Effect of electric field on the electronic and magnetic properties of a graphene nanoribbon/aluminium nitride bilayer system

Abstract

The effect of an external electric field on the electronic and magnetic properties of the heterostructure of zigzag graphene nanoribbons (ZGNRs) placed on an aluminium nitride nanosheet (AlNNS) is studied using density functional theory (DFT). DFT calculations show that the local magnetic moments and total magnetization of the edge carbon atoms in the 4-ZGNR/AlNNS with a spin up electron subsystem are strongly dependent on a transverse electric field. We can control the band gap of the 4-ZGNR/AlNNS by using an external electric field. We established the critical values of the transverse electric field providing for semiconductor–metal phase transition in spin down electron configuration, which opens up potential opportunities for applications in spintronics devices. The effect of an external electric field (both amplitude and direction) on the effective charge and formation of the interface state energy is also studied and discussed.

---

1 Introduction

One-dimensional (1D) carbon nanostructures, such as graphene nanoribbons (GNRs),^1,2 have been recently studied in a large number of works due to their remarkable properties and predicted tempting prospects for spintronics applications.^3–7 It is well-known that the electronic properties of GNRs depend strongly on their topological shapes⁸ and that their band gap is sensitive to quantum confinement effects.^5,9,10 An armchair graphene nanoribbon (AGNR) can be metallic or semiconducting depending on its width. Zigzag graphene nanoribbons (ZGNRs) are semiconductors^5,11 which display remarkable edge ferromagnetism^3,9 and susceptibility to control of the band gap using a transverse electric field (E_t).^12,13

The placement of GNRs on a dielectric substrate may result in a change in their structure and physical properties⁵ caused by the nonlocal dielectric screening of the electron–electron interaction in the GNRs.¹⁴ The substrate substantially modulates the electronic, magnetic and transport properties of GNRs,^9,14 especially in the case of short nanoribbons.^9,15 The properties of the substrate in bilayer heterostructures are important, as it can provide new properties in graphene devices. The interest in the electronic, magnetic and optical properties of aluminium nitride nanoribbons (AlNNRs) has been increasing in recent years.^16–19 A monocrystalline aluminium nitride nanosheet (AlNNS) has been successfully synthesized on a Si substrate using the vapour-phase transport method with Al powder and NH₃ gas as the source materials.²⁰ By using transmission electron microscopy (TEM) and scanning electron microscopy (SEM), it has been shown that the produced AlNNS has a uniform structure and smooth surface.²⁰ The AlNNS has a hexagonal structure and it is characterized by a covalent bond. The AlNNS with perfect crystalline structure turned out to be just 6% less stable than its wurtzite-AlN (w-AlN) crystalline modification.²¹ In recent work,¹⁷ the electronic and magnetic properties of AlNNSs and AlNNRs were studied using density functional theory (DFT). Most importantly, the energy gap and carrier mobility in the 2D AlN nanomembrane can be tuned by applying a perpendicular electric field (E_p) to the 2D nanostructure surface.²² Thus, an AlNNS can be used as a substrate model in the ZGNR/AlNNS bilayer.

The spin moments of two ZGNR edges in the singlet ground state are known to show antiferromagnetic (AF) ordering.^6,23 The AF ordering of edge states in ZGNRs has recently been disputed in previous works.^24,25 In our opinion, the discussion of that question remains open. The quantum confinement effects and inter-edge super-exchange interactions in ZGNRs offer an opportunity to alter their electronic and magnetic properties. The AF-ZGNRs are semiconducting.^1,11,26,27 The magnetism characteristics in systems such as the 8-ZGNR/h-BN revealed that the edge C atoms in ZGNRs have the highest local magnetic moments (LMMs) in comparison to the other C atoms.^11,15 Based on DFT calculations,^13,27 it has been shown that an electric field crossing the width of ZGNRs can effectively cancel the spin energy degeneracy of two edges and makes ZGNRs spin-selective. The E_t can be employed to control and regulate the carrier transport through spin-polarized ZGNRs, for example, in a spin filter or a field-effect transistor.²⁸

The study of the 4-ZGNR/AlNNS model is significant for the understanding of quantum confinement effects (width and edge geometry) and for finding out a way to control the electronic, magnetic and other properties of 1D ZGNR channels by using an external electric field. The nature of the mentioned effects and the opportunities for modulation of the energy gap and magnetism in a ZGNR/AlNNS heterostructure have not been thoroughly investigated. Therefore, in the present work, we use DFT-D2 to calculate the electronic and transport properties of the ZGNR/AlNNS heterostructure. We examine the potential for tuning the spin-polarized electron structure and the magnetic properties of the ZGNR/AlNNS bilayer with an external electric field. The effect of the supercell size and external electric field on the band structure, the LMMs and the effective charge of atoms are also studied.

