Quantum Hall studies of a Semi-Dirac Nanoribbon

In the past few decades, graphene has attracted much attention due to its peculiar dispersion relation at low energies, similar to the spectrum of relativistic particles described by the Dirac theorywallace ; neto . More precisely, the spectrum has two cones, the so-called “Dirac cones” in the vicinity of two non-equivalent points $K_{1}$ and $K_{2}$ in the reciprocal space. Anisotropy in graphene was another interesting aspect which was discussed long ago by Paulingpauling , and could be induced by uniaxial stress or bending of a graphene sheet. The main motive is to tune the hopping energy between neighbouring carbon atoms with precision, which was later found to be feasible in optical lattices tarruell via controlling the lattice potential in order to have a handle on the effective mass in a honeycomb lattice. In a tight-binding model for graphene, if one of the three nearest-neighbor hopping energies is tuned, the two Dirac points with opposite chiralities approach each other and merge into one forming the so-called semi-Dirac point. The band dispersion simultaneously exhibits massless Dirac (linear) and massive fermionic (quadratic) features along two different directions, thereby producing a highly anisotropic electronic dispersiondietl ; pickett . The materials that host such anisotropic dispersion are phosphorene under pressure and doping rodin ; guan , electric fields rudenko ; dut , TiO2/VO2 superlattices pickett ; pardo1 ; pardo2 , graphene under deformation mon , and BEDT-TTF₂I₃ salt under pressure konno ; pichon . Experimentally, semi-Dirac dispersion has been observed in a few-layer black phosphorene by means of the in situ deposition of potassium atoms kim . A straightforward approach to realize semi-Dirac materials can be achieved by breaking the hexagonal symmetry of the honeycomb lattice, e.g, by strain. However, directly applying strain to realize the transition in materials, such as graphene or silicene is prohibited by the exorbitant magnitude of the strain required, which would eventually disintegrate them si ; zhao . Some successfully synthesized graphene-like honeycomb materials, such as silicene vogt ; meng , germanene li and stanene zhu , are found to be easily oxidized or they chemically absorb other atoms because of their buckling geometries molle ; spencer ; houselt ; ciraci . It is these absorbed atoms that will modify the hopping energies in the honeycomb lattice which is essentially applying a strain that creates a differential hopping.

On the other hand, behaviors of electrons in graphene, exposed to a strong perpendicular magnetic field played an important role not only for the discovery of quantum Hall effectsvp ; zheng , but also for proving the existence of massless Dirac particlesfirsov ; zhang . The unconventional Hall conductivity was found to be quantized as $\sigma_{xy}=2(2n+1)e^{2}/h$firsov ; zhang , where both the spin and the valley degeneracies are taken into account. Experimental measurements confirm that the Landau levels of a monolayer graphene obey the relation, $E_{n}=sgn(n)\sqrt{2\hbar v_{F}e|n|B}$, where $v_{F}$=$10^{6}$ m/s is the Fermi velocity, $B$ is the magnetic field and $n$ denote Landau level indicessadowski1 ; sadowski2 .

Recently, quite a few studies on Landau levels and transport properties in presence of a magnetic field in phosphorene have been reported chang ; roldan . More precisely, they have found that the anisotropic band structure that leads to Hall quantization in presence of a perpendicular magnetic field is similar to that of a conventional two-dimensional electron gas (2DEG). Since phosphorene may be considered as a realistic material that possesses semi-Dirac properties, it is necessary to pursue quantum Hall studies on the semi-Dirac systems. As discussed above, the energy dispersion of phosphorene is similar to that of the semi-Dirac systems, it is likely that other properties too show similar characteristics.

