Josephson current induced by spin-mixing Cooper pairs

Meng Hao; Wu Xiu-Qiang

doi:10.7498/aps.72.20231008

Abstract

Based on the Bogoliubov-de Gennes equations, we investigate the transport of the Josephson current in a one-dimensional S/F_L-F-F_R/S junction, where S and F are superconductor and ferromagnet, and F_L,R are the left and right interfaces with noncollinear magnetizations. It is found that the F_L and F_R interfaces can induce spin-mixing and spin-flip effects, which can transform a part of spin-singlet pairs in the S into equal-spin triplet pairs in the F. For the short S/F_L-F-F_R/S junction, the spin-singlet pairs and the equal-spin triplet pairs can survive in the F layer. Therefore, with the increase of the ferromagnetic exchange field and the angle difference of interface magnetization rotation, the critical current oscillates on a base level. If the F is transformed into half-metal, only the equal-spin triple pairs exist in the F layer, and the oscillation characteristic of critical current disappears. In addition, the F_L and F_R interfaces can work as conventional potential barriers. As a result, the critical current exhibits double oscillation behaviors with the increase of ferromagnetic thickness, in which the long-wave oscillation arises from the phase change of the spin-singlet pairs in the ferromagnetic layer, and the short-wave oscillation is caused by the resonant tunneling effect when the spin-singlet pairs and the equal-spin triplet pairs pass through two interfacial barriers.

Keywords:

Authors and contacts

Meng Hao¹,
Wu Xiu-Qiang²

1.
School of Physics and Telecommunication Engineering, Shaanxi University of Technology, Hanzhong 723001, China
2.
Department of Physics, Yancheng Institute of Technology, Yancheng 224051, China

Corresponding author: Meng Hao, menghao2021@163.com

Funds: Project supported by the National Natural Science Foundation of China (Grant No. 12174238), the Natural Science Basic Research Program of Shaanxi Province, China (Grant Nos. 2020JM-597, 2023-JC-YB-025), the Scientific Research Foundation of Shaanxi University of Technology, China (Grant No. SLGKY2006), the Innovation Team of Shaanxi Universities, China (Grant No. 2022-94), and the School-level Youth Innovation Team of Shaanxi University of Technology, China.

