Electronic structures and Aharonov–Bohm effect in higher-order topological Dirac Semimetal nanoribbons with strong confinements

Electronic structures and magnetotransport properties of topological Dirac semimetal (TDSM) nanoribbons are studied by adopting the tight-binding lattice model and the Landauer–Büttiker formula based on the non-equilibrium Green’s function. For concreteness, the TDSM material Cd3As2 grown along the experimentally accessible [110] crystallographic direction is taken as an example. We found that the electronic structures of the TDSM nanoribbon depend on both the strength and direction of the magnetic field (MF). The transversal local charge density (LCD) distribution of the electronic states in the TDSM nanoribbon is moved gradually from the center toward the hinge of each surface as a [010] direction MF strength is increased, forming the two-sided hinge states. However, one-sided surface states are generated in the TDSM nanoribbon when a [001] direction MF is applied. As a result, one-sided hinge states can be achieved once a tilted MF is placed to the TDSM nanoribbon. The underlying physical mechanism of the desired one-sided hinge states is attributed to both the orbital and Zeeman effects of the MF, which is given by analytical analyses. In addition, typical Aharonov–Bohm interference patterns are observed in the charge conductance of the two-terminal TDSM nanoribbon with a tilted MF. This conductance behaviour originates from the unique interfering loop shaped by the one-sided hinge states. These findings may not only further our understanding on the external-field-induced higher-order (HO) topological phases but also provide an alternative method to probe the HO boundary states.


Introduction
In the past decades, topological insulators (TIs) have garnered enormous attentions since the pioneering predications of the z 2 topological phase in two-dimensional (2D) materials including graphene [1] and HgTe/CdTe quantum well [2]. This kind of 2D materials is featured by the 1D helical spin-momentumlocked edge states, which are protected by the nontrivial topology of the bulk band and will not vanish unless the bulk gap is closed. Up to now, HgTe/CdTe [3] and InAs/GaSb quantum wells [4,5] are two typical 2D TIs that have been verified by transport experiments. Later, TIs were extended to 3D material such as Bi 2 Se 3 [6,7], which possesses 2D spin-momentum-locked surface states. Besides insulators, the topological classification are also generalized to metals. Differing from those of the TIs, topological metals are characterized by the crystal-symmetry-protected Fermi arc states as there is no band gap for metals. Na 3 Bi [8,9] and Cd 3 As 2 [10,11] are two representative 3D topological Dirac semimetals (TDSMs) with 2D Fermi arc surface states, which have been predicted theoretically and confirmed experimentally.
Very recently, topological materials are developed up to a higher level, which began with the proposal of quantized electric multipole insulators [12]. In contrast to the aforementioned TIs where d-dimensional system has (d − 1)-dimensional boundary states, the quantized electric multipole insulators harbor (d − n)-dimensional boundary states (n ⩾ 2). In general, the insulators with first-order lower boundary states are dubbed as first-order TIs while those with high-order lower boundary states as higher-order (HO) TIs. Related subjects about the HOTIs were investigated extensively and various schemes were put forward to realize the HO topological insulating phases. For example, hinge or corner states are predicted to emerge in topological systems with particular symmetries such as time-reversal and fourfold rotation symmetries [13,14], spacetime-inversion symmetry [15,16], and chiral symmetry [17]. In addition, HO topological insulating phases can also be generated and modulated by external means including the Zeeman field [18,19], the stacking of antiferromagnetic TI multilayers [20], and even structural [21,22] or impurity/defect [23,24] disorders. Similar to the developing routine of the first-order topological materials, HO topology classification is also broadened from insulators to metals. The first proposed HO topological metal is the topological quadrupolar semimetal, which is a 3D extension of the 2D quadrupole TI, giving rise to 1D hinge states [25]. Subsequently, 3D HO topological Dirac [26,27], Weyl [28,29], and nodal-Line [30] semimetals were raised consecutively, which own 1D Fermi arc hinge states. Moreover, HO topological phase can also be transited from the first-order topological phase in semimetals by an magnetic field (MF) [31,32], the periodic driving [33], and disorder [34]. When a perpendicular MF along the y-axis direction is applied to the first-order TDSM Cd 3 As 2 , Landau bands of the Fermi arc surface states and band gap are induced by the orbital and Zeeman effects of the MF, respectively, leading to the hinge states on both sides of the top and bottom surfaces [32]. More interestingly, as the MF is tilted towards the z-axis direction, one-sided hinge states are achieved due to the broken twofold degeneracy of the energy spectra, which may be the physical origin of the 3D quantum Hall effect observed in Cd 3 As 2 [35].
