Electronic and transport properties of noncollinear magnetic monatomic Mn chains: Fano resonances in the superlattice of noncollinear magnetic barriers and magnetic anisotropic bands

Journal of Magnetism and Magnetic Materials 379 (2015) 167–178

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Electronic and transport properties of noncollinear magnetic monatomic Mn chains: Fano resonances in the superlattice of noncollinear magnetic barriers and magnetic anisotropic bands C.J. Dai a, X.H. Yan a,b,n, Y. Xiao a, Y.D. Guo b a b

College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, People's Republic of China College of Electronic Science and Engineering, Nanjing University of Posts and Telecommunications, Nanjing 210046, People's Republic of China

art ic l e i nf o

a b s t r a c t

Article history: Received 30 August 2014 Received in revised form 10 November 2014 Accepted 15 December 2014 Available online 16 December 2014

By means of the density functional theory combined with non-equilibrium Green's function method, ballistic transport properties of one-dimensional noncollinear magnetic monatomic chains were investigated using the single-atomic Mn chains as a model system. Fano resonances are found to exist in the monatomic Mn chains with spin-spiral structure. Furthermore, in the monatomic Mn chains with magnetic soliton lattice, Fano resonances are enhanced and cause the conductance splitting in the transmission spectra. The Fano resonances in the noncollinear magnetic single-atomic Mn chains are arising from the coupling of the localized d-states and the extended states of the quantum channels. By constructing a theoretical model and calculating its conductance, it is found that the phenomena of Fano resonances and the accompanying conductance splitting exist universally in the superlattice of onedimensional noncollinear magnetic barriers, due to the interference of the incident waves and reflected waves by the interfaces between the neighboring barriers. Moreover, the band structures of the ferromagnetic and spin-spiral monatomic Mn chains exhibit a strong dependence on the spatial arrangement of the magnetic moments of Mn atoms when spin–orbit coupling is considered. & 2014 Elsevier B.V. All rights reserved.

Keywords: Noncollinear magnetism Monatomic chains Fano resonances Magnetic anisotropy

1. Introduction One-dimensional (1D) transition metal (TM) monatomic chains and nanowires have received much attention for both the meaning of fundamental physics [1–6] and the potential applications in atomic-scale magnetic devices and spintronics in the future [7–9]. According to the recent theoretical and experimental results, many 1D TM monatomic chains exhibit a noncollinear magnetism, especially the spin-spiral ground state [8, 10–12]. For example, Co singe-atomic chains on monatomic layer of Mn on W(110) were investigated by spin-polarized scanning tunneling microscope and found to have a spin-spiral structure (SS) [8]. From the work of Tung et al. and Schubert et al., stable SSs are also predicted to exist in both the free standing V, Mn, and Fe monatomic chains [10] and the single-atomic alloying chains of Mn, Fe, and Cr [11]. On the other hand, under the magnetic field perpendicular to the helical axis, the periodic SS will turn into another noncollinear magnetic order which is called magnetic soliton lattice (MSL) [13–15]. This n Corresponding author at: College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, People's Republic of China. E-mail address: [email protected] (X.H. Yan).

http://dx.doi.org/10.1016/j.jmmm.2014.12.024 0304-8853/& 2014 Elsevier B.V. All rights reserved.

transition from SS to MSL has been studied experimentally in magnetic crystals of Cr1/3NbS2 [16]. Recently, an experimental result revealed that the magnetoresistance (MR) along the axis in the chiral magnetic crystal of Cr1/3NbS2 can be attributed to the magnetic scattering of conduction electrons by the MSL [17]. This result suggests that noncollinear magnetism, such as SS and MSL, may result in some nontrival transport properties for the 1D or quasi-1D atomic-scale structures. From this point of view, it is necessary to study the scattering effect of the noncollinear magnetism on the electronic transport properties of 1D TM single-atomic chains for the promise of the spintronics devices and atomic-scale devices in the future application. As a typical 1D TM single-atomic chain, monatomic Mn chains (MMCs) have been studied both experimentally and theoretically in recent years [9–12]. MMCs of up to ten atoms were created on an insulating CuN/Cu(001) surface using scanning tunneling microscope experimentally [9]. On the other hand, both freestanding MMCs and supported MMCs on Cu(110) and Ag(110) are predicted to have a spin-spiral ground state from previous computational results [10–12]. Based on above results, MMCs, as a model system of 1D monatomic chains, constitute a unique research category for

