Landau levels and magneto-transport property of monolayer phosphorene

We investigate theoretically the Landau levels (LLs) and magneto-transport properties of phosphorene under a perpendicular magnetic field within the framework of the effective k·p Hamiltonian and tight-binding (TB) model. At low field regime, we find that the LLs linearly depend both on the LL index n and magnetic field B, which is similar with that of conventional semiconductor two-dimensional electron gas. The Landau splittings of conduction and valence band are different and the wavefunctions corresponding to the LLs are strongly anisotropic due to the different anisotropic effective masses. An analytical expression for the LLs in low energy regime is obtained via solving the decoupled Hamiltonian, which agrees well with the numerical calculations. At high magnetic regime, a self-similar Hofstadter butterfly (HB) spectrum is obtained by using the TB model. The HB spectrum is consistent with the LL fan calculated from the effective k·p theory in a wide regime of magnetic fields. We find the LLs of phosphorene nanoribbon depend strongly on the ribbon orientation due to the anisotropic hopping parameters. The Hall and the longitudinal conductances (resistances) clearly reveal the structure of LLs.

The group V element phosphorus can exist in several allotropes and black phosphorus (BP) is the most stable phase under normal conditions 1 . Recently, layered BP has attracted intensive attention because of its unique electronic properties and potential applications in nanoelectronics [2][3][4][5][6][7][8] . In the bulk form, BP is a van der Waals-bonded layered material where each layer forms a puckered surface due to sp 3 hybridization 2,3 . BP possesses a direct band gap 0.3 eV located at Z point 3,4 . This direct gap increases to 1.5-2 eV when the thickness decreases from bulk to few layers and eventually monolayer via mechanical exfoliation 3,5,9 . Hence, BP is an appealing candidate for tunable photodetection from the visible to the infrared part of the spectrum 10 . Further, the field-effect-transistor (FET) based on few layer BP is found to have an on/off ratio of 10 5 and a carrier mobility at room temperature as high as 10 3 cm 2 /V·s 3,5 , which make BP a favorable material for next generation electronics.
The low energy physics of monolayer BP (phosphorene) around Γ point can be well described by an anisotropic two band k·p model 2 , which agrees well with a tight binding (TB) model 11 . To date, various interesting properties for phosphorene have been predicted theoretically and verified experimentally, including those related to strain induced gap modification 2 , tunable optical properties 12 , layer controlled anisotropic excitons 13 , quantum oscillations in few layers BP [14][15][16] etc. However, the Landau levels (LLs) and magneto-transport (MT) properties of this unique anisotropic system remain unexplored.
In this work, we study the LL spectra and MT properties of phosphorene under a perpendicular magnetic field. By using an effective k·p Hamiltonian, we find that the LLs linearly depend both on energy index n and magnetic field B at low-field regime, which means the LLs in phosphorene are similar with that in conventional semiconductor two dimensional gases (2DEGs). Interestingly, owing to the anisotropic energy dispersions, i.e., the effective masses, the Landau splittings of conduction and valence band are different for a fixed magnetic field, and the wavefunctions corresponding to the LLs show strong anisotropic behavior. We obtain an analytical expression for the LLs in low energy regime via solving a decoupled Hamiltonian, which agrees well with the numerical data in low energy regime. At high-field regime, magneto-level spectrum, i.e., the Hofstader butterfly (HB) spectrum, is obtained by using a tight binding (TB) model. We find that the results obtained by the effective k·p Hamiltonian and TB model agree with each other in weak magnetic field cases. Further, we find the LLs of phosphorene nanoribbon depend strongly on the ribbon orientation due to the anisotropic hopping parameters. In order to detect those interesting magneto energy spectra, we calculate MT properties of phosphorene within the framework of the linear response theory. By using Kubo formula, we find the Hall and the longitudinal conductances (resistances) clearly reveal the structure of LLs.

Results
Tight binding and Low energy k·p model. In the top view of phosphorene, as shown in Figure 1(a), a 1 = 3.32 Å and a 2 = 4.38 Å are the primitive vectors, a = 2.22 Å and θ = 96.79° are the in-plane bond length and bond angle 11 , respectively. The unit cell of phosphorene contains four atoms (see the solid rectangle) with two phosphorus atoms in the lower layer and the other two atoms in the upper layer. Very recently, a tight binding (TB) model of phosphorene has been proposed and is given by 11 where the summation runs over all the lattice sites of phosphorene, ( ) † c c j i is the creation (annihilation) operator of electron on the site j(i), and t ij are the hopping parameters. It has been shown that five hopping links (see Fig. 1(a)) are enough to describe the electronic band structure of phosphorene 11 . The related hopping parameters are: t 1 = − 1.22 eV, t 2 = 3.665 eV, t 3 = − 0.205 eV, t 4 = − 0.105 eV, and t 5 = − 0.055 eV.