2 Model and computational method

We consider the graphene heterostructure consisting of a ZGNR placed on an AlNNS substrate. The ZGNR was the basic cell of the atomic structure used in the calculations. The ZGNR was placed in the xy-plane. All dangling bonds of the edge C atoms were passivated with H atoms. In this study, we consider two configurations (stacking) of the arrangement of the graphene nanoribbons on the substrate. In the first configuration, the edge carbon atom C1 in the first sublattice graphene was located directly on the Al atom (see Fig. 1). In the second one, the carbon atom C1 in graphene is located directly on the N atom. However, total energy calculations show that the configuration in which the edge carbon atom C1 in the first sublattice graphene is located directly on the Al atom is the most stable. Therefore, in the present study, we show only calculations for this configuration. The 4-ZGNR/AlNNS bilayer model of the first configuration is shown in Fig. 1. The atomic and the band structure calculations were performed using the Quantum Espresso software suite²⁹ based on DFT.^30–32 We used the Löwdin population analysis³³ to determine the effective charges Q_a of the edge carbon atoms and the nearest Al, N, and H atoms.

Fig. 1 (a and b) Relaxed atomic structure of 4-ZGNR/AlNNS. (c) Model of the sawtooth potential. (d) Scheme of the ZGNR/AlNNS heterostructure under an external electric field. E_t and E_p stand for the transverse and perpendicular electric fields.

For the AlNNS substrate, we cut out a 2D unit cell in the (0001) direction from a w-AlN structure (see also ref. 17). The 2D AlNNS relaxation was carried out until the sum of all of the forces in the system was reduced below 10⁻⁴ eV Å⁻¹. In order to study the surface problem, we used the Monkhorst–Pack method to generate a flat k-point mesh of (6 × 6 × 4). The band structure and the projected density of states (PDOS) of the 4-ZGNR are calculated using DFT.

To study the 4-ZGNR/AlNNS bilayer, we used a supercell containing unit cells of (3 × 1) AlNNS and (4 × 1) GNRs. The supercell parameter was selected to be divisible by the equilibrium value of the graphene unit cell parameter. Graphene cells were used to build 4-ZGNR. The surface and the 4-ZGNR/AlNNS interface were simulated as a slab consisting of one atomic layer of AlNNS and a monolayer of 4-ZGNR. The basic supercell consisted of 88 atoms, and each slab was isolated from the others by 25 Å vacuum space. Plane waves and pseudo-potentials were used in the basic calculations. The influence of core electrons on the physical properties of the heterostructure was considered using ultrasoft pseudo-potentials. A nonlocal exchange correlation functional was used in a Perdew–Burke–Ernzerhof (PBE, PBEsol) parametrization.³⁴ The plane wave cutoff energy for the self-consistent field calculation was 410 eV.³⁵

We established the equilibrium parameters for the lattices, atomic positions of the ZGNR and the AlNNS, and the distance d between the nanoribbon and the substrate atomic layers. The structures were relaxed until the energy and the force on each atom were less than 10⁻⁵ eV and 0.01 eV Å⁻¹, respectively. A (25 × 1 × 1) k-point mesh with the Monkhorst–Pack scheme was used to sample the 1D Brillouin zone. The intervals between the ribbons were maintained at 15 Å for both layer–layer and edge–edge distances to simulate isolated ribbons. In all our calculations, spin-restricted wave functions and calculations upon a singlet ground state were considered. Symmetry breaking in the spin-densities yielded an overestimated band gap and an exact treatment of the electron correlation in singlet wave functions could cancel this symmetry breaking in the spin-densities.^24,25 In the present study, we considered van-der-Waals interactions in our system within the DFT framework using a semi-empirical potential introduced in the total energy functional (DFT-D2) as follows:³⁶ E_DFT-D2 = E_DFT + E_disp. In the DFT-D2 calculations we used a PBE exchange correlation functional with a dispersion correction (PBE-D2).

The interaction of electrons with an external electrostatic field along the y-axis is determined by the following expression^13,37

U_ext(r) = |e|E × r, (1)

where U is used to distinguish the potential energy in the electrostatic field from all other potential terms of the Kohn–Sham equation The Kohn–Sham equation can be interpreted as:

The applied electrostatic potential changes linearly along the entire unit cell. Therefore, to apply periodic boundary conditions, the original value should be restored at the cell boundary. Thus, the sawtooth potential shape with a period equal to the unit cell period along the y-axis is most suitable [Fig. 1(c)]. The scheme of the relative placement of the slab and the external electrostatic potential is shown in Fig. 1(d). If E_ext exists in the range from 0 to l and the computational cell size L is along the y-axis, then the gauge field E* is defined as E* = −E_ext(l/L), where the minus sign indicates that E* and E_ext are always opposite in direction.