Motivated by the previous works of the MF-driving topological phase transition in TDSMs [31,32] and the special magnetotransport phenomena in 3D HOTIs [56,63], in this paper we investigated the electronic structures and magnetotransport properties of a strongly confined TDSM nanoribbon, since the quantum confinement effects are inevitable in the practical applications. The TDSM material Cd 3 As 2 grown along the experimentally accessible [110] crystallographic direction is selected as an example for the concrete calculations. It is found that the local charge density (LCD) distribution of Fermi arc surface states in the TDSM nanoribbon is sensitive to both the strength and direction of the MF. For the TDSM nanoribbon with a MF along the [010] direction, the LCD distributions of the Fermi arc states on both the top and bottom surfaces are moved from their centers toward hinges, leading to the hinge states on both sides. However, for the TDSM nanoribbon with a MF along the [001] direction, one-sided Fermi arc surface state is generated and its LCD distribution keeps almost unchanged as the MF strength is varied. Therefore, one-sided hinge states can be achieved in the TDSM nanoribbon with a tilt MF, which is the combination effects of both the [010]-and [001]-component MF. The underlying physical mechanism of the MF-tunable topological hinge states is analyzed by analytical calculations. In addition, the two-terminal magnetoconductance of the tilt-MF-modulated TDSM shows typical AB interference patterns, which are attributed to the interfering loops formed by the one-sided hinge states. These results may be helpful for the further understanding and detecting the external-field-driving HO boundary states in topological materials.
The rest of this paper is arranged as follows. In section 2, theoretical model of the considered system and calculation methods are presented. In section 3, numerical results and discussions are demonstrated. Finally, section 4 concludes the paper.

Model and methods
The results obtained in this paper are not limited to one particular topological metal material. However, for concreteness, 3D TDSM Cd 3 As 2 grew along the [110] direction is selected as an example, which has been synthesized successfully in experiments [64]. Its low-energy physics can be described by the four-band model around the Γ-point of the Brillouin zone, namely |s + , − 3 2 ⟩, in which s and p denote the electron and hole subbands, respectively [10]. For the MF B = (0, B y , B z = B y tanθ) shown in figure 9(a), which is always perpendicular to the movement direction of electrons, the Landau gauge is chosen so that its vector potential is given as A = (B y z − B z y, 0, 0). Considering both the orbital effect and Zeeman effect of the MF, the 4 × 4 Hamiltonian of the 3D TDSM Cd 3 As 2 grew along the [110] direction with a tilt MF takes the form [32] with , σ 0 and σ i (i = x, y, or z) are the 2 × 2 unit and Pauli matrices, respectively. ∆ y = −ie −iα B y µ B G * /4 and ∆ z = B z µ B G * /4 are the y-and z-component Zeeman energies of the MF, respectively. Here G * = g s (σ 0 + σ z ) + g p (σ 0 − σ z ), µ B is the Bohr magneton, g s and g p are the effective g factors. h ↓ k = (h ↑ −k ) * , in which ↑ / ↓ denotes the spin up/down state along the z-axis direction for one of the Weyl semimetals.
In the following numerical calculations, we model the Hamiltonian in equation (1) by discretizing it onto a 3D simple cubic lattice with the lattice constant a according to the standard finite-difference method and the nearest tight-binding approximation [65], where The nearest-neighbor hopping matrix elements T x,y,z read where ϕ i,i+1 =´i +1 i A · dx/ϕ 0 with the flux quanta ϕ 0 =h/e. Hereh and −e are the reduced planck constant and the electron charge, respectively.
