Anisotropic magnetoelectronic structures and magneto-transport properties of topological Dirac semimetal nanowires

Journal of Magnetism and Magnetic Materials 484 (2019) 373–381

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Research articles

Anisotropic magnetoelectronic structures and magneto-transport properties of topological Dirac semimetal nanowires

T

Xianbo Xiaoa, , Zhengfang Liub, Qingping Wub, Yuan Lic, Fei Lid, Yan Dua ⁎

a

School of Computer Science, Jiangxi University of Traditional Chinese Medicine, Nanchang 330004, China School of Basic Science, East China Jiaotong University, Nanchang 330013, China c Department of Physics, Hangzhou Dianzi University, Hangzhou 310018, China d Department of Scientific Research, Jiangxi University of Traditional Chinese Medicine, Nanchang 330004, China b

ABSTRACT

The magnetoelectronic structures, and the charge and spin conductance spectra of topological Dirac semimetal (TDS) nanowires are calculated by adopting the tightbinding Hamiltonian model in combination with the Green’s-function-based Landauer-Büttiker formula. Anisotropic electronic structures and magneto-transport properties are found in TDS nanowires with different confinement and magnetic field directions. With respect to in-plane-confinement TDS nanowires, trivial surface/corner states are observed and they are split into two spin-polarized states with oppositive signs when a weak out-of-plane magnetic field is applied. Moreover, the spin-polarized surface states will be transformed into bulk and corner states as the magnetic field strength is increased. Therefore, quantized spin conductance energy windows and fully-spin-polarized conductance can be achieved in the normal/out-of-plane-magnetic-field-modulated/normal TDS system in the case of the weak and strong fields, respectively. However, chiral surface states are generated when an in-plane magnetic field is applied, resulting in the perfect charge conductance steps of the normal/in-plane-magnetic-field-modulated/normal TDS system. With respect to the TDS nanowire with both the in-plane and out-ofplane confinements, nontrivial spin-momentum-locked helical surface states are observed. However, the helical character of the surface states will be destroyed and nontrivial-trivial topological phase transition happens as a high in-plane magnetic field is applied. In addition, the energy position of the surface subband inside the bulk energy gap is shifted so that the surface state transport can be switched on or off by tuning the magnetic field strength. Interestingly, the surface subband situated inside the bulk energy gap is split into two subbands with opposite spin signs as an out-of-plane magnetic field is added. Therefore, the topological phase transition between the quantum spin Hall and spin-polarized quantum Hall phases in the TDS nanowire, and the spin-polarized transport in the normal/out-of-planemagnetic-field-modulated/normal TDS system, can be controlled by varying the magnetic field strength. These effects may benefit both the fundamental understanding of the magneto-electronic characters of the TDS nanostructures and the design of low-dissipation (spin) electronic devices.

1. Introduction Since the theoretical predictions [1,2] and experimental verification [3] of topological insulators (TIs), also named quantum spin Hall (QSH) insulators, topological materials have been paid much attention. These kinds of materials are helpful to understand topological states of matter as well as may find applications in low-dissipation (spin) electronic devices [4–6]. Recently, a novel topological material, namely threedimensional (3D) topological Dirac semimetal (TDS), has been proposed [7]. This type of topological material has crystal-symmetry-protected Dirac points and the energy dispersions are linear along all three dimensions in the bulk, which can be considered as a 3D version of graphene. Typically, A3Bi (A = Na, K, Rb) [8] and Cd3 As2 [9] have been predicted by the first-principles calculations to be 3D TDSs. Shortly after the theoretical prediction, Na3Bi [10–12] and Cd3 As2 [13–16] have been confirmed successfully their TDS character by experiments. Moreover, TDSs can be used as idea platforms to investigate the

⁎

quantum phase transitions between different topological states through breaking some symmetries. Later, Weyl semimetal [17–19], node-line semimetal [20], and topological metal with multi-fermion coexistence [21–23] are proposed. From the viewpoint of practical applications, manipulation of electronic structures and edge/surface states transport in topological nanostructures is one of the prime requirements. Edge and surface band gaps are opened around Dirac points when 2D and 3D TIs are confined into nanoribbon [24] and film [25], respectively. However, the band gaps can be eliminated by applying a transversal electric field, restoring the ideal QSH effect [26,27]. Moreover, spin-momentum-locked ringlike boundary states are discovered when the 2D TIs are confined further into quantum dots [28]. Normal-topological insulator phase transition is found in GaN/InN/GaN [29] and GaAs/Ge/GaAs [30] quantum well, which originates from the intrinsic-electric-field-induced band gap reduction and spin–orbit coupling enhancement. In addition, surface magnetism of 3D TIs can be tuned electrically through the RKKY

Corresponding author. E-mail address: [email protected] (X. Xiao).

https://doi.org/10.1016/j.jmmm.2019.04.015 Received 16 December 2018; Received in revised form 1 April 2019; Accepted 4 April 2019 Available online 13 April 2019 0304-8853/ © 2019 Elsevier B.V. All rights reserved.