3 Atomic structure and electronic properties of the 4-ZGNR/AlNNS bilayer system

We first consider the electronic properties of the 2D AlNNS. Based on the atomic structure as shown in Fig. 2(a), we can see that the 2D AlNNS transforms from a corrugated surface into a planar graphite-like mesh structure as presented in previous work.¹⁷ The total energies before and after relaxation of the selected 2D AlNNS fragment are E_tot = −32.4392 Ry per cell and E_tot = −32.4991 Ry per cell, respectively. This indicates that the relaxed 2D AlNNS structure is stable. For a relaxed 2D AlNNS nanostructure, our calculations show that the Al–N bond length is 0.17989 nm. This result is in good agreement with earlier DFT calculations (0.1797 nm) and is lower than the one in the w-AlN crystalline structure (0.1872 nm).¹⁷ In the 2D AlNNS, the bond between neighboring Al and N atoms has an sp²-configuration. This is different from the tetragonal sp³-configuration typical of the 3D w-AlN structure.³⁸ The DFT calculations for the total energy also show that the sp²-configuration in the 2D AlNNS corresponds to the stable atomic structure. The bond between the Al and N atoms is formed by the combination of the p_z orbitals in Al and N with charge transferring from Al to N due to the different electronegativity of Al and N atoms. This charge transfer between the Al and N atoms in the 2D AlNNS results in ionic bonding. The trend in the cohesive energy can be correlated with the nature of chemical bonds as demonstrated by the contour plots of the charge density (Fig. 2(c)). The bonds have more ionic character in AlN due to a charge transfer from Al to N, which leads to localized wavefunctions (shown as concentric spheres in Fig. 2(c)) centered at the N-atoms with no significant overlap of the two neighboring spherical islands. The 2D AlNNS has an indirect-gap and is a nonmagnetic semiconductor characterized by an energy gap of 2.06 eV (from the K point to the Γ point) as shown in Fig. 2(b). The PDOS as shown in Fig. 2(b) indicate that when the donor level is in the range of 1.4 to 3.8 eV, the band gap leads to E_g = 4.5 eV.

Fig. 2 (a) Atomic structure of the AlNNS fragment before and after relaxation, (b) the band structure and PDOS of s- (red) and p-states (blue) of the electrons of Al and N atoms for both spins after relaxation. (c) Distribution maps of the total charge density for the Al and N atoms in AlNNS.

In Fig. 3, we show the band structure of 4-ZGNR with different supercell sizes. The lowest energy of the conduction band and highest energy of the valence band are located near the X point. The band gap does not depend on the ZGNR supercell size. Our calculations show that the LMMs do not depend on the supercell size. The LMM of the edge C atoms passivated with hydrogen atoms is 0.27 μ_B per atom. Our result is in good agreement with available data.¹⁵

Fig. 3 DFT-D2 calculation of the band structure for 4-ZGNR with different supercell sizes: (a) (1 × 1), (b) (2 × 1), (c) (3 × 1), and (d) (4 × 1). The red solid and blue dotted lines refer to spin up and spin down, respectively.

In the 4-ZGNR/AlNNS bilayer system, our calculations give the distance between the nanoribbon and the substrate as d = 0.320 nm. This distance is smaller than the interlayer distances in graphite (0.335 nm)³⁹ but is larger than the distance between GNR and hexagonal boron nitride.^11,35,40 For the highlighted 4-ZGNR/AlNNS fragment shown in Fig. 1(a), the total energy is E_tot = −1153.7098 Ry per cell. Our DFT-D2 calculations also show that the length of the σ-bonds in a planar (0001) arrangement can be described by using the PBE-D2 functional. In the 4-ZGNR/AlNNS, the C–C bond length for the internal atoms of the nanoribbon is 1.4407 Å and the C–C bond length between the two edge C1 and C2 carbon atoms is 1.3896 Å. The difference in the bond length between the internal carbon atoms and between the edge carbon atoms in the 4-ZGNR/AlNNS is quite large. This deformation is expressed by the appearance of peaks in the electron density states near the Fermi level and induces the LMM at the ZGNR edges.

The band structure and PDOS of the 4-ZGNR/AlNNS for different spins are shown in Fig. 4. In comparison to the isolated 4-ZGNR and AlNNS cases, when the 4-ZGNR is located on the AlNNS substrate, the orbital of the edge carbon atom shifts towards the higher energy area with an amplitude of 370 meV. Besides this, a splitting of 45 meV between the spin up and spin down electron subsystems is observed. Near the Dirac point (k = 2π/3), the flat bands of the 4-ZGNR/AlNNS are closer to the Fermi level than that of the suspended ZGNR case.