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fundamental physics and applications in the atomic-scale devices. In this paper, ballistic transport properties of SS and MSL in MMCs were investigated by means of first-principle calculations combined with the non-equilibrium Green's function (NEGF) method. Owing to the existence of SS, the Fano resonances [18,19] are found in the transmission spectra of MMCs with SS. Moreover, the Fano resonances are enhanced in the MMCs with MSL and bring the conductance splitting in their transmission spectra. By constructing a theoretical model and calculating its conductance, Fano resonances and accompanying conductance splitting are found to exist universally in the superlattice of one-dimensional noncollinear magnetic barriers, therefore are expected to occur in other 1D or quasi-1D nanoscale systems with SS and MSL. Finally, we consider the effect of spin–orbit coupling (SOC) on the band structure (BS) and transmission spectrum of MMCs with SS, for that SOC usually leads to unique phenomena, such as magnetic anisotropic BS and MR [20–26]. This paper is organized as follows. In Section 2, we introduce the geometry structures (Section 2.1) and the details of firstprinciple calculations (Section 2.2). In Section 3, we discuss the transport properties of the noncollinear magnetic MMCs. In Section 4, a theoretical model of noncollinear magnetic MMCs is constructed to study the electron transport in the superlattice of noncollinear magnetic barriers. The BS and transmission spectra of spin-spiral MMCs are discussed in Section 5, with considering the effect of SOC. We summarize our work and give the conclusions in Section 6.

2. Junction structures and computational details 2.1. Junction structures In this work, we consider the transverse spin-spiral MMCs, where all the spins rotate in a plane perpendicular to the chain axis. Therefore the number of total Mn atoms in each period (n) and the angle between the magnetic moments of adjacent atoms (θ) have a simple numerical relation of n ¼360 m/θ (m is the smallest integer to keep n is an integer) and θ can be used to distinguish the chains. The distance of nearest Mn atoms is fixed to be 2.4 Å, which is the bond length of the ground state of freestanding MMCs [10,12]. In order to calculate the transmission spectra of SS in MMCs, we construct spin-spiral junctions (SSJs) by attaching the spin-spiral MMCs to two semi-infinite ferromagnetic (FM) MMCs. Fig. 1(a) presents a SSJ with θ ¼ 90° (90°-SSJ), which contains a SS with θ ¼ 90° (90°-SS) in the effective scattering region. As shown in Fig. 1(a), the magnetic moments of two FM electrodes are parallel to each other. In order to improve the matching of the SS and the electrodes, two matching regions (L0 and R0 in Fig. 1) are included in the effective scattering region and used to connect the SS and two leads. Each matching region contains four Mn atoms and the magnetization of the matching regions is parallel to that of the electrodes. For all the junction structures in this work, the electrodes and the matching regions are the same as that are shown in Fig. 1(a). We change θ from 0° to 162°, in steps of 18°, to investigate the scattering effect of SS in MMCs. In particular, when θ equals to 0°, all magnetic moments of the Mn atoms have the same directions, therefore the 0°-SSJ is a FM monatomic junction structure (FMJ). For comparison, we also

Fig. 1. Noncollinear magnetic junction structures. (a) The 90°-SSJ, a SS with θ¼ 90° is in the effective scattering region. Considering that the matching regions and the electrodes of all the junctions in this work are the same as that are shown in (a), only the effective scattering regions of AFMJ and MSLJ are shown here. (b) The AFMJ, where the magnetic moments of central five Mn atoms are antiparallel to that of other Mn atoms. (c) The MSLJ, which contains a forced FM domain constituted by five Mn atoms.

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consider the antiferromagnetic junction structure (AFMJ), which contains 21 Mn atoms in the effective scattering region and the magnetic moments of central five atoms are antiparallel to that of other atoms, as shown in Fig. 1(b). Comparing to spin-spiral MMCs, MMCs with MSL contains a forced FM domain consisting of multiple Mn atoms, the magnetic moments of which have the same directions. Here, all the MSL structures are constructed by adding a FM domain to the central part of spin-spiral MMC with θ ¼90° (90°-MMC). Similarly, we can obtain MSL junctions (MSLJs) by attaching these MSL structures to two semi-infinite FM MMCs. The effective scattering region of a MSLJ which contains a FM domain of five Mn atoms is presented in Fig. 1(c). This junction structure is named as MSLJ04, where “04” means that it contains four Mn atoms in the effective scattering region more than its corresponding SSJ [see Fig. 1(a)]. To study the scattering effect of MSL in MMCs, transport properties of from MSLJ01 to MSLJ12 are investigated. 2.2. Computational details Transport properties of MMCs are calculated by means of NEGF [27–29] coupled with a generalized gradient approximation (GGA) [30–33] in the density functional theory (DFT) [34,35]. All calculations of electronic and transport properties were performed by an ab initio DFT code, OPENMX [36]. It is necessary to point out that both GGA and local-density approximation (LDA) [37–39] produce the same results for our systems, due to that the

169

electronic and transport properties are dependent on their magnetic order mainly. The pseudoatomic orbitals (PAOs) located at atoms are used to be basis functions. The PAO basis functions are specified by Mn8.0-s2p2d2, which represents that Mn stands for atomic symbol, 8.0 indicates the cutoff radius (Bohr), and s2p2d2 means that two primitive orbital for s, p and d orbitals are employed. A cutoff energy of 250 Ry is used in the numerical integrations and the solution of Poisson equation. For all junction configurations, the transport direction is along x axis and the transport properties were calculated in three-dimensional supercells, with an interchain separation of 12 Å in y–z plane. In addition, the DFT software, Atomistix Toolkit (ATK) [40,41] is used to calculate the spatially resolved localized density of states (LDOS) at some energy points for several junction configurations. In the ATK calculations, a double-ζ plus polarization basis set is employed and the cutoff energy is chosen as 250 Ry. The noncollinear localdensity approximation (NCLDA) to the exchange-correlation functional is used to deal with the noncollinear magnetism in MMCs.

