We present and experimentally verify a universal theoretical framework for the description of spin-charge interconversion in non-magnetic metal/insulator structures with interfacial spin–orbit coupling (ISOC). Our formulation is based on drift-diffusion equations supplemented with generalized boundary conditions. The latter encode the effects of ISOC and relate the electronic transport in such systems to spin loss and spin-charge interconversion at the interface. We demonstrate that the conversion efficiency depends solely on these interfacial parameters. We apply our formalism to two typical spintronic devices that exploit ISOC: a lateral spin valve and a multilayer Hall bar, for which we calculate the non-local resistance and the spin Hall magnetoresistance, respectively. Finally, we perform measurements on these two devices with a BiOx/Cu interface and verify that transport properties related to the ISOC are quantified by the same set of interfacial parameters.
References
1.
I.
Žutić
, J.
Fabian
, and S. D.
Sarma
, Rev. Mod. Phys.
76
, 323
(2004
).2.
G.
Vignale
and J.
Supercond
, Nov. Magn.
23
, 3
(2010
).3.
J.
Sinova
, S. O.
Valenzuela
, J.
Wunderlich
, C. H.
Back
, and T.
Jungwirth
, Rev. Mod. Phys.
87
, 1213
(2015
).4.
S. O.
Valenzuela
and M.
Tinkham
, Nature
442
, 176
(2006
).5.
T.
Kimura
, Y.
Otani
, T.
Sato
, S.
Takahashi
, and S.
Maekawa
, Phys. Rev. Lett.
98
, 156601
(2007
).6.
A.
Aronov
and Y. B.
Lyanda-Geller
, JETP Lett.
50
, 431
(1989
), available at http://www.jetpletters.ac.ru/ps/1132/article_17140.shtml.7.
V. M.
Edelstein
, Solid State Commun.
73
, 233
(1990
).8.
Y.
Ando
and M.
Shiraishi
, J. Phys. Soc. Jpn.
86
, 011001
(2017
).9.
A.
Soumyanarayanan
, N.
Reyren
, A.
Fert
, and C.
Panagopoulos
, Nature
539
, 509
(2016
).10.
I. M.
Miron
, K.
Garello
, G.
Gaudin
, P.-J.
Zermatten
, M. V.
Costache
, S.
Auffret
, S.
Bandiera
, B.
Rodmacq
, A.
Schuhl
, and P.
Gambardella
, Nature
476
, 189
(2011
).11.
L.
Liu
, C.-F.
Pai
, Y.
Li
, H.
Tseng
, D.
Ralph
, and R.
Buhrman
, Science
336
, 555
(2012
).12.
C.
Safeer
, E.
Jué
, A.
Lopez
, L.
Buda-Prejbeanu
, S.
Auffret
, S.
Pizzini
, O.
Boulle
, I. M.
Miron
, and G.
Gaudin
, Nat. Nanotechnol.
11
, 143
(2016
).13.
V. T.
Pham
, I.
Groen
, S.
Manipatruni
, W. Y.
Choi
, D. E.
Nikonov
, E.
Sagasta
, C.-C.
Lin
, T. A.
Gosavi
, A.
Marty
, L. E.
Hueso
, I. A.
Young
, and F.
Casanova
, Nat. Electron.
3
, 309
(2020
).14.
S.
Manipatruni
, D. E.
Nikonov
, C.-C.
Lin
, T. A.
Gosavi
, H.
Liu
, B.
Prasad
, Y.-L.
Huang
, E.
Bonturim
, R.
Ramesh
, and I. A.
Young
, Nature
565
, 35
(2019
).15.
S.
Karube
, K.
Kondou
, and Y.
Otani
, Appl. Phys. Express
9
, 033001
(2016
).16.
J.
Kim
, Y.-T.
Chen
, S.
Karube
, S.
Takahashi
, K.
Kondou
, G.
Tatara
, and Y.
Otani
, Phys. Rev. B
96
, 140409
(2017
).17.
H.
Nakayama
, Y.
Kanno
, H.
An
, T.
Tashiro
, S.
Haku
, A.
Nomura
, and K.
Ando
, Phys. Rev. Lett.
117
, 116602
(2016
).18.
H.
Tsai
, S.
Karube
, K.
Kondou
, N.
Yamaguchi
, F.
Ishii
, and Y.
Otani
, Sci. Rep.
8
, 5564
(2018
).19.