Fig. 4 DFT-D2 spin-polarized calculations of the band structures for (a) a suspended supercell of 4-ZGNR; (b) the AlNNS substrate; and (c) the 4-ZGNR/AlNNS. (d) LMMs of the four edge C atoms (passivated with H atoms) in the suspended supercell of 4-ZGNR and with the AlNNS substrate. (e) Dependence of the band gap of 4-ZGNR and ZGNR/AlNNS on the wave vector for both spins. The PDOS of the 2p_z orbitals of the C1, C2, Al, and N edge atoms in (f) the 4-ZGNR/AlNNS and (g) the suspended ZGNR and AlNNS substrate. The solid and dotted lines refer to spin up and spin down, respectively. The k1, k2 and k3 stand for the wave vector where k = 0, k = 2π/3a, and k = π/a, respectively.

The dependence of the LMMs on the number of edge carbon atoms is shown in Fig. 4(d). Our calculations show that the change in the LMMs is due to the charge transfer between the C atoms and those of the substrate, in particular, the Al atoms. The dependence of the band gap of the 4-ZGNR and the ZGNR/AlNNS on the wave vector is shown in Fig. 4(e). The PDOS of the two edge C1 and C2 atoms in the ZGNR/AlNNS are shown in Fig. 4(f) and (g) for the spin up and spin down electron subsystems, respectively. Here, only the left edge C atoms are examined.

For the edge C atoms, the PDOS shows a shift of 0.80 eV near the Fermi level in both of the cases, with and without the substrate. In our DFT calculations, the LMMs of the edge C atoms passivated with hydrogen atoms in the bilayer heterostructure are smaller than those in the suspended GNR. A stronger bond between C and Al is indicated by the PDOS of the Al atoms [see Fig. 4(f)]. The difference in the charge transfer may be caused by the different distances between the edge C atoms (passivated with hydrogen atoms) and the nearest substrate atoms.

The dependence of the effective charges and LMMs of the edge C atoms passivated with hydrogen atoms (for spin up and spin down) on the edge carbon atoms are shown in Fig. 5. To study the charge transfer process between the C atoms and the nearest substrate atoms in detail, we calculated the effective charges of the C1 atoms from the left (spin up) and right (spin down) sides of a ZGNR supercell (4 × 4). The results of the DFT calculation of the effective charges are summarized in Table 1. As shown in Table 1, we can see that the main charge transfer occurs between the Al and N atoms in the substrate. This determines the bond ionicity. In addition, there is also a small charge transfer from the Al atoms to the C1 atom. Our result is in good agreement with previous work.¹⁷

Fig. 5 Dependence of the LMMs (red) and effective charges (blue) of the edge C atoms passivated with hydrogen on their atomic positions in the 4-ZGNR/AlNNS.

Table 1 Effective charges of the edge C1 atoms (spin up, spin down) and the nearest Al and N atoms, and distances between C1 and Al (N) atoms in the 4-ZGNR/AlNNS interface. C^α_edge and C^β_edge stand for edge C atoms with spin up and spin down, respectively; H^α_edge, H^β_edge are passivated H atoms on C^α_edge and C^β_edge atoms, respectively. Al^α/β and N^α/β are the closest Al and N atoms to the C^α/β_edge edge atom

	C^αedge	Al^α	N^α	H^αedge	C^βedge	Al^β	N^β	H^βedge
Qe	−0.132	1.119	−1.048	0.201	−0.123	1.117	−1.046	−0.198
dC–Al(N), Å	—	3.471	3.3141	—	—	3.3249	3.2833	—
dAl–N, Å	—	—	1.6438	—	—	—	1.8401	—

---

In the 4-ZGNR/AlNNS system, according to our DFT calculations, the distance d between the ZGNR and the AlNNS substrate is equal to 0.320 nm. The adsorption energy of C atoms on the AlNNS is E_ads = 0.060 eV per atom. We note that the energy of 0.060 eV per C atom is typical of physical adsorption.³⁵ Our estimation for the adsorption bond energy and interplanar spacing is in good agreement with experimental data.^41–43

Fig. 6 illustrates the distribution of the charge density in spin up π regions near the Fermi level, which allows estimation of the spatial localization of the spin density for C atoms in the ZGNRs/AlNNS at a wave vector k of 0, 2π/3, or π. The partial charge density is relevant to the valence band. In the absence of an electric field, the wave function squares of the occupied bands of the PDOS of the C 2p_z-orbitals in the range of −0.36 to 0 eV are localized exclusively on the C atoms. However, the shape of these orbitals depends on the wave vector. In particular, the PDOS is localized exclusively on the left C1 atoms (spin up) at k = 0. At k = 2π/3, the partial redistribution of the electron density on the C atoms occurs. Finally, at k = π a complex redistribution of electron density follows with the formation of hybrid p orbitals for the C5–C6 and C7–C8 atoms.