As the 3D TDSM is confined into an infinitely long nanoribbon along the x-axis direction, the tight-binding Hamiltonian in equation (2) can be divided into two parts in the plane cell where H i is the Hamiltonian of the ith isolated plane cell, H i,i+1 and H i+1,i are intercell Hamiltonian between the ith plane cell and the (i + 1)th plane cell with H i+1,i = H † i,i+1 . According to the time-independent Shrödinger equation based on the Hamiltonian in equation (7), the energy band, transversal wave function of the TDSM nanoribbons can be obtained under the open boundary conditions. Therefore, the transversal LCD and the z-component local spin-polarized density (LSPD) distributions are defined accordingly as and respectively. Here m/n is the index of the subband along y/z axis and o denotes the orbital degree (s or p), ϕ ↑/↓ (y, z) is the spin-resolved transversal wave-function of the TDSM nanoribbon. The transport properties of the two-terminal TDSM nanoribbon are studied by using the Landauer-üttiker formalism combined with non-equilibrium Green's function method. The charge conductance between the two leads is given by [65] where G r/a is the retarded/advanced Green's function of the whole two-terminal TDSM nanoribbon, Γ L/R is the linewidth function describing the coupling between the lead and TDSM nanoribbon, and the trace takes over both the spatial and spin degrees of freedom. The linewidth functions can be obtained from L/R is the retarded/advanced self-energy for the left or right lead. In the modeling, we assume that these two leads are semi-infinite and have the same widths as the TDSM nanoribbon. The self-energies of the leads Σ r/a L/R as well as the Green's function G r/a can be computed by using a recursive method based on the Hamiltonian in equation (7) [66].

Electronic structures of the TDSM nanoribbon with different MFs
Firstly, we consider an infinitely long TDSM nanoribbon with a MF along the y-axis direction. Figure 1 shows its energy bands as a function of k x at different MF strengths. The color bar denotes the total charge density around the four corners, which is defined as ρ corners = ∑ y,z ρ L (y, z) with y/z ∈ [0, 4] and [11,15]. For the TDSM nanoribbon without the MF, namely B y = 0.0, energy subbands are lifted due to the strong quantum confinement effect, as shown by the thick lines in figure 1(a). Moreover, a band gap E g ≈ 10.13 meV is opened at k x = 0.0 and parabolic dispersion instead of linear dispersion is found near the bottom of each subband, which implies massive electron will be generated when the Fermi energy is located at these regions. The effect of the quantum confinement on the energy band of the TDSM nanoribbon can be clearly seen by comparing it with that of the 3D TDSM at In order to show the evolution of the energy subbands of the TDSM nanoribbon with increasing MF strength more clearly, the energy subbands at k x = 0.0 as a function of the MF strength is demonstrated in figure 2(a). As the MF along the y-axis direction is applied to the TDSM nanoribbon, each energy subband is split into two subbands and the energy gap between them increases monotonously when the MF strength is increased first. This effect is attributed to the Zeeman coupling terms ∆ y and ∆ † y in equation (1), which couples the two Weyl semimetals h ↑ k and h ↓ k and open the gap, as indicated by the Landau band of the 3D  TDSM at k y = 0.0 as a function of the MF strength shown in figure 2(b). However, as the MF strength is increased further, energy subband crossings are found in the energy subbands of the TDSM nanoribbon. Therefore, the quantum-confinement-induced band gap of the TDSM nanoribbon decreases with the increase of the MF strength first (see the shaded area in figure 2(a)). The band gap will be closed when the MF strength is increased to the critical magnitude B C y = 20.9 T. After that, the band gap will be reopened and increased as the MF strength is increased further, as shown by the thick blue line in figure 2(c). However, the MF dependent band gap of the 3D TDSM is quite different from that of TDSM nanoribbon (see the shaded area in figure 2(b)). The energy difference, i.e. the band gap, between the lowest split subbands at k y = 0.0 increases monotonously as the MF strength is increased, as shown by the thin olive line in figure 2(c).