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References

[1]	Golubov A A, Kupriyanov M Y, Il’ichev E 2004 Rev. Mod. Phys. 76 411 Google Scholar
[2]	Buzdin A I 2005 Rev. Mod. Phys. 77 935 Google Scholar
[3]	Bergeret F S, Volkov A F, Efetov K B 2005 Rev. Mod. Phys. 77 1321 Google Scholar
[4]	Eschrig M 2011 Phys. Today 64 43 Google Scholar
[5]	Blamire M G, Robinson J W A 2014 J. Phys. Condens. Matter 26 453201 Google Scholar
[6]	Linder J, Robinson J W A 2015 Nat. Phys. 11 307 Google Scholar
[7]	Eschrig M 2015 Rep. Prog. Phys. 78 104501 Google Scholar
[8]	Mel'nikov A S, Mironov S V, Samokhvalov A V, Buzdin A I 2022 Physics-Uspekhi 65 1248 Google Scholar
[9]	Kimura T, Otani Y, Hamrle J 2006 Phys. Rev. Lett. 96 037201 Google Scholar
[10]	Keizer R S, Goennenwein S T B, Klapwijk T M, Miao G, Xiao G, Gupta A 2006 Nature 439 825 Google Scholar
[11]	Anwar M S, Czeschka F, Hesselberth M, Porcu M, Aarts J 2010 Phys. Rev. B 82 100501(R Google Scholar
[12]	Kontos T, Aprili M, Lesueur J, Genêt F, Stephanidis B, Boursier R 2002 Phys. Rev. Lett. 89 137007 Google Scholar
[13]	Jiang J S, Davidović D, Reich D H, Chien C L 1995 Phys. Rev. Lett. 74 314 Google Scholar
[14]	Blum Y, Tsukernik A, Karpovski M, Palevski A 2002 Phys. Rev. Lett. 89 187004 Google Scholar
[15]	Ryazanov V V, Oboznov V A, Rusanov A Yu, Veretennikov A V, Golubov A A, Aarts J 2001 Phys. Rev. Lett. 86 2427 Google Scholar
[16]	Oboznov V A, Bol’ginov V V, Feofanov A K, Ryazanov V V, Buzdin A I 2006 Phys. Rev. Lett. 96 197003 Google Scholar
[17]	Yamashita T, Takahashi S, Maekawa S 2006 Appl. Phys. Lett. 88 132501 Google Scholar
[18]	Bergeret F S, Volkov A F, Efetov K B 2001 Phys. Rev. Lett. 86 4096 Google Scholar
[19]	Kadigrobov A, Shekhter R I, Jonson M 2001 Europhys. Lett. 54 394 Google Scholar
[20]	Eschrig M, Kopu J, Cuevas J C, Gerd Schön 2003 Phys. Rev. Lett. 90 137003 Google Scholar
[21]	Houzet M, Buzdin A I 2007 Phys. Rev. B 76 060504(R Google Scholar
[22]	Asano Y, Tanaka Y, Golubov A A 2007 Phys. Rev. Lett. 98 107002 Google Scholar
[23]	Sosnin I, Cho H, Petrashov V T, Volkov A F 2006 Phys. Rev. Lett. 96 157002 Google Scholar
[24]	Robinson J W A, Witt J D S, Blamire M G 2010 Science 329 59 Google Scholar
[25]	Khaire T S, Khasawneh M A, Pratt W P, Birge N O 2010 Phys. Rev. Lett. 104 137002 Google Scholar
[26]	Klose C, Khaire T S, Wang Y X, et al. 2012 Phys. Rev. Lett. 108 127002 Google Scholar
[27]	Martinez W M, Pratt W P, Birge N O 2016 Phys. Rev. Lett. 116 077001 Google Scholar
[28]	Sanchez-Manzano D, Mesoraca S, Cuellar F A, et al. 2022 Nat. Mater. 21 188 Google Scholar
[29]	De Gennes P G 1966 Superconductivity of Metals and Alloys (New York: Benjamin) pp137–170
[30]	Bagwell P F 1992 Phys. Rev. B 46 12573 Google Scholar
[31]	Beenakker C W J 1991 Phys. Rev. Lett. 67 3836 Google Scholar
[32]	Bardeen J, Kümmel R, Jacobs A E, Tewordt L 1969 Phys. Rev. 187 556 Google Scholar
[33]	Cayssol J, Montambaux G 2004 Phys. Rev. B 70 224520 Google Scholar
[34]	Bulaevskii L N, Buzdin A I, Kulić M L, Panyukov S V 1985 Adv. Phys. 34 175 Google Scholar
[35]	Meng H, Wu X Q, Ren Y J, Wu J S 2022 Phys. Rev. B 106 174502 Google Scholar
[36]	Markos P, Soukoulis C M 2008 Wave Propagation: From Electrons to Photonic Crystals and Left-Handed Materials (Princeton: Princeton University Press) pp56–73
[37]	Cohen-Tannoudji C, Diu B, Laloë F 2019 Quantum Mechanics (Vol. 1) (Weinheim: Wiley-VCH) pp2–32
[38]	Iogansen L V 1964 Sov. Phys.-JETP 18 146
[39]	Meng H, Wu X Q, Zheng Z M 2013 Europhys. Lett. 104 37003 Google Scholar
[40]	Meng H, Wu J S, Wu X Q, Ren M Y, Ren Y J 2016 Sci. Rep. 6 21308 Google Scholar

Cited By

图 1 (a) S/F_L-F-F_R/S结的结构示意图, 其中F内的粗箭头表示交换场方向, F_L和F_R为具有非共线磁矩的自旋活性界面; (b) F_j界面处磁矩$ {{\boldsymbol{\rho}}}_{j} $的磁化方向, 其中$ {j}=\rm{L} $, $ \rm{R} $分别对应左侧和右侧界面. $ \theta_{j} $和$ \chi_{j} $分别表示F_j界面的极化角和方位角