Journal of Magnetism and Magnetic Materials 484 (2019) 373–381

X. Xiao, et al.

interaction [31]. On the other hand, the edge and surface states transport in TI nanostructures can be controlled by using a single barrier [32], nano-constriction structures [33–35], a transversal electric field [36], or magnetic fields [37,38]. As the electronic structures and topological characters of TDSs are different from those of TIs, they exhibit intriguing magneto-transport features [39]. The Landau energy levels and Berry phase of the 3D TDSs under strong magnetic fields have been studied and found that they can be transformed into the Weyl semimetal phase since the time-reversal symmetry is broken [40,41]. A huge negative magneto-resistance effect has been found in TDS microribbons in the presence of parallel magnetic fields even at room temperature, which results from chiral anomaly [42,43]. However, a linear magnetoresistance [41,44,45] and an anomalously quadratic magnetoresistance [46] have been observed in 3D Cd3 As2 and Na3Bi thin films under perpendicular magnetic fields, respectively. Aharonov-Bohm oscillations [47] and quantum Hall effect (QHE) [48,49] are also found in TDS nanofilms, indicating that the decreasing bulk carrier concentration and the surface states mainly contribute to the magneto-transport. Moreover, anisotropic magnetotransport and electronic properties have been observed in 3D TDSs because of their inhomogeneous energy dispersion along the in-plane and out-of-plane directions [50–52]. Thickness-dependent quantum oscillations are observed in TDS thin films [53]. Enhanced transmission oscillations and finite transmission within the bulk band gap are found in TDS thin films, which are attributed to the influence of inter-valley mixing and Fermi arc states, respectively [54]. In our recent papers, anisotropic size effects have been revealed when the 3D TDSs are confined into nanostructures [55] and the surface state transport in the TDS nanostructures can be manipulated by electrical means [56]. According to the works above, one may naturally ask questions what will happen when the 3D TDSs are confined further into nanowires. Moreover, what are the effects of external magnetic fields on their electronic structures and quantum transport properties. In this work, the magnetoelectronic structures and magneto-transport properties of TDS nanowires irradiated by magnetic fields are studied in detail. It is found that the magnetoelectronic structures and magnetoconductances of the considered system depend strongly on both the confinement and magnetic field directions. i) For the TDS nanowires with the in-plane [set as the plane with the x and y axes; see the inset of Fig. 5(c)] confinements, trivial energy subbands, and charge-density distribution located at all four boundaries/corners of the confined plane are found. Interestingly, when an out-of-plane [set along the z axis; see also the inset of Fig. 5(c)] weak magnetic field is applied, each energy subband and surface/corner state is spin split with opposite signs, respectively. Therefore, quantized spin conductance energy windows can be achieved in a hybrid normal/out-of-plane-magneticfield-modulated/normal TDS system. Furthermore, spin-polarized energy minibands with the same sign emerge as the magnetic field becomes strong. As a result, fully-spin-polarized current can be achieved in the output lead of the hybrid TDS system within a wide energy range. However, Landau energy levels are generated in the TDS nanowires in the presence of an in-plane magnetic field (set along the x axis). In addition, the surface state density distribution is obviously modified and it only resides at one of the four boundaries. ii) For TDS nanowires with both in-plane and out-of-plane [set along y and z axes; see the inset of Fig. 12(a)] confinements, nontrivial surface energy subbands and spin-momentum locked surface states are found. However, the helical character of the surface states is destroyed and nontrivial-trivial topological phase transition happens when a strong in-plane magnetic field is applied. Furthermore, the upper surface energy subband that penetrates throughout the entire bulk energy gap is moved up to the higherenergy area. Consequently, the surface state transport in the hybrid normal/in-plane-magnetic-field-modulated/normal TDS system can be turned on or off by tuning the external magnetic field strength.

However, spin-non-degenerate surface energy subbands are generated when the TDS nanowires are modulated by an out-of plane magnetic field. Therefore, both the quantum spin Hall and spin-polarized quantum Hall phases can be achieved within the bulk energy gap. Moreover, spin-polarized surface state transport in the hybrid normal/ out-of-plane-magnetic-field-modulated/normal TDS system can be controlled by tuning the external magnetic field strength. These findings may not only facilitate our further understanding of TDS nanostructures, but also provide useful guidance for designing topological (spin) electronics devices. The rest of this paper is arranged as follows. In Section 2, the theoretical model of the system and the calculation methods are presented. In Section 3, the numerical results and discussions are demonstrated. Finally, Section 4 concludes the paper. 2. Model and methods The low-energy effective model of the topological Dirac materials A3Bi (A = Na,K,Rb) [8] and Cd3As2 [9] is obtained from ab initio calculation results. The low-energy states near the Fermi level including four orbital bases, i.e., s 1 , 2

1 2

, p3 , 2

3 2

, s1 , 2

1 2

, and p3 , 2

3 2

. Near

the point in the first Brillouin zone, the low-energy effective Hamiltonian of the 3D TDSs is given by:

H (k ) =

0 (k )

M (k ) Ak 0 0

+

Ak+ 0 M (k ) 0 0 M (k ) 0 Ak+

0 0 , Ak M (k )

(1)

k x ± ik y , and in which 0 (k ) = M1, and M2 < 0 M (k ) = M0 M1 kz2 to guarantee the band inversion feature. These parameters A, Ci , and Mi can be achieved by fitting the fist-principles calculation results for a particular material. By solving the Schrödinger equation based on the Hamiltonian (1), the energy band of the 3D TDSs can be obtained. Two Dirac points locate at the k z axis and around which the linear dispersion along the in-plane and out-of-plane directions are anisotropic in the energy band [55]. When the 3D TDSs are confined into nanowires, the energy band and transverse wave functions can be obtained numerically by solving the Schrödinger equation based on the discrete version of the Hamiltonian as follows. By using the standard tight-binding formalism, the discrete Hamiltonian of the electron in TDS nanowires on a cuboid lattice reads:

C0 + C1 kz2 + C2 (k x2 + k y2) , k± = M2 (k x2 + k y2 ) with parameters M0 ,

H TB = l, m, n

vl, m, n cl†, m, n cl, m, n +

l, m , n

t x cl†+ 1, m, n cl, m, n + ty cl†, m + 1, n cl, m, n

+ t z cl†, m, n + 1 cl, m, n + H. c. ,

(2)

is the electron creation(annihilation) operator on the where site (l, m , n). The on-site matrix element vl, m, n reads:

cl†, m, n (cl, m, n ) +

0 0 0

vl, m, n =

0 0 0

0 0 +

0

0 0 , 0

(3)

= C0 ± M0 + 2(C2 + 2(C2 + 2(C1 M1)/ az2 where with the lattice constants ax , a y , and az along the three orthogonal directions. The nearest-neighbor hopping matrix elements t x , ty and t z read: M2)/ ax2

±

t x / y/ z =

374

t+x / y / z

I0

I0

t x / y/ z

,

M2)/ a y2

(4)

Journal of Magnetism and Magnetic Materials 484 (2019) 373–381

X. Xiao, et al.

in which

( C2 + M2)/ a x2

t±x =

M2)/a x2

( C2

( C2 + M2)/ a y2

t±y =

t±z

iA/(2a x ) )

iA/(2ax )

A/(2a y ) )

A/(2a y )

M2)/ a y2

( C2

( C1 + M1)/ az2

=

0

0

(5)

,

C

(6)

,

M1)/ az2

( C1

,

quantized axis. Therefore, > = (1, 0) T denotes the spin-up state and > = (0, 1) T means the spin-down one, where T represents transposition. Owing to the fact that spin flipping does not occur in the investigated system, the transverse charge- and spin-density distributions are defined, respectively, as:

1/ 0

S

2/ 0

C

i , j, E =

s

i, j, E

+

p

i , j, E

2

+

s

i, j, E

GS =

p

2 s

i , j, E

2

+

p

i, j, E

i , j, E

2 s

i , j, E

2 p

i, j, E

,

(11) where i (j)=x , y , or z. and denote the different spin states, respectively. Moreover, the magneto-transport in the two-terminal TDS nanowires can be calculated by adopting the recursive Green’s-function (GF) method [58]. Taking the effects of the two semi-infinite leads into account, the spin-resolved conductance in the TDS system is given by the Landauer-Bu¨ ttiker formula [59,60]

G

=

e2 h

2

N

tµ

,

µ, = 1

2

+

p

i, j, E

s

i, j, E

2

+

p

i , j, E

2 s

i, j, E

2 p

i , j, E

.

+G ,

(15)

e G G , 4 e 2 /h

(16)

We first consider a TDS nanowire confined along both the x- and yaxis directions. The transverse widths of the TDS nanowire are set at Wx = Wy = 20 ax / y . The energy band of the TDS nanowire is shown in Fig. 1(a). The bulk energy gap is opened around the two Dirac points with k z = ±0.903 nm 1 and the continuous bulk energy band is discretized into hump-like subbands, which are attributed to the finite-size effect [54,55]. For clarity, only the highest bulk valence subband is presented, as shown by the thick black line in the figure. Interestingly, the continuous surface band is also lifted into subbands, as shown by the thin red lines, since both the x- and y-axis directions are confined. In addition, the bottom of each surface subband is located inside the bulk energy gap, which is different from that of TDS film [54,55]. When a weak ( 1 = 0.01) out-of-plane magnetic field along the z-axis direction is applied, each subband is split into two sets of subbands, as displayed blue and green lines in Fig. 1(b). However, the salient profile of each set of subbands is the same as that of Fig. 1(a). The split energy gap between each pair of subbands is enlarged with the increasing magnetic field strength. Therefore, each set of subbands tends to come together, forming minibands, as shown in Fig. 1(c). The magnetic field strength is taken as 1 = 0.1. This effect can also be observed in Fig. 1(d), where the subbands at k z = 0.0 as a function of the magnetic field strength are given. Furthermore, the hump-like bulk energy subbands will be transformed into parabola-like subbands one by one as the magnetic field becomes strong. The hump-parabola transformation of the bulk valence subbands is attributed to the effect of the out-of-plane magnetic field. The spin–orbit coupling induced subband invert is suppressed, restoring the normal bulk valence subbands.

(10)

i, j, E =

i , j, E

3.1. TDS nanowires with confinement along in-plane directions

and S

s

respectively. The material parameters in this work are chosen to be those for a Na3Bi obtained from the first-principles calculations [8], namely, C0 = 63.82 meV, C1 = 87.536 meV·nm2, C2 = 84.008 meV·nm2 , M0 = 86.86 meV, M1 = 106.424 meV·nm2, M2 = 103.610 meV·nm2 , and A = 245.98 meV·nm . The lattice constants are set at ax = a y = 0.3 nm and az = 0.5 nm. The magnetic flux is in units of 0 . In the calculation of the transport properties of the hybrid normal/magnetic-field-modulated/ normal TDS system, the lengths of the magnetic-field-modulated nanowires are taken to be L = 50 az and 50 ax in the following subsections A and B, respectively. In order to decrease the interface scattering effect between the nanowire and leads, the smooth interface case such as that in Ref. [62] is considered. The length of the interfaces equals that of the nanowire. In addition, the charge and spin conductances are in the unit of e 2/ h and e/(4 ), respectively.