According to figure 1, the color of each energy subband varies as the MF strength is increased, which means the electronic state in the TDSM nanoribbon is sensitive to the MF strength. In order to show the evolvement of electronic state in the TDSM nanoribbon with the increasing MF strength more intuitively, the transversal LCD and LSPD distributions of the TDSM nanoribbon are displayed in figure 3. For simplicity and clarity, the electron energy is set at E = −4.5 meV so that only the highest valence subband is occupied. Due to the time-reversal symmetry, the LCD distributions corresponding to the two degenerate states of point A in figure 1(a) are mainly localized to the centers of the top and bottom surfaces of TDSM nanoribbon, respectively, as shown in figure 3(a). In addition, they are completely spin polarized with oppositive signs, as shown in figure 3(e), indicating the helical spin-momentum-locked surface states in the TDSM nanoribbon. Therefore, the transversal LCD and LSPD distributions of the TDSM nanoribbon without a MF are symmetrical with respect to both the y-axis and z-axis directions. As the MF is applied to the TDSM nanoribbon, the time-reversal symmetry is broken and the off-diagonal terms, i.e. the Zeeman coupling terms ∆ y and ∆ † y , in equation (1) are nonzero so that the state with respect to the point B/C/D in figure 1(b)/(c)/(d) is the superposition state of the top and bottom Fermi arc states. Therefore, the LCD distribution for each point is also localized at both the top and bottom surfaces. However, it is moved from the centers to the right hinges with the increasing MF strength, as shown in figures 3(b)-(d), leading to the formation of the so-called hinge state. As a result, the transversal LCD distribution of the TDSM nanoribbon along the z-axis direction is asymmetrical while that along the y-axis direction is still symmetrical. The movement of the LCD distribution coincides with the color evolution of the points A-D in figure 1. Furthermore, the maximum magnitude of the LCD increases while that of the z-component LSPD decreases as the MF strength is increased, as shown in figures 3(f)-(h), which is attributed to the enhancement of the coupling between the top and bottom Fermi arc surface states (namely h ↑ k and h ↓ k ) by the increasing Zeeman coupling terms in equation (1).
Secondly, we consider an infinitely long TDSM nanoribbon with a MF along the z-axis direction. Figures 4(a)-(d) show its energy bands as a function of k x at different MF strengths B z = 0.0, 3.64, 7.28, and 10.92 T, respectively. Similar to those of the TDSM nanoribbon with a perpendicular MF along the y-axis direction, each energy subband in figure 4(a) is also split into two subbands and the energy gap between them increases as the MF strength is increased, as shown in figures 4(b)-(d). However, the mechanism of the split subbands is different from that of the TDSM nanoribbon with a y-axis MF, which originates from the Zeeman splitting effect, i.e. the term ∆ z in equation (1). Therefore, each split subband is spin non-degenerate. Moreover, each energy subband is just shifted wholly up or down but its dispersion almost keep unchange as the MF strength is increased.
For the sake of understanding the evolution of the energy subbands of the TDSM nanoribbon with increasing z-axis MF strength, the energy subbands at k x = 0.0 as a function of the MF strength is shown in figure 5(a). Each spin-degenerated subband is lifted as the z-axis MF is added to the TDSM nanoribbon. Further, the energy gap between the split subbands increases monotonously as the MF strength is increased, leading to the subband crossings, which is similar to the case of y-axis MF shown in figure 2(a). Consequently, the quantum-confinement-induced band gap of the TDSM nanoribbon decreases monotonously with increasing MF strength first (see the shaded area in figure 5(a)) and it can be closed completely at the critical MF strength B C z = 72.5 T. After that, the band gap will reopened and increased with the further increase of the MF strength, as shown in figure 5(c). Compared with the energy subbands of the TDSM nanoribbon, the Landau band of the 3D TDSM as a function of the z-axis MF at k z = 0.0 shows quite different behaviours, as shown in figure 5(b). It is very sensitive to the variety of MF strength. Furthermore, the band crossings in the Landau bands cannot be removed because k z is a good quantum number in this  case. Therefore, the whole band gap can not be opened in the Landau band of the 3D TDSM as a function of k z , as shown by the inset in figure 5(c).
