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Asymmetric bandgaps and Landau levels in a Bernal-stacked hexagonal boron-nitride bilayer
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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 015304 (7pp)
doi:10.1088/0953-8984/26/1/015304
Asymmetric bandgaps and Landau levels in a Bernal-stacked hexagonal boron-nitride bilayer Xuechao Zhai and Guojun Jin National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing, 210093, People’s Republic of China E-mail: [email protected] Received 17 September 2013, in final form 29 October 2013 Published 25 November 2013 Abstract
A Bernal-stacked hexagonal boron-nitride (h-BN) bilayer is a two-dimensional polar crystal. Within the tight-binding approximation, we investigate the band structure of a gated h-BN bilayer by analyzing the density of states and the behavior of the charge transfer. We find that the bandgaps of the h-BN bilayer vary asymmetrically under two opposite biases due to asymmetric changes of the interlayer and intralayer polarities. We also find that the bias-driven net charge transfer between layers can be up to 0.2 electron per unit cell. Under the bias along one direction, the system exhibits quantum phase transitions from a semiconductor to a semimetal and then to a semiconductor again, whereas under the reverse bias, the system is always semiconducting. Besides, asymmetric Landau levels under opposite biases arise in the presence of a magnetic field. Moreover, dispersive edge states are found to exist in the bulk bandgap for an h-BN bilayer nanoribbon under the bias along one direction, which does not happen when the bias is reversed. All these properties of h-BN bilayers are measurable in transport experiments. (Some figures may appear in colour only in the online journal)
1. Introduction
effective mass in bulk bilayers [5]. This fact has an impact on the advantage of graphene nanoribbons and bilayers in electronics but does not affect their superiority in photonics and optoelectronics. Meanwhile, some new two-dimensional semiconducting materials have also attracted researchers’ attention, such as graphane [6], MoS2 [7], and hexagonal boron nitride (h-BN) [8]. Experimentally, by micromechanical exfoliation [9] and chemical vapor deposition [10], mono- or few-layer h-BN sheets can be prepared. An obvious difference from graphene is that the BN bond in h-BN materials has a polarity which results in large bandgaps. Both experimental [11] and theoretical [12] studies have reported a bandgap of about 4.6 eV for two-dimensional single-layer h-BN and about 5.97 eV for three-dimensional h-BN single crystals [13]. These large bandgaps are beneficial for h-BN materials to work as good dielectric spacer layers in magnetic tunneling
In the past decade, there have been many studies on two-dimensional materials, because of their relatively easy production, convenient integration, and potential applications in future solid-state devices. The most widely studied two-dimensional material is graphene [1], which has a particularly simple structure but very rich physics. The fact that pristine graphene has no band gap, however, hampers its applications in electronics, photonics, and optoelectronics. Thus, the means of modulating the bandgap becomes an important issue. It has been demonstrated that the band gap can be experimentally tuned by controlling the width of graphene nanoribbons [2] or varying the external interlayer bias on graphene bilayers [3, 4]. Nevertheless, the carrier mobility may decrease significantly, due to the edge-enhanced scattering in narrow ribbons [2] and the bias-increased 0953-8984/14/015304+07$33.00
1
c 2014 IOP Publishing Ltd Printed in the UK
J. Phys.: Condens. Matter 26 (2014) 015304
X Zhai and G Jin
can be described by X X † H1(2) = (εi ± 1)c†i ci − t ci cj ; i
H12 = −γ
hiji
X
c†i cj ;
† H21 = H12 .
(2)
i∈1,j∈2
Figure 1. The lattice structure of a gated Bernal-stacked h-BN
In H1(2) , the first term indicates the original on-site energy εi , in addition to the electrostatic potential energy of the top (bottom) layer +1 (−1), induced by a bias voltage, as done in a graphene bilayer [3, 4, 22]. The second term is the nearest-neighbor intralayer coupling with the hopping integral t. The interlayer coupling strength (van der Waals interaction) is described by γ in H12(21) . The parameters t, γ , εi are extracted from first-principle calculations [20, 19]: t = 2.37 eV, γ = 0.6 eV, and εB = −εN ≡ ε0 = 2.095 eV. We choose the atomic wavefunctions localized at sites B1 , B2 , N1 , N2 as the basis, ordered in the right inset of figure 1, where 1 (2) labels the top (bottom) layer respectively. Then the Hamiltonian of the h-BN bilayer in the momentum space reads ε0 + 1 0 f (k) γ 0 ε0 − 1 0 f (k) H(k) = ∗ (3) , f (k) 0 −ε0 + 1 0 γ f ∗ (k) 0 −ε0 − 1 √ where f (k) = −teikx a [1 + 2e−i(3kx a/2) cos( 3ky a/2)]. Expanding equation (3) in the vicinity√of two valleys K = √ (0, −4π/3 3a) and K0 = (0, 4π/3 3a), we obtain the low-energy four-band Hamiltonian, γ H(k) = v(ησx px + σy py ) ⊗ Iτ + (σx ⊗ τx 2 − σy ⊗ τy ) + 1Iσ ⊗ τz + ε0 τz ⊗ Iσ , (4)
bilayer in Cartesian coordinates (x, y, z). γ is the interlayer hopping and 1, −1 describe the effective electrostatic potentials. The right inset denotes the atoms in a unit cell, where B (N) marks the boron (nitride) atom and the subscript 1 (2) labels the top (bottom) layer.
junctions [14]. According to the latest first-principle [15–17] and tight-binding [18] calculations, the bandgaps of few-layer h-BN sheets can be remarkably reduced by applying an interlayer bias voltage to decrease their polarities effectively. These tunable bandgaps enable h-BN materials to be good media for light generation and detection in optoelectronic devices [13, 16]. In this paper, we show that the bandgap of an energetically stable Bernal-stacked h-BN bilayer [19–21] without an inversion symmetry varies asymmetrically under two opposite biases due to the asymmetric changes of the interlayer and intralayer polarities. It is argued that the bias-induced net charge transfer between layers can be up to 0.2 of an electron per unit cell. Under the bias along one direction, the system exhibits phase transitions from a semiconductor to a semimetal and then to a semiconductor again. On the other hand, the system is always semiconducting under the reverse bias. Also, under the two opposite biases, by applying a perpendicular magnetic field, the Landau levels are found to be asymmetric and can show up in the magneto-electronic spectra of a confined h-BN bilayer. Furthermore, dispersive edge states are stable in the bulk bandgap under a bias only along one direction, but do not exist along the opposite direction.
where η = −1(+1) labels K (K0 ), v = 3ta/2h¯ = 7.82 × 105 m s−1 is the Fermi velocity, σ and τ are Pauli matrices representing the sublattice pseudospin and layer degrees of freedom, respectively, and I is a 2 × 2 identity matrix. The first term indicates the Dirac linear dispersion. The second term is the interlayer coupling term. The third term represents the interlayer polarity controlled by an external bias. The last term represents the intralayer polarity of the BN covalent bond. Without the last term, equation (4) goes back to the Hamiltonian of a gated graphene bilayer [1].
