and the other is to apply voltage to a single ferromagnetic barrier.It is worth mentioning that the effect of a single voltage applied to either ferromagnetic strip is the same.Therefore, in this paper, we only show the case of a single voltage applied to the first ferromagnetic strip.The rest of the article is listed below.In the second part, we will give the theoretical form of calculating the electron transport probability and tunneling conductance in the monolayer black phosphorene with the double magnetic barriers by the modulated electrostatic barrier.In the third part, the numerical results of the electron transport properties are given and discussed.The fifth part gives a summary of the content of the article.

Derivation of the theoretical formula
The tunneling characteristic of monolayer black phosphorene through double magnetic barriers modulated by the electrostatic barrier is studied.We use a two-band model of monolayer black phosphorene [10].
where w is the width of the barrier, d is the distance between the barriers, ζequal to ±1 corresponds to parallel and antiparallel structures of the double magnetic barriers, and  is the estimated magnetic field, and the experimental magnetic field can reach 3.
where r and t are the reflected and transmitted amplitude respectively.The transmission probability  =  *  is obtained by continuous wave function on the boundary and transition matrix method [11].To illustrate the tunneling effect of magnetic barrier on electrons in black phosphorene, the ballistic conductance is calculated by Landauer − B ttiker formalism [12]  =  T(,  ) where  = ,  represents the longitudinal width of the monolayer black phosphorene.In general, the tunneling magnetoresistance is expressed as TMR = ( −  )/ ,  represents the conductance through the parallel magnetic structure and  represents the conductance through the antiparallel magnetic structure.

Numerical results and discussion
In this section, we will use the above formula to discuss the tunneling conductance of monolayer black phosphorene electrons through the parallel and antiparallel double magnetic barriers by the modulated electrostatic barrier.In this paper, we set as =d=5 nm and  =  ℏ = 1 nm ⁄ .The model we studied is shown in Figure 1, the black arrow represents the δ function magnetic field (solid lines and dashed lines represent parallel and antiparallel configurations, respectively), and the corresponding vector potential contour is shown by the green lines.In addition, the electrostatic barrier is represented by the blue solid lines.Figure 1 of (a) shows the magnetic structure modulated by a double electrostatic barrier.Since the single electrostatic barrier in the first barrier is consistent with that in the second barrier, we only give the outline of the single electrostatic barrier in the second barrier, as shown in Figure 1(b).
In Figure 2, when double electrostatic barriers are applied to the system and the incident electron energy E=0.75 eV, the transmission of the system is shown in Figure 2  As the double electrostatic barrier increases, the antiparallel transmission of the system gradually forms two symmetrical transmission peaks.Although parallel transmission is suppressed, the resonance peak still exists, which is determined by Klein tunneling.As the electrostatic barrier continues to increase, the Fermi level is in the large band gap of the monolayer black phosphorus, resulting in the transmission being prohibited.For a system modulated by a single electrostatic barrier, under the same size electrostatic barrier, antiparallel transmission exhibits a peak in the positive angle direction, and parallel transmission is strongly suppressed.As the electrostatic barrier increases, the antiparallel transmission peak shifts from a positive angle to a negative angle, as shown in Figure 2(c) and 2(d).For a single electrostatic barrier modulated system, the transmission of a single electrostatic barrier modulated system still exists when the transmission of a double electrostatic barrier modulated system is zero due to the different positions of fermi levels in different barrier regions.
In Figure 3, we give the corresponding conductance and energy relationship diagram.The conductivity of the system is more significantly suppressed under the double electrostatic barrier modulation.It is worth noting that under the modulation of the double electrostatic barrier, the first peak of the conductivity moves towards a larger energy with the decrease of the peak value becoming more and more obvious, as shown in Figure 3(b).For antiparallel conductance, when the electrostatic barrier  =1.2 eV, the conductance between the two peaks is zero, and the parallel conductance has a certain value, which implies a large tunneling magnetoresistance.By comparing parallel and antiparallel conductance, we can find a filter effect of antiparallel conductance, and the application of an electrostatic barrier increases the filter effect.Although this situation also occurs under the modulation of a single electrostatic barrier, it is not obvious, as shown in Figure 3(c) and 3(d).The reason for this difference comes from the difference in Fermi energy levels in the vector barrier region, resulting in the difference in the transmitted wave vector.In double electrostatic barriers, the transmission wave vector  in the first barrier, the region is equal to the transmission wave vector in the second barrier region, but under a single electrostatic barrier,   , as shown in Figure 1.To further understand this difference between a double and a single electrostatic barrier modulation, The relationship between conductance and energy in the case of the double electrostatic and single electrostatic barrier  =3 eV is given.In this case, the Fermi level of the system is in the valence band from the region near the conductance to the region across the band gap.We find that the main difference in conductance for different electrostatic barrier modulated is the range of energies with antiparallel conductance values greater than the parallel conductance values, as shown in Figure 4 of (a) and 4(b).This is not present in the conduction band.Only under the double magnetic barrier modulation, the antiparallel conductance of the system will be filtered when E=0.5 eV, as shown in Figure 3. Therefore, we give the relation diagram of the conductance and the electrostatic barrier when the incident energy E=0.5 eV.In a double barrier system, the peak of parallel and antiparallel conductance appears at approximately  =2.6 eV, while in a single electrostatic barrier system, the peak of parallel conductance still appears at  =2.6 eV, but the peak of antiparallel conductance appears at  =2.9 eV, as shown in Figure 4 of (c) and 4(d).This is the effect of the asymmetry of the transmitted wave vector caused by the change in the Fermi level mentioned earlier.Here, the modulation of a single electrostatic barrier implies magnetoresistance with positive and negative oscillations.

Summary
In summary, we have explored the tunneling conductance of monolayer black phosphorene through parallel and antiparallel δ-function magnetic fields modulated by the electrostatic barrier.Our results show that double electrostatic barriers can adjust the antiparallel conductance of the system to be zero except for sharp peaks, and the parallel conductance is not zero, resulting in a large tunneling magnetoresistance.Compared with the two electrostatic barrier modulations, although the same situation can occur with the modulation of a single electrostatic barrier, it is not obvious.Interestingly, the tunneling magnetoresistance exhibits oscillations of positive to negative concerning a single electrostatic barrier, which does not occur under double electrostatic barrier modulation.The unusual characteristic of tunneling magnetoresistance results from the mismatch of the transmission wave vectors in the antiparallel configuration and the change in Fermi energy levels modulated by the electrostatic barrier.This property is useful in magnetic memory based on two-dimensional electronic gas materials and may be better realized in two-dimensional superlattice structures.

Figure 1 .
Figure 1.Profiles of the vector potential and the electrostatic barrier.(a) Double electrostatic barriers; (b) a single electrostatic barrier.

Figure 2 .
Figure 2. The relation between transmission probability and incidence angle with E=0.75 eV and U 0 =0, 0.6, 1.2 eV.(a) and (b) double barriers (c) and (d) single barriers with P and AP structures; (a) and 2(b).

Figure 3 .
Figure 3.The relation between conductance and incidence angle with E=0.75 eV and  =0, 0.6, 1.2 eV.(a) and (b) double barriers (c) and (d) single barriers with P and AP structures;

Figure 4 .
Figure 4.The relation between conductance and incident energy with  =3 eV.(a) Double electrostatic barrier; (b) A single electrostatic barrier.The relation between conductance and electrostatic barrier with E =0.5 eV.(c) Double electrostatic barrier; (d) A single electrostatic barrier.