Differing from that of the TDSM nanoribbon with a perpendicular MF along the y-axis direction, the color of each energy subband shown in figure 4 almost keeps unchange as the strength of the MF is increased. To understand this effect more clearly, the transversal LCD and LSPD distributions of the TDSM nanoribbon with a perpendicular MF along the z-axis direction are presented in figure 6. The electron energy is also fixed at E = −4.5 meV, as indicated by the points A-D in figure 4 and only the electronic state of the highest valence subband is demonstrated. Once the MF is added to the TDSM nanoribbon, the LCD distribution is only localized at the center of the top surface, as shown in figure 6(b), which indicates that one-sided surface state is achieved. Further, the maximum magnitude of the LCD is increased and its position is moved toward the top surface (namely y-axis direction) with the increasing MF strength, as shown in figures 6(c) and (d). Consequently, the transversal LCD of the TDSM nanoribbon along the y-axis direction is asymmetrical while that along the z-axis direction preserves symmetry. On the contrary, the magnitude of the LSPD everywhere almost maintain a constant, as shown in the figures 6(f)-(h), which originates from the difference between the diagonal terms in equation (1) due to the Zeeman splitting effect.
Finally, we consider an infinitely long TDSM nanowire with a MF along the arbitrary direction inside the plane yoz. We call the MF in this case as the tilt MF for the sake of discrimination. The tilted angle relative to the y-axis is fixed at θ = 20 • , as shown in figure 9(a). For comparison, the y-component MF strength in each  subfigure is set as the same as that in figure 1, which also guarantees that the z-component MF strength is consistent with that in figure 4, respectively. Similar to that in figure 1, the whole energy band gap is decreased with the increasing MF strength first and then it will be increased as the MF strength is increased further, as shown in figures 7(b)-(d), which is attributed to both the Zeeman coupling and splitting effects of the MF. Moreover, the color of each subband is also varied gradually, indicating the change of the charge density near the hinges of the TDSM nanoribbons. Therefore, there is few differences between the energy bands in figures 1 and 7. However, the transversal LCD and LSPD distributions of the TDSM nanoribbon are quite different for these two cases, as shown in figure 8. As the tilt MF is applied to the TDSM nanoribbon, the transversal LCD and LSPD distributions are only localized near the upper surface. In addition, the position of the maximum magnitudes of the LCD and LSPD is moved towards the right corner with the In order to understand the underlying mechanism of the emergence of the hinge states in the TDSM nanoribbon, an analytical deduction is given as follows. Here we take the 3D TDSM with a MF along the y-axis direction as an example. Its electron Hamiltonian around the Dirac point can be expressed approximately as where ± denotes the chirality of the electron and v F is the Fermi velocity. Therefore, the Schrödinger equation based on the above Hamiltonian is in which ϕ ↑/↓ (z) is the spin-resolved wave-function along the z-axis direction. Here we only consider the chirality +1 and ignore the y-component wave-function for simplicity. Set l B = √h /(eB y ), ξ = l B k x + z/l B , and ε = El B /(hv F ), the above Schrödinger equation can be decoupled into two equations: Solving these two equations, we obtain the eigenvalues ε = √ 2n + 1 (n = 0, 1, 2, · · · ) and the corresponding eigenfunctions ϕ ↑ (ξ) = c n H n (ξ)e −ξ 2 /2 , where c n = 1/(n!2 n √ π) 1 2 and H n is Hermite polynomials. According to the eigenfunctions, the center of the wave-function along the z-axis direction is shifted a quantity −hk x /(eB y ), which depends on both the x-component electron momentum k x and the MF strength B y . This conclusion is in accord with the numerical results shown in figures 3 and 8.

Magnetotransport properties of the TDSM nanoribbon with a tilt MF
Next, we consider the magnetotransport in the TDSM nanoribbon. Here the TDSM nanoribbon with a tilt MF is chosen as an example. The setup of the two-terminal TDSM system is shown in figure 9(a), where the TDSM nanoribbon with a tilt MF is sandwiched two semi-infinite leads. The TDSM nanoribbon length is taken as L x = 60a except that in figure 9(d). The tilt angle is the same as that in figure 7. In general, the leads can be any metals since the essential physics discussed below is independent of the particular leads. However, in order to minimize the contact resistance, we model the leads by using the same material Cd 3 As 2 too.