3. The DFT results and discussion 3.1. Transmission spectra of SSJs In order to investigate the scattering effect of SS in MMCs, we calculated the transmission spectra of FMJ, SSJs (θ ¼ 18–162°) and

Fig. 2. Transmission spectra of the (a) FMJ, (b) 36°-SSJ, (c) 72°-SSJ, (d) 90°-SSJ, (e) 144°-SSJ, and (f) AFMJ.

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Fig. 4. Transmission spectrum and the LDOSs at resonance and off-resonance energies from the ATK calculations. (a) The transmission spectrum of 90°-SSJ in the energy range of (  2.5 eV,  1.0 eV), where a typical asymmetric conductance peak is presented. The LDOS isosurface plots at (b) E ¼  1.39 eV, (c) E¼  2.16 eV and (d) E¼  2.46 eV. The LDOSs in (b) and (d) are at off-resonance energies and the LDOS in (c) is at resonance energy. The regions between two black vertical lines in (b)(d) denote the SS in 90°-SSJ. The value of f near each isosurface plots represents the maximum value of LDOS, in units of states/(Å3eV). These f values indicate the relative magnitudes of all the plots.

Fig. 3. The s- and d-DOS of (a) FMJ and (b) 90°-SSJ.

transmission spectra. 3.2. Fano resonances in SSJs

AFMJ. Transmission spectra of these junction configurations are shown in Fig. 2. For the FMJ, the majority- and minority-bands are independent, resulting in a transmission spectrum with clear steps, as shown in Fig. 2(a). When θ increases from 0°, the two spin bands are hybridized and the transmission curves decrease gradually. This result suggests that the scattering effect of SS becomes stronger by the increasing of θ due to the hybridization of majority- and minority-spins. For AFMJ, the contribution of dstates to the conductance disappears owing to the spin filtering of the AFM domain [the central part in Fig. 1(b)] and therefore AFMJ exhibit a nearly unchanged conductance of 2 G0 (G0 ¼e2/h is the conductance quantum) near the Fermi level, dominated only by sstates. In Fig. 3(a) and (b), the density of states (DOS) of s- and d-states (s- and d-DOS) of FMJ and 90°-SSJ are presented. When θ increases from 0° to 90°, s-DOS remains unchanged nearly while the d-DOS changes greatly. The peaks in d-DOS of 90°-SSJ can be divided into two types, one of which has the corresponding peaks at the same positions in the d-DOS of FMJ (such as the peaks at E¼  2.33 eV, 1.36 eV and 1.78 eV et al.) and another type has no (such as the peaks at E¼  2.66 eV, 2.16 eV and 1.19 eV et al.). The peaks of latter type are newly generated in d-DOS of 90°-SSJ. Comparing the d-DOS and transmission spectrum of 90°-SSJ, it can be found that each newly generated peak in d-DOS has a corresponding conductance peak in the transmission spectrum. Additionally, the dips at E ¼  2.9 eV and 0.9 eV in the d-DOS of 90°-SSJ exhibit the same one to one correspondence to the sharp reductions at the same positions in the transmission spectrum of 90°-SSJ. We calculated the transmission spectra and d-DOS of SSJs with other θ and all the transmission spectra have the same dependence on the d-DOS. Due to the hybridization of majority- and minority- spins in SSJs, the newly generated peaks and valleys in d-DOS are the key factors for the conductance peaks and valleys in the

From the transmission spectra of SSJs in Fig. 2 and the DOS curves in Fig. 3(b), the typical asymmetric peaks of Fano resonance can be observed. This resonance feature is arising from the coupling of the localized d-states and the extended states of the quantum channels. We infer about the physical pictures of these resonances by calculating the transmission spectrum and LDOS of 90°-SSJ at the resonance and off-resonance energies, using the software ATK. In Fig. 4(a), we present transmission spectrum around the characteristic resonance peak at E¼  2.16 eV and this curve is consistent with the result from OPENMX in Fig. 2(d). From the isosurface plots in Fig. 4(b)–(d), LDOS is confined at the spinspiral domain at the resonance energy (E¼  2.16 eV) while LDOSs exhibit the character of extended states at the off-resonance energies (E¼  1.39 eV and 2.46 eV). Considering the results in Section 3.1, the newly generated peak at E ¼  2.16 eV in the d-DOS of 90°-SSJ [Fig. 3(b)] corresponds to the localized LDOS in Fig. 4 (c) and can be attributed to the existence of SS in 90°-SSJ. Therefore the appearance of Fano resonance in 90°-SSJ is caused by the noncollinear magnetism of SS. 3.3. Transport properties of MSLJs In order to investigate the scattering effect of MSL in MMCs, transmission spectra of MSLJs containing 2–13 Mn atoms in the forced FM domain, i.e., from MSLJ01 to MSLJ12, were calculated and shown in Fig. 5. With the increase of the length of the forced FM domain, the number of resonant conductance peaks and valleys increases along with the conductance peaks becoming sharper in the energy ranges of about (  3 eV,  1 eV) and (0.5 eV, 2 eV). As the length of forced FM domain tends to infinity, the peaks will fill in the energy ranges continuously. Similarly, the dDOS exhibits the same peak splitting when the FM domain becomes longer, as shown in Fig. 6. The peak splitting in transmission spectra (Fig. 5) and DOS