2. Theoretical formulation
Figure 1 shows the lattice configuration of a Bernal-stacked h-BN bilayer. The electrostatic potentials, +1 and −1, from an effective interlayer bias voltage, are applied to the top and bottom layers of the h-BN bilayer respectively. Each layer of the h-BN bilayer is constructed of two inequivalent trigonal sublattices, which are associated by strong polar covalent σ -bonds. In Cartesian coordinates (x, y, z), the Bravais lattice √ is built from the primitive vectors a± = (a/2)(3x ± 3y), ˚ is the BN-bond length [20]. where a = 1.45 A The Hamiltonian of this system can be written in four parts, H = H1 + H2 + H12 + H21 ,
3. Results and discussion 3.1. Asymmetric bandgaps under opposite biases
By diagonalizing the Hamiltonian in equation (3), we derive the electronic dispersion q p (5) E = ± 12 + ε02 + 12 γ 2 + |f (k)|2 ± h(k),
(1)
where H1(2) is the Hamiltonian of the top (bottom) layer, and H12(21) represents the interlayer coupling. In the π -electron tight-binding approximation [18, 19], which can reproduce consistent results from density functional theory, these parts
where h(k) is expressed as h(k) = (412 + γ 2 )|f (k)|2 + (21ε0 + 12 γ 2 )2 . 2
J. Phys.: Condens. Matter 26 (2014) 015304
X Zhai and G Jin
figure 2(a), the bandgap closes at 1 = ε0 . As 1 continues to increase, the bandgap reopens. Compared with the more than 4 eV bandgap in the natural-state h-BN bilayer [19], the bandgap here is significantly reduced, which is consistent with the previous conclusion from first-principle calculations [15, 16]. Figure 2(d) plots the different band structures induced by the two opposite bias voltages at 1 = ±ε0 . From figures 2(c) and (d), it can be seen that the bandgaps are always direct. To get a large reduction of the bandgaps, just as given in figures 2(a) and (b), an effective interlayer bias of about 4 V needs to be applied between the top and bottom layers of the h-BN bilayer. There are two feasible ways to apply such a high electric field: one is to utilize the interface polarization to inject huge electric charges [23], the other is to apply double gates as done in a graphene bilayer [3], where a large electric field 3 V nm−1 is used. To elaborate on the asymmetric changes of the bandgaps in figure 2, it is necessary to quantitatively analyze the bias-driven variation of the interlayer and intralayer polarities of the h-BN bilayer. With the Green function method, the total density of states (TDOS) per unit cell is given by
Figure 2. (a), (b) Bandgaps of the h-BN bilayer under opposite
biases 1 > 0 and 1 < 0. The shaded areas in the insets show the intrinsic interlayer built-in field without a bias voltage. (c), (d) Electronic dispersions in the h-BN bilayer under the different chosen 1.
g(E) =
if 0 < 1 < ε0 ,
gi (E) = −
i=1
4 2X Im Gii (E), π i=1
(7)
where gi (E) is the partial density of states (PDOS) contributed by the ith atom. The factor 2 is from the spin degeneracy. The symbol Gii (E) is the diagonal matrix element of the Green function G(E), which can be calculated from the Hamiltonian in equation (3) by Z 1 eik·(ri −rj ) . (8) Gij (E) = dk BZ E + iδ − H(k)
The bandgap can be analytically given in the form Eg = 2(1 − ε0 ) ( −1, q × γ / γ 2 + 412 ,
4 X
(6)
else.
The expression in equation (6) agrees with the numerical results in figures 2(a) and (b), where the adjustable bias potential 1 takes values in the vicinity of ε0 and −ε0 respectively. In the case of 1 > 0, the system exhibits phase transitions from a semiconductor to a semimetal and then to a semiconductor again. However, in the case of 1 < 0, the bandgap decreases monotonically as |1| increases. In particular, at 1 = −ε0 , the bandgap is approximately 2γ , but is zero at 1 = ε0 . Thus the bandgaps of the h-BN bilayer vary asymmetrically under the biases along the two opposite directions. This does not happen in a Bernal-stacked graphene bilayer with inversion symmetry [1] or an AA-stacked h-BN bilayer with mirror symmetry [18, 19]. Therefore, the asymmetric bandgaps in figures 2(a) and (b) should be attributed to the lack of intrinsic mirror and inversion symmetries in the structure even in the absence of voltage, as seen from figure 1. Corresponding to these asymmetries, there exist γ -induced charge transfers and hence interlayer built-in fields, as sketched in the insets of figures 2(a) and (b), which prevent charge compensation between the top and bottom h-BN bilayer surfaces. In figure 2(c), we plot the band structures when 1 takes three positive values: ε0 − 0.5 eV, ε0 and ε0 + 0.5 eV. It can be seen that the Fermi energy always stays at E = 0. The valence (conduction) band is driven up (down) as 1 increases from 1 = ε0 − 0.5 eV to ε0 . In accordance with
Herein, BZ is the volume of the first Brillouin zone, i and j denote the atom positions in a unit cell, and δ takes an infinitesimal positive value. Using equation (7), we first check the TDOS curves at 1 = 0 (a large bandgap) and 1 = ε0 (zero bandgap) in figure 3(a). To display the TDOS variation near 1 = ε0 , we then plot the TDOS curves at 1 = ε0 − 0.2 and 1 = ε0 + 0.2 in figure 3(b). The bandgap exhibits the phenomenon of firstly closing and then reopening itself as 1 increases, which agrees with the result in figure 2(a) and is analogous to the previously proposed metal–insulator transition induced by increasing the hybridization [24]. Additionally, under the two opposite biases 1 = ±ε0 , the PDOS curves are plotted in figures 3(c) and (d). At 1 = ε0 , the PDOS in figure 3(c) takes non-zero values at E = 0 for N1 and B2 , which leads to a closing of the bandgap (non-zero TDOS) in figure 3(a). In figure 3(d), although the PDOS of N2 as well as B1 moves close to the Fermi energy driven by the bias, the bandgap is not closed at 1 = −ε0 . Using the calculated PDOSs, the electron number localized at the ith atom is given by Z 0 dE gi (E). (9) ni = −∞
By calculating equation (9), one can verify that about 0.429 of an electron is transferred from B to N for an isolated 3
J. Phys.: Condens. Matter 26 (2014) 015304
X Zhai and G Jin
Figure 3. (a) Total density of states (TDOS) per unit cell at 1 = 0 (solid line) and 1 = ε0 (dashed line). (b) TDOS at 1 = ε0 − 0.2 eV (short-dashed line), ε0 (dashed line), and ε0 + 0.2 eV (solid line). (c) Partial density of states (PDOS) for sublattices B1 , B2 , N1 , N2 in the occupied valence bands at 1 = ε0 . (d) PDOS for B1 , B2 , N1 , N2 at 1 = −ε0 . (e) The electron number ni versus the bias under 1 > 0. (f) The electron number ni versus the bias under 1 < 0.