Fig. 6 Spin charge density distribution on the C atoms near the Fermi level in the 4-ZGNR/AlNNS for the spin up electron subsystems at: (a) k = 0, (b) k = 2π/3, and (c) k = π.

4 The 4-ZGNR/AlNNS bilayer system in an electric field

4.1 The transverse electric field effect E_t

In this part, we use DFT calculations to study the effect of the transverse electric field E_t on the band structure and the LMMs at the GNR edges in the 4-ZGNR/AlNNS bilayer system. In Fig. 7, we show the dependence of the LMMs of the edge C atoms and the total magnetization of the 4-ZGNR/AlNNS (for both spin up and spin down) on the transverse electric field E_t applied along the nanoribbon width. We can see that the LMM of the edge C atoms and the total magnetization of the computational cell of the 4-ZGNR/AlNNS for the spin down almost do not depend on the transverse electric field E_t. For the spin up, however, the LMM and the total magnetization change dramatically due to the increasing E_t. They increase when the transverse electric field E_t increases from zero to 1 V nm⁻¹. After that, they quickly decrease with increasing E_t.

Fig. 7 Dependence of the LMMs of the edge C atoms (red, left vertical axis) and the magnetization of the computational cell (blue, right vertical axis) of the 4-ZGNR/AlNNS on E_t. The filled and empty symbols stand for spin up and spin down, respectively.

In comparison to available data, we see that the LMM in the 4-ZGNR/AlNNS is 1.1 times lower than that in the 8-ZGNR/h-BN system.^11,13,15 With further increase of the E_t, the LMM decreases quickly. At E_t = 3 V nm⁻¹, the LMM is equal to 0.23 μ_B. The dependence of the total magnetization and the LMM on E_t are similar to each other (see Fig. 7). Remarkably, the effect of the substrate on the LMM is manifested differently when low E_t (from −1 V nm⁻¹ to 1 V nm⁻¹) is applied to the 4-ZGNR/AlNNS and 8-ZGNR/h-BN. The LMM observed is about 0.3 μ_B for the 8-ZGNR/h-BN.¹³ Essentially, the difference can be explained by two tendencies: exchange splitting in the edge states of the C atoms (according to the Stoner model⁴⁴) and the tendency for the contraction of the valence and conduction bands.⁶ This behavior may also be caused by another mechanism,¹⁴ which can be explained by the varied modification of the electron energy owing to the dielectric screening of the electron–electron interaction in graphene heterostructures with different substrates. The dielectric permittivity value⁴⁵ of the examined w-AlN substrate (ε = 9.14) is 1.8 times that of the h-BN substrate (ε = 5.06). This difference may affect the character of electron–electron interactions, according to previous work.⁴⁶

The band structure near the Fermi level of the 4-ZGNR/AlNNS under applied E_t is shown in Fig. 8. In the absence of an electric field, the 4-ZGNR/AlNNS has a band gap of E_g = 0.67 eV or 0.65 eV opening at the Dirac point (k = 2π/3) corresponding to the α- or spin down subsystems, respectively. When the E_t increases, the band gap for spin up increases and the band gap for spin down quickly decreases. At a certain critical value of E_t, it is possible to predict the band gap closure for spin down, which corresponds to the semiconductor-metal transition for this electron subsystem. This effect is illustrated in Fig. 8(d). Generally, a semimetallic character in the 4-ZGNR/AlNNS is accomplished when the critical field value of 3 V nm⁻¹ is applied. The dependence of the energy gap on the external electric field is also shown in Fig. 9. We can see an energy gap opening at k = 2π/3a for both the spin up and spin down electron subsystems.

Fig. 8 DFT-D2 calculations for the band structure of the 4-ZGNR/AlNNS and the PDOS for the 2p_z-orbitals of the C1 edge atoms as well as the Al (green) and N atoms (pink) under the E_t: (a) E_t = 0 V nm⁻¹, (b) E_t = 1 V nm⁻¹, (c) E_t = 2 V nm⁻¹, and (d) E_t = 3 V nm⁻¹. Red (solid line) and blue (dashed line) stand for spin up and spin down, respectively.

Fig. 9 Dependence of the energy gap at different wave vector values on the external electric field: (a) a transverse electric field and (b) a perpendicular electric field. The k1, k2 and k3 stand for the wave vector k = 0, k = 2π/3a, and k = π/a, respectively (solid and dotted lines refer to spin up and down).