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171

Fig. 5. Transmission spectra of from MSLJ01 to MSLJ12.

(Fig. 6) indicates that the Fano resonances are enhanced in MSLJs due to the increasing of the length of the FM domain. By using ATK, we calculated the transmission spectrum and LDOS of MSLJ12

to show the characters of Fano resonances in MSLJs. In Fig. 7(a), the transmission spectrum from ATK exhibits the same feature of conductance splitting with the results from OPENMX in Fig. 5. We

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Fig. 8. The superlattice of the noncollinear magnetic multiple-barriers structure. The orange arrows (the horizontal solid arrows) represent the incoming waves in region 1 and the outgoing waves in region 3. The two blue dashed arrows denote the reflected waves for up-spin and down-spin, respectively. In region 2, the purple arrows (tilt solid arrows) denote the magnetic moments in the Mn barriers. The angle θj represents the direction of magnetic moment in each Mn barrier, with respect to the positive direction of z-axis.

Fig. 6. d-DOS of MSLJ12, which show clearly the phenomenon of peak splitting.

present the LDOS isosurface plots at resonance and off-resonance energies in Fig. 7(b)–(h). Similarly, the Fano resonances in MSLJ12 also originate from the coupling of the localized d-states [Fig. 7 (c) and (g)] and their nearby continuous states of quantum channels [Fig. 7(b), (d), (f), and (h)]. From the LDOS isosurface plots at the resonance energies of  1.81 eV [Fig. 7(c)],  1.89 eV [Fig. 7(e)], and  2.29 eV [Fig. 7(g)], we find that the localized d-states distribute in several separate regions in the FM domain. Furthermore, the number of the separate regions is increasing continuously with the increase of |E|. For example, there are 3, 4, and 5 separate regions in the LDOSs in Fig. 7(c), (e), and (g), respectively. The interesting features of the Fano resonances in MSLJs reveal that there may be more profound physical pictures behind above results of SSJs and MSLJs. In next section, we try to explain the origin of the Fano resonances in noncollinear magnetic MMCs and further study the electron transport in the superlattice of

noncollinear magnetic barriers.

4. Theoretical model and numerical results From above results, the localized resonant d-states are the key factor to the Fano resonances in the transmission spectra of SSJs and MSLJs. These resonant d-states localize in the spin-spiral domain in SSJs while distribute in the FM domain mainly in MSLJs. Especially, the LDOS at resonance energies in MSLJs exhibit a spatial distribution with strong dependence on |E|. It is the existence of SS and MSL in the junction structures give rise to the Fano resonances and the resonant d-states with nontrivial spatial distribution. In this section, we construct a model of noncollinear magnetic multiple-barriers structure as an approximate description of the noncollinear magnetic Mn chains. By calculating the transmission spectrum of this model by the transfer matrix method, we try to give out an explanation of the appearance of the Fano resonance based on the numerical results and study the

Fig. 7. Transmission spectrum and LDOSs of MSLJ12 calculated using ATK. (a) The transmission spectrum of MSLJ12 in the energy range of (  3.0 eV,  0.5 eV). This curve also exhibits the phenomenon of conductance splitting. The LDOS isosurface plots at (b) E¼  1.75 eV, (c) E¼  1.81 eV, (d) E¼  1.86 eV, (e) E¼  1.89 eV, (f) E¼  2.18 eV, (g) E¼  2.29 eV, and (h) E¼  2.38 eV are shown. The LDOSs in (c), (e), and (g) are at resonance energies and the LDOSs in (b), (d), (f), and (h) are at off-resonance energies. All f values are in units of states/(Å3eV).

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electron transport in the superlattice of noncollinear magnetic barriers.