2

+

2

+

and

(9)

2

2

i, j, E =

GC = G

where 2 = B0 a y az is the magnetic flux through an elementary out-ofplane rectangular lattice. The Zeeman effect from the external magnetic field is not included here, since it does not change the fundamental 1 results of the present work. Moreover, the Zeeman spin split 2 gµB B0 , where g is the Landé g factor and µB the Bohr magneton, is a relatively small term in the Hamiltonian of TDSs in the presence of a magnetic field. For example, it is about 0.925 meV for Cd3As2 with g = 32 and B0 = 1.0 T [57]. Using the discrete Hamiltonian (2), the energy band and transverse wave functions of electrons in the TDS nanowires can be obtained conveniently. Then, the transverse charge- and spin-density distributions are defined, respectively, as: 2

i, j, E

Similarly, the charge and spin conductances are defined as [61]:

(8)

= (Tlm (n + 1), lmn) ,

p

(14)

where 1 = B0 ax a y is the magnetic flux through an elementary in-plane square lattice, 0 = h /e is the flux quanta with the electron charge ( e ) and h the Planck constant. Similarly, when an in-plane magnetic field B = (B0, 0, 0) is applied, the vector potential A = (0, 0, B0 y ) is taken, and the nearest-neighbor hopping energy along the z axis is transformed as:

Tlmn, lm (n + 1) = t z e 2im

2

+

(13)

(7)

= (T(l + 1) mn, lmn) ,

i , j, E

s

and

and I0 is a (2 × 2) zero matrix. When an out-of-plane magnetic field B = (0, 0, B0) is introduced, its effects could be incorporated into the nearest-neighbor hopping energy by the Peierl’s phase factor. To keep the transitional symmetry of system along the x-axis direction, the vector potential A = (B0 y , 0, 0) is selected. Therefore, the nearestneighbor hopping energy along the x axis is taken as:

Tlmn,(l + 1) mn = t x e 2im

2

i , j, E =

(12)

where tµ is the spin-dependent transmission coefficient of electrons injected from the mode µ with spin to the output mode with spin , and N is the number of the propagating modes. 3. Numerical results and discussions In the next numerical calculations, z axis is taken as the spin375

Journal of Magnetism and Magnetic Materials 484 (2019) 373–381

X. Xiao, et al.

Fig. 3. Same as Fig. 2, but with weak ( 1 = 0.01) out-of-plane magnetic field. Electron energies are set at those of points A3 and A 4 in Fig. 1(b), respectively. Fig. 1. Energy band of the in-plane-confinement TDS nanowire (a) without and with a (b) weak ( 1 = 0.01) and (c) strong ( 1 = 0.1) out-of-plane magnetic field. (d) Energy subbands at k z = 0.0 as a function of the magnetic field strength.

the transverse charge- and spin-density distributions of points A3 and A 4 [as indicated in Fig. 1(b)] are displayed in the upper and lower panels of Fig. 3, respectively. The energies corresponding to this two point are the minima of the lowest pair of surface subbands, respectively. One can observe that the salient features of the transverse charge- density distributions AC3 (x , y , E ) and AC4 (x , y , E ) are the same as those of the TDS nanowire without a magnetic field, as shown in Figs. 3(a) and 3(c). However, the spin-density distributions with respect to points A3 and A 4 are fully polarized with opposite signs, as indicated in Figs. 3(b) and 3(d), respectively. This phenomenon means that the split subbands in Fig. 1(b) are Zeeman-like ones, although the Zeeman term induced by the external magnetic field is not incorporated in Hamiltonian (1). The Zeeman-like spin split effect can be interpreted by analyzing the symmetries of the considered systems. The spin parity Px x is conserved for the TDS nanowires, where Px and x are the inversion operator and Pauli matrix for the x-component, respectively [63]. However, the spin parity is broken when an out-of-plane magnetic field is introduced, namely the off-diagonal terms in Eq. (5) are modified by the magnetic field effects, as shown in Eq. (8). These energyband characteristics are similar to those of the Rashba semiconductor nanowire with a perpendicular magnetic field [63–65], since the terms Ak ± in the Hamiltonian (1) resemble the Rashba spin-orbit coupling term. However, the energy gaps between each split subbands at k z = 0.0 are about 20.0 meV in the case of weak magnetic field 1 = 0.01, i.e. they are independent of the subband index. Further, no Landau energy levels are formed in this case. This two features are different from that of Rashba wire with a perpendicular magnetic field. Fig. 4 shows the transverse charge- and spin-density distributions of the in-plane-confinement TDS nanowire under the modulation of a high ( 1 = 0.1) out-of-plane magnetic field. For comparison, points A5 and A 6 in Fig. 1(c)/1(d) is taken as examples, since they situate the same pair of surface subbands of A3 and A 4 , respectively. Interestingly, differing from those of the TDS nanowire without [see Fig. 2(a)] and with a weak magnetic field [see Figs. 3(a) and 3(c)], the transverse chargedensity distribution of the point A5 is focused at the center of the confined plane, as shown in Fig. 4(a). However, the transverse chargedensity distribution of the point A 6 is transited to the corners of the confined plane, as shown in Fig. 4(c). In addition, the spin-density distributions of the surface state corresponding to the points A5 and A 6 are also completely polarized with the opposite signs, as displayed, respectively, in Figs. 4(b) and 4(d), indicating that each miniband in Fig. 1(c) is spin non-degenerate and each subband within the miniband has the same spin sign. This effect results from the increasing Zeemantype spin split energy gaps as the magnetic field strength is increased. Therefore, these subbands with the same spin sign tend to come together, forming the minibands.