Generally, the energy dispersion of phosphorene should be described by a four band model 11,17 in the TB framework. However, it can be also expressed by a two-band model due to the D 2h point group invariance 17 . In the two-band model, the unit cell contains two phosphorus atoms (see the dashed rectangle in Fig. 1(a)), where one is in the upper layer and the other in the lower layer. Expanding the TB model around Γ point with a coordinate rotation 17 (τ y → τ x , τ x → τ z ), one obtains the low energy k·p model for phosphorene, which reads where E c = 0.34 eV (E υ = − 1.18 eV) is the conduction (valence) band edge, γ = − 5.2305 eV·Å describes the interband coupling between the conduction and valence band, parameters α, β, λ , η are related to the effective masses with α = ħ 2 /2m cx , β = ħ 2 /2m cy , λ = ħ 2 /2m υx , η = ħ 2 /m υy . Here m cx = 0.793m e , m cy = 0.848m e , m υx = 1.363m e , m υy = 1.142m e , and m e is the free electron mass. The eigenvalue of this Hamiltonian is where + /− is for conduction/valence band, respectively. Similar with other low energy k·p models 2,12 , the dispersion described by Eq. (3) is strongly anisotropic. The energy gap E g is E c − E υ = 1.52 eV, which is consistent with the first principle calculations (an intrinsic energy gap around 2 to 2.2 eV minus the excition binding energy) 13 and the recently measured optical gap 5 1.45 eV. Figure 1 presents the dispersion of TB (the black solid line) and the k·p models (the red dashed line), from which we find they agree well with each other in a quite wide energy regime. It seems the energy dispersion is linear along Γ − X direction (see Fig. 1(c)). However, it is actually parabolic. In the long wave limit, we have . Hence, we can expand Eq. (3) around Γ point and obtain the energy dispersion of conduction and valence band, which reads From Eq. (4), one can easily find that the dispersion near Γ point is quadratic. Owing to the interband coupling, the effective masses around Γ point along k x direction are modified as  ( ) . However, the effective masses around Γ point along k y remain unchanged with m cy = 0.848m e , m υy = 1.142m e .
Landau levels in monolayer phosphorene. When a perpendicular magnetic filed B = (0, 0, B) is applied, taking the Landau gauge A = (− By, 0, 0), we define the creation and annihilation operators as is the magnetic length. One finds Hamiltonian (2) turns to cy cx 1 4 , r x = m cx /2m υx and r y = m cy /2m υy . Interestingly, the second (third) term in Eq. (6) looks like the Rasshba (Dressehaus) spin-orbit interaction in conventional semiconductor 2DEG 18 . In order to understand how the non-diagonal element h R and h D couple the Landau levels (LLs) in conduction and valence band, we firstly simplify the Hamiltonian by ignoring the third term in h υ (see Eq. (7)) since it is a second-order perturbation. It will be included in numerical calculation. In this approximation, we see that the term h R couples the LL φ c n with φ υ are wave functions of the harmonic oscillator corresponding to h c . When only the term h R exits, we obtain = ( Here Ω n R and ϑ n are defined from The h R induces the coupling of the LL's, which is schematically shown in Fig. 2(a). We see that both ϕ +  The h D induces the coupling of the LL's as schematically shown in Fig. 2(b). We see that both ϕ + . A particular eigenstate is the lowest LL in conduction band (φ c 0 ), which is independent of h D . Therefore, when both h R and h D exist, the LLs are coupled into the following two groups

Figure 2. Schematic illustration of the inter-LL coupling induced by
The two groups are schematically illustrated in Fig. 2(c,d). The eigenvalues and eigenvectors can be evaluated numerically by taking the eigenvectors of h c in Eq. (7) as basis functions (see methods for details).
From the Hamiltonian (2), the LLs can be solved analytically in low energy regime. Although the dispersion is dominated by the off-diagonal element, we can decouple the conduction and valence band in low energy regime due to the large optical gap (1.52 eV), i.e., the weak interband coupling. The role of the off-diagonal elements can be taken into account perturbatively. The decoupled Hamiltonian reads where α′ = α + γ 2 /E g , λ ′ = λ + γ 2 /E g . The dispersion of this Hamiltonian is presented by the blue dash-dotted lines in Fig. 1(d). We see that , and ω e = eB/m e . Our formulism for the low-energy LLs was used in the studies about the magneto-optical 19 and transport properties 14,15 . Figure 3 presents the LLs versus (a) LL index n with different magnetic fields and (b) magnetic field B. The number of basis function used in the calculation is 200 to get convergent numerical results. As shown in Fig. 3(b), we find the analytical LLs (the blue dashed lines) are in good agreement with the numerical results (the red solid lines), which means the decouple Hamiltonian (12) is a good approximation in low energy regime. However, the Landau splittings of conduction and valence band are different for a fixed magnetic field (see Eq. (13)) due to the different anisotropic effective masses at zero field. Further, the Landau energies linearly depend both on LL index n (see Fig. 2(a)) and magnetic field B (see Fig. 2(b)), which is similar with that of conventional semiconductor 2DEGs since the zero field dispersion is quadratic (see Eq. (4)). Meanwhile, we find the LLs are equally spaced which can also been seen clearly in Eq. (13).