Figure 1. (a) Schematic diagram of the S/F_L-F-F_R/S junction, where thick arrow in F indicates the directions of the exchange field, and the F_L and F_R are spin-active interfaces with non-collinear magnetic moments; (b) the direction of magnetization $ {{\boldsymbol{\rho}}}_{j} $ at the F_j interface, where $ {j}=\rm{L} $ and $ \rm{R} $ correspond to the left and right interfaces, respectively. $ \theta_{j} $ and $ \chi_{j} $ denote polar and azimuthal angles of the F_j interface, respectively

DownLoad: Full-Size Img PowerPoint

图 2 S/F/S结中Josephson电流随铁磁特征的变化　(a)临界电流$ I_{{\rm{c}}} $随铁磁交换场$ h_{z} $和铁磁厚度d的变化特征; (b)不同交换场下$ I_{{\rm{c}}} $随d的变化特征; (c)不同铁磁厚度时$ I_{{\rm{c}}} $随$ h_{z} $的变化特征; $ h_{z}/E_{{\rm{F}}} = 0.15 $时的(d)Andreev能谱$ E_{\rm{A}}(\phi) $和(e)电流-位相关系$ I(\phi) $. 在所有图形中, 界面极化强度取$ P_{{\rm{L}}}=P_{{\rm{R}}} = 0 $

Figure 2. Variation of Josephson current with ferromagnetic characteristics in the S/F/S junction: (a) Critical current $ I_{{\rm{c}}} $ versus exchange field $ h_z $ and ferromagnetic thickness d; (b) dependence of $ I_{{\rm{c}}} $ on d for different exchange fields; (c) dependence of $ I_{{\rm{c}}} $ on $ h_z $ for different ferromagnetic thicknesses; (d) Andreev energy spectrum $ E_{\rm{A}}(\phi) $ and (e) current-phase relation $ I(\phi) $ for $ h_{z}/E_{{\rm{F}}} = $$ 0.15 $. In all panels, the strengths of interfacial polarization are taken as $ P_{{\rm{L}}}=P_{{\rm{R}}} = 0 $

DownLoad: Full-Size Img PowerPoint

图 3 临界电流$ I_{\rm{c}} $随铁磁交换场$ h_z $和界面磁矩偏转角度差$ {\text{δ}}\chi $的变化特征　(a) k_Fd = 25; (b) k_Fd = 50; (c) k_Fd = 70. 在所有图形中, 界面极化强度取$ P_{{\rm{L}}}=P_{{\rm{R}}} = 1 $

Figure 3. The critical current $ I_{{\rm{c}}} $ versus exchange field $ h_z $ and angle difference of interface magnetization rotation $ {\text{δ}}\chi $: (a) k_Fd = 25; (b) k_Fd = 50; (c) k_Fd = 70. In all panels, the strengths of interfacial polarization are taken as $ P_{{\rm{L}}}=P_{{\rm{R}}} = 1 $

DownLoad: Full-Size Img PowerPoint

图 4 临界电流$ I_{\rm{c}} $随铁磁层厚度d和界面磁矩偏转角度差$ {\text{δ}}\chi $的变化特征　(a), (c), (e) 电流变化的侧视图; (b), (d), (f) 电流变化的俯视图. (a), (b) $ h_z/E_{{\rm{F}}} = 0.1 $; (c), (d) $ h_z/E_{{\rm{F}}} = 0.5 $; (e), (f) $ h_z/E_{\rm{F}} = 1.01 $. 在所有图形中, 界面极化强度取$ P_{{\rm{L}}}=P_{{\rm{R}}} = 1 $

Figure 4. Variation of the critical current with ferromagnetic thickness d and angle difference of interface magnetization rotation $ {\text{δ}}\chi $: (a), (c), (e) Side view of current changes; (b), (d), (f) top view of current changes. (a), (b) $ h_z/E_{{\rm{F}}} = 0.1 $; (c), (d) $ h_z/E_{{\rm{F}}} = 0.5 $; (e), (f) $ h_z/E_{\rm{F}} = 1.01 $. In all panels, the strengths of interfacial polarization are taken as $ P_{{\rm{L}}}=P_{{\rm{R}}} = 1 $.