To clarify the characteristics of the energy band of the in-planeconfinement TDS nanowire without an out-of-plane magnetic field, Fig. 2(a) shows the transverse charge-density distribution in the TDS nanowire. The electron energy is taken as the same as that of point A1 in the first surface subband in Fig. 1(a), where the wave vector k z = 0.0 . Interestingly, the charge-density distribution AC1 (x , y , E ) is localized at the four boundaries, which originates from the band-inversion feature of Hamiltonian (1). Note that Hamiltonian (1) is homogeneous along the x and y axes, resulting in the isotropic charge-density distribution inside the transverse plane of the TDS nanowire. Moreover, due to the time-reversal and inversion symmetries of the TDS nanowire, point A1 , in fact, has two degenerate states localized on each side. Therefore, the spin-density distribution AS1 (x , y , E ) is zero everywhere, as shown in Fig. 2(b). Interestingly, the charge-density distribution will be transformed gradually from the boundaries to corners of the transverse plane as the surface subband index is increased. For simplicity, only the transverse charge-density distribution with respect to the point A2 in the fourth surface subband of Fig. 1(a) are plotted in Fig. 2(c). Obvious corner state is found in this case and it is also spin degenerate, which can be verified by the transverse spin-density distribution shown in Fig. 2(d). Owing to the split subbands of the in-plane-confinement TDS nanowire irradiated under a weak ( 1 = 0.01) out-of-plane magnetic field,

Fig. 2. Transverse (a, c) charge- and (b, d) spin-density distributions of inplane-confinement TDS nanowire. Electron energies of the upper and lower panels are set at those of points A1 and A2 in Fig. 1(a), respectively. 376

Journal of Magnetism and Magnetic Materials 484 (2019) 373–381

X. Xiao, et al.

demonstrating that this system may be used to design a spin-filtering device. The underlying physics of this device originates from the lowest surface spin-polarized energy miniband with the same sign is occupied. However, the step structures of the surface charge and spin conductance are destroyed when the electron energy E > 7.0 meV, since the second miniband is occupied as well, so that inter-miniband mixing is inevitable. Moreover, the amplitude of the charge conductance is increased, while that of the spin conductance is decreased with increasing electron energy because the second miniband has the oppositive spin sign with respect to the lowest miniband. Interestingly, the amplitude of the bulk charge conductance equals the bulk spin conductance within the energy range [ 299.0, 168.0] meV, as shown by the black solid and dashed lines in Fig. 5(c), since the hump-like bulk energy subbands are transformed into the parabola-like ones. The energy band of the in-plane-confinement TDS nanowire in the presence of an in-plane magnetic field along the x axis is displayed in Fig. 6(a). The magnetic field strength is set at 1 = 0.025. Differing from the effects of the out-of-plane magnetic field on the energy subband of the TDS nanowire, the surface energy subbands around k z = 0.0 become flat one by one as the strength of the in-plane magnetic field is increased, as shown by the red solid lines in Fig. 6(a). This phenomenon indicates that the surface Landau levels (LLs) are formed in the considered system. However, the energy gaps around k z = 0.0 between the neighbor LLs are subbands index dependent, which are different from the normal LLs in semiconductors nanowires. The underlying physics is attributed that only the diagonal terms in Eq. (7) are modified by the magnetic vector potential, as shown in Eq. (9). The LLs at k z = 0.0 as a function of the magnetic field strength are shown in Fig. 6(b). Both the bulk and surface LLs are lifted monotonously with increasing magnetic field strength, as displayed by the black and red solid lines, respectively. Furthermore, the energy widths between the LLs are widened with the increasing magnetic field strength, which means the confinement effect enhanced by the in-plane magnetic field [64]. To understand the LLs formed in the in-plane-confinement TDS nanowire with an in-plane magnetic field, the transverse charge-density distributions with respect to points C1 and C2 are shown in Fig. 7(a) and (c), respectively. The electron energy is set at E = 26.0 meV, the same as that of points C1 and C2 in Fig. 6(a). The magnetic field strength is the same as that in Fig. 6(a). The transverse charge-density distributions in the TDS nanowires are modified by the in-plane magnetic field. They are localized at opposite surfaces, indicating the chiral character of the LLs, since the electron states of C1 and C2 with opposite velocity directions. Furthermore, the LLs are spin degenerate as the transverse spin-density distribution is always zero, as shown in Figs. 7(b) and (d). Owing to the spin degeneracy of the LLs, only the charge conductance of the normal/in-plane-magnetic-field-modulated/normal TDS system with the in-plane confinements as a function of the electron energy is shown in Fig. 8. The strengths of the in-plane magnetic field are taken as 1 = 0.0 (solid line), 0.025 (dashed line), and 0.1 (dotted line). The energy width of the charge conductance gap (GC = 0 ) is widened as the strength of the magnetic field is increased, as shown by the dashed and dotted lines, which means that the energy gap of the

Fig. 4. Same as Fig. 2, but with strong ( 1 = 0.1) out-of-plane magnetic field. Electron energies are taken as those of points A5 and A 6 in Fig. 1(c), respectively.