⎛ψ j ⎞ ⎛ϕ j ⎞ ⎜ ↑⎟ ⎜ ↑⎟ U ( ) = θ j ⎜ j⎟ ⎜ j ⎟, ⎝ ψ↓ ⎠ ⎝ ϕ↓ ⎠

(1 ≤ j ≤ N) (4)

4.1. Details of the theoretical model

here, rotation matrix U (θ j ) is

The theoretical model of 1D noncollinear magnetic multiplebarriers structure is presented in Fig. 8. The whole model can be divided into three regions, the regions 1 and 3 are incident and outgoing regions, respectively, and region 2 represents the SS or MSL domain. This model contains N Mn atoms and the magnetic moment of each Mn atom is set to be a certain direction to construct a SSJ or MSLJ, such as shown in region 2 in Fig.8. The transport direction is also along x axis and the distance between two neighboring Mn atoms is d ¼2.4 Å. With any loss of generality, it is assumed that there are no magnetic barriers in the incident region (region 1) and outgoing region (region 3). In a free electron approximation of the spin-polarized conduction electrons localized at each Mn atom, the longitudinal part of the effective one-electron Hamiltonian in the j-th segment can be written as

⎛ θj ⎜ cos 2 ⎜ U (θ j ) = ⎜ θj ⎜− sin ⎝ 2

Hxj = −

2 1⎛ d ⎞ ⎜ ⎟ + U j (x) − h j (x) ⋅σ j . 2 ⎝ dx ⎠

(1)

The system of units we used incorporates unit electron mass and unit Plank constant. Eq. (1) contains three terms contributed by kinetic energy − 1/2(d/dx)2, potential U (x) and magnetic exchange energy −h(x) ⋅σ , where −h(x) is the intrinsic molecular field and σ is Pauli spin operator. In our theoretical model, potential U is assumed to be a constant of U0 . Within each Mn barrier, the magnitude of magnetic moment is also set to be a constant, with h j = h0 (j = 1, ⋯ , N). However, the directions of each h j are determined by the magnetic order of a SSJ or MSLJ and the axis of spin quantization is parallel to the polarization axis of the atomic magnetic field in each Mn barrier in region 2. Therefore, all the Mn barriers have the identical material properties except the directions of magnetic moments. In each Mn barrier, the value of the magnetic exchange energy is also a constant due to the uncharged h j and can be represented as

−σh0 (σ = ± 1), where σ = + 1, − 1 means that the spin quantization direction is parallel and antiparallel to the direction of atomic magnetic field. Owing to above discussions, the potential U0 and magnetic exchange energy −σh0 are not dependent on the order number of the barrier (j). For all Mn barriers, the one-electron energy is

E=

1 2 k σ + U0 − σh0 , 2

(2)

where k σ is electron momentum. Noticing that only real values of k σ need to be considered, U0 ¼  8 eV and h0 ¼ 5 eV are chosen to make k σ a real value in the following discussions. Consider a spin-up incident plane wave with a unit conductance in region 1 (x < 0). Due to the reflection of the interface between regions 1 and 2, the wave function in region 1 will contain spin-down component and can be expressed by plane wave

⎧ L −ik L x ik L x ⎪ ψ↑ = e ↑ + r↑ e ↑ ⎨ , ⎪ ψ L = r e−ik↓L x ↓ ⎩ ↓

(x < 0) (3)

where k↑L = k↓L = 2(E − U0 ) and r↑, r↓ are reflection amplitudes of spin-up and spin-down wave functions. In region 2, which contains N Mn barriers with different magnetic moment directions, the wave function in the j-th Mn barrier can be written as

173

θj ⎞ ⎟ 2 ⎟ , θj⎟ cos ⎟ 2⎠

sin

(5)

where the θj indicates the angle between the magnetic moment direction in j-th Mn barrier and the positive direction of z-axis. Wave functions ϕ↑j and ϕ↓j represent the spin-up and spin-down components in the xyz coordinate system and have following form

⎧ j j ik j x j −ik j x ⎪ ϕ↑ = A↑ e ↑ + B↑ e ↑ ⎨ ⎪ ϕ j = A j eik↓j x + B j e−ik↓j x ↓ ↓ ⎩ ↓

(6)

So the wave functions in the j-th Mn barrier can be expressed as

⎧ ⎛ j j ⎞ ⎪ ψ↑j = ⎜A↑j eik↑ x + B↑j e−ik↑ x⎟ ⎝ ⎠ ⎪ ⎪ θ j ⎛ j ik j x θj j ⎞ cos + ⎜A↓ e ↓ + B↓j e−ik↓ x⎟ sin ⎪ ⎪ ⎝ ⎠ 2 2 ⎨ , θ j ⎛ j ik j x ⎞ j ⎪ j j −ik↑ x ⎟ ⎜A e ↑ sin B e ψ = − + ↑ ⎪ ↓ ⎝ ↑ ⎠ 2 ⎪ θj ⎛ j ik j x j x⎞ ⎪ j − ik + ⎜A↓ e ↓ + B↓ e ↓ ⎟ cos ⎪ ⎝ ⎠ ⎩ 2

(0 ≤ x ≤ Nd)

(7)

where k σj = 2(E − U0 + σh0 ) . In region 3, there are only the transmitted waves

⎧ R ik R x ⎪ ψ↑ = t↑ e ↑ ⎨ , ⎪ ψ R = t eik↓R x ↓ ⎩ ↓

(x > L) (8)

with transmission amplitudes t↑, t↓ to be determined. By matching above wave functions and their derivatives at the interfaces between adjacent Mn barriers, we can get an equation