On the other hand, we find the corresponding eignevectors are anisotropic due to different effective masses along Γ − X and Γ − Y direction. In Landau gauge A = (− By, 0, 0), the eigenvectors are  Figure 4 presents the contour plot of spatial density distributions (SDDs) corresponding to the first two LLs in conduction band. As plotted in Fig. 4, unlike the isotropic case, we find the SDDs for the first two LL are ellipses, which show strong anisotropy. The decay length of the SDDs along x direction is larger than that along y direction as the effective masses along Γ − X direction is smaller than that in Γ − Y direction (see Eq. (4)) in conduction band. The same conclusion can be drawn for SDDs corresponding to LLs in valence band.
Hofstadter Butterfly spectrum. Adopting the TB model, we plot the Hofstadter Butterfly (HB) spectrum as a function of Φ /Φ 0 (magnetic field B) with q = 199 in Fig. 5(a). As shown in Fig. 5(a), we find two gapped self-similar HB spectrum coming from the conduction and valence orbitals, respectively. Moreover, the LL energies linearly depend on magnetic field B at low field region, which is in line with the results obtained from the k·p model (see Eq. (13)). The band width of the HB spectrum in conduction and valence band is different because of the different band widths at zero field. Figure 5(b) depicts the magneto-levels i.e., the HB spectrum (the blue dots) and the LLs (the red solid lines) calculated from the k·p theory as a function of magnetic field at low field regime with q = 10007. As shown in the figure, we find they agree well with each other in wide regime of magnetic fields.

Landau levels in phophorene nanoribbons.
Owing to the anisotropic hopping parameters in phosphorene, one can expect the edge dependent LLs in phophorene nanoribbons (PNRs). Figure 6 shows the energy spectra of a zigzag-edged PNR (ZPNR) with and without an external magnetic field. When a strong magnetic field B = 30 T is applied perpendicular to the ZPNR, one can clearly see the LLs. While for an armchair-edged phosphorene nanoribbon (APNR) with the same with, the LLs show different energy spacing with that in the ZPNR for the higher LLs. Comparing the energy spectra of the ZPNR and APNR, an important difference between them is that there is a topological quasi-flat band located in the bulk gap of the ZPNR 17 . There are two kinds of edges states in ZPNRs. The one is the edge states arising from the LLs in ZPNR, the other come from the topological quasi-flat band. The degeneracy of the topological qusi-flat band lifts under the influence of the magnetic field. However, since the topological quasi-flat band are mainly localized near the edges, and the decay length (~1.2 nm) is less ψ ψ  than the magnetic length (l B = 25.6 nm/ B = 4.67 nm), the edge states arising from the topological quasi-flat band are almost independent of magnetic fields, i.e., no Landau quantization (see Fig. 6(b)). The LLs of PNRs depend strongly on the ribbon orientation due to the anisotropic band structure of bulk phosphorene (see Fig. 6(d)). This anisotropy of the LLs can be observed in the conductance (see Fig. 6(e)) as a function of Fermi energy (E f ) for ZPNR (red solid line) and APNR (blue dash-dotted line).