The charge and spin conductances of the hybrid normal/out-ofplane-magnetic-field-modulated/normal TDS system with in-plane confinement, as sketched in the inset of Fig. 5(c), as a function of the electron energy are plotted in Fig. 5. Perfect quantized charge conductance steps with the value 2n (n = 0, 1, 2, …) are observed in the hybrid TDS system without the magnetic field, as shown by the solid line in Fig. 5(a). The position and energy range of each quantized charge conductance step is determined by the energy subbands in Fig. 1(a). Moreover, the spin conductance of the system remains zero, as shown by the dashed line in Fig. 5(a), since the energy subbands of the TDS nanowires are spin degenerate. For the hybrid TDS system with a weak ( 1 = 0.01) out-of-plane magnetic field in the middle region, the quantized charge conductance steps with the value 2n + 1 (n = 0, 1, 2, …) are found in addition to those with the value 2n , as shown by the solid line in Fig. 5(b). Interestingly, surface spin conductances with the value 1 are generated within the energy range of the odd surface charge conductance steps; otherwise, the surface spin conductance will vanish, as shown by the red dashed line in Fig. 5(b). However, for the bulk spin conductance, in addition to the value 1 within the energy range of the odd bulk charge step, the value 2 can also be achieved within the energy range of the even bulk charge conductance step next to the conductance gap, as indicated by the black dashed line in Fig. 5(b). This effect is attributed to the spin split humplike bulk energy subbands of the TDS nanowire with a weak out-ofplane magnetic field shown by the black solid line in Fig. 1(b). As the magnetic field strength is increased, the surface spin conductance energy windows disappear. Furthermore, the amplitudes of the surface spin conductance steps equal those of the surface charge conductance steps within the energy range of the lowest surface energy miniband [ 100.0, 7.0] meV, as shown, respectively, by the red solid and dashed lines in Fig. 5(c). Here, the magnetic field strength is taken as 1 = 0.1. Therefore, the current in the output lead is completely polarized,

Fig. 5. Charge and spin conductance spectra as function of electron energy of normal/out-of-plane-magnetic-field-modulated/normal TDS system with inplane confinements. Magnetic field strengths in (a)–(c) are set as same as those in Fig. 1(a)–(c), respectively.

Fig. 6. (a) Energy band of the in-plane-confinement TDS nanowire in presence of in-plane magnetic field. Magnetic field strength is set at 1 = 0.025. (b) Landau levels as function of in-plane magnetic field strength at k z = 0.0 . 377

Journal of Magnetism and Magnetic Materials 484 (2019) 373–381

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Fig. 7. Same as Fig. 2, but with in-plane magnetic field 1 = 0.025. Electron energy is set at E = 26.0 meV, corresponding to the points C1 and C2 .

Fig. 9. Energy bands of TDS nanowire with both in-plane and out-of-plane confinements in presence of different in-plane magnetic field strengths (a) 2 = 0.0 , (b) 0.008 , and (c) 0.1. (d) Energy subbands as a function of in-plane magnetic field strength at k x = 0.0 . The red solid lines with arrows mean that the hopping between this two points is forbidden, while the green ones mean allowing.

surface conduction subband. This effect originates from the finite-sizeinduced surface gaps also can be tuned by an in-plane magnetic field, which is similar to that of the TI films [66,67]. However, this effect can not be observed in 3D TIs and TDSs, where only the positions of Dirac points are shifted by the in-plane magnetic field [66,67]. Moreover, the upper surface conduction subband around k x = 0.0 becomes flat. However, the V-like upper surface conduction subband becomes a Wlike one as the magnetic field strength is increased, as shown in Fig. 9(c). Here the magnetic field strength 2 = 0.1. More interestingly, the lowest bulk conduction subband becomes linearly dispersed and almost touches the upper surface conduction subband. This effect can be seen more clearly in Fig. 9(d), where the energy subbands at k x = 0.0 as a function of the magnetic field strength are presented. The upper surface conduction subband moves upwards monotonously, while the lowest bulk conduction subband moves downwards first and then upwards as the strength of the magnetic field is increased. Finally, they almost touch each other. Fig. 10(a) and (c) show the transverse charge-density distributions in the TDS nanowire with both in-plane and out-of-plane confinements for points D1 and D2 in Fig. 9(a), respectively. The electron energy E = 0.0 meV. It can be seen that the transverse charge-density distributions of this two points are the same, namely they are localized near the centers of the left and right boundaries (0,5az ) and (30a y ,5az ). Furthermore, the charge-density distribution of point D1 are completely spin polarized with oppositive signs, as shown in Figs. 10(b). However, the spin-density distribution of point D2 is contrary to that of point D1, as shown in Fig. 10(d). This character confirms the topological and spin-momentum-locked helical characters of the nontrivial surface states. Therefore, the backscattering between points D1 and D2 induced by nonmagnetic impurities is forbidden, as shown by the red solid line in Fig. 9(a). The transverse charge- and spin-density distributions in both inplane and out-of-plane confined TDS nanowire with strong in-plane magnetic field are plotted in Fig. 11. The magnetic field strength is set at 2 = 0.1. The electron energy is taken as E = 40.0 meV, which locates within the W-like surface conduction subband in Fig. 9(c). Therefore, four different electron states E1, E2 , E3 , and E4 are occupied in this case, as shown in Fig. 9(c). For conciseness, only the charge- and spin-density distribution of points E1 and E2 are given here. The surface states can be maintained even the strong magnetic field is applied, as shown in Figs. 11(a) and 11(c), indicating that they are robust against the in-