electromagnetic momentum eA, where e is the absolute value of the electron charge and A is the vector potential satisfying the Landau gauge (−By, 0, 0). Then the low-energy Hamiltonian can be expressed as γ H(k) = v[ησx (px − eBy) + σy py ] ⊗ Iτ + (σx ⊗ τx 2 − σy ⊗ τy ) + 1Iσ ⊗ τz + ε0 τz ⊗ Iσ ε+ 0 ηvO γ 0 −ε 0 ηvO − = (11) . ηvO† 0 ε− 0 γ ηvO† 0 −ε+
monolayer h-BN sheet, which is consistent with the previous calculation [12]. For a gated h-BN bilayer, the electron number ni is given in figure 3(e). Under 1 ≤ ε0 , ni does not change. In this case, about 0.572 of an electron is transferred from B to N in each BN covalent bond, and thus 0.572 − 0.429 = 0.143 of an electron is dragged from B1 to N2 in a unit cell. Under 1 > ε0 , the localized charges in B1 and N2 have small changes, but there exist charges transferred from N1 to B2 . This leads to a reopening of the gap. Because N1 is not directly coupled with B2 here, this process is completed through the intermediates B1 and N2 . The bias-driven net charge transfer can be up to 0.1 of an electron per unit cell at 1 = ε0 + 0.4 eV. One can further check that the total number of electrons satisfies Z 0 4 X nT = ni = dE g(E) = 4, (10) i=1
Here, we define the two energies ε± = ε0 ± 1 and two operators O† = px + ipy − eBy, O = px − ipy − eBy, which are the raising and lowering operators acting on the Landau functions ϕ(x, y) = eipx x/h¯ φN (y).√ N is a nonnegative integer satisfying O† φN = i(h¯ /`c ) 2(N + 1)φN +1 , OφN = √ −i(h¯ /`c ) 2N φN −1 , and Oφ0 = 0. It obeys the commutation √ relation [O, O† ] = −2/`2c , where `c = h¯ /eB is the magnetic length. The four-component eigenstates of H can be reconstructed as ψ = (cB1 φN −1 , cB2 φN , cN1 φN , cN2 φN +1 ), which has been used in graphene bilayers [26, 27]. For simplicity, the dimensionless transformation is adopted here with all the energy terms rescaled by h¯ v/`c ≡ C, and then ˜ = 1/C, ε˜ ± = ε± /C, γ˜ = γ /C, E˜ = E/C. By solving the 1 eigenvalue from Hψ = Eψ, a quartic equation is derived as
−∞
which is independent of the externally applied interlayer bias. If a negative 1 is applied, the electron number ni versus the bias behaves as is plotted in figure 3(f). Differently from the results in figure 3(e), there exist charges transferred from B1 to N2 as long as |1| increases, while the charges of B2 or N1 change little. The bandgap is not closed because the intralayer polarity cannot be canceled out by the bias. According to our calculation, at 1 = −ε0 , about 0.516 of an electron is transferred from B to N in each intralayer BN bond, and thus only 0.087 of an electron is transferred from B1 to N2 in a unit cell. In addition, the bias-driven net charge transfer is up to 0.2 of an electron per unit cell at 1 = −ε0 − 0.4 eV.
2 2 E˜ 4 − (4N + 2 + ε˜ + + ε˜ − + γ˜ 2 )E˜ 2 + 2(˜ε+ − ε˜ − )E˜ 2 2 = 0. + 4N (N + 1) + (4N + 2)˜ε+ ε˜ − + (˜ε+ + γ˜ 2 )˜ε−
3.2. Landau levels for low-energy electrons
(12) Since the band structure of the h-BN bilayer has been engineered by the interlayer bias, a natural problem is how the low-energy electrons in a gated h-BN bilayer behave when a perpendicular magnetic field is applied. In this case, the electrons described by equation (4) move with an extra
Then we can get ˜ 2 − 1) 2N = (E˜ 2 − ε˜ 02 + 1 q ˜ 2 + γ˜ 2 )E˜ 2 − 41 ˜ E˜ + ω(1), ± (41 4
(13)
J. Phys.: Condens. Matter 26 (2014) 015304
X Zhai and G Jin
˜ 2 . The presence of −41 ˜ E˜ + where ω(1) = 1 − γ˜ 2 (˜ε0 − 1) ω(1) in the square root leads to the asymmetry of the Landau levels under the two opposite biases. Specifically, at the Fermi energy E = 0, the Landau index N is zero when 1 takes the value ε0 in equation (13). The zero-energy Landau levels have fourfold degeneracy, including the spin degrees of freedom of electrons and holes. Actually, at 1 = ε0 , by solving the eigenvalues of equation (4), the low-energy dispersion without magnetic field can be given as Ev = s u u γ2 t 2 2 γ2 2 ± v p + + 2ε0 ± (γ 2 + 4ε02 ) v2 p2 + + ε02 , 2 4 (14) where p is the momentum, which satisfies p2 = p2x + p2y . The two subbands closest to the Fermi energy E = 0 can be simplified to E = ±q
v2 p2 γ 2 + 4ε02
(p → 0),
(15)
which reveals the same parabolic dispersion relation as that in a graphene bilayer [25]. An essential difference is that the h-BN bilayer electronic states have no chirality due to the existence of polarities. Thus, under a magnetic field, the zero-energy Landau level with fourfold degeneracy differs from that with eightfold degeneracy in a graphene bilayer [25]. Moreover, due to the strong polarities of the h-BN bilayer, the effective mass of an electron m∗ = 0.61 me is much larger than the estimated value 0.054 me in a graphene bilayer [27, 28].
Figure 4. (a) The two-chain model for the calculation of the energy
bands of the √ h-BN bilayer nanoribbon, where the parameter cm = cos( 3ky a/2 − mφ) comes from the magnetic field B and is obtained by the Fourier transformation. (b)–(e) Low-energy spectra of tight-binding electrons in a 43.5 nm-width nanoribbon tuned by the interlayer bias under a magnetic flux φ = 0.005, where (b) and (c) are for an h-BN bilayer nanoribbon, and (d) and (e) are for a graphene bilayer nanoribbon.