DownLoad: Full-Size Img PowerPoint

图 5 (a)$ {\text{δ}}\chi $取3个特殊值时临界电流$ I_{\rm{c}} $随铁磁层厚度d的变化特征; (b), (c) $ {\text{δ}}\chi = 0 $时的Andreev能谱$ E_{\rm{A}}(\phi)$和电流-位相关系$ I(\phi) $ (图(b)插图描述了$ I_{\rm{c }}$随$ {\text{δ}}\chi $的变化); (d), (e) $ k_{{\rm{F}}}d = 29.9 $时的Andreev能谱$ E_{\rm{A}}(\phi) $和电流-位相关系$ I(\phi) $. 在所有图形中, 其他参数为$ h_{z}/E_{{\rm{F}}} = 0.1 $和$ P_{{\rm{L}}}=P_{{\rm{R}}} = 1 $

Figure 5. (a) Variation of the critical current with ferromagnetic thickness d when $ {\text{δ}}\chi $ takes three special values; (b), (c) Andreev energy spectrum $ E_{\rm{A}}(\phi) $ and current-phase relation $ I(\phi) $ for $ {\text{δ}}\chi = 0 $ (the inset in panel (b) illustrates the dependence of $ I_{\rm{c}} $ on $ {\text{δ}}\chi $); (d), (e) Andreev energy spectrum $ E_{\rm{A}}(\phi) $ and current-phase relation $ I(\phi) $ for $ k_{{\rm{F}}}d = 29.9 $. In all panels, other parameters are $ h_{z}/E_{{\rm{F}}} = 0.1 $ and $ P_{{\rm{L}}}=P_{{\rm{R}}} = 1 $.

DownLoad: Full-Size Img PowerPoint

图 6 $ {\text{δ}}\chi = 0 $时不同铁磁交换场下临界电流$ I_{\rm{c}} $随界面极化强度$ P_{{\rm{L}}} $和$ P_{{\rm{R}}} $的变化特征　(a) $ h_z/E_{{\rm{F}}} $ = 0.1; (b) $ h_z/E_{{\rm{F}}} $ = 0.5; (c) $ h_z/E_{{\rm{F}}} $ = 1.01. 所有图形中, F层厚度取$ k_{{\rm{F}}}d = 50 $

Figure 6. Variation of the critical current with the strengths of interfacial polarization $ P_{{\rm{L}}} $ and $ P_{{\rm{R}}} $ for three different exchange fields in the case of $ {\text{δ}}\chi = 0 $: (a) $ h_z/E_{{\rm{F}}} $ = 0.1; (b) $ h_z/E_{{\rm{F}}} $ = 0.5; (c) $ h_z/E_{{\mathrm{F}}} $ = 1.01. In all panels, the thickness of the F layer is taken as $ k_{{\rm{F}}}d = 50 $