In this work, we have explored the influence of magnetic field for a semi-Dirac system using a tight-binding Hamiltonian on a honeycomb lattice. We study the Landau level spectrum and Hofstadter butterfly using a nanoribbon in order to show that the semi-Dirac system has quite distinct properties as compared to Dirac fermions. We also calculate the density of states (DOS) via the tight-binding propagation methodyuan ; yuan2 , which is a sophisticated numerical tool used in large-scale calculations for any realistic system. We have implemented the recently developed real-space order-$N$ quantum transport approach to calculate the Kubo conductivities as a function of the Fermi energy for moderate as well as very high values of the magnetic fieldrappoprt . The Hall conductivity in a semi-Dirac system shows the standard quantization, namely, $\sigma_{xy}\propto 2n$ as compared to the previously observed anomalous quantization, that is, $\sigma_{xy}\propto 4(n+1/2)$ for a Dirac system. The longitudinal conductivities show highly anisotropic behavior in one direction compared to the other, which is obviously absent for Dirac systems.

The paper is organized as follows. The low energy tight-binding Hamiltonian is described in sec. II. We have further studied the Landau level spectra and the Hofstadter butterfly for a nanoribbon in presence of a magnetic field in sec. III. The transport properties are investigated by computing the Hall and the longitudinal conductivities in sec. IV. We conclude with a brief summary in sec. V.

We study the tight-binding model on the honeycomb lattice with anisotropic hopping that leads to semi-Dirac electronic spectra at low energy. More precisely, the hopping energy to one of the neighbours ($t_{2}$) is different than the other two ($t$) as shown in Fig. 1(a). It is also instructive to look at the full dispersion with the following three nearest neighbor vectors in real space, $\vec{\delta_{1}}=\big{(}0,a\big{)}$; $\vec{\delta_{2}}=\Big{(}\frac{\sqrt{3}a}{2},-\frac{a}{2}\Big{)}$ and $\vec{\delta_{3}}=\Big{(}-\frac{\sqrt{3}a}{2},-\frac{a}{2}\Big{)}$, where $a$ is the lattice constant. The dispersion relation for a semi-Dirac system can be written as,

The above expression in Eq. (1) is plotted in Fig. 2(a). The Brillouin zone with high-symmetry points for $t_{2}=t$ is shown in Fig. 2(b). For the Dirac case (that is, $t_{2}=t$), the dispersion shows that the Dirac points touch at the $K_{1}$ and $K_{2}$ points at the Brillouin zone corners as shown in Fig. 2(c). With increasing the strength of the parameter $t_{2}$, the two Dirac points originally located at $K_{1}$ ($\frac{2\pi}{3a}$,$\frac{2\pi}{\sqrt{3}a}$) and $K_{2}$ ($-\frac{2\pi}{3a}$,$\frac{2\pi}{\sqrt{3}a}$) move closer till they merge at the $M$ point resulting in a semi-Dirac spectrum (see Fig. 2(d)). As mentioned earlier, such manipulation of the Dirac points and their eventual merger have been achieved in honeycomb optical latticestarruell . Thus by fixing $t_{2}=2t$ and focusing on the $M$ point (0,$\frac{2\pi}{\sqrt{3}a}$), the low-energy effective Hamiltonian based on the tight-binding model for a semi-Dirac system, apart from a constant term, can be written askush ; chen ; firoz ,

where $p_{x}$ and $p_{y}$ are the momenta along the $x$ and the $y$ directions respectively. $\sigma_{x}$ and $\sigma_{y}$ are the Pauli spin matrices in the pseudospin space. The velocity along the $p_{y}$ direction, $v_{F}$, and the effective mass, $m^{*}$ corresponding to the parabolic dispersion along $p_{x}$ are expressed as $v_{F}={3ta}/{\hbar}$ and $m^{*}=2\hbar/3ta^{2}$. Henceforth we set $a=1$. The dispersion relation corresponding to Eq. (2) ignoring a constant shift in energy can be written as,

where ‘+’ denotes for the conduction band and ‘-’ stands for the valence band. Equation (3) shows that the dispersion is linear (Dirac-like) along $y$-direction, whereas the dispersion along the $x$-direction is quadratic (non-relativistic), the combination of which results in the semi-Dirac dispersion. The three-dimensional plot in Fig. 2(a) indicates the anisotropic band structure in a semi-Dirac system.