Then, the Hamiltonian reads H1(2)
3.3. Confinement effects in the magneto-electronic spectra
ε+ −2tc1 0 0 ··· 0 0 −2tc −ε −t 0 ··· 0 0 − 1 0 −t ε+ −2tc2 · · · 0 0 0 −2tc2 −ε− · · · 0 0 = 0 , . .. .. .. .. .. .. .. . . . . . . 0 0 0 0 · · · ε− −2tcN 0 0 0 0 · · · −2tcN −ε+
Considering that a real h-BN bilayer system is always finite, we turn to study the edge effects on the Landau levels in a confined zigzag-edged h-BN bilayer nanoribbon. In the presence of a perpendicular magnetic field [29–31], the Hamiltonian in equation (2) reads X X H1(2) = (εi ± 1)c†iσ ciσ − t eiφij c†iσ cjσ , (16) iσ
hijiσ
where the phase factor φij in the hopping term comes from Rj the magnetic field, i.e. φij = i A · dl/φ0 and φ0 = h/e. ¯ √ If we define φ = (3 3/4)a2 B/φ0 , the magnetic flux in a honeycomb lattice is 2φ. Using the Fourier transformation, the Hamiltonian of the h-BN bilayer nanoribbon can be simplified to the equivalent two-chain model illustrated in figure 4(a), which is developed by extending the one-chain model used previously [29]. Without loss of generality, the width of the h-BN bilayer nanoribbon is fixed at 43.5 nm (N = 200), and a typical magnetic flux [30] φ = 0.005 is used here. The Zeeman splitting is ignored because it is small [1, 30, 28].
(17)
where √ the diagonal matrix elements are antisymmetric, cm = cos( 3ky a/2 − mφ), and H1(2) is a 2N × 2N matrix. For the interlayer coupling matrix, the non-zero matrix elements satisfy 2i−1,2i 2i,2i−1 H12 = H21 = −γ ,
i = 1, 2, 3, . . . , N. (18)
Now, we plot the low-energy magneto-electronic spectra of the h-BN bilayer nanoribbon in figures 4(b) and (c) when 1 takes the positive values ε0 − 0.05 eV and ε0 respectively. For comparison, the Landau-level spectra of a graphene bilayer 5
J. Phys.: Condens. Matter 26 (2014) 015304
X Zhai and G Jin
nanoribbon are also plotted in figure 4(d) at 1 = 0 and in figure 4(e) at 1 = 0.05 eV. From figures 4(b) to (e), the formation of the Landau levels can be identified by the presence of flat energy subbands, following the spectra of bulk systems. The dispersive parts of the spectra are caused by the edges of the h-BN bilayer nanoribbon. According to the filling of the Landau levels, the quantum Hall conductivity can be found. Comparing the results in figures 4(c) and (d), neither of these two systems has bulk band gaps, while the zero-energy Landau levels are fourfold degenerate for the h-BN bilayer nanoribbon, but eightfold degenerate for the graphene bilayer nanoribbon. On the other hand, comparing the results in figures 4(b) and (e), although both the h-BN bilayer nanoribbon and the graphene bilayer nanoribbon have bulk bandgaps, no edge states exist in the bandgap of the h-BN bilayer nanoribbon while edge states (denoted by dashed lines) exist within the 21 bandgap of the graphene bilayer nanoribbon [31]. This difference in Landau levels between the h-BN bilayer nanoribbon and the graphene bilayer nanoribbon is another result of symmetry breaking driven by polarities. We further investigate the energy spectra of the h-BN bilayer nanoribbon when 1 takes some negative values. In figure 5(a), we first plot the spectrum (left panel) at 1 = −ε0 without magnetic field, then plot the probability density distribution of the states (right panel) at given energies E = ±0.256 eV. It is shown that the states in the bulk bandgap are localized at the edges. In the presence of an applied perpendicular magnetic field, the low-energy magneto-electronic spectra are plotted in figure 5(b) at 1 = −ε0 (left panel) and 1 = −ε0 − 0.5 eV (right panel). The results display that the Landau levels far from the Fermi energy are almost indistinguishable, due to the very small level spacing caused by the large effective mass of electrons. In the low-energy bulk bandgap region, the system still exhibits fourfold degenerate edge states, which are different from the Hall edge states and may be viewed as a feature of the h-BN bilayer nanoribbon in the case of 1 < 0. Thus, these dispersive edge states are robust against magnetic field, and can be controlled by the interlayer bias. By comparing the results in figures 4(b) and (c) with those in figure 5(b), we can see an obvious characteristic that the magneto-electronic spectra of the h-BN bilayer nanoribbon are different under opposite biases. In other words, dispersive edge states exist in the bulk bandgap only in the case of 1 < 0. These obtained Landau levels and edge states have been further demonstrated to be robust against different kinds of edge conditions and weak roughness, and may be measured by high-magnetic field experiments, as done in graphene nanoribbons [32, 33].
Figure 5. (a) Low-energy band structure (left) of tight-binding
electrons without magnetic field for the h-BN bilayer nanoribbon at 1 = −ε0 and the probability distribution |ψ|2 of the edge states (right) on the top (bottom) layer at E = ±0.256 eV. (b) Energy spectra of the gated h-BN bilayer nanoribbon tuned by 1 in the presence of a magnetic flux φ = 0.005. The quantity 2e2 /h in (a) and (b) refers to the non-equilibrium conductance of the edge channels.
semiconductor to a semimetal and then to a semiconductor again under the bias along one direction, but it is always semiconducting under the bias along the other direction. Also, it is obtained that the Landau levels change asymmetrically under opposite biases. Moreover, in a confined zigzag-edged h-BN bilayer, dispersive edge states are found to be stable in the bulk bandgap under a bias along one direction but do not exist when the bias is reversed. Because the Bernal-stacked h-BN bilayer with macroscopic dimension has been proved to be stable [19, 20] and can be synthesized using chemical exfoliation [9] or chemical vapor deposition [10], the widely tunable bandgaps discussed here, by applying an interlayer bias, may provide possibilities for applications in electronic and optoelectronic devices based on h-BN bilayers. Acknowledgments
4. Summary
This work was supported by the State Key Program for Basic Research of China (Grants Nos 2009CB929504 and 2011CB922102), the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD), and the National Natural Science Foundation of China (Grants Nos 60876065 and 11074108).