Fig. 8. Charge conductance spectra as function of electron energy of normal/inplane-magnetic-field-modulated/normal TDS system with in-plane confinements. Magnetic field strengths are set at 1 = 0.0 (solid line), 0.025 (dashed line), and 0.1 (dotted line), respectively.

considered system becomes wide. In addition, both the bulk and surface charge conductance steps can be maintained even as a high magnetic field is applied, resulting from the energy widths between the LLs being enlarged with increasing magnetic field strength, so that subband intermixing is negligible [64,65]. 3.2. TDS nanowires with confinement along both in-plane and out-of-plane directions We then consider the TDS nanowires confined along both the y and z axis directions. The transverse widths of the TDS nanowire are set at Wy = 30a y and Wz = 10az . The energy band of the TDS nanowire with both in-plane and out-of-plane confinements is plotted in the Fig. 9(a). Since the topological character of the considered TDS nanowire is periodically varied with increasing width along the out-of-plane direction and the period is 7az [55], nontrivial surface energy subbands can be observed, as indicated by the red dotted lines. The upper surface conduction subband penetrates throughout the bulk energy gap and the lower valence one is buried in the bulk valence subbands (as indicated by black solid lines), which is different from that of TIs [1,2]. However, similar to that of the TIs, a small surface energy gap is opened because of the interaction between the upper and down surface states [24]. The energy band of the weak in-plane-magnetic-field-modulated TDS nanowire with both in-plane and out-of-plane confinements is shown in Fig. 9(b). The strength of the in-plane magnetic field is taken as 2 = 0.008. It can be seen that the surface energy subbands move upwards to the higher-energy region while the bulk energy subbands move downward to the lower-energy region, especially the upper 378

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Fig. 12. (a) Charge conductance spectra as a function of electron energy of normal/in-plane-magnetic-field-modulated/normal TDS system with both inplane and out-of-plane confinements. Magnetic field strengths are set at 2 = 0.0 (solid line), 0.008 (dashed line), and 0.1 (dotted line), respectively. For clarity, solid and dashed lines are lifted entirely by 2.0 and 1.0, respectively. (b) Surface charge conductance spectrum of the hybrid TDS system as function of magnetic field strength. Electron energy is taken as E = 0.0 meV. Fig. 10. Transversal (a, c) charge- and (b, d) spin-density distributions of TDS nanowire with both in-plane and out-of-plane confinements. Electron energy is set at E = 0.0 meV, corresponding to points D1 and D2 in Fig. 9(a).

magnetic field, as shown by the dashed and dotted lines. At the same time, the energy width of the surface conductance is narrowed, as shown by the red part of each line. This phenomenon means that the charge conductance of the hybrid TDS system can be controlled by the in-plane magnetic field. Moreover, the charge conductance steps are destroyed when the magnetic field becomes strong, resulting from the huge energy band mismatch between the normal lead and TDS nanowire with a strong magnetic field. The in-plane-magnetic-field-controlled surface state transport effect can be seen more clearly in Fig. 12(b), where the surface charge conductance of the hybrid TDS system as a function of magnetic field strength is plotted. The electron energy is set at E = 0.0 meV. A sharp step-shaped curve is observed, namely the surface charge conductance is switched off when the magnetic field strength is increased to 2 = 0.04 . This effect indicates that the considered system may be used to design a topological field-effect transistor. The physics mechanism of the device is attributed that the surface conduction band moving upwards to the higher-energy region as the magnetic field is increased, as shown in Fig. 9(a)–(c). The energy bands of the out-of-plane-magnetic-field-modulated TDS nanowire with both in-plane and out-of-plane confinements are displayed in Fig. 13(a) and (b). The magnetic field strengths are taken as (a) 2 = 0.004 and (b) 2 = 0.016 . The main characters of both the bulk and surface subbands keep unchange except each subband is split. Moreover, the energy width between the two split subbands is enlarged as the magnetic field strength is increased. This phenomenon is clarified by Fig. 13(c), in which the split subbands at k x = 0.0 as a function of the magnetic strength are presented. For the surface conduction subband, one of the split subbands (as indicated by the green dotted line) increases while another (as indicated by the blue dotted line) decreases monotonously as the magnetic field strength is increased. However, for the surface valence subband, one of the split subband (as indicated by the green dotted line) first increases and then decreases so that a peak is observed. The transverse charge- and spin-density distributions of the out-ofplane-magnetic-field-modulated TDS nanowire with both in-plane and out-of-plane confinements are shown in Fig. 14. The electron energy is

Fig. 11. Same as Fig. 10, but with strong ( 2 = 0.1) in-plane magnetic field. Electron energy is set at E = 40.0 meV, corresponding to points E1, E2 , E3 , and E4 in Fig. 9(c).

plane magnetic field. However, the spin-density distributions of point E2 is similar to that of point E1, as shown in Fig. 11(b) and (d), respectively. Therefore, the backscattering between this two electron states caused by the nonmagnetic impurities is allowed, as indicated the green solid lines in Fig. 9(c). This effect means that nontrivial-trivial topological phase transition will happen when the surface subband are transformed from V-like to W-like shape with increasing magnetic field strength. In addition, the charge-density distributions of E3 and E4 are the same as that of E2 and E1, while their spin-density distributions are opposite to that of E2 and E1, respectively, which are not shown here. As a result, the electron state hopping between points E3 and E4 is also allowed. The charge conductance of the normal/in-plane-magnetic-fieldmodulated/normal TDS system with both in-plane and out-of-plane confinements as a function of the electron energy is plotted in Fig. 12(a). The strengths of the in-plane magnetic field of the solid, dashed, and dotted lines are taken as the same as those in Fig. 9(a)–(c), respectively. For clarity, the solid and dashed lines are lifted entirely by 2 and 1, respectively. No charge conductance gap (GC = 0 ) is found in the charge conductance curve (solid line) of the TDS system without an in-plane magnetic field, which means that the transport of the surface states can not be turned off. However, the charge conductance gap is opened when an in-plane magnetic field is applied in the TDS nanowire and its energy width is widened with increasing strength of the