⎛t↑ ⎞ ⎛T11 ⎛1 ⎞ ⎜ ⎟ ⎜ ⎜r ⎟ ⎜ ↑ ⎟ = T ⎜0 ⎟ = ⎜T21 ⎜t↓ ⎟ ⎜T31 ⎜0 ⎟ ⎜ ⎟ ⎜⎜ ⎜r ⎟ ↓ ⎝ ⎠ ⎝0 ⎠ ⎝T41 ⎛1 ⎜ ⎜2 ⎜ ⎜1 ⎜2 =⎜ ⎜ ⎜0 ⎜ ⎜ ⎜⎜0 ⎝

0

1 2ik↑L

0 −

1 2ik↑L

1 0 2 1 0 2

⎛ ik R Nd ⎜e ↑ ⎜ ⎜0 TM ⎜ R ik↑R Nd ⎜ ik↑ e ⎜⎜ ⎝0

T12 T13 T14 ⎞ ⎛t↑ ⎞ ⎟⎜ ⎟ T22 T23 T24 ⎟ ⎜0 ⎟ T32 T33 T34 ⎟⎟ ⎜t↓ ⎟ ⎟⎜ ⎟ T42 T43 T44 ⎠ ⎝0 ⎠ ⎞ ⎟ ⎟ ⎟ ⎟ 0 ⎟ ⎟ 1 ⎟ 2ik↓L ⎟⎟ 1 ⎟ − ⎟ 2ik↓L ⎟⎠ 0

0 0 R

0 eik↓ Nd 0 0 R

0 ik↓R eik↓ Nd

⎞ 0⎟ ⎛t ⎞ ↑ ⎟⎜ ⎟ 0 ⎟ ⎜0 ⎟ ⎟ 0⎟ ⎜t↓ ⎟ ⎜ ⎟ ⎟⎟ ⎝0 ⎠ 0⎠

Here, the matrix TM can be expressed as

(9)

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N

TM =

∏ Tj j=1

⎡ j 2 2 ⎢ cos φ cos k↑ d + sin φ ⎢ j ⎢ cos k↓ d ⎢ ⎢ ⎢ ⎢ − sin φ cos φ ⎢ cos k↑j d − ⎢ ⎢ cos k↓j d N ⎢ ⎢ =∏⎢ j k↑ cos2φ sin k↑j d j=1 ⎢ ⎢ + k↓j sin2φ sin k↓j d ⎢ ⎢ ⎢ ⎢ ⎢ j ⎢ −k↑ sin φ cos φ ⎢ j j ⎢ sin k↑ d + k↓ sin φ ⎢ j ⎣ cos φ sin k↓ d

− sin φ cos φ

( cos k↑j d − cos k↓j d) sin2φ cos k↑j d

(

+ cos2φ cos k↓j d

)

−

cos2φ sin k↑j d k↑j

−

−k↑j sin φ cos φ sin k↑j d + k↓j sin φ cos φ sin k↓j d

+ k↓j sin2φ 4.2. Numerical results

sin k↓j d

sin2φ sin k↓j d

⎤ ⎥ ⎥ k↑j ⎥ sin φ cos φ sin k↓j d ⎥ − ⎥ k↓j ⎥ ⎥ j 2 sin φ sin k↑ d ⎥ − ⎥ k↑j ⎥ ⎥ cos2φ sin k↓j d ⎥ − ⎥ k↓j ⎥ ⎥ − sin φ cos φ ⎥ j ⎥ cos k↑ d − ⎥ ⎥ cos k↓j d ⎥ ⎥ sin2φ cos k↑j d ⎥ ⎥ + cos2φ cos k↓j d ⎦ sin φ cos φ sin k↑j d

k↓j

sin φ cos φ sin k↑j d k↑j −

Fig. 9. The transmission spectra of 45°-SSJ and 135°-SSJ from the theoretical model.

k↑j cos2φ sin k↑j d

sin φ cos φ sin k↓j d k↓j

cos2φ cos k↑j d + sin2φ cos k↓j d

(

)

(

− sin φ cos φ cos

k↑j d

− cos

k↓j d

)

(10)

where Tj is the transfer matrix for the j-th Mn barrier. Transmission amplitudes t↑, t↓ can be solved based on Eqs. (9) and (10) and can be written as

⎧ T33 ⎪t↑ = ⎪ T11T33 − T13 T31 ⎨ . T31 ⎪ = t ↓ ⎪ T13 T31 − T11T33 ⎩

(11)

Based on the values of t↑, t↓in Eq. (11), the transmission coefficient T (E) for electron tunneling through noncollinear magnetic multiple Mn barriers can be expressed as

T (E) =

(

t↑⁎

= t↑ =

0

2

+ t↓

2

T33 T11T33 − T13 T31

n

π = d, kσ

2

2

+

T33 . T11T33 − T13 T31

(12)