To include a magnetic field, we shall work with a semi-Dirac nanoribbon which is infinitely long along $x$, but has a finite width along $y$. We apply a uniform magnetic field, $\mathbf{B}=B\hat{z}$ perpendicular to the plane of the ribbon. Owing to the presence of the vector potential $\vec{A}$, each tight-binding wave-function picks up an extra phase term. We have chosen the Landau gauge as $\vec{A}=(-By,0,0)$ such that the translational invariance along the $x$-direction remains unaltered under the choice of the gauge. Hence, the momentum along the $x$-direction is conserved and acts as a good quantum number. To make $k_{x}$ a dimensionless quantity, we have absorbed the lattice spacing $a$ into the definition of $k_{x}$. The ribbon width is such that it has $N$ unit cells along the $y$-axis (where the index $n$ for the unit cells takes $\in 0$…..$N-1$) as shown in Fig. 3(a). The tight-binding Hamiltonian in the presence of magnetic field has the form,

where $a^{\dagger}_{i}$ ($b_{j}$) creates (annihilates) an electron on sublattice $A$ ($B$). $t_{ij}$ is the hopping amplitude between nearest neighbor sites, which obtain a phase due to the magnetic field by the Peierls substitution, namely, $t_{ij}=t\rightarrow te^{2i\pi\phi_{ij}}$ (here $t$ denotes both $t$ or $t_{2}$). $\phi_{ij}$ is the magnetic flux and is given by the line integral of the vector potential from a site $i$ to a site $j$, namely, $\phi_{ij}=e/h\int_{i}^{j}\vec{A}.d\vec{l}$. The flux is usually denoted in terms of the flux quantum $\phi_{0}=h/e$ ($h$ is Planck^′s constant and $e$ is the magnitude of the electron charge).

Thus the tight-binding Hamiltonian in presence of the perpendicular magnetic field can be written in terms of $m$ and $n$ (where $m$ increases along the $x$-direction and $n$ increases along the negative $y$-direction) (see Fig. 3(a))castro ,

where the summation $\langle{mn}\rangle$ is over the nearest neighbors. $a^{\dagger}(m,n)$ and $b(m,n)$ denote the creation and annihilation operators at the ($m$,$n$) site, respectively. Equation (5) for a zigzag semi-Dirac ribbon ($\alpha=1$) reduces to

Using the above Hamiltonian as mentioned in Eq. (6) we have numerically calculated the Hofstadter butterflyrammal as well as the Landau level spectrum for the number of unit cells $N=100$. Figure 4(a) shows fractal spectra plotted as a function of magnetic flux, $\phi/\phi_{0}$ for a semi-Dirac nanoribbon. It can be seen clearly that there occurs opening of a central gap with a flat band at zero energy. The gap gets larger along with the two identical spectra that emerge from the conduction and the valence bands by tuning $t_{2}$. For comparison, the same is plotted for the Dirac system ($t_{2}=t$) as shown in Fig. 4(b). We can see that there is no gap at zero energy with the flat band when one goes from $t_{2}=2t$ to $t_{2}=t$.