From Fig. 10 we can estimate the spatial localization of the spin density for C atoms (both spins) in the 4-ZGNR/AlNNS at k = 2π/3. In the absence of E_t, wave functions (squares) of occupied and unoccupied bands of the carbon PDOS in the range from −0.25 to 0.45 eV are localized exclusively on the C1 atoms [Fig. 10(a)]. The spherical shape of these orbitals is slightly deformed in the direction perpendicular to the interface surface. When E_t is applied, the charge transfer occurs from the right edge (spin down) to the left edge (spin up), as illustrated by the density distributions shown in Fig. 10(b) and (c).

Fig. 10 Electron density distribution of the carbon 2p_z-orbitals in the 4-ZGNR/AlNNS (both spins) for the valence band (left panel) and the conduction band (right panel): (a) E_t = 0, (b) E_t = 3 V nm⁻¹, and (c) E_p = 3 V nm⁻¹.

The regularity is confirmed by the effective charge calculation data for the edge C1 atoms and their nearest Al and N atoms as listed in Table 2. In particular, the charge transfer from the C^α atom to a C^β atom is approximately Δq = 0.052 at E_t = 3 V nm⁻¹. The results of DFT calculations of the total energy E_tot for the 4-ZGNR/AlNNS are summarized in Table 2 (for the computational cell). It should be noted that the total energy of the interface in question increases slightly with increasing E_t. This increase of the total energy in the 4-ZGNR/AlNNS system is consistent with the changing tendency in its band structure when the p_z orbitals of edge C atoms and the orbitals of Al and N atoms are mixed. The mixing between the p_z orbitals of the edge C atoms and Al and N atoms in the 4-ZGNR/AlNNS leads to the formation of an interface state energy.

Table 2 Effective charges of the edge C1 atoms and the nearest Al and N atoms in the 4-ZGNR/AlNNS interface, and total cell energy E_tot in the presence of E_t. The α and β symbols stand for the spin up and spin down, respectively

Et, V nm^−1	Effective charge per atom, e						Etot, Ry per cell
	Al^α	Al^β	N^α	N^β	C^α	C^β
0	1.1189	1.1168	−1.0479	−1.0459	−0.1323	−0.1227	−1153.7098
0.5	1.1182	1.1163	−1.0473	−1.0462	−0.1346	−0.1218	−1153.4620
1	1.1173	1.1161	−1.0477	−1.0465	−0.1375	−0.1177	−1153.2214
1.5	1.1168	1.1151	−1.0473	−1.0460	−0.1417	−0.1145	−1152.9887
2	1.1166	1.1148	−1.0481	−1.0453	−0.1452	−0.1115	−1152.7636
2.5	1.1166	1.1149	−1.0465	−1.0457	−0.1470	−0.1088	−1152.5474
3	1.1173	1.1154	−1.0458	−1.0457	−0.1602	−0.1087	−1152.3412

---

4.2 The perpendicular electric field effect E_p

Our DFT calculations show that the LMM at the edge of the ZGNR is 0.26 μ_B and that the LMM is independent of the electric field E_p applied perpendicularly to the surface of the 4-ZGNR/AlNNS. The magnetization is 3.4 μ_B and it remains constant within the range of E_p. The electron density distribution corresponding to the highest energy of the valence band and the lowest energy of the conduction band is at k = 2π/3. Under the electric field E_p, the orbitals of the C atoms are deformed along the direction of the nearest Al atom. Similar to the case of E_t, the interface state energy is observed when the E_p is applied.

The effective charge calculation data for the edge C atoms (passivated with hydrogen) and their nearest Al and N atoms are summarized in Table 3 confirming charge transfer from Al to the N and C atoms. The results of the DFT calculations of the total energy for the 4-ZGNR/AlNNS are also listed in Table 3 (for the computational cell). It should be noted that the total energy of the interface in question decreases with the increase of E_p. This decrease of the total energy in the 4-ZGNR/AlNNS system correlates quite well with the tendency towards modification of its band structure when E_p is applied.

Table 3 Effective charges of the edge C1 atoms and the nearest Al and N atoms in the 4-ZGNR/AlNNS interface, and the total cell energy E_tot, in the presence of E_p

Et, V nm^−1	Effective charge per atom, e						Etot, Ry per cell
	Al^α	Al^β	N^α	N^β	C^α	C^β
0	1.1189	1.1168	−1.0479	−1.0459	−0.1323	−0.1227	−1153.7098
1	1.1173	1.1151	−1.0477	−1.0459	−0.1299	−0.1211	−1153.7154
2	1.1156	1.1134	−1.0475	−1.0456	−0.1282	−0.1189	−1153.7254
3	1.1137	1.1116	−1.0471	−1.0453	−0.1262	−0.1177	−1153.7398