(13)

a standing-wave-like interference occurs in the SSJs. This standing-wave-like interference means that the frequencies of incoming and reflected waves are the same while the amplitudes are different. Therefore the interference is enhanced if the vibration directions of incoming and reflected waves are same while the interference is weakened when the vibrations of two waves are in the opposite directions. Combined with the Eq. (2), the interferences for spin-up or spin-down components take place at the energies of

E=

⎛t↑ ⎞ ⎜ ⎟ 0 ) ⎜0 ⎟ ⎜t↓ ⎟ ⎜ ⎟ ⎝0 ⎠

t↓⁎

Based on Eq. (12), we calculated the transmission spectra of SSJs and MSLJs. In order to show the scattering effect of SS in MMCs, transmission spectra of two SSJs which contain the same number of Mn atoms, i.e., 45°- and 135°-SSJs, were calculated and the results are shown in Fig. 9. When θ increases from 45° to 135°, transmission spectrum exhibits an obvious reduction, which indicates clearly that the scattering effect becomes stronger with increase of the separate angle θ. From the transmission spectra in Fig. 9, characteristic asymmetry of Fano resonances can be observed clearly. From a point view of semi-classical physical picture, we try to explain the origin of the resonances. At the interface between two adjacent barriers, the incoming waves are reflected and the interference of the incoming and reflected waves occurs in each Mn barrier. Usually, the interference of incoming and reflected waves will introduce some changes in the conductance, such as the reductions in the transmission spectra in Fig. 9. If the distance between the two nearest-neighboring Mn atoms is the integer multiple of the half-wavelength (π /k σ ), that is

1 ⎛⎜ π ⎞⎟2 2 n + U0 − σh0 , 2 ⎝d ⎠

(14)

where the energy E is determined by d , U0 , h0 and not associated with θ . This result can be confirmed by the fact that the most significant conductance reduction occurs in the same energy ranges, of (1.07 eV, 1.74 eV) and (3.9 eV, 4.73 eV) for both SSJs with θ ¼ 45° and 135°. Based on above discussions, both the resonant peaks and valleys in transmission spectra can be attributed to the interference of the incoming and reflected waves in the Mn barriers. For a real MMC with SS, the incoming waves are reflected by the noncollinear magnetic moments of Mn atoms and the

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175

Fig. 11. The 90°-MMCs with different chirality. (a) Left-handed 90°-MMC. (b) Righthanded 90°-MMC. (c) Cycloid 90°-MMC.

Fig. 10. The transmission spectra of (a) 90°-SSJ, (b) MSLJ10 and (c) MSLJ20.

interference occur in the spin-spiral region, resulting in that the resonant localized states distribute in the spin-spiral domain [see Fig. 4(c)]. These resonant localized states are coupled with extended states of quantum channels, therefore contribute partially to the conductance. Using the theoretical model, the transmission spectra of the MSLJs with θ ¼ 90° were calculated. The transmission spectrum of 90°-SSJ is shown in Fig. 10(a). In Fig. 10(b) and (c), the transmission spectra are corresponding to the MSLJs containing FM domain with 11 Mn atoms (MSLJ10) and 21 Mn atoms (MSLJ20), respectively. With the increase of the length of FM domain, the conductance splitting occurs in the transmission spectra. Noticing that the tunneling channels of up-spin and down-spin are independent in FM domain, the FM domain can be treated as a single Mn barrier with a length of D = NFM d , where NFM means the number of Mn atoms in FM domain. Therefore, the energy E where the interference of incident and reflected ones occur and the standing waves are formed in the FM domain can be determined by

E=

1 ⎛⎜ π ⎞⎟2 2 n + U0 − σh0 . 2 ⎝D⎠

(15)

This equation provides more interference energy points due to

that D is larger than d, leading to the conductance splitting in transmission spectra in Fig. 10(b) and (c). Furthermore, when |E| increases, n is also increasing, which means there will be more regions with enhanced and destructive interference appearing in the FM domain. Reflected in the real MMCs with MSL, this result indicates that more separate regions, where the resonant d-states distribute in, will appear in the FM domain, as shown in Fig. 7(c), (e), and (g). Based on the numerical results from the theoretical model, the appearance of the Fano resonances can be ascribed to the interference of the incoming and reflected waves in each Mn barrier. These numerical results exhibit the same features to the results from the first-principles calculations, indicateing that the phenomena of Fano resonances and resonant peak splitting in SSJs and MSLJs exist universally in the superlattice of noncollinear magnetic barriers and are expected to be found in other 1D or quasi-1D noncollinear magnetic nanoscale structures.

