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The quantum transport behavior of the normal-superconductor-normal (NSN) junction is studied theoretically based on a type-II Weyl semimetal which is rotated a certain angle. The calculation results show that the orientation angle determines the scattering mechanism of the system. In the NSN junction, there exist simultaneously two local Andreev reflections (ARs) (retro AR and specular AR) and two local election transmissions (ETs) (retro ET and specular ET) when the orientation angle is small. Moreover, the retro AR is gradually suppressed with the further increase of the orientation angle. When the orientation angle exceeds the critical angle, the scattering mechanism in NSN junction is the same as that of the NSN junction in normal mental, i.e. the normal electron reflection, normal electron transmission, retro Andreev reflection and crossed Andreev reflection take place simultaneously. In addition, the total conductance of the system is unaffected by the chemical potential, nor by the incident angle when the orientation angle is smaller than the critical angle, but decreases with the increase of the incident angle when the orientation angle is greater than the critical angle. The conductance of crossed Andreev reflection increases with incident angle increasing under some conditions.
[1] Chang G Q, Xu S Y, Sanchez D S, Huang S M, Lee C C, Chang T R, Bian G, Zheng H, Belopolski I, Alidoust N, Jeng H T, Bansil A, Lin H, Hasan M Z 2016 Sci. Adv. 2 e1600295
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图 1 (a)第二类Weyl半金属的NSN结倾斜θ角度的示意图. 在(b)
$ \theta<\theta_{{\rm{c}}} $ 和(c)$ \theta>\theta_{{\rm{c}}} $ 两种情况下$ k_y $ 和$ k_z $ 一定时NSN结的能谱, 其中红色(蓝色)表示导带(价带), 实线(虚线)表示电子(空穴)Figure 1. (a) Schematic diagram of the NSN junction based on type-II WSM when changing the orientation angle θ. Energy spectra with finite
$ k_y $ and$ k_z $ of the NSN junction for (b)$ \theta<\theta_{{\rm{c}}} $ and (c)$ \theta>\theta_{{\rm{c}}} $ , where the conduction (valence) bands are colored with red (blue), and the solid (dashed) lines denote electrons (holes)
图 2
$\theta < \theta_{{\rm{c}}}$ 时, 在入射能分别为(a)$ E=0 $ 和(b)$E= 0.6$ 时Andreev反射和电子隧穿的系数随倾斜角θ的变化. 相关参数为v1 = 1.1, v2 = 1, µ = 0.5, UL = 0, UR = 0, α = 30°, U = 100,$\varDelta=1$ , L = 10ξFigure 2. Andreev reflection and electron transmission coefficients as functions of orientation angle θ for (a)
$ E=0 $ and (b)$ E=0.6 $ when$ \theta<\theta_{{\rm{c}}} $ . Parameters: v1 = 1.1, v2 = 1, µ = 0.5, UL = 0, UR = 0, α = 30°, U = 100,$ \varDelta=1 $ , L = 10ξ
图 3
$\theta < \theta_{{\rm{c}}}$ 时, Andreev反射和电子隧穿的系数随入射能E的变化 (a), (c)$ v_1=1.1 $ ; (b), (d)$ \mu=0.5 $ . 相关参数取值为$\theta=3^{\circ}, $ $ \alpha=30^{\circ},\; v_2=1, \;U_{{\rm{L}}}=0, \; U_{{\rm{R}}}=0, \; U=100, \; \varDelta=1, \; L=10\xi$ Figure 3. Andreev reflections and electron transmission coefficients as a function of the incident energy E when