---

In Fig. 11, we show the PDOS of the 2p_z orbitals of the edge C1 atoms (both spins), and Al and N atoms under the applied E_p. For the edge C atoms, the size of the PDOS peaks near the Fermi level does not depend on the amplitude of E_p for either electron configuration. The shift of the localized edge states near the Fermi level is 0.8 eV. The response of the edge magnetism to E_p in the 4-ZGNR/AlNNS has not been established. The peak in the occupied PDOS of the N atoms nearest to the Fermi level is characterized by the bond energies of −1.0 eV and −1.6 eV at E_p = 0 V nm⁻¹ and E_p = 3 V nm⁻¹, respectively. Minor PDOS peaks with an energy of −0.3 eV, associated with the surface states of the N atoms for both spin up and spin down, shift to the position corresponding to the energy of −0.6 eV at E_p = 3 V nm⁻¹. The unoccupied DOS of the AlNNS substrate mostly contains the contributions of the 2p_z electrons of the Al atoms and it is located above the Fermi level of 1.5 eV. The effect of E_p on these crystalline states is manifested in the strengthening of the interaction between the 2p_z orbitals of the C, Al, and N atoms. In addition, the shifting of 0.7 eV of the unoccupied states towards the Fermi level is observed.

Fig. 11 Dependence of the PDOS of the 2p_z orbitals of Al (green) and N (pink) (nearest to the edge C atoms passivated with hydrogen) with both spins on E_p: (a and c) E_p = 0 V nm⁻¹ and (b and d) E_p = 3 V nm⁻¹. The insets show the zoomed-in images of the unoccupied PDOS of Al and N.

5 Conclusions

In conclusion, we used DFT calculations to study the electronic and magnetic properties of the 4-ZGNR/AlNNS heterostructure under an external electric field. The 4-ZGNR/AlNNS with a spin up electron subsystem is sensitive to the external electric field, especially the electric field E_t applied in parallel to the surface of the 4-ZGNR/AlNNS. However, the LMM at the edge of the ZGNR and the magnetization are independent of the E_p applied perpendicularly to the surface of the heterostructure. Moreover, we can control the band gap of the 4-ZGNR/AlNNS by using the external electric field. With the semiconductor–metal–semiconductor transition, the 4-ZGNR/AlNNS becomes a promising material for applications in nanospintronic devices.

Acknowledgements

This research was funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant no. 103.01-2014.04.