The charge conductance of the TDSM nanoribbon as a function of the electron energy is displayed in figure 9(b). The y-component strength of the tilt MF is taken as B y = 5.0 T. The charge conductance is determined by both the energy bands of the TDSM nanoribbon in the middle region and the two lead. Therefore, a conductance gap (G = 0.0) is found in the charge conductance spectrum. It is worth noting that the energy window corresponding to the conductance gap are the same as the band gap of the lead at k x = 0.0, as shown in figure 1(a), but not that of the TDSM nanoribbon in the middle region shown in the inset of figure 9(b). The underlying mechanism of this effect is that the band gap of the TDSM nanoribbon at k x = 0.0 is located within that of the lead. In addition, conductance steps emerge in the charge conductance spectrum due to the occupation of the energy subbands one by one as the electron energy is varied, as shown by the energy band in the inset. Interestingly, obviously aperiodic oscillating behaviors especially near the threshold of each subband can be found in the conductance spectrum. We take three resonance points with negative energies R 1 -R 3 , as indicated in figure 9(b), to give detail analyses. The electron energies for these three points are E = −4.8, −6.0, and − 7.8 meV, corresponding to the wave vectors k x = 0.0519, 0.10336, and 0.1594 nm −1 , respectively, as shown by the points R 1 -R 3 in the inset. In addition, another set of resonance points with the positive energies is also observed in the charge conductance spectra, as shown by R 4 -R 6 in figure 9(b). The electron energies and wave vectors corresponding to these three points are (11.2 meV, −0.053 nm −1 ), (13.8 meV, −0.104 nm −1 ), and (18.0 meV, −0.155 nm −1 ), respectively. Surprisingly, all these two sets of wave vectors satisfy well to the resonance condition Therefore, the conductance oscillations are typical AB interference patterns [69,70]. The origin of the AB effect is sketched in the inset of figure 9(d), where the forward hinge state is reflected at the right interface and return from the opponent hinge and reflected at the left interface again, forming an effective interfering loop. Further, owing to the small slope near the threshold of each subband, the energy intervals between the nearest conductance resonances there are small too, resulting in the aperiodic conductance vibrations. The charge conductance of the TDSM nanoribbon as a function of the y-component MF strength is demonstrated in figure 9(c). The electron energy is taken as E = 2.5 meV. To ensure the conducting of the two leads, top-gates with voltage V g = 7.5 mV is applied to the leads. As the MF strength B y < 8.1 T, the charge conductance of the system is close to zero, which results from that no subband is occupied in the TDSM nanoribbon, as shown by the wave vector as a function of the MF strength in the inset. However, as the MF strength B y > 8.1 T, the highest valence subband band is occupied, leading to the conducting of the whole system. This effect means that the charge conductance of the system can be switched on or off by tuning the MF strength. Moreover, the charge conductance also displays an aperiodic oscillations. We found that the resonance points S 1 -S 5 , whose wave vectors are k x = −0.0488, −0.1011, −0.1492, −0.1985, and − 0.2472 nm −1 , respectively, also meet the Aharonov-Bohm interference resonance condition |∆k x | = π/L x = 0.052 nm −1 , as shown in the inset. The charge conductance of the TDSM nanoribbon as a function of the length of the scattering region is given in figure 9(d). The electron energy and the y-component MF strength are taken as −4.5 meV and 20.0 T, respectively, corresponding the point C in figure 7(c). Differing from those in figures 9(b) and (c), the charge conductance shows a periodic oscillation. The period of the oscillation satisfies well the condition ∆L x = |π/k x | ≈ 11.7 nm.

Conclusions
In conclusion, electronic structures and magnetotransport properties of the TDSM nanoribbon are investigated by using the tight-binding discretized Hamiltonian model and non-equilibrium-Green-function-based Landauer-Büttiker equation. According to the evolution of the transversal LCD and LSPD distributions of the TDSM nanoribbon with the increasing MF strength, we found that one-sided hinge states can be achieved in the TDSM nanoribbon with a tilt MF. The underlying physics of the hinge states is analyzed by an analytical calculation and found that it is the combination result of the orbital effect of the MF along the y-axis direction and the Zeeman effect of the MF along the z-axis direction. Furthermore, typical AB interference patterns are found in the charge conductance of the two-terminal TDSM system due to the presence of the unique one-sided-hinge-state-formed interfering loop. These findings may benefit to further understand the external-field-driving HO topological phases and provide an alternative means to detect the HO topological states as well.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).