We have systematically studied the influence of polarities changed by an interlayer bias on the band structure of a Bernal-stacked h-BN bilayer with the effective tight-binding approximation. It is found that the bandgaps of the h-BN bilayer vary asymmetrically under opposite biases. The system exhibits quantum phase transitions from a 6
J. Phys.: Condens. Matter 26 (2014) 015304
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[1] Castro Neto A H, Guinea F, Peres N M R, Novoselov K S and Geim A K 2009 Rev. Mod. Phys. 81 109 [2] Li X, Wang X, Zhang L, Lee S and Dai H 2008 Science 319 1229 [3] Zhang Y, Tang T-T, Girit C, Hao Z, Martin M C, Zettl A, Crommie M F, Shen Y R and Wang F 2009 Nature 459 820 [4] Weitz R T, Allen M T, Feldman B E, Martin J and Yacoby A 2010 Science 330 812 [5] Sławi´nska J, Zasada I and Klusek Z 2010 Phys. Rev. B 81 155433 [6] Elias D C et al 2009 Science 323 610 [7] Radisavljevic B, Radenovic A, Brivio J, Giacometti V and Kis A 2011 Nature Nanotechnol. 6 147 [8] Pakdel A, Zhi C, Bando Y and Golberg D 2012 Mater. Today 15 256 ¨ and Zettl A 2008 Appl. Phys. [9] Pacil´e D, Meyer J C, Girit C ¸ O Lett. 92 133107 [10] Song L et al 2010 Nano Lett. 10 3209 [11] Nagashima A, Tejima N, Gamou Y, Kawai T and Oshima C 1995 Phys. Rev. B 51 4606 [12] Topsakal M, Akt¨urk E and Ciraci S 2009 Phys. Rev. B 79 115442 [13] Watanabe K, Taniguchi T and Kanda H 2004 Nature Mater. 3 404 [14] Lukaszew R A, Sheng Y, Uher C and Clarke R 1999 Appl. Phys. Lett. 75 1941 [15] Balu R, Zhong X, Pandey R and Karna S P 2012 Appl. Phys. Lett. 100 052104
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Asymmetric bandgaps and Landau levels in a Bernal-stacked hexagonal boron-nitride bilayer
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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 015304 (7pp)
doi:10.1088/0953-8984/26/1/015304
Asymmetric bandgaps and Landau levels in a Bernal-stacked hexagonal boron-nitride bilayer Xuechao Zhai and Guojun Jin National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing, 210093, People’s Republic of China E-mail: [email protected] Received 17 September 2013, in final form 29 October 2013 Published 25 November 2013 Abstract
A Bernal-stacked hexagonal boron-nitride (h-BN) bilayer is a two-dimensional polar crystal. Within the tight-binding approximation, we investigate the band structure of a gated h-BN bilayer by analyzing the density of states and the behavior of the charge transfer. We find that the bandgaps of the h-BN bilayer vary asymmetrically under two opposite biases due to asymmetric changes of the interlayer and intralayer polarities. We also find that the bias-driven net charge transfer between layers can be up to 0.2 electron per unit cell. Under the bias along one direction, the system exhibits quantum phase transitions from a semiconductor to a semimetal and then to a semiconductor again, whereas under the reverse bias, the system is always semiconducting. Besides, asymmetric Landau levels under opposite biases arise in the presence of a magnetic field. Moreover, dispersive edge states are found to exist in the bulk bandgap for an h-BN bilayer nanoribbon under the bias along one direction, which does not happen when the bias is reversed. All these properties of h-BN bilayers are measurable in transport experiments. (Some figures may appear in colour only in the online journal)
1. Introduction
effective mass in bulk bilayers [5]. This fact has an impact on the advantage of graphene nanoribbons and bilayers in electronics but does not affect their superiority in photonics and optoelectronics. Meanwhile, some new two-dimensional semiconducting materials have also attracted researchers’ attention, such as graphane [6], MoS2 [7], and hexagonal boron nitride (h-BN) [8]. Experimentally, by micromechanical exfoliation [9] and chemical vapor deposition [10], mono- or few-layer h-BN sheets can be prepared. An obvious difference from graphene is that the BN bond in h-BN materials has a polarity which results in large bandgaps. Both experimental [11] and theoretical [12] studies have reported a bandgap of about 4.6 eV for two-dimensional single-layer h-BN and about 5.97 eV for three-dimensional h-BN single crystals [13]. These large bandgaps are beneficial for h-BN materials to work as good dielectric spacer layers in magnetic tunneling
In the past decade, there have been many studies on two-dimensional materials, because of their relatively easy production, convenient integration, and potential applications in future solid-state devices. The most widely studied two-dimensional material is graphene [1], which has a particularly simple structure but very rich physics. The fact that pristine graphene has no band gap, however, hampers its applications in electronics, photonics, and optoelectronics. Thus, the means of modulating the bandgap becomes an important issue. It has been demonstrated that the band gap can be experimentally tuned by controlling the width of graphene nanoribbons [2] or varying the external interlayer bias on graphene bilayers [3, 4]. Nevertheless, the carrier mobility may decrease significantly, due to the edge-enhanced scattering in narrow ribbons [2] and the bias-increased 0953-8984/14/015304+07$33.00
1
c 2014 IOP Publishing Ltd Printed in the UK
J. Phys.: Condens. Matter 26 (2014) 015304
X Zhai and G Jin
can be described by X X † H1(2) = (εi ± 1)c†i ci − t ci cj ; i
H12 = −γ
hiji
X
c†i cj ;
† H21 = H12 .
(2)
i∈1,j∈2
Figure 1. The lattice structure of a gated Bernal-stacked h-BN
In H1(2) , the first term indicates the original on-site energy εi , in addition to the electrostatic potential energy of the top (bottom) layer +1 (−1), induced by a bias voltage, as done in a graphene bilayer [3, 4, 22]. The second term is the nearest-neighbor intralayer coupling with the hopping integral t. The interlayer coupling strength (van der Waals interaction) is described by γ in H12(21) . The parameters t, γ , εi are extracted from first-principle calculations [20, 19]: t = 2.37 eV, γ = 0.6 eV, and εB = −εN ≡ ε0 = 2.095 eV. We choose the atomic wavefunctions localized at sites B1 , B2 , N1 , N2 as the basis, ordered in the right inset of figure 1, where 1 (2) labels the top (bottom) layer respectively. Then the Hamiltonian of the h-BN bilayer in the momentum space reads ε0 + 1 0 f (k) γ 0 ε0 − 1 0 f (k) H(k) = ∗ (3) , f (k) 0 −ε0 + 1 0 γ f ∗ (k) 0 −ε0 − 1 √ where f (k) = −teikx a [1 + 2e−i(3kx a/2) cos( 3ky a/2)]. Expanding equation (3) in the vicinity√of two valleys K = √ (0, −4π/3 3a) and K0 = (0, 4π/3 3a), we obtain the low-energy four-band Hamiltonian, γ H(k) = v(ησx px + σy py ) ⊗ Iτ + (σx ⊗ τx 2 − σy ⊗ τy ) + 1Iσ ⊗ τz + ε0 τz ⊗ Iσ , (4)
bilayer in Cartesian coordinates (x, y, z). γ is the interlayer hopping and 1, −1 describe the effective electrostatic potentials. The right inset denotes the atoms in a unit cell, where B (N) marks the boron (nitride) atom and the subscript 1 (2) labels the top (bottom) layer.
junctions [14]. According to the latest first-principle [15–17] and tight-binding [18] calculations, the bandgaps of few-layer h-BN sheets can be remarkably reduced by applying an interlayer bias voltage to decrease their polarities effectively. These tunable bandgaps enable h-BN materials to be good media for light generation and detection in optoelectronic devices [13, 16]. In this paper, we show that the bandgap of an energetically stable Bernal-stacked h-BN bilayer [19–21] without an inversion symmetry varies asymmetrically under two opposite biases due to the asymmetric changes of the interlayer and intralayer polarities. It is argued that the bias-induced net charge transfer between layers can be up to 0.2 of an electron per unit cell. Under the bias along one direction, the system exhibits phase transitions from a semiconductor to a semimetal and then to a semiconductor again. On the other hand, the system is always semiconducting under the reverse bias. Also, under the two opposite biases, by applying a perpendicular magnetic field, the Landau levels are found to be asymmetric and can show up in the magneto-electronic spectra of a confined h-BN bilayer. Furthermore, dispersive edge states are stable in the bulk bandgap under a bias only along one direction, but do not exist along the opposite direction.
where η = −1(+1) labels K (K0 ), v = 3ta/2h¯ = 7.82 × 105 m s−1 is the Fermi velocity, σ and τ are Pauli matrices representing the sublattice pseudospin and layer degrees of freedom, respectively, and I is a 2 × 2 identity matrix. The first term indicates the Dirac linear dispersion. The second term is the interlayer coupling term. The third term represents the interlayer polarity controlled by an external bias. The last term represents the intralayer polarity of the BN covalent bond. Without the last term, equation (4) goes back to the Hamiltonian of a gated graphene bilayer [1].
