Figure 4 - available from: Scientific Reports
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Amplitude of the MZMs wave-functions (a) in the vortex core; and (b) at the edge. The horizontal axis indicates the lattice site along (0,1) direction with the vortex core centered at (0,0), the center of the square lattice. The wave-functions of electron and hole components are identical for each spin, consistent with the requirement for Majorana fermion.
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Sau, Lutchyn, Tewari and Das Sarma (SLTD) proposed a heterostructure consisting of a semiconducting thin film sandwiched between an s-wave superconductor and a magnetic insulator and showed possible Majorana zero mode. Here we study spin polarization of the vortex core states and spin selective Andreev reflection at the vortex center of the SLTD mo...
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... large system. In a finite size system, the vortex core state and the edge state have a hybridization, leading to a pair of the MZMs with energies ±E 0 very close to zero. Our numerical calculations agree with this analysis and the calculated E 0 ∼ 10 −6 . By linear recombination of the two MZMs, we find a MZM localized in the vortex core, [see Fig. 4(a)], and the other one is localized at the edge [see Fig. 4(b)]. Both satisfy the MZM condition γ † = γ in Eq. (6) to a high accuracy. Note that we only consider a single vortex in our model calculation, since the edge MZM will be easily destroyed in real system, where there are many vortices. Below we will only focus on the bound ...
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... and the edge state have a hybridization, leading to a pair of the MZMs with energies ±E 0 very close to zero. Our numerical calculations agree with this analysis and the calculated E 0 ∼ 10 −6 . By linear recombination of the two MZMs, we find a MZM localized in the vortex core, [see Fig. 4(a)], and the other one is localized at the edge [see Fig. 4(b)]. Both satisfy the MZM condition γ † = γ in Eq. (6) to a high accuracy. Note that we only consider a single vortex in our model calculation, since the edge MZM will be easily destroyed in real system, where there are many vortices. Below we will only focus on the bound states in the vortex core. As shown in Fig. 4(a), one can see ...
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... is localized at the edge [see Fig. 4(b)]. Both satisfy the MZM condition γ † = γ in Eq. (6) to a high accuracy. Note that we only consider a single vortex in our model calculation, since the edge MZM will be easily destroyed in real system, where there are many vortices. Below we will only focus on the bound states in the vortex core. As shown in Fig. 4(a), one can see that the MZM's wave-function at the center of the vortex is fully polarized with spin-up: |u ↑ | = 0 and |u ↓ | = 0. As a comparison, the wave function of the first excited state is shown in Appendix A, which is spin-down at the center of the vortex. ...
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Citations
... In addition, in the course teaching process, the course team will introduce some cutting-edge developments and topics into the teaching process to promote students to understand the connection between basic knowledge and frontier and enhance their interest in exploring frontier. Examples include gravitational wave detection, CCD technology, topological superconductivity, spin flow, and so on [15][16][17][18]. ...
Many aspects of college physics teaching can be combined with moral education. The article integrates ideological and political elements into the teaching of college physics courses from four aspects, which can stimulate students’ spirit of exploration and innovation, improve students’ scientific literacy, cultivate students’ scientific attitude of seeking truth from facts and establish a dialectical materialist world view. Take the derivation of wave function as an example, guide students to analyze from the perspective of dialectical materialism methodology, analyze the essence of wave from the micro level, and finally get the wave function. At the same time, from the physical meaning of wave function, students should be guided not to be tempted by foreign objects and adhere to their own hearts. Help them establish a correct outlook on life, values and world outlook. Then various educational elements are cleverly integrated into the teaching content to enhance the effect of education.
... The ef ciency of these algorithms is sensitive to the matrix size. In constructing the tight-binding Hamiltonian matrix, the discretization is normally performed by using nite-difference schemes on uniform grids [41] where the inter-grid distance must be much smaller than any characteristic length in the system. In our device, the MBS wavefunctions would be strongly localized in a tiny fraction of the device, such as the vortex core and the device edges. ...
... The elements of G R S (i j; i j) is solved by the above linear equation with H in sparse matrix form. This can be numerically obtained by high-performance algorithms such as the Intel MKL PARDISO Solver [41,46]. ...
Scanning tunneling microscopy (STM) is an indispensable tool in detecting Majorana bound states (MBSs) in vortices of topological superconductors. By reducing the computational complexity via non-uniform grids, we systematically study the tunnel coupling as well as the temperature dependence of the differential conductance of MBSs in two dimensional devices. Numerical results show that the conductance peak approaches the quantized value 2e 2/h in strong coupling limit at low temperatures which are characteristic features of MBSs. More interestingly, a conductance local minimum in the spatially scanning is observed when the STM tip is placed at the vortex center. The dip structure can be enhanced with increased temperature or enlarged vortex size. We ascribe this observation to the sensitivity of the Andreev reflection processes of carriers at the vortex center where the thermal energy could be comparable to the vanishing pair potential. We also investigate the STM of two-vortex systems where the hybridization of the vortices can lead to oscillatory behavior of the state energy. With small inter-vortex distances, the original MBSs in vortices can merge into topologically trivial states and the conductance peak can be significantly suppressed.
... Over the years, AR has been employed as a tool in a wide variety of problems ranging from distinguishing between singlet and triplet states [4] to topological phase transitions [5] to experimental signatures [8,9] of Majorana fermions [14]. Also, intriguing transport properties of topological superconductors [15,16] and junctions of superconductors with topological insulators [17] have been understood in terms of AR. ...