J.-C.
Rojas-Sánchez
, S.
Oyarzún
, Y.
Fu
, A.
Marty
, C.
Vergnaud
, S.
Gambarelli
, L.
Vila
, M.
Jamet
, Y.
Ohtsubo
, A.
Taleb-Ibrahimi
, P.
Le Fèvre
, F.
Bertran
, N.
Reyren
, J.-M.
George
, and A.
Fert
, Phys. Rev. Lett.
116
, 096602
(2016
).20.
K.
Kondou
, R.
Yoshimi
, A.
Tsukazaki
, Y.
Fukuma
, J.
Matsuno
, K.
Takahashi
, M.
Kawasaki
, Y.
Tokura
, and Y.
Otani
, Nat. Phys.
12
, 1027
(2016
).21.
E.
Lesne
, Y.
Fu
, S.
Oyarzun
, J.
Rojas-Sánchez
, D.
Vaz
, H.
Naganuma
, G.
Sicoli
, J.-P.
Attané
, M.
Jamet
, E.
Jacquet
et al, Nat. Mater.
15
, 1261
(2016
).22.
D. C.
Vaz
, P.
Noël
, A.
Johansson
, B.
Göbel
, F. Y.
Bruno
, G.
Singh
, S.
Mckeown-Walker
, F.
Trier
, L. M.
Vicente-Arche
, A.
Sander
et al, Nat. Mater.
18
, 1187
(2019
).23.
Y.
Niimi
, Y.
Kawanishi
, D. H.
Wei
, C.
Deranlot
, H. X.
Yang
, M.
Chshiev
, T.
Valet
, A.
Fert
, and Y.
Otani
, Phys. Rev. Lett.
109
, 156602
(2012
).24.
J.
Linder
and T.
Yokoyama
, Phys. Rev. Lett.
106
, 237201
(2011
).25.
I. V.
Tokatly
, E. E.
Krasovskii
, and G.
Vignale
, Phys. Rev. B
91
, 035403
(2015
).26.
J.
Borge
and I. V.
Tokatly
, Phys. Rev. B
96
, 115445
(2017
).27.
J.
Borge
and I. V.
Tokatly
, Phys. Rev. B
99
, 241401
(2019
).28.
C.
Sanz-Fernández
, J.
Borge
, I. V.
Tokatly
, and F. S.
Bergeret
, Phys. Rev. B
100
, 195406
(2019
).29.
V. P.
Amin
, J.
Zemen
, and M. D.
Stiles
, Phys. Rev. Lett.
121
, 136805
(2018
).30.
V. P.
Amin
and M. D.
Stiles
, Phys. Rev. B
94
, 104420
(2016
).31.
V. P.
Amin
and M. D.
Stiles
, Phys. Rev. B
94
, 104419
(2016
).32.
Symmetry arguments alone cannot fix the relation between and .27 However, we will see by contrasting theory with experiment, that reciprocity requires .
33.
Notice that, in principle, an additional divergenceless term may appear in the rhs of Eq. (5). Indeed, as demonstrated in Ref. 27, symmetry allows for a term proportional to the out-of-plane component of the spin ECP. In the present work, we only consider spin polarization parallel to the I/N interface and, hence, we neglect that term.
34.
S.
Takahashi
and S.
Maekawa
, Phys. Rev. B
67
, 052409
(2003
).35.
Y.
Niimi
, H.
Suzuki
, Y.
Kawanishi
, Y.
Omori
, T.
Valet
, A.
Fert
, and Y.
Otani
, Phys. Rev. B
89
, 054401
(2014
).36.
37.
V. T.
Pham
, L.
Vila
, G.
Zahnd
, A.
Marty
, W.
Savero-Torres
, M.
Jamet
, and J.-P.
Attané
, Nano Lett.
16
, 6755
(2016
).38.
T.
Kimura
, J.
Hamrle
, and Y.
Otani
, Phys. Rev. B
72
, 014461
(2005
).39.
T.
Kimura
, J.
Hamrle
, and Y.
Otani
, Appl. Phys. Lett.
85
, 3501
(2004
).40.
M.
Isasa
, M. C.
Martínez-Velarte
, E.
Villamor
, C.
Magén
, L.
Morellón
, J. M.
De Teresa
, M. R.
Ibarra
, G.
Vignale
, E. V.
Chulkov
, E. E.
Krasovskii
, L. E.