2. Theoretical formulation
Figure 1 shows the lattice configuration of a Bernal-stacked h-BN bilayer. The electrostatic potentials, +1 and −1, from an effective interlayer bias voltage, are applied to the top and bottom layers of the h-BN bilayer respectively. Each layer of the h-BN bilayer is constructed of two inequivalent trigonal sublattices, which are associated by strong polar covalent σ -bonds. In Cartesian coordinates (x, y, z), the Bravais lattice √ is built from the primitive vectors a± = (a/2)(3x ± 3y), ˚ is the BN-bond length [20]. where a = 1.45 A The Hamiltonian of this system can be written in four parts, H = H1 + H2 + H12 + H21 ,
3. Results and discussion 3.1. Asymmetric bandgaps under opposite biases
By diagonalizing the Hamiltonian in equation (3), we derive the electronic dispersion q p (5) E = ± 12 + ε02 + 12 γ 2 + |f (k)|2 ± h(k),
(1)
where H1(2) is the Hamiltonian of the top (bottom) layer, and H12(21) represents the interlayer coupling. In the π -electron tight-binding approximation [18, 19], which can reproduce consistent results from density functional theory, these parts
where h(k) is expressed as h(k) = (412 + γ 2 )|f (k)|2 + (21ε0 + 12 γ 2 )2 . 2
J. Phys.: Condens. Matter 26 (2014) 015304
X Zhai and G Jin
figure 2(a), the bandgap closes at 1 = ε0 . As 1 continues to increase, the bandgap reopens. Compared with the more than 4 eV bandgap in the natural-state h-BN bilayer [19], the bandgap here is significantly reduced, which is consistent with the previous conclusion from first-principle calculations [15, 16]. Figure 2(d) plots the different band structures induced by the two opposite bias voltages at 1 = ±ε0 . From figures 2(c) and (d), it can be seen that the bandgaps are always direct. To get a large reduction of the bandgaps, just as given in figures 2(a) and (b), an effective interlayer bias of about 4 V needs to be applied between the top and bottom layers of the h-BN bilayer. There are two feasible ways to apply such a high electric field: one is to utilize the interface polarization to inject huge electric charges [23], the other is to apply double gates as done in a graphene bilayer [3], where a large electric field 3 V nm−1 is used. To elaborate on the asymmetric changes of the bandgaps in figure 2, it is necessary to quantitatively analyze the bias-driven variation of the interlayer and intralayer polarities of the h-BN bilayer. With the Green function method, the total density of states (TDOS) per unit cell is given by
Figure 2. (a), (b) Bandgaps of the h-BN bilayer under opposite
biases 1 > 0 and 1 < 0. The shaded areas in the insets show the intrinsic interlayer built-in field without a bias voltage. (c), (d) Electronic dispersions in the h-BN bilayer under the different chosen 1.
g(E) =
if 0 < 1 < ε0 ,
gi (E) = −
i=1
4 2X Im Gii (E), π i=1
(7)
where gi (E) is the partial density of states (PDOS) contributed by the ith atom. The factor 2 is from the spin degeneracy. The symbol Gii (E) is the diagonal matrix element of the Green function G(E), which can be calculated from the Hamiltonian in equation (3) by Z 1 eik·(ri −rj ) . (8) Gij (E) = dk BZ E + iδ − H(k)
The bandgap can be analytically given in the form Eg = 2(1 − ε0 ) ( −1, q × γ / γ 2 + 412 ,
4 X
(6)
else.
The expression in equation (6) agrees with the numerical results in figures 2(a) and (b), where the adjustable bias potential 1 takes values in the vicinity of ε0 and −ε0 respectively. In the case of 1 > 0, the system exhibits phase transitions from a semiconductor to a semimetal and then to a semiconductor again. However, in the case of 1 < 0, the bandgap decreases monotonically as |1| increases. In particular, at 1 = −ε0 , the bandgap is approximately 2γ , but is zero at 1 = ε0 . Thus the bandgaps of the h-BN bilayer vary asymmetrically under the biases along the two opposite directions. This does not happen in a Bernal-stacked graphene bilayer with inversion symmetry [1] or an AA-stacked h-BN bilayer with mirror symmetry [18, 19]. Therefore, the asymmetric bandgaps in figures 2(a) and (b) should be attributed to the lack of intrinsic mirror and inversion symmetries in the structure even in the absence of voltage, as seen from figure 1. Corresponding to these asymmetries, there exist γ -induced charge transfers and hence interlayer built-in fields, as sketched in the insets of figures 2(a) and (b), which prevent charge compensation between the top and bottom h-BN bilayer surfaces. In figure 2(c), we plot the band structures when 1 takes three positive values: ε0 − 0.5 eV, ε0 and ε0 + 0.5 eV. It can be seen that the Fermi energy always stays at E = 0. The valence (conduction) band is driven up (down) as 1 increases from 1 = ε0 − 0.5 eV to ε0 . In accordance with
Herein, BZ is the volume of the first Brillouin zone, i and j denote the atom positions in a unit cell, and δ takes an infinitesimal positive value. Using equation (7), we first check the TDOS curves at 1 = 0 (a large bandgap) and 1 = ε0 (zero bandgap) in figure 3(a). To display the TDOS variation near 1 = ε0 , we then plot the TDOS curves at 1 = ε0 − 0.2 and 1 = ε0 + 0.2 in figure 3(b). The bandgap exhibits the phenomenon of firstly closing and then reopening itself as 1 increases, which agrees with the result in figure 2(a) and is analogous to the previously proposed metal–insulator transition induced by increasing the hybridization [24]. Additionally, under the two opposite biases 1 = ±ε0 , the PDOS curves are plotted in figures 3(c) and (d). At 1 = ε0 , the PDOS in figure 3(c) takes non-zero values at E = 0 for N1 and B2 , which leads to a closing of the bandgap (non-zero TDOS) in figure 3(a). In figure 3(d), although the PDOS of N2 as well as B1 moves close to the Fermi energy driven by the bias, the bandgap is not closed at 1 = −ε0 . Using the calculated PDOSs, the electron number localized at the ith atom is given by Z 0 dE gi (E). (9) ni = −∞
By calculating equation (9), one can verify that about 0.429 of an electron is transferred from B to N for an isolated 3
J. Phys.: Condens. Matter 26 (2014) 015304
X Zhai and G Jin
Figure 3. (a) Total density of states (TDOS) per unit cell at 1 = 0 (solid line) and 1 = ε0 (dashed line). (b) TDOS at 1 = ε0 − 0.2 eV (short-dashed line), ε0 (dashed line), and ε0 + 0.2 eV (solid line). (c) Partial density of states (PDOS) for sublattices B1 , B2 , N1 , N2 in the occupied valence bands at 1 = ε0 . (d) PDOS for B1 , B2 , N1 , N2 at 1 = −ε0 . (e) The electron number ni versus the bias under 1 > 0. (f) The electron number ni versus the bias under 1 < 0.