5. Effect of SOC In above discussions, we just focus on the scattering effect of SS and MSL in MMCs, without considering the SOC. However, SOC may introduce some important changes to the conductance of noncollinear magnetic MMCs. In this section, we investigate the

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x . The inset in (b) is the Fig. 12. BSs of FM MMC and 90°-MMC. (a) BS of FM MMC without SOC. (b) BS of FM MMC with SOC for magnetization parallel to the chain axis, M∥^ amplification of the region in the small square. (c) BS of FM MMC with SOC for magnetization perpendicular to the chain axis, M⊥^ x . The area within the rectangle in upper left corner in (c) is manifested and shown in (d). (e) BS of 90°-MMC without SOC. BSs of (f) Left-handed and (g) right-handed 90°-MMC with SOC. (h) BS of the cycloid 90°MMC with SOC.

effect of SOC on the electronic properties and conductance of the FM and spin-spiral MMCs. We consider the band structures of FM MMC and left-handed, right-handed and cycloid spin-spiral MMCs with θ ¼ 36°, 72°, 90°, 108°, 144° first. As examples, the structures of left-handed, righthanded and cycloid 90°-MMC are shown in Fig. 11. In Fig. 12(a)–(c), we present the BSs of FM MMC. Energy bands A1 and A2 are doubly degenerate with absence of SOC in Fig. 12(a). When SOC is considered, the band structures of FM MMCs exhibit some dramatic results. If the magnetization is parallel to the wire axis, M∥^ x, two doublets A1 and A2 split into two singlets, respectively. However, when the magnetization is perpendicular to the chain axis, M⊥^ x , the splitting of bands is much smaller than M∥^ x from the band structure in Fig. 12(c), which shows little differences compared to the band structure in Fig. 12(a). The area within red rectangle in Fig. 12(c) is manifested and shown in Fig. 12(d), where the splitting of bands only occurs at the crossing points of energy bands. The BSs of 90°-MMC are shown in Fig. 12(e)–(h). Compared to the band structure without SOC in Fig. 12(e), splitting of energy

bands are clearly seen in the band structures of left- and righthanded 90°-MMC in Fig. 12(f) and (g). For cycloid 90°-MMC, the band structure exhibits splitting only near the band crossing points [Fig. 12(h)]. The BSs of the spin-spiral MMCs with other θ exhibit the same results. These results reveal that the BSs of FM MMC and spin-spiral MMCs exhibit a strong dependence on the spatial arrangement of the magnetic moments of Mn atoms. Different from the results in the work of Velel et al., the splitting of bands cannot bring any change to the conductance at the Fermi level due to that both two nondegenerate subbands obtained from the doubly degenerate band cross over the Fermi level, such as shown in Fig. 12(b), (f), and (g). For FMJ and SSJs, the effect of SOC comes from the splitting of bands at the band crossing points. It can be predicted that these spitting of bands will bring conductance reduction within a very narrow energy range, i.e., conductance dips, in the transmission spectra. In Fig. 13, we give out the transmission spectra of FMJ with M⊥^ x and righthanded 90°-SSJ. The positions of conductance dips are consistent with the band crossing points where the splitting of bands occurs.

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of FMJ and SSJs. All these results may be useful to understand the electronic and transport properties of some 1D or quasi-1D noncollinear magnetic nanoscale structures. On the other hand, they also may be helpful for the preparation of atomic-scale devices.

Acknowledgements This work was supported by the National Natural Science Foundation of China (NSFC51032002, NSFC11247033, NSFC11374162 and NSFC11347129), the key project of National High Technology Research and Development Program (“863” Program) of China (2011AA050526), the Science and Technology Support Plan of Jiangsu Province (BE2011191).

References

Fig. 13. Transmission spectra of (a) FMJ with M⊥^ x and (b) right-handed 90°-SSJ with SOC.

These slight changes in transmission spectra of FMJ and SSJs are arising from the weak SOC for Mn atoms. However, according above results, the SS may be an effective way to control the conductance at the Fermi level for other 1D or quasi-1D nanoscale structures with stronger SOC.

6. Conclusions We investigated the scattering effect of SS and MSL on the ballistic transport properties of 1D single-atomic chains using the MMCs as a model system. For SSJs, conductance curves reduce when θ increases, which indicates that the scattering effect becomes stronger with the increasing of θ. Furthermore, the Fano resonance is found to exist in the SSJs owing to the existence of SS. We find that these Fano resonances can be attributed to the coupling of localized d-states and the extended states of quantum channels, from the LDOSs at resonance and off-resonance energies. For MSLJs, the Fano resonances are enhanced and the transmission spectra exhibit a dramatic conductance splitting when the length of the FM domain increases. From the LDOS, the localized d-states distribute in some separate regions in the FM domain and the number of these separate regions increases continuously with the increasing of the |E|. By means of the theoretical model based on the free electron approximation, the formation of the resonant states can be attributed to the interference of the incident wave functions and reflected ones. Considering the universality of the superlattice of noncollinear magnetic barriers, the phenomena of Fano resonances and conductance splitting are expected to occur in other 1D atom-scale structures with SS and MSL. In addition, we also consider the effect of SOC on the electronic and transport properties of FM MMC and spin-spiral MMCs. The BSs of FM MMC and spin-spiral MMCs exhibit a strong dependence on the spatial arrangements of the magnetic moments of Mn atoms. Moreover, SOC will bring some conductance dips in the transmission spectra

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