Figure 5 shows the Landau level spectra for different values of the magnetic flux (such as $\phi/\phi_{0}=1/100,1/200,1/500,1/1600$) for the semi-Dirac and the Dirac systems. Figures 5(a)-5(d) show the evolution of the energy levels in presence of the magnetic field for a semi-Dirac nanoribbon ($t_{2}=2t$). For comparison, we have also plotted the Landau level spectrum for the Dirac case ($t_{2}=t$) using the same values of the magnetic flux as shown in Fig. 5(e)-5(h). It is to be noted that in a semi-Dirac system, there is no zero energy bulk state, which implies that the zero energy state in Fig. 5 is an edge state. On the other hand, zero energy bulk states exist in a Dirac system. Further, for $t_{2}=2t$, the Landau levels are not equidistant, since their energies vary as $(n+\frac{1}{2})^{2/3}$ ($n$ being the Landau level index)pickett which lies intermediate to the behavior of the Dirac system and the conventional 2DEG. As a consequence, the gap between the Landau levels shrinks as one considers larger $n$. In the case of a Dirac system, since the energies of the Landau levels go as $\sqrt{n}$, its non-equidistant Landau spectra can have a different quantitative behavior from a semi-Dirac system. For a large value of the flux, $\phi$ such as $\phi=\phi_{0}/100$, the flatness of the energy bands are observed for both the semi-Dirac and the Dirac systems owing to shrinking of the cyclotron radius (see Fig. 5(a) and 5(e)). The energy bands become parabolic for $\phi=\phi_{0}/200$ as seen from Fig. 5(b). The flatness of the Landau spectrum in the bulk completely vanishes in the semi-Dirac system as compared to a Dirac one (see Fig. 5(b) and 5(f)). With lower values of $\phi/\phi_{0}$, the Landau levels demonstrate a dispersive behavior and start getting flatter for large values of $n$ for $t_{2}=2t$ (see Fig. 5(c)). In the case of a Dirac system ($t_{2}=t$), the Landau levels show quite a distinct behavior, where the flat bands become dispersive in the bulk corresponding to larger values of $n$ and lower values of $\phi/\phi_{0}$ (see Fig. 5(g)). For a small value of the magnetic field, such that the flux is given by, $\phi=\phi_{0}/1600$, the energy bands eventually become flat in the bulk for the semi-Dirac case as shown in Fig. 5(d). This is not the case for a Dirac system (see Fig. 5(h)). For all values of $\phi/\phi_{0}$, the zero-energy mode is completely separated from the bulk bands for a semi-Dirac system as compared to the Dirac case (see any of Fig. 5).

To study the transport properties in the presence of a perpendicular magnetic field, we consider a large sample of a lattice model of the semi-Dirac system consisting of millions of atoms. The contribution in transport comes from both the off-diagonal and the diagonal terms as appear in the Kubo formulakubo1 . The former contributes to the Hall conductivity ($\sigma_{xy}$), whereas the latter leads to individual longitudinal conductivity in different directions ($\sigma_{xx}$ and $\sigma_{yy}$).

In this work, we have studied the influence of a perpendicular magnetic field with a semi-Dirac dispersion within the framework of a tight-binding model of a honeycomb lattice. In order to compare and contrast with a prototype Dirac material, such as, graphene, we have presented our results for both cases. We consider a semi-Dirac nanoribbon with a finite width and study the Hofstadter butterfly and properties of the Landau level spectra. We have observed two identical gapped spectra symmetrically placed above and a below $E=0$ and along with a zero-energy mode in the Hofstadter butterfly spectrum in contrast to what is observed for a Dirac system. The Landau level becomes fully dispersive in the bulk for moderate values of the magnetic flux which is not true for the Dirac case. Furthermore, we explore the magneto-transport properties using the Kubo formula. We observed that the Hall conductivity shows standard quantization similar to that of a conventional semiconductor 2DEG with a parabolic band, which is highly contrasted with respect to a Dirac system. The zero Landau level peak is absent in the case of a semi-Dirac system. The longitudinal conductivities, $\sigma_{xx}$ and $\sigma_{yy}$ show anisotropic behaviour due to the distinct dispersion in two longitudinal directions.

PS acknowledges Sim$\tilde{a}$o M. Jo$\tilde{a}$o, SK Firoz Islam, Abbout Adel for useful discussions and Rafiqul Rahaman for helping in computation. SB thanks SERB, India for financial support under Grant No. EMR/2015/001039. SM is supported by JSPS KAKENHI under Grant No. JP18H03678.