electromagnetic momentum eA, where e is the absolute value of the electron charge and A is the vector potential satisfying the Landau gauge (−By, 0, 0). Then the low-energy Hamiltonian can be expressed as γ H(k) = v[ησx (px − eBy) + σy py ] ⊗ Iτ + (σx ⊗ τx 2 − σy ⊗ τy ) + 1Iσ ⊗ τz + ε0 τz ⊗ Iσ ε+ 0 ηvO γ 0 −ε 0 ηvO − = (11) . ηvO† 0 ε− 0 γ ηvO† 0 −ε+
monolayer h-BN sheet, which is consistent with the previous calculation [12]. For a gated h-BN bilayer, the electron number ni is given in figure 3(e). Under 1 ≤ ε0 , ni does not change. In this case, about 0.572 of an electron is transferred from B to N in each BN covalent bond, and thus 0.572 − 0.429 = 0.143 of an electron is dragged from B1 to N2 in a unit cell. Under 1 > ε0 , the localized charges in B1 and N2 have small changes, but there exist charges transferred from N1 to B2 . This leads to a reopening of the gap. Because N1 is not directly coupled with B2 here, this process is completed through the intermediates B1 and N2 . The bias-driven net charge transfer can be up to 0.1 of an electron per unit cell at 1 = ε0 + 0.4 eV. One can further check that the total number of electrons satisfies Z 0 4 X nT = ni = dE g(E) = 4, (10) i=1
Here, we define the two energies ε± = ε0 ± 1 and two operators O† = px + ipy − eBy, O = px − ipy − eBy, which are the raising and lowering operators acting on the Landau functions ϕ(x, y) = eipx x/h¯ φN (y).√ N is a nonnegative integer satisfying O† φN = i(h¯ /`c ) 2(N + 1)φN +1 , OφN = √ −i(h¯ /`c ) 2N φN −1 , and Oφ0 = 0. It obeys the commutation √ relation [O, O† ] = −2/`2c , where `c = h¯ /eB is the magnetic length. The four-component eigenstates of H can be reconstructed as ψ = (cB1 φN −1 , cB2 φN , cN1 φN , cN2 φN +1 ), which has been used in graphene bilayers [26, 27]. For simplicity, the dimensionless transformation is adopted here with all the energy terms rescaled by h¯ v/`c ≡ C, and then ˜ = 1/C, ε˜ ± = ε± /C, γ˜ = γ /C, E˜ = E/C. By solving the 1 eigenvalue from Hψ = Eψ, a quartic equation is derived as
−∞
which is independent of the externally applied interlayer bias. If a negative 1 is applied, the electron number ni versus the bias behaves as is plotted in figure 3(f). Differently from the results in figure 3(e), there exist charges transferred from B1 to N2 as long as |1| increases, while the charges of B2 or N1 change little. The bandgap is not closed because the intralayer polarity cannot be canceled out by the bias. According to our calculation, at 1 = −ε0 , about 0.516 of an electron is transferred from B to N in each intralayer BN bond, and thus only 0.087 of an electron is transferred from B1 to N2 in a unit cell. In addition, the bias-driven net charge transfer is up to 0.2 of an electron per unit cell at 1 = −ε0 − 0.4 eV.
2 2 E˜ 4 − (4N + 2 + ε˜ + + ε˜ − + γ˜ 2 )E˜ 2 + 2(˜ε+ − ε˜ − )E˜ 2 2 = 0. + 4N (N + 1) + (4N + 2)˜ε+ ε˜ − + (˜ε+ + γ˜ 2 )˜ε−
3.2. Landau levels for low-energy electrons
(12) Since the band structure of the h-BN bilayer has been engineered by the interlayer bias, a natural problem is how the low-energy electrons in a gated h-BN bilayer behave when a perpendicular magnetic field is applied. In this case, the electrons described by equation (4) move with an extra
Then we can get ˜ 2 − 1) 2N = (E˜ 2 − ε˜ 02 + 1 q ˜ 2 + γ˜ 2 )E˜ 2 − 41 ˜ E˜ + ω(1), ± (41 4
(13)
J. Phys.: Condens. Matter 26 (2014) 015304
X Zhai and G Jin
˜ 2 . The presence of −41 ˜ E˜ + where ω(1) = 1 − γ˜ 2 (˜ε0 − 1) ω(1) in the square root leads to the asymmetry of the Landau levels under the two opposite biases. Specifically, at the Fermi energy E = 0, the Landau index N is zero when 1 takes the value ε0 in equation (13). The zero-energy Landau levels have fourfold degeneracy, including the spin degrees of freedom of electrons and holes. Actually, at 1 = ε0 , by solving the eigenvalues of equation (4), the low-energy dispersion without magnetic field can be given as Ev = s u u γ2 t 2 2 γ2 2 ± v p + + 2ε0 ± (γ 2 + 4ε02 ) v2 p2 + + ε02 , 2 4 (14) where p is the momentum, which satisfies p2 = p2x + p2y . The two subbands closest to the Fermi energy E = 0 can be simplified to E = ±q
v2 p2 γ 2 + 4ε02
(p → 0),
(15)
which reveals the same parabolic dispersion relation as that in a graphene bilayer [25]. An essential difference is that the h-BN bilayer electronic states have no chirality due to the existence of polarities. Thus, under a magnetic field, the zero-energy Landau level with fourfold degeneracy differs from that with eightfold degeneracy in a graphene bilayer [25]. Moreover, due to the strong polarities of the h-BN bilayer, the effective mass of an electron m∗ = 0.61 me is much larger than the estimated value 0.054 me in a graphene bilayer [27, 28].
Figure 4. (a) The two-chain model for the calculation of the energy
bands of the √ h-BN bilayer nanoribbon, where the parameter cm = cos( 3ky a/2 − mφ) comes from the magnetic field B and is obtained by the Fourier transformation. (b)–(e) Low-energy spectra of tight-binding electrons in a 43.5 nm-width nanoribbon tuned by the interlayer bias under a magnetic flux φ = 0.005, where (b) and (c) are for an h-BN bilayer nanoribbon, and (d) and (e) are for a graphene bilayer nanoribbon.