Hueso
, and F.
Casanova
, Phys. Rev. B
93
, 014420
(2016
).41.
E.
Sagasta
, Y.
Omori
, M.
Isasa
, Y.
Otani
, L. E.
Hueso
, and F.
Casanova
, Appl. Phys. Lett.
111
, 082407
(2017
).42.
H.
Tsai
, K.
Kondou
, and Y.
Otani
, Jpn. J. Appl. Phys., Part 1
58
, 110907
(2019
).43.
In principle, any transport measurement in a system with a linear response will fulfill the Onsager reciprocity.57–59 Within our theory, the global reciprocity is guarantied by . Importantly, the global reciprocity does not imply a point-wise identity of the local current densities.
44.
J. R.
Sánchez
, L.
Vila
, G.
Desfonds
, S.
Gambarelli
, J.
Attané
, J.
De Teresa
, C.
Magén
, and A.
Fert
, Nat. Commun.
4
, 2944
(2013
).45.
Y.-T.
Chen
, S.
Takahashi
, H.
Nakayama
, M.
Althammer
, S. T. B.
Goennenwein
, E.
Saitoh
, and G. E. W.
Bauer
, Phys. Rev. B
87
, 144411
(2013
).46.
H.
Nakayama
, M.
Althammer
, Y.-T.
Chen
, K.
Uchida
, Y.
Kajiwara
, D.
Kikuchi
, T.
Ohtani
, S.
Geprägs
, M.
Opel
, S.
Takahashi
et al, Phys. Rev. Lett.
110
, 206601
(2013
).47.
X.-P.
Zhang
, F. S.
Bergeret
, and V. N.
Golovach
, Nano Lett.
19
, 6330
(2019
).48.
A.
Brataas
, Y. V.
Nazarov
, and G. E.
Bauer
, Phys. Rev. Lett.
84
, 2481
(2000
).49.
Following our convention, the vector normal to the interface, points from the Cu towards the insulating YIG layer.
50.
N.
Vlietstra
, J.
Shan
, V.
Castel
, J.
Ben Youssef
, G. E. W.
Bauer
, and B. J.
van Wees
, Appl. Phys. Lett.
103
, 032401
(2013
).51.
M.
Althammer
, S.
Meyer
, H.
Nakayama
, M.
Schreier
, S.
Altmannshofer
, M.
Weiler
, H.
Huebl
, S.
Geprägs
, M.
Opel
, R.
Gross
, D.
Meier
, C.
Klewe
, T.
Kuschel
, J.-M.
Schmalhorst
, G.
Reiss
, L.
Shen
, A.
Gupta
, Y.-T.
Chen
, G. E. W.
Bauer
, E.
Saitoh
, and S. T. B.
Goennenwein
, Phys. Rev. B
87
, 224401
(2013
).52.
T.
Kosub
, S.
Vélez
, J. M.
Gomez-Perez
, L. E.
Hueso
, J.
Fassbender
, F.
Casanova
, and D.
Makarov
, Appl. Phys. Lett.
113
, 222409
(2018
).53.
E.
Villamor
, M.
Isasa
, S.
Vélez
, A.
Bedoya-Pinto
, P.
Vavassori
, L. E.
Hueso
, F. S.
Bergeret
, and F.
Casanova
, Phys. Rev. B
91
, 020403
(2015
).54.
K. S.
Das
, F. K.
Dejene
, B. J.
van Wees
, and I. J.
Vera-Marun
, Appl. Phys. Lett.
114
, 072405
(2019
).55.
L. J.
Cornelissen
, K. J. H.
Peters
, G. E. W.
Bauer
, R. A.
Duine
, and B. J.
van Wees
, Phys. Rev. B
94
, 014412
(2016
).56.
N.
Vlietstra
, J.
Shan
, V.
Castel
, B. J.
van Wees
, and J.
Ben Youssef
, Phys. Rev. B
87
, 184421
(2013
).57.
H. B. G.
Casimir
, “On Onsager's principle of microscopic reversibility
,” Rev. Mod. Phys.
17
, 343
–350
(1945
).58.
L.
Onsager
, “Reciprocal relations in irreversible processes. I
,” Phys. Rev.
37
, 405
(1931
).59.
L.
Onsager
, “Reciprocal relations in irreversible processes. II
,” Phys. Rev.
38
, 2265
(1931
).© 2020 Author(s).
2020
Author(s)
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