$ \theta<\theta_{{\rm{c}}} $ : (a), (c)$ v_1= $ 1.1; (b), (d)$ \mu=0.5 $ . Parameters:$ \theta=3^{\circ} $ ,$ \alpha=30^{\circ}, \; v_2=1, \; U_{{\rm{L}}}=0, \; U_{{\rm{R}}}=0, \; U=100, \; \varDelta=1, \; L=10\xi$ 
图 4
$\theta < \theta_{{\rm{c}}}$ 时, Andreev反射和电子隧穿的系数随超导体长度L的变化 (a)$ E=0; $ (b)$ E=0.6 $ . 相关参数:$v_1= $ $ 1.1,$ $ v_2=1, $ $ \mu=0.5, $ $U_{{\rm{L}}}=0, $ $U_{{\rm{R}}}=0, $ $\theta=3^{\circ}, $ $\alpha= $ $ 30^{\circ},$ $ U=100, \; \varDelta=1$ Figure 4. Andreev reflections and electron transmissions coefficients as a function of the SC region length L with incident energy (a)
$ E=0 $ and (b)$ E=0.6 $ when$ \theta<\theta_{{\rm{c}}} $ . Parameters:$v_1=1.1, $ $ v_2=1, $ $\mu=0.5, $ $U_{{\rm{L}}}=0, $ $U_{{\rm{R}}}=0, $ $\theta=3^{\circ}, $ $\alpha=30^{\circ}, $ $U=100, $ $ \varDelta=1 $ 
图 5
$\theta > \theta_{{\rm{c}}}$ 时, Andreev反射和电子隧穿的系数随倾斜角θ的变化. 插图为散射系数随入射能E的变化. 相关参数:$ E=0, $ $v_1=1.3, $ $v_2=1, $ $\mu=0, $ $U_{{\rm{L}}}=0,$ $U_{{\rm{R}}}=0,$ $\alpha=85^{\circ}, $ $U=100, $ $\varDelta=1,$ $ L=\xi $ Figure 5. Andreev reflections (
$ R_{{\rm{ee}}} $ and$ R_{{\rm{eh}}} $ ) and electron transmissions coefficients ($ T_{{\rm{ee}}} $ and$ T_{{\rm{eh}}} $ ) as a function of orientation angle θ when$ \theta>\theta_{{\rm{c}}} $ . The inset shows the scattering coefficients with the increment of the incident energy E. Parameters:$ E=0, $ $v_1=1.3, $ $v_2=1, $ $\mu=0, $ $U_{{\rm{L}}}=0, $ $ U_{{\rm{R}}}=0, $ $\alpha=85^{\circ}, $ $U=100, $ $\varDelta=1, $ $ L=\xi $ 
图 6 总电导G和交叉Andreev反射电导
$ G_{{\rm{CAR}}} $ 随(a), (b)倾斜角θ和(c), (d)偏压$ eV $ 的变化. 相应参数: (a), (b)$\mu=0.5,\; L= $ $ 10\xi$ ; (c)$ \; \theta=59^{\circ}, \; \alpha=70^{\circ}, L=\xi $ ; (d)$\; \theta=59^{\circ}, \; \alpha=70^{\circ},\; L=10\xi.$ 其他参数:$v_1=2/\sqrt{3}, \; v_2=1, \; U_{{\rm{L}}}=0, \; U_{{\rm{R}}}=0, \; U=100, \; $ $ \varDelta=1, \; \tilde{q}_m=10$ Figure 6. Conductance G and its CAR component
$ G_{{\rm{CAR}}} $ dependence of (a), (b) orientation angle θ and (c), (d) bias votage$ eV $ . Parameters: (a), (b)$ \mu=0.5, L=10\xi $ ; (c)$ \; \theta=59^{\circ}, \; \alpha=70^{\circ}, L=\xi $ ; (d)$ \; \theta=59^{\circ}, \; \alpha=70^{\circ}, L=10\xi $ . Other parameters:$v_1=2/\sqrt{3}, $ $ \; v_2=1, \; U_{{\rm{L}}}=0, \; U_{{\rm{R}}}=0, \; U=100, \; \varDelta=1, \; \tilde{q}_m=10$ 表 1 各参数对散射系数的影响
Table 1. Influence of various parameters on scattering coefficient
倾斜角