Fig. 13. Energy bands of out-of-plane-magnetic-field-modulated TDS nanowire with both in-plane and out-of-plane confinements. Magnetic field strengths are taken as 2 = 0.004 (a) and 0.016 (b), respectively. (c) Energy subbands as function of magnetic field strength at k x = 0.0 . 379

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of-plane magnetic field resides in quantum spin Hall phase within this energy range. The charge and spin conductances of the hybrid TDS system as a function of the magnetic field strength is plotted in Fig. 15(b). The electron energy is taken as E = 0.0 meV. The magnitude of the charge conductance transitions from 2 to 1, while that of the spin conductance transitions from 0 to 1 as the magnetic field strength is increased to 0.016. This effect demonstrates that a topological phase transition from the quantum spin Hall phase to the spin-polarized quantum Hall phase occurs in the out-of-plane-magnetic-field-modulated TDS nanowire with both in-plane and out-of-plane confinements. 4. Conclusions In conclusion, the magnetoelectronic structures and magnetotransport properties of TDS nanowires were investigated in this paper, and it was found that they depend strongly on both the confinement and magnetic field directions. For the TDS nanowire with in-plane confinements, trivial spin split subbands and surface/corner states are observed in the nanowire with a weak out-of-plane magnetic field. However, spin-polarized minibands, bulk and corner states are generated in the nanowire with a strong magnetic field. Therefore, ideal spin conductance energy windows and completely spin-polarized current can be achieved in the hybrid TDS system in the presence of the weak and strong magnetic fields, respectively. LLs and chiral surface states are formed in the in-plane-magnetic-field-modulated TDS nanowire with in-plane confinements. Consequently, perfect quantized charge conductance plateaus can be obtained in the hybrid TDS system even in the presence of a high in-plane magnetic field. For the TDS nanowires with both the in-plane and out-of-plane confinements, nontrivial surface states can also be observed in the nanowires. However, they will be damaged and nontrivial-trivial topological phase transition happens when a strong in-plane magnetic is applied. Furthermore, the positions of the surface energy subbands are modified so that the surface state transport in the hybrid TDS system can be switched on or off by tuning the magnetic field strength. Spin split surface energy subbands are observed in the nanowire with an out-of-plane magnetic field. As a result, topological phase transitions from the quantum spin Hall phase to the spin-polarized quantum Hall phase can be achieved by adjusting the magnetic field strength. The results presented here may be important for utilizing novel topological nanostructures for nanoelectronics and spintronics applications.

Fig. 14. Same as Fig. 10, but with out-of-plane magnetic field ( 2 = 0.04 ). Electron energy is set at E = 0.0 meV, corresponding to points F1 and F2 in Fig. 13(a).

Fig. 15. (a) Charge and spin conductance spectra as function of electron energy of normal/out-of-plane-magnetic-field-modulated/normal TDS system with both in-plane and out-of-plane confinements. Magnetic field strength is set at 2 = 0.016 , same as that in Fig. 13(b). (b) Charge and spin conductance spectra of hybrid TDS system as function of magnetic field strength. Electron energy is taken as E = 0.0 meV.

set at E = 0.0 meV, which ensures that only the lowest two split surface conduction subbands are occupied, as indicated by points F1 and F2 in Fig. 13(a). Differing from those of the TDS nanowire without the magnetic field, the transverse charge-density distributions corresponding to these two points only reside at the left- and right-hand surfaces of the TDS nanowire, as shown in Fig. 14(a) and (c), respectively. Furthermore, they are completely spin polarized, but with oppositive signs, as shown in Fig. 14(b) and (d), demonstrating that the subbands in Fig. 13 are spin non-degenerate. The charge and spin conductances of the normal/out-of-planemagnetic-field-modulated/normal TDS system with both in-plane and out-of-plane confinements as a function of the electron energy are displayed in Fig. 15(a). The magnetic field strength is taken as the same as that in Fig. 13(b), namely 2 = 0.016 . The red solid and purple dashed lines represent the charge and spin conductances of the surface state transport inside the bulk energy gap [−38.0,35.0] meV, respectively. Two different topological phases are revealed by the charge and spin conductances. Within the energy range [−38.0,−2.0] meV, only one of the split surface states is conducted, so that the charge conductance magnitude equals 1. In addition, the magnitude of the spin conductance within this energy range equals 1 as well. Therefore, the considered TDS nanowire with the out-of-plane magnetic field locates in the spin-polarized quantum Hall phase when the electron energy is within this energy range. Moreover, the considered system can be used to design a topological spin filter device, since the current in the output lead is completely spin polarized. However, within the other energy range, [-2.0,35.0] meV, both split surface states are transported, leading to the charge conductance magnitude being equal to 2, but with vanishing spin conductance. As a result, the TDS nanowire with the out-

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