Magneto-transport properties of monolayer phosphorene. In order to detect the calculated magneto energy spectrum, we study the magneto-transport properties of phosphorene. In the presence of a perpendicular magnetic field, there are two contributions to magneto-conductance 20 : the Hall and collisional conductance. The former is from the non-diagonal contribution and the later from the localized states which contribute to the Shubnikov-de Haas (SdH) oscillation. In order to calculate the electrical conductance in the presence of a magnetic field, we follow the formulation of the general Liouville where j is the filling factor. This result is the same as that for a conventional semiconductor 2DEG 18,20 , since the zero field dispersion in low energy regime is quadratic (see Eq. (4)). To obtain the longitudinal conductance, we assume that electrons are elastically scattered by randomly distributed charged impurities since this type of scattering is dominant at low temperatures. The longitudinal conductance is where U 0 = e 2 /4πε 0 ε r , n i is the impurity concentration, k s is the screening wavevector, ε r is the dielectric constant, and ε 0 is the dielectric permittivity. Figure 7 shows (a) the Hall (σ xy ) and (b) the longitudinal conductances (σ xx ) as a function of Fermi energy (E f ) for two different magnetic fields B = 4 T and 8 T, respectively. Other parameters used are temperature T = 1 K, impurity concentration n i = 2 × 10 8 cm −2 , screen potential vector k s = 5 × 10 7 m −1 , Boltzman constant k B = 1.38 × 10 23 J/K, and dielectric constant ε r = 10.2 23 . As plotted in Fig. 7(a), we find that Hall conductance is strictly quantized due to the quantized LLs. It increases one by one in the unit of G 0 = 2e 2 /h with the increasing of Fermi energy since the LLs are filled one by one. Therefore, we observe the integer Hall plateaus at 0, ± 2, ± 4, ± 6, … in Hall conductance. This is similar with that in conventional semiconductor 2DEG 18 . Moreover, the Hall conductance reveals the LLs clearly since the transitions of the plateaus happen to be the energy value of LLs (see Fig. 3(a)). Further, for a fixed magnetic field, the width of the plateaus is equal since the LL spacings of two adjacent LLs are equal according to Eq. (13). As depicted in Fig. 7(b), the longitudinal conductance shows that (i) pronounced peaks appear when the Fermi energy coincides with the LLs, and (ii) a well splitting SdH oscillation can be observed, which corresponds to the LLs (see Fig. 3(a)). Meanwhile, the amplitude of longitudinal conductance increases with the increasing of the Fermi energy because of the larger scattering rate of LLs with higher index. Moreover, the intervals between the peaks are equal since the LLs are equally spaced according to Eq. (13).  Figure 8 presents the Fermi energy spectra and resistances as a function of magnetic field for a given electron concentration n e = 1.45 × 10 12 cm −2 . Generally, the Hall (ρ xy ) and the longitudinal resistances (ρ xx ) can be detected directly via Hall measurement 3,14 . As shown in Fig. 8(b), at low magnetic field, the Hall resistance linearly depend on the magnetic field and the longitudinal one is a constant. However, at the high magnetic field regime, the Hall resistance is strictly quantized with Hall plateaus due to Landau quantization. It increases (in unit of ρ 0 = h/e 2 ) one by one with increasing magnetic field since the LLs leak out of the Fermi level one by one (see Fig. 8(a)). This is also reflected in the transitions of filling factor (see Fig. 8(b)). Therefore, we observe plateaus at 1/8, 1/10, 1/12, 1/14, …, in Hall resistance corresponding to filling factor j = 4, 5, 6, 7, …, with the decreasing of magnetic field. Meanwhile, we find a clear SdH oscillation in longitudinal resistance. The amplitude of longitudinal resistance increases with the magnetic field since it is proportional to B 2 . However, this oscillation is quenched in low magnetic field due to tiny LL splittings in weak field cases.

Discussion
Our results about the LL spectrum in phosphorene can be also applied to multilayer BPs since the k·p Hamiltonians for the multilayer ones are similar with that for phosphorene. A very recent paper 24 demonstrates that the low energy LLs in bulk phosphorus also depend linearly both on the LL index n and magnetic field B. Meanwhile, this result has been verified in several recent magneto transport experiments [14][15][16] . Our results have been employed to illustrate the absence of non-trivial Berry's phase of LLs in multilayer BPs 14 .
In summary, we studied theoretically the Landau levels and magneto-transport properties of phosphorene under a perpendicular magnetic field within the framework of an effective k·p Hamiltonian and TB model. In the low field regime, we found that the LLs linearly depend both on the LL index n and magnetic field B, which is similar with that of conventional semiconductor two-dimensional electron gas. For a fixed magnetic field, the Landau splittings of conduction and valence band are different and the wavefunctions corresponding to the LLs show strong anisotropic behavior due to the anisotropic effective masses. We obtained an analytical expression for the LLs in low energy regime via solving a decoupled Hamiltonian. This analytical solution agrees well with the numerical results. At high magnetic regime, a self-similar Hofstadter butterfly (HB) spectrum was obtained by using the TB model. The HB spectrum is in good agreement with the LLs calculated from the effective k·p theory in a wide regime of magnetic fields.
Further, we found the LLs of phosphorene nanoribbons (PNRs) depend strongly on the ribbon orientation due to the anisotropic hopping parameters. There are two kinds of edge states in ZPNRs under a perpendicular magnetic field. The one is the edge states arising from the LLs, the other comes from the topological flat band. The second edge states are almost independent of magnetic fields because their decaying length is less than the magnetic length l B . Moreover, the Hall and the longitudinal conductances (resistances) clearly reveal the structure of LLs in phosphorene sheet. and U q ≈ U 0 /k s . As the collision is elastic and the eigenvalue is independent on k x , only the transitions n→ n are allowed. Then, one can obtain Eq. (19) directly by using Eq. (14).
Moreover, one can obtain the Hall resistance and the longitudinal one with the conductances (σ xy and σ xx ) via expressions of ρ xy = σ xy /S and ρ xx = σ xx /S, where σ σ σ σ σ = − ≈ = / S n e B xx yy xy yx xy e 2 2 2