（$0<\theta<90^{\circ}$）参数 散射系数的变化 $0^{\circ} < \theta\leqslant\theta_{ {\rm{c} } }$ 倾斜角$(\theta)$ θ较小时, Andreev反射占主导; 随着θ的增加, Andreev反射逐渐被抑制; 当$\theta=\theta_{{\rm{c}}}$ 时, 只发生径向电子隧穿$T_1$, 其他散射过程被完全抑制. 入射能$(E)$ $E < \varDelta$ 时, 只发生Andreev反射, 电子隧穿被完全抑制. $E > \varDelta$ 时, Andreev反射随E的增加逐渐被抑制, 其类型取决于$E, \mu, v_1$的关系. 化学势$(\mu)$ $E<\mu$时, 只发生径向Andreev反射($A_1$); $E>\mu$时, 只发生镜面Andreev反射($A_2$), $E=\mu$是两种Andreev 反射互相转换的临界点. 倾斜系数$(v_1)$ $v_1\neq 1.1$, 两种Andreev反射共存; $v_1=1.1$, 只存在一种Andreev反射, 并且$A_1$随着$v_1$的增加而增大, $A_2$的变化趋势与$A_1$正相反. 超导体长度$(L)$ L较小时, Andreev 反射系数单调递增; $L>3\xi$以后, Andreev反射系数稳定在1. $\theta_{{\rm{c}}}<\theta<90^{\circ}$ 倾斜角$(\theta)$ θ较小时, 交叉Andreev反射($T_{{\rm{eh}}}$)被完全抑制; 随着θ的增加, 电子正常反射($R_{\rm ee}$)和局域Andreev反射($R_{{\rm{eh}}}$)增强, 电子隧穿($T_{\rm ee}$)减小, 交叉Andreev反射($T_{{\rm{eh}}}$)先增大后减小, 散射系数而呈振荡型变化. 入射能$(E)$ 交叉Andreev反射($T_{{\rm{eh}}}$)效果最好时, 随着E的增加, $R_{\rm ee}$单调递增, $T_{\rm ee}$和$T_{{\rm{eh}}}$先增大后减小, 而$R_{{\rm{eh}}}=0$, 其中$T_{{\rm{eh}}}$的最大值可达到0.43. 
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[1] Chang G Q, Xu S Y, Sanchez D S, Huang S M, Lee C C, Chang T R, Bian G, Zheng H, Belopolski I, Alidoust N, Jeng H T, Bansil A, Lin H, Hasan M Z 2016 Sci. Adv. 2 e1600295
Google Scholar
[2] Xu S Y, Belopolski I, Alidoust N, Neupane M, Bian G, Zhang C, Sankar R, Chang G, Yuan Z, Lee C C, Huang S M, Zheng H, Ma J, Sanchez D S, Wang B K, Bansil A, Chou F, Shibayev P P, Lin H, Jia S, Hasan M Z 2015 Science 349 613
Google Scholar
[3] Xu S Y, Belopolski I, Sanchez D S, Zhang C, Chang G, Guo C, Bian G, Yuan Z, Lu H, Chang T R, Shibayev P P, Prokopovych M L, Alidoust N, Zheng H, Lee C C, Huang S M, Sankar R, Chou F, Hsu C H, Jeng H T, Bansil A, Neupert T, Strocov V N, Lin H, Jia S, Zahid Hasan M 2015 Sci. Adv. 1 e1501092
Google Scholar
[4] Zheng H, Hasan M Z 2018 Adv. Phys. X 3 1466661
Google Scholar
[5] Zhang K X, Du Y P, Wang P D, Wei L M, Li L, Zhang Q, Qin W, Lin Z Y, Cheng B, Wang Y F, Xu H, Fan X D, Sun Z, Wan X G, Zeng C G 2020 Chin. Phys. Lett. 37 090301
Google Scholar
[6] Zhou Z Z, Liu H J, Wang G Y, Wang R, Zhou X Y 2021 Chin. Phys. Lett. 38 077101
Google Scholar
[7] Morishima K, Kondo K 2021 J. Appl. Phys. 129 125104
Google Scholar
[8] Meng W, Zhang X, He T, Jin L, Dai X, Liu Y, Liu G 2020 J. Adv. Res. 24 523
Google Scholar
[9] Huang L, McCormick T M, Ochi M, Zhao Z, Suzuki M T, Arita R, Wu Y, Mou D, Cao H, Yan J, Trivedi N, Kaminski A 2016 Nat. Mater. 15 1155
Google Scholar
[10] Jiang J, Liu Z K, Sun Y, Yang H F, Rajamathi C R, Qi Y P, Yang L X, Chen C, Peng H, Hwang C C, Sun S Z, Mo S K, Vobornik I, Fujii J, Parkin S S P, Felser C, Yan B H, Chen Y L 2017 Nat. Commun. 8 13973
Google Scholar
[11] Tamai A, Wu Q S, Cucchi I, Bruno F Y, Riccò S, Kim T K, Hoesch M, Barreteau C, Giannini E, Besnard C, Soluyanov A A, Baumberger F 2016 Phys. Rev. X 6 031021
[12] Xu S Y, Alidoust N, Chang G, Lu H, Singh B, Belopolski I, Sanchez D S, Zhang X, Bian G, Zheng H, Husanu M A, Bian Y, Huang S M, Hsu C H, Chang T R, Jeng H T, Bansll A, Neupert T, Strocov V N, Lin H, Jia S, Hasan M Z 2017 Sci. Adv. 3 e1603266
Google Scholar