Then, the Hamiltonian reads H1(2)
3.3. Confinement effects in the magneto-electronic spectra
ε+ −2tc1 0 0 ··· 0 0 −2tc −ε −t 0 ··· 0 0 − 1 0 −t ε+ −2tc2 · · · 0 0 0 −2tc2 −ε− · · · 0 0 = 0 , . .. .. .. .. .. .. .. . . . . . . 0 0 0 0 · · · ε− −2tcN 0 0 0 0 · · · −2tcN −ε+
Considering that a real h-BN bilayer system is always finite, we turn to study the edge effects on the Landau levels in a confined zigzag-edged h-BN bilayer nanoribbon. In the presence of a perpendicular magnetic field [29–31], the Hamiltonian in equation (2) reads X X H1(2) = (εi ± 1)c†iσ ciσ − t eiφij c†iσ cjσ , (16) iσ
hijiσ
where the phase factor φij in the hopping term comes from Rj the magnetic field, i.e. φij = i A · dl/φ0 and φ0 = h/e. ¯ √ If we define φ = (3 3/4)a2 B/φ0 , the magnetic flux in a honeycomb lattice is 2φ. Using the Fourier transformation, the Hamiltonian of the h-BN bilayer nanoribbon can be simplified to the equivalent two-chain model illustrated in figure 4(a), which is developed by extending the one-chain model used previously [29]. Without loss of generality, the width of the h-BN bilayer nanoribbon is fixed at 43.5 nm (N = 200), and a typical magnetic flux [30] φ = 0.005 is used here. The Zeeman splitting is ignored because it is small [1, 30, 28].
(17)
where √ the diagonal matrix elements are antisymmetric, cm = cos( 3ky a/2 − mφ), and H1(2) is a 2N × 2N matrix. For the interlayer coupling matrix, the non-zero matrix elements satisfy 2i−1,2i 2i,2i−1 H12 = H21 = −γ ,
i = 1, 2, 3, . . . , N. (18)
Now, we plot the low-energy magneto-electronic spectra of the h-BN bilayer nanoribbon in figures 4(b) and (c) when 1 takes the positive values ε0 − 0.05 eV and ε0 respectively. For comparison, the Landau-level spectra of a graphene bilayer 5
J. Phys.: Condens. Matter 26 (2014) 015304
X Zhai and G Jin
nanoribbon are also plotted in figure 4(d) at 1 = 0 and in figure 4(e) at 1 = 0.05 eV. From figures 4(b) to (e), the formation of the Landau levels can be identified by the presence of flat energy subbands, following the spectra of bulk systems. The dispersive parts of the spectra are caused by the edges of the h-BN bilayer nanoribbon. According to the filling of the Landau levels, the quantum Hall conductivity can be found. Comparing the results in figures 4(c) and (d), neither of these two systems has bulk band gaps, while the zero-energy Landau levels are fourfold degenerate for the h-BN bilayer nanoribbon, but eightfold degenerate for the graphene bilayer nanoribbon. On the other hand, comparing the results in figures 4(b) and (e), although both the h-BN bilayer nanoribbon and the graphene bilayer nanoribbon have bulk bandgaps, no edge states exist in the bandgap of the h-BN bilayer nanoribbon while edge states (denoted by dashed lines) exist within the 21 bandgap of the graphene bilayer nanoribbon [31]. This difference in Landau levels between the h-BN bilayer nanoribbon and the graphene bilayer nanoribbon is another result of symmetry breaking driven by polarities. We further investigate the energy spectra of the h-BN bilayer nanoribbon when 1 takes some negative values. In figure 5(a), we first plot the spectrum (left panel) at 1 = −ε0 without magnetic field, then plot the probability density distribution of the states (right panel) at given energies E = ±0.256 eV. It is shown that the states in the bulk bandgap are localized at the edges. In the presence of an applied perpendicular magnetic field, the low-energy magneto-electronic spectra are plotted in figure 5(b) at 1 = −ε0 (left panel) and 1 = −ε0 − 0.5 eV (right panel). The results display that the Landau levels far from the Fermi energy are almost indistinguishable, due to the very small level spacing caused by the large effective mass of electrons. In the low-energy bulk bandgap region, the system still exhibits fourfold degenerate edge states, which are different from the Hall edge states and may be viewed as a feature of the h-BN bilayer nanoribbon in the case of 1 < 0. Thus, these dispersive edge states are robust against magnetic field, and can be controlled by the interlayer bias. By comparing the results in figures 4(b) and (c) with those in figure 5(b), we can see an obvious characteristic that the magneto-electronic spectra of the h-BN bilayer nanoribbon are different under opposite biases. In other words, dispersive edge states exist in the bulk bandgap only in the case of 1 < 0. These obtained Landau levels and edge states have been further demonstrated to be robust against different kinds of edge conditions and weak roughness, and may be measured by high-magnetic field experiments, as done in graphene nanoribbons [32, 33].
Figure 5. (a) Low-energy band structure (left) of tight-binding
electrons without magnetic field for the h-BN bilayer nanoribbon at 1 = −ε0 and the probability distribution |ψ|2 of the edge states (right) on the top (bottom) layer at E = ±0.256 eV. (b) Energy spectra of the gated h-BN bilayer nanoribbon tuned by 1 in the presence of a magnetic flux φ = 0.005. The quantity 2e2 /h in (a) and (b) refers to the non-equilibrium conductance of the edge channels.
semiconductor to a semimetal and then to a semiconductor again under the bias along one direction, but it is always semiconducting under the bias along the other direction. Also, it is obtained that the Landau levels change asymmetrically under opposite biases. Moreover, in a confined zigzag-edged h-BN bilayer, dispersive edge states are found to be stable in the bulk bandgap under a bias along one direction but do not exist when the bias is reversed. Because the Bernal-stacked h-BN bilayer with macroscopic dimension has been proved to be stable [19, 20] and can be synthesized using chemical exfoliation [9] or chemical vapor deposition [10], the widely tunable bandgaps discussed here, by applying an interlayer bias, may provide possibilities for applications in electronic and optoelectronic devices based on h-BN bilayers. Acknowledgments
4. Summary
This work was supported by the State Key Program for Basic Research of China (Grants Nos 2009CB929504 and 2011CB922102), the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD), and the National Natural Science Foundation of China (Grants Nos 60876065 and 11074108).
We have systematically studied the influence of polarities changed by an interlayer bias on the band structure of a Bernal-stacked h-BN bilayer with the effective tight-binding approximation. It is found that the bandgaps of the h-BN bilayer vary asymmetrically under opposite biases. The system exhibits quantum phase transitions from a 6
J. Phys.: Condens. Matter 26 (2014) 015304
X Zhai and G Jin
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