We study nonlocal transport in a two-leg Kitaev ladder connected to
two normal metals. The coupling between the two legs of the ladder
when the legs are maintained at a (large) superconducting phase
difference, results in the creation of subgap Andreev states. These
states in turn are responsible for the enhancement of crossed Andreev
reflection. We find that tuning the different parameters of the system
suitably leads to enhancement of crossed Andreev reflection signalled
by transconductance acquiring the most negative value
possible. Furthermore, subgap states cause oscillations of the
transconductance as a function of various system parameters such as
chemical potential and ladder length, which are seen to be a
consequence of Fabry-Pérot resonance.
... As the authors demonstrate, it can be reversed by the change of the direction of magnetic field. Recently it has also been demonstrated theoretically, that the Majorana spin polarization in the vortex is totally parallel to the external magnetic field [32]. ...
... It is instructive to analyze the contributions to the density of states originating from different ABS levels, and in particular from the Majorana bound states. To do so we start from the Lehmann representation of the QD Green's function [40,41], which in T = 0 has the particle and hole parts: (32) where the subscript g denotes ground state (E g = 0), and the set of |φ i and E ABS i (i = 1, .., 12) constitutes the eigen-spectrum of the Hamiltonian. In our discussion it is convenient to rewrite (33) and (34) in the terms of the second quantized operators γ i corresponding to the E ABS i levels. ...
We consider a model of a Josephson junction of two topological superconducting wires mediated by an interacting quantum dot. An additional normal electrode coupled to the dot from the top allows to probe its density of states. The Majorana states adjacent to the dot hybridize across the junction and from a bound state in the dot. The dot is subjected to the effective magnetic field arising from the superposition of the fields driving each wire into topological states, which, dependent on the angle between the fields, introduces variable Zeeman splitting of the dot active level. We show that electron interactions in the dot diminish the characteristic for Majoranas zero bias peak arising in the transverse conductance through the dot and introduce an overall asymmetry of the conductance. They also renormalize the hybridization between the end-state Majoranas in shorter wires. The Majorana spin polarization is determined by the effective magnetic field in the dot. Phase-biased Josephson current exhibits spin polarization in thermal equilibrium, which possesses characteristic [Formula: see text] periodicity, and its sign can be switched when an unpaired Majorana state is present in the junction. We also observe spin-dependent Majorana state 'leaking', which can be controlled by the position of the dot level in energy scale.
... The interaction between ferromagnetism and superconductivity can also stimulate the creation of odd-frequency (or odd-time) spin-triplet Cooper pairs with m = 0, ±1 spin projections on the local quantization axis [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32]. The generation of these triplet correlations has a few experimental signatures, including a nonmonotonic variation of the critical temperature when the magnetization vectors in SF 1 F 2 hybrids undergo incommensurate rotations [33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48]. ...
We theoretically study self-consistent proximity effects in finite-sized systems consisting of ferromagnet (F) layers coupled to an s-wave superconductor (S). We consider both SF1F2 and SH nanostructures, where the F1F2 bilayers are uniformly magnetized and the ferromagnetic H layer possesses a helical magnetization profile. We find that when the F1F2 layers are weakly ferromagnetic, a hard gap can emerge when the relative magnetization directions are rotated from parallel to antiparallel. Moreover, the gap is most prominent when the thicknesses of F1 and F2 satisfy dF1≤dF2, respectively. For the SH configuration, increasing the spatial rotation period of the exchange field can enhance the induced hard gap. Our investigations reveal that the origin of these findings can be correlated with the propagation of quasiparticles with wave vectors directed along the interface. To further clarify the source of the induced energy gap, we also examine the spatial and energy-resolved density of states, as well as the spin-singlet and spin-triplet superconducting correlations, using experimentally accessible parameter values. Our findings could be beneficial for designing magnetic hybrid structures where a tunable superconducting hard gap is needed.
... The interaction between ferromagnetism and superconductivity can also stimulate the creation of odd-frequency (or oddtime) spin-triplet Cooper pairs having m = 0, ±1 spin projections on the local quantization axis. [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32] The generation of these triplet correlations have a few experimental signatures, including a non-monotonic variation of the critical temperature when the magnetization vectors in SF 1 F 2 hybrids undergo incommensurate rotations. [43][44][45][46][47][48][49][52][53][54][55][56][57][58][59][60] For both the SH and SF 1 F 2 structures, all three triplet components can be induced simultaneously. ...
We theoretically study self-consistent proximity effects in finite-sized systems consisting of ferromagnet ($\rm F$) layers coupled to an $s$-wave superconductor ($\rm S$). We consider both $\rm SF_1F_2$ and $\rm SH$ nanostructures, where the $\rm F_1 F_2$ bilayers are uniformly magnetized, and the ferromagnetic $\rm H$ layer possesses a helical magnetization profile. We find that when the $\rm F_1 F_2$ layers are weakly ferromagnetic, a hard gap can emerge when the relative magnetization directions are rotated from parallel to antiparallel. Moreover, the gap is most prominent when the thicknesses of $\rm F_1$ and $\rm F_2$ satisfy $\rm d_{F1}\leq d_{F2}$, respectively. For the $\rm SH$ configuration, increasing the spatial rotation period of the exchange field can enhance the induced hard gap. Our investigations reveal that the origin of these findings can be correlated with the propagation of quasiparticles with wavevectors directed along the interface. To further clarify the source of the induced energy gap, we also examine the spatial and energy resolved density of states, as well as the spin-singlet, and spin-triplet superconducting correlations, using experimentally accessible parameter values. Our findings can be beneficial for designing magnetic hybrid structures where a tunable superconducting hard gap is needed.