[13] Bruno F Y, Tamai A, Wu Q S, Cucchi I, Barreteau C, De La Torre A, McKeown Walker S, Riccò S, Wang Z, Kim T K, Hoesch M, Shi M, Plumb N C, Giannini E, Soluyanov A A, Baumberger F 2016 Phys. Rev. B 94 121112(R
Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
[18] Afanasiev A N, Greshnov A A, Svintsov D 2021 Phys. Rev. B 103 205201
Google Scholar
[19] Zyuzin A A, Tiwari R P 2016 JETP Lett. 103 717
Google Scholar
[20] Duan H J, Zheng S H, Wang R Q, Deng M X, Yang M 2019 Phys. Rev. B 99 165111
Google Scholar
[21] Zhang M, Yang Z, Wang G 2018 J. Phys. Chem. C 122 3533
Google Scholar
[22] Autès G, Gresch D, Troyer M, Soluyanov A A, Yazyev O V 2016 Phys. Rev. Lett. 117 066402
Google Scholar
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Google Scholar
[26] Li D, Rosenstein B, Shapiro B Y, Shapiro I 2017 Phys. Rev. B 95 094513
Google Scholar
[27] Das K, Agarwal A 2019 Phys. Rev. B 99 085405
Google Scholar
[28] Sharma G, Goswami P, Tewari S 2017 Phys. Rev. B 96 045112
Google Scholar
[29] Hou Z, Sun Q F 2017 Phys. Rev. B 96 155305
Google Scholar
[30] Niu Z P 2019 J. Phys. Condens. Matter 31 485701
Google Scholar
[31] Cheng Q, Sun Q F 2021 Phys. Rev. B 103 144518
Google Scholar
[32] Beenakker C W J 2006 Phys. Rev. Lett. 97 067007
Google Scholar
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Google Scholar
[34] Soori A, Sahu M R, Das A, Mukerjee S 2018 Phys. Rev. B 98 075301
Google Scholar
[35] Wang C, Zou Y, Song J, Li Y X 2018 Phys. Rev. B 98 035403
Google Scholar
[36] Wei M, Zhou M, Zhang Y T, Xing Y 2020 Phys. Rev. B 101 155408
Google Scholar
[37] Zhang S, Zhang T 2021 Phys. Status Solidi B 258 2100192
Google Scholar
[38] Uchida S, Habe T, Asano Y 2014 J. Phys. Soc. Jpn. 83 064711
Google Scholar
[39] Azizi A, Abdollahipour B 2020 Phys. Rev. B 102 024512
Google Scholar
[40] Mohammadpour H, Asgari A 2011 Phys. Lett. A 375 1339
Google Scholar
[41] Nehar R, Bhakuni D S, Sharma A, Soori A 2019 J. Phys. Condens. Matter 31 345304
Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
[46] Chen W, Shen R, Sheng L, Wang B G, Xing D Y 2011 Phys. Rev. B 84 115420
Google Scholar
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Google Scholar
[48] Li X S, Zhang S F, Sun X R, Gong W J 2018 New J. Phys. 20 103005
Google Scholar
[49] O’Brien T E, Diez M, Beenakker C W J 2016 Phys. Rev. Lett. 116 236401
Google Scholar
[50] Burkov A A, Balents L 2011 Phys. Rev. Lett. 107 127205
Google Scholar
[51] Wehling T O, Black-Schafferc A, Balatsky A V 2014 Adv. Phys. 63 1
Google Scholar
[52] De Gennes P G 1966 Superconductivity of Metals and Alloys (New York: Benjamin) pp137–160
[53] Li H, Ouyang G 2019 Phys. Rev. B 100 085410
Google Scholar
[54] Blonder G E, Tinkham M, Klapwijk T M 1982 Phys. Rev. B 25 4515
Google Scholar
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