References

1. Q. Tang and Z. Zhou, Prog. Mater. Sci., 2013, 58, 1244–1315.
2. I.-S. Byun, D. Yoon, J. S. Choi, I. Hwang, D. H. Lee, M. J. Lee, T. Kawai, Y.-W. Son, Q. Jia, H. Cheong and B. H. Park, ACS Nano, 2011, 5, 6417–6424.
3. J. Baringhaus, M. Ruan, F. Edler, A. Tejeda, M. Sicot, A. Taleb-Ibrahimi, A.-P. Li, Z. Jiang, E. H. Conrad, C. Berger, C. Tegenkamp and W. A. de Heer, Nature, 2014, 506, 349–354.
4. D. Krychowski, J. Kaczkowski and S. Lipinski, Phys. Rev. B: Condens. Matter Mater. Phys., 2014, 89, 035424.
5. J. Yu, Z. Zhang and W. Guo, J. Appl. Phys., 2013, 113, 133701.
6. L. F. Huang, G. R. Zhang, X. H. Zheng, P. L. Gong, T. F. Cao and Z. Zeng, J. Phys.: Condens. Matter, 2013, 25, 055304.
7. Q. Tang, Z. Zhou and Z. Chen, Nanoscale, 2013, 5, 4541–4583.
8. L. Brey and H. A. Fertig, Phys. Rev. B: Condens. Matter Mater. Phys., 2006, 73, 235411.
9. V. V. Ilyasov, B. C. Meshi, V. C. Nguyen, I. V. Ershov and D. C. Nguyen, J. Appl. Phys., 2014, 115, 053708.
10. F. J. Martin-Martinez, S. Fias, G. V. Lier, F. D. Proft and P. Geerlings, Chem.–Eur. J., 2012, 18, 6183–6194.
11. V. Ilyasov, V. Nguyen, I. Ershov and D. Nguyen, J. Struct. Chem., 2014, 55, 191–200.
12. Z. F. Wang, S. Jin and F. Liu, Phys. Rev. Lett., 2013, 111, 096803.
13. V. V. Ilyasov, B. C. Meshi, V. C. Nguyen, I. V. Ershov and D. C. Nguyen, J. Chem. Phys., 2014, 141, 014708.
14. X. Jiang, N. Kharche, P. Kohl, T. B. Boykin, G. Klimeck, M. Luisier, P. M. Ajayan and S. K. Nayak, Appl. Phys. Lett., 2013, 103, 133107.
15. V. V. Ilyasov, B. C. Meshi, V. C. Nguyen, I. V. Ershov and D. C. Nguyen, AIP Adv., 2013, 3, 092105.
16. S. Valedbagi, A. Fathalian and S. Mohammad Elahi, Opt. Commun., 2013, 309, 153–157.
17. C.-W. Zhang, J. Appl. Phys., 2012, 111, 043702.
18. A. Du, Z. Zhu, Y. Chen, G. Lu and S. C. Smith, Chem. Phys. Lett., 2009, 469, 183–185.
19. C.-W. Zhang and F.-B. Zheng, J. Comput. Chem., 2011, 32, 3122–3128.
20. X. Zhang, Z. Liu and S. Hark, Solid State Commun., 2007, 143, 317–320.
21. E. F. de Almeida Jr, F. de Brito Mota, C. de Castilho, A. Kakanakova-Georgieva and G. Gueorguiev, Eur. Phys. J. B, 2012, 85, 1–9.
22. R. G. Amorim, X. Zhong, S. Mukhopadhyay, R. Pandey, A. R. Rocha and S. P. Karna, J. Phys.: Condens. Matter, 2013, 25, 195801.
23. F. Withers, T. H. Bointon, M. Dubois, S. Russo and M. F. Craciun, Nano Lett., 2011, 11, 3912–3916.
24. M. Huzak, M. S. Deleuze and B. Hajgato, J. Chem. Phys., 2011, 135, 104704.
25. B. Hajgato, M. Huzak and M. S. Deleuze, J. Phys. Chem. A, 2011, 115, 9282–9293.
26. R. Qin, J. Lu, L. Lai, J. Zhou, H. Li, Q. Liu, G. Luo, L. Zhao, Z. Gao, W. N. Mei and G. Li, Phys. Rev. B: Condens. Matter Mater. Phys., 2010, 81, 233403.
27. Y.-W. Son, M. L. Cohen and S. G. Louie, Nature, 2006, 444, 347–349.
28. J. Guo, D. Gunlycke and C. T. White, Appl. Phys. Lett., 2008, 92, 163190.
29. P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni, I. Dabo, A. D. Corso, S. de Gironcoli, S. Fabris, G. Fratesi, R. Gebauer, U. Gerstmann, C. Gougoussis, A. Kokalj, M. Lazzeri, L. Martin-Samos, N. Marzari, F. Mauri, R. Mazzarello, S. Paolini, A. Pasquarello, L. Paulatto, C. Sbraccia, S. Scandolo, G. Sclauzero, A. P. Seitsonen, A. Smogunov, P. Umari and R. M. Wentzcovitch, J. Phys.: Condens. Matter, 2009, 21, 395502.
30. P. Hohenberg and W. Kohn, Phys. Rev., 1964, 136, B864–B871.
31. W. Kohn and L. J. Sham, Phys. Rev., 1965, 140, A1133–A1138.
32. A. Dal Corso, A. Pasquarello and A. Baldereschi, Phys. Rev. B: Condens. Matter Mater. Phys., 1997, 56, R11369–R11372.
33. P.-O. Lowdin, Int. J. Quantum Chem., 1982, 21, 275–353.
34. K. Burke, J. P. Perdew and M. Ernzerhof, Int. J. Quantum Chem., 1997, 61, 287–293.
35. V. Ilyasov and I. Ershov, Phys. Solid State, 2012, 54, 2335–2343.
36. S. Grimme, J. Comput. Chem., 2004, 25, 1463–1473.
37. C.-W. Chen, M.-H. Lee and S. J. Clark, Nanotechnology, 2004, 15, 1837.
38. H. Sahin, S. Cahangirov, M. Topsakal, E. Bekaroglu, E. Akturk, R. T. Senger and S. Ciraci, Phys. Rev. B: Condens. Matter Mater. Phys., 2009, 80, 155453.
39. D. Chung, J. Mater. Sci., 2002, 37, 1475–1489.
40. G. Giovannetti, P. A. Khomyakov, G. Brocks, P. J. Kelly and J. van den Brink, Phys. Rev. B: Condens. Matter Mater. Phys., 2007, 76, 073103.
41. S. Akcöltekin, M. El Kharrazi, B. Köhler, A. Lorke and M. Schleberger, Nanotechnology, 2009, 20, 155601.
42. T. Tsukamoto and T. Ogino, Appl. Phys. Express, 2009, 2, 075502.
43. M. Stephens, R. Rhodes and C. Wieman, J. Appl. Phys., 1994, 76, 3479–3488.
44. E. Stoner, Proc. R. Soc. A, 1938, 165, 372.
45. The Ioffe Institute, Basic parameters for crystal structure (http://www.ioffe.rssi.ru/SVA/NSM/Semicond/BN/basic.html). Access date: 05 April 2015.
46. O. Artamonov, S. Samarin and J. Williams, Tech. Phys., 2009, 54, 1034–1040.

---