DownLoad: Full-Size Img PowerPoint

[1]	Golubov A A, Kupriyanov M Y, Il’ichev E 2004 Rev. Mod. Phys. 76 411 Google Scholar
[2]	Buzdin A I 2005 Rev. Mod. Phys. 77 935 Google Scholar
[3]	Bergeret F S, Volkov A F, Efetov K B 2005 Rev. Mod. Phys. 77 1321 Google Scholar
[4]	Eschrig M 2011 Phys. Today 64 43 Google Scholar
[5]	Blamire M G, Robinson J W A 2014 J. Phys. Condens. Matter 26 453201 Google Scholar
[6]	Linder J, Robinson J W A 2015 Nat. Phys. 11 307 Google Scholar
[7]	Eschrig M 2015 Rep. Prog. Phys. 78 104501 Google Scholar
[8]	Mel'nikov A S, Mironov S V, Samokhvalov A V, Buzdin A I 2022 Physics-Uspekhi 65 1248 Google Scholar
[9]	Kimura T, Otani Y, Hamrle J 2006 Phys. Rev. Lett. 96 037201 Google Scholar
[10]	Keizer R S, Goennenwein S T B, Klapwijk T M, Miao G, Xiao G, Gupta A 2006 Nature 439 825 Google Scholar
[11]	Anwar M S, Czeschka F, Hesselberth M, Porcu M, Aarts J 2010 Phys. Rev. B 82 100501(R Google Scholar
[12]	Kontos T, Aprili M, Lesueur J, Genêt F, Stephanidis B, Boursier R 2002 Phys. Rev. Lett. 89 137007 Google Scholar
[13]	Jiang J S, Davidović D, Reich D H, Chien C L 1995 Phys. Rev. Lett. 74 314 Google Scholar
[14]	Blum Y, Tsukernik A, Karpovski M, Palevski A 2002 Phys. Rev. Lett. 89 187004 Google Scholar
[15]	Ryazanov V V, Oboznov V A, Rusanov A Yu, Veretennikov A V, Golubov A A, Aarts J 2001 Phys. Rev. Lett. 86 2427 Google Scholar
[16]	Oboznov V A, Bol’ginov V V, Feofanov A K, Ryazanov V V, Buzdin A I 2006 Phys. Rev. Lett. 96 197003 Google Scholar
[17]	Yamashita T, Takahashi S, Maekawa S 2006 Appl. Phys. Lett. 88 132501 Google Scholar
[18]	Bergeret F S, Volkov A F, Efetov K B 2001 Phys. Rev. Lett. 86 4096 Google Scholar
[19]	Kadigrobov A, Shekhter R I, Jonson M 2001 Europhys. Lett. 54 394 Google Scholar
[20]	Eschrig M, Kopu J, Cuevas J C, Gerd Schön 2003 Phys. Rev. Lett. 90 137003 Google Scholar
[21]	Houzet M, Buzdin A I 2007 Phys. Rev. B 76 060504(R Google Scholar
[22]	Asano Y, Tanaka Y, Golubov A A 2007 Phys. Rev. Lett. 98 107002 Google Scholar
[23]	Sosnin I, Cho H, Petrashov V T, Volkov A F 2006 Phys. Rev. Lett. 96 157002 Google Scholar
[24]	Robinson J W A, Witt J D S, Blamire M G 2010 Science 329 59 Google Scholar
[25]	Khaire T S, Khasawneh M A, Pratt W P, Birge N O 2010 Phys. Rev. Lett. 104 137002 Google Scholar
[26]	Klose C, Khaire T S, Wang Y X, et al. 2012 Phys. Rev. Lett. 108 127002 Google Scholar
[27]	Martinez W M, Pratt W P, Birge N O 2016 Phys. Rev. Lett. 116 077001 Google Scholar
[28]	Sanchez-Manzano D, Mesoraca S, Cuellar F A, et al. 2022 Nat. Mater. 21 188 Google Scholar
[29]	De Gennes P G 1966 Superconductivity of Metals and Alloys (New York: Benjamin) pp137–170
[30]	Bagwell P F 1992 Phys. Rev. B 46 12573 Google Scholar
[31]	Beenakker C W J 1991 Phys. Rev. Lett. 67 3836 Google Scholar
[32]	Bardeen J, Kümmel R, Jacobs A E, Tewordt L 1969 Phys. Rev. 187 556 Google Scholar
[33]	Cayssol J, Montambaux G 2004 Phys. Rev. B 70 224520 Google Scholar
[34]	Bulaevskii L N, Buzdin A I, Kulić M L, Panyukov S V 1985 Adv. Phys. 34 175 Google Scholar
[35]	Meng H, Wu X Q, Ren Y J, Wu J S 2022 Phys. Rev. B 106 174502 Google Scholar
[36]	Markos P, Soukoulis C M 2008 Wave Propagation: From Electrons to Photonic Crystals and Left-Handed Materials (Princeton: Princeton University Press) pp56–73
[37]	Cohen-Tannoudji C, Diu B, Laloë F 2019 Quantum Mechanics (Vol. 1) (Weinheim: Wiley-VCH) pp2–32
[38]	Iogansen L V 1964 Sov. Phys.-JETP 18 146
[39]	Meng H, Wu X Q, Zheng Z M 2013 Europhys. Lett. 104 37003 Google Scholar
[40]	Meng H, Wu J S, Wu X Q, Ren M Y, Ren Y J 2016 Sci. Rep. 6 21